Tutorial
Open Access

Hybrid Oscillator-Qubit Quantum Processors: Instruction Set Architectures, Abstract Machine Models, and Applications

Yuan Liu ^1,2,3,*, Shraddha Singh ^4,5,*, Kevin C. Smith ^4,5,6,*, Eleanor Crane ¹, John M. Martyn ¹, Alec Eickbusch ^4,5,†, Alexander Schuckert ⁷, Richard D. Li^4,5, Jasmine Sinanan-Singh¹ et al.

Micheline B. Soley ⁸, Takahiro Tsunoda ^4,5, Isaac L. Chuang ^1,9, Nathan Wiebe ^10,11,12, and Steven M. Girvin ^4,5,‡

¹Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
²Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, North Carolina 27606, USA
³Department of Computer Science, North Carolina State University, Raleigh, North Carolina 27606, USA
⁴Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA
⁵Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
⁶Brookhaven National Laboratory, Upton, New York 11973, USA
⁷Joint Quantum Institute and QuICS, National Institute of Standards and Technology (NIST) and University of Maryland, College Park, Maryland 20742, USA
⁸Department of Chemistry, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
⁹Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
¹⁰Department of Computer Science, University of Toronto, Toronto, ON M5S1J7, Canada
¹¹Canadian Institute for Advanced Research, Toronto, ON Canada
¹²Pacific Northwest National Laboratory, Richland, Washington, USA

^*These authors contributed equally to this work (ordered alphabetically).
^†Present address: Google Quantum AI, Santa Barbara, California 93111, USA.
^‡Contact author: steven.girvin@yale.edu

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PRX Quantum 7, 010201 – Published 7 January, 2026

DOI: https://doi.org/10.1103/4rf7-9tfx

Abstract

This tutorial offers a pedagogical guide to hybrid quantum processors that integrate discrete-variable (DV) qubits and continuous-variable (CV) oscillators. Aimed at computer scientists, engineers, and physicists, it provides an overview of the experimental, algorithmic, and architectural aspects of this novel and rapidly developing hardware model. Experimental realizations of this model include superconducting, trapped-ion, and neutral-atom platforms. By combining DV and CV components, hybrid oscillator-qubit processors enable a powerful new paradigm that offers complementary strengths for quantum control, error correction, computation, and simulation. Working toward the goal of a full-stack system connecting applications to CV-DV hardware, we define and formulate abstract machine models and instruction set architectures. These essential abstractions enable codesign of hardware and software, and resource estimation for exploring the potential of current and future hardware for computational and simulation tasks. Using these abstractions, we present both new and existing examples that illustrate the benefits of hybrid CV-DV processors relative to traditional DV-only hardware in computation as well as quantum simulation of physical models. Examples include algorithms for transferring states between DV and CV systems, performing the quantum Fourier transform, and simulation of lattice gauge theories. Relative to qubit-only hardware, the bosonic degrees of freedom natively available in hybrid architectures can substantially reduce the circuit complexity of simulations for physical models containing bosons. A key technique is the extension of quantum signal processing ideas to CV-DV systems. This work is intended to serve as a timely and comprehensive guide to this relatively unexplored yet promising approach to quantum computation and to provide a road map to guide future development.

Physics Subject Headings (PhySH)

Popular Summary

A massive effort is now under way to build quantum computers which store and process information in radically new ways using the properties of a variety of objects acting as two-level quantum bits (qubits). In this work we investigate the significant advantages of a novel hybrid architecture that includes both discrete-variable (DV) qubits and continuous-variable (CV) harmonic oscillators, the latter of which stores quantum information in a formally infinite number of excitation levels. In addition to being useful for general quantum computation, this architecture is especially advantageous for quantum simulations of physical models involving boson degrees of freedom (e.g., molecular and lattice vibrations in chemical and condensed-matter systems, and lattice gauge theories describing nuclear and particle physics). The ideas and techniques we explain and develop are already being pursued in multiple platforms including neutral atoms, trapped ions, and superconducting circuits.

This hybrid architecture combines the best features of gate-based qubit quantum computation and photonic quantum computation, and we develop new compilation methods, including a novel extension of the quantum signal processing technique, to take advantage of these features. We define three abstract machine models and several instruction set architectures that standardize methodologies for universal operations on the qubits, the oscillators, and joint operations between them.

This tutorial provides pedagogical material intended to catalyze collaboration between the physics and computer science communities and bridge experimental and theoretical pursuits for realizing large-scale full-stack systems from this hybrid architecture. We conclude with a description of some of the remaining outstanding questions and open research directions. Altogether, this work paves the way for a new hybrid CV-DV paradigm of quantum computation.

Article Text

References (733)

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010).
Y. Alexeev, et al., Quantum computer systems for scientific discovery, PRX Quantum 2, 017001 (2021).
L. M. K. Vandersypen and I. L. Chuang, NMR techniques for quantum control and computation, Rev. Mod. Phys. 76, 1037 (2005).
J. A. Jones, Controlling NMR spin systems for quantum computation, Prog. Nucl. Magn. Reson. Spectrosc. 140-141, 49 (2024).
J. Dijkema, X. Xue, P. Harvey-Collard, M. Rimbach-Russ, S. L. de Snoo, G. Zheng, A. Sammak, G. Scappucci, and L. M. K. Vandersypen, Cavity-mediated iswap oscillations between distant spins, Nat. Phys. 21, 168 (2025).
D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, Fast quantum gates for neutral atoms, Phys. Rev. Lett. 85, 2208 (2000).
N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, Sub-Poissonian loading of single atoms in a microscopic dipole trap, Nature 411, 1024 (2001).
D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, S. Ebadi, M. Kalinowski, A. Keesling, N. Maskara, H. Pichler, M. Greiner, V. Vuletić, and M. D. Lukin, A quantum processor based on coherent transport of entangled atom arrays, Nature 604, 451 (2022).
T. M. Graham, et al., Multi-qubit entanglement and algorithms on a neutral-atom quantum computer, Nature 604, 457 (2022).
J. I. Cirac and P. Zoller, Quantum computations with cold trapped ions, Phys. Rev. Lett. 74, 4091 (1995).
C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, Demonstration of a fundamental quantum logic gate, Phys. Rev. Lett. 75, 4714 (1995).
D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Quantum dynamics of single trapped ions, Rev. Mod. Phys. 75, 281 (2003).
H. Häffner, C. Roos, and R. Blatt, Quantum computing with trapped ions, Phys. Rep. 469, 155 (2008).
J. P. Home, D. Hanneke, J. D. Jost, J. M. Amini, D. Leibfried, and D. J. Wineland, Complete methods set for scalable ion trap quantum information processing, Science 325, 1227 (2009).
C. Monroe and J. Kim, Scaling the ion trap quantum processor, Science 339, 1164 (2013).
C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Trapped-ion quantum computing: Progress and challenges, Appl. Phys. Rev. 6, 021314 (2019).
S. Moses, et al., A race-track trapped-ion quantum processor, Phys. Rev. X 13, 041052 (2023).
K. Hémery, K. Ghanem, E. Crane, S. L. Campbell, J. M. Dreiling, C. Figgatt, C. Foltz, J. P. Gaebler, J. Johansen, M. Mills, S. A. Moses, J. M. Pino, A. Ransford, M. Rowe, P. Siegfried, R. P. Stutz, H. Dreyer, A. Schuckert, and R. Nigmatullin, Measuring the Loschmidt amplitude for finite-energy properties of the Fermi-Hubbard model on an ion-trap quantum computer, PRX Quantum 5, 030323 (2024).
O. Katz, M. Cetina, and C. Monroe, Programmable $n$ -body interactions with trapped ions, PRX Quantum 4, 030311 (2023).
M. Iqbal, N. Tantivasadakarn, R. Verresen, S. L. Campbell, J. M. Dreiling, C. Figgatt, J. P. Gaebler, J. Johansen, M. Mills, S. A. Moses, J. M. Pino, A. Ransford, M. Rowe, P. Siegfried, R. P. Stutz, M. Foss-Feig, A. Vishwanath, and H. Dreyer, Non-Abelian topological order and anyons on a trapped-ion processor, Nature 626, 505 (2024).
S. Pezzagna and J. Meijer, Quantum computer based on color centers in diamond, Appl. Phys. Rev. 8, 011308 (2021).
G. Burkard, T. D. Ladd, A. Pan, J. M. Nichol, and J. R. Petta, Semiconductor spin qubits, Rev. Mod. Phys. 95, 025003 (2023).
M. Reagor, H. Paik, G. Catelani, L. Sun, C. Axline, E. Holland, I. M. Pop, N. A. Masluk, T. Brecht, L. Frunzio, M. H. Devoret, L. Glazman, and R. J. Schoelkopf, Reaching 10 ms single photon lifetimes for superconducting aluminum cavities, Appl. Phys. Lett. 102, 192604 (2013).
M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, Quantum memory with millisecond coherence in circuit QED, Phys. Rev. B 94, 014506 (2016).
A. Romanenko, R. Pilipenko, S. Zorzetti, D. Frolov, M. Awida, S. Belomestnykh, S. Posen, and A. Grassellino, Three-dimensional superconducting resonators at $t = 20$ mk with photon lifetimes up to $τ = 2$ s, Phys. Rev. Appl. 13, 034032 (2020).
O. Milul, B. Guttel, U. Goldblatt, S. Hazanov, L. M. Joshi, D. Chausovsky, N. Kahn, E. Çiftyürek, F. Lafont, and S. Rosenblum, Superconducting cavity qubit with tens of milliseconds single-photon coherence time, PRX Quantum 4, 030336 (2023).
A. Copetudo, C. Y. Fontaine, F. Valadares, and Y. Y. Gao, Shaping photons: Quantum information processing with bosonic cQED, Appl. Phys. Lett. 124, 080502 (2024).
A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
A. Blais, S. M. Girvin, and W. D. Oliver, Quantum information processing and quantum optics with circuit quantum electrodynamics, Nat. Phys. 16, 247 (2020).
Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems, Rev. Mod. Phys. 85, 623 (2013).
U. L. Andersen, J. S. Neergaard-Nielsen, P. Van Loock, and A. Furusawa, Hybrid discrete- and continuous-variable quantum information, Nat. Phys. 11, 713 (2015).
G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, Quantum technologies with hybrid systems, Proc. Natl. Acad. Sci. 112, 3866 (2015).
S. Lloyd and S. L. Braunstein, Quantum computation over continuous variables, Phys. Rev. Lett. 82, 1784 (1999).
R. Dutta, B. Allen, N. P. Vu, C. Xu, K. Liu, F. Miao, B. Wang, A. Surana, C. Wang, Y. Ding, and V. S. Batista, Solving constrained optimization problems using hybrid qubit-qumode quantum devices, ArXiv:2501.11735.
S. B. Bravyi and A. Y. Kitaev, Fermionic quantum computation, Ann. Phys. 298, 210 (2002).
T. E. O’Brien, P. Rożek, and A. R. Akhmerov, Majorana-based fermionic quantum computation, Phys. Rev. Lett. 120, 220504 (2018).
D. González-Cuadra, D. Bluvstein, M. Kalinowski, R. Kaubruegger, N. Maskara, P. Naldesi, T. V. Zache, A. M. Kaufman, M. D. Lukin, H. Pichler, B. Vermersch, J. Ye, and P. Zoller, Fermionic quantum processing with programmable neutral atom arrays, PNAS 120, e2304294120 (2023).
A. Rad, A. Schuckert, E. Crane, G. Nambiar, F. Fei, J. Wyrick, R. M. Silver, M. Hafezi, Z. Davoudi, and M. J. Gullans, Analog quantum simulator of a quantum field theory with fermion-spin systems in silicon, ArXiv:2407.03419.
A. Schuckert, E. Crane, A. V. Gorshkov, M. Hafezi, and M. J. Gullans, Fermion-qubit fault-tolerant quantum computing, ArXiv:2411.08955.
S. Girvin, in Proc. Int. Sch. Phys. “Enrico Fermi,” Vol. 209: Quantum Fluids of Light and Matter (IOS Press Ebook, Amsterdam, 2025), p. 23.
E. Crane, K. C. Smith, A. Schuckert, T. Tomesh, A. Eickbusch, J. M. Martyn, S. Kühn, L. Funcke, N. Wiebe, M. A. DeMarco, I. L. Chuang, and S. Girvin, Hybrid oscillator-qubit quantum processors: Simulating fermions, bosons, and gauge fields, ArXiv:2409.03747.
E. Chiari, W. Makhlouf, L. Pepe, E. Koridon, J. Klein, B. Senjean, B. Lasorne, and S. Yalouz, Ab initio polaritonic chemistry on diverse quantum computing platforms: Qubit, qudit, and hybrid qubit-qumode architectures, ArXiv:2506.12504.
J. Hennessy, D. Patterson, and K. Asanović, Computer Architecture: A Quantitative Approach (Elsevier Science, Amsterdam, 2012).
M. Davis, R. Sigal, and E. J. Weyuker, Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science (Elsevier, Amsterdam, 1994).
M. E. Beverland, P. Murali, M. Troyer, K. M. Svore, T. Hoeffler, V. Kliuchnikov, G. H. Low, M. Soeken, A. Sundaram, and A. Vaschillo, Assessing requirements to scale to practical quantum advantage, ArXiv:2211.07629.
S. Bravyi, O. Dial, J. M. Gambetta, D. Gil, and Z. Nazario, The future of quantum computing with superconducting qubits, J. Appl. Phys. 132, 160902 (2022).
D. Hall, Microprocessors and Interfacing: Programming and Hardware, McGraw-Hill Computer Science Series (McGraw-Hill, Inc., Glencoe, 1992).
W. Stallings, Computer Organization and Architecture: Designing for Performance, William Stallings Books on Computer and Data Communications Technology (Pearson Education, 2003.)
J. M. Martyn, Z. M. Rossi, A. K. Tan, and I. L. Chuang, Grand unification of quantum algorithms, PRX Quantum 2, 040203 (2021).
N. Killoran, J. Izaac, N. Quesada, V. Bergholm, M. Amy, and C. Weedbrook, strawberry fields: A software platform for photonic quantum computing, Quantum 3, 129 (2019).
J. Zhou, Y. Liu, Y. Shi, A. Javadi-Abhari, and G. Li, Bosehedral: Compiler Optimization for Bosonic Quantum Computing, in 2024 ACM/IEEE 51st Annual International Symposium on Computer Architecture (ISCA) (Buenos Aires, Argentina, 2024), p. 261.
T. J. Stavenger, E. Crane, K. C. Smith, C. T. Kang, S. M. Girvin, and N. Wiebe, C2QA - Bosonic Qiskit, in 2022 IEEE High Performance Extreme Computing Conference (HPEC) (Waltham, MA, USA, 2022), p. 1.
bosonic qiskit Github Repository, 2022, https://github.com/C2QA/bosonic-qiskit.
K. C. Smith, E. Crane, T. Stavenger, and S. M. Girvin, Introducing bosonic qiskit: A package for simulating bosonic and hybrid qubit-bosonic circuits (2023), https://medium.com/qiskit/introducing-bosonic-qiskit-a-package-for-simulating-bosonic-and-hybrid-qubit-bosonic-circuits-1e1e528287bb.
Z. Liu and K. C. Smith, Simulating a bosonic error correcting code using bosonic qiskit (2024), https://www.ibm.com/quantum/blog/bosonic-error-correction.
Z. Chen, J. Li, M. Guo, H. Chen, Z. Li, J. Bierman, Y. Huang, H. Zhou, Y. Liu, and E. Z. Zhang, genesis: A compiler framework for Hamiltonian simulation on hybrid CV-DV quantum computers, ArXiv:2505.13683.
E. Decker, E. Gustafson, E. McKinney, A. K. Jones, L. Goetz, A. Li, A. Schuckert, S. Stein, G. Li, and E. Crane, symbolic hamiltonian compiler for hybrid qubit-boson processors, ArXiv:2506.00215.
X. Pan, P. Song, and Y. Y. Gao, Continuous-variable quantum computation in circuit QED, Chin. Phys. Lett. 40, 110303 (2023).
S. Krastanov, V. V. Albert, C. Shen, C.-L. Zou, R. W. Heeres, B. Vlastakis, R. J. Schoelkopf, and L. Jiang, Universal control of an oscillator with dispersive coupling to a qubit, Phys. Rev. A 92, 040303(R) (2015).
A. Eickbusch, V. Sivak, A. Z. Ding, S. S. Elder, S. R. Jha, J. Venkatraman, B. Royer, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, Fast universal control of an oscillator with weak dispersive coupling to a qubit, Nat. Phys. 18, 1464 (2022).
B. Mischuck and K. Mølmer, Qudit quantum computation in the Jaynes-Cummings model, Phys. Rev. A 87, 022341 (2013).
Y. Liu, J. Sinanan-Singh, M. T. Kearney, G. Mintzer, and I. L. Chuang, Constructing qudits from infinite-dimensional oscillators by coupling to qubits, Phys. Rev. A 104, 032605 (2021).
N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Extending the lifetime of a quantum bit with error correction in superconducting circuits, Nature 536, 441 (2016).
P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys-Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, L. Frunzio, M. Mirrahimi, and M. H. Devoret, Quantum error correction of a qubit encoded in grid states of an oscillator, Nature 584, 368 (2020).
V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsioutsios, S. Ganjam, A. Miano, B. L. Brock, A. Z. Ding, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, Real-time quantum error correction beyond break-even, Nature 616, 50 (2023).
B. L. Brock, S. Singh, A. Eickbusch, V. V. Sivak, A. Z. Ding, L. Frunzio, S. M. Girvin, and M. H. Devoret, Quantum error correction of qudits beyond break-even, Nature 641, 612 (2025).
Y. Ma, Y. Xu, X. Mu, W. Cai, L. Hu, W. Wang, X. Pan, H. Wang, Y. P. Song, C.-L. Zou, and L. Sun, Error-transparent operations on a logical qubit protected by quantum error correction, Nat. Phys. 16, 827 (2020).
Z. Ni, S. Li, X. Deng, Y. Cai, L. Zhang, W. Wang, Z.-B. Yang, H. Yu, F. Yan, S. Liu, C.-L. Zou, L. Sun, S.-B. Zheng, Y. Xu, and D. Yu, Beating the break-even point with a discrete-variable-encoded logical qubit, Nature 616, 56 (2023).
J. M. Gertler, B. Baker, J. Li, S. Shirol, J. Koch, and C. Wang, Protecting a bosonic qubit with autonomous quantum error correction, Nature 590, 243 (2021).
Y. Lu, A. Maiti, J. W. O. Garmon, S. Ganjam, Y. Zhang, J. Claes, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, High-fidelity parametric beamsplitting with a parity-protected converter, Nat. Commun. 14, 5767 (2023).
J. Leppäkangas, P. Stadler, D. Golubev, R. Reiner, J.-M. Reiner, S. Zanker, N. Wurz, M. Renger, J. Verjauw, D. Gusenkova, S. Pogorzalek, F. Vigneau, P. Yang, W. Kindel, H.-S. Ku, F. Deppe, and M. Marthaler, Quantum algorithms for simulating systems coupled to bosonic modes using a hybrid resonator-qubit quantum computer, ArXiv:2503.11507.
R. T. Sutherland and R. Srinivas, Universal hybrid quantum computing in trapped ions, Phys. Rev. A 104, 032609 (2021).
H. C. J. Gan, G. Maslennikov, K.-W. Tseng, C. Nguyen, and D. Matsukevich, Hybrid quantum computing with conditional beam splitter gate in trapped ion system, Phys. Rev. Lett. 124, 170502 (2020).
D. J. Gorman, P. Schindler, S. Selvarajan, N. Daniilidis, and H. Häffner, Two-mode coupling in a single-ion oscillator via parametric resonance, Phys. Rev. A 89, 062332 (2014).
P.-Y. Hou, J. J. Wu, S. D. Erickson, D. C. Cole, G. Zarantonello, A. D. Brandt, S. Geller, A. Kwiatkowski, S. Glancy, E. Knill, A. C. Wilson, D. H. Slichter, and D. Leibfried, Coherent coupling and non-destructive measurement of trapped-ion mechanical oscillators, Nat. Phys. 20, 1636 (2024).
O. Katz and C. Monroe, Programmable quantum simulations of bosonic systems with trapped ions, Phys. Rev. Lett. 131, 033604 (2023).
A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation, Phys. Rev. A 69, 062320 (2004).
A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics, Nature 431, 162 (2004).
A. Sørensen and K. Mølmer, Quantum computation with ions in thermal motion, Phys. Rev. Lett. 82, 1971 (1999).
K. Mølmer and A. Sørensen, Multiparticle entanglement of hot trapped ions, Phys. Rev. Lett. 82, 1835 (1999).
G. Milburn, S. Schneider, and D. James, Ion trap quantum computing with warm ions, Fortschr. Phys. 48, 801 (2000).
A. Sørensen and K. Mølmer, Entanglement and quantum computation with ions in thermal motion, Phys. Rev. A 62, 022311 (2000).
O. Katz, M. Cetina, and C. Monroe, $n$ -body interactions between trapped ion qubits via spin-dependent squeezing, Phys. Rev. Lett. 129, 063603 (2022).
O. Băzăvan, S. Saner, D. Webb, E. Ainley, P. Drmota, D. Nadlinger, G. Araneda, D. Lucas, C. Ballance, and R. Srinivas, Squeezing, trisqueezing, and quadsqueezing in a spin-oscillator system, ArXiv:2403.05471.
O. Katz, L. Feng, A. Risinger, C. Monroe, and M. Cetina, Demonstration of three- and four-body interactions between trapped-ion spins, Nat. Phys. 19, 1452 (2023).
J. Whitlow, Z. Jia, Y. Wang, C. Fang, J. Kim, and K. R. Brown, Quantum simulation of conical intersections using trapped ions, Nat. Chem. 15, 1509 (2023).
A. Serafini, A. Retzker, and M. B. Plenio, Manipulating the quantum information of the radial modes of trapped ions: Linear phononics, entanglement generation, quantum state transmission and non-locality tests, New J. Phys. 11, 023007 (2009).
S. C. Burd, R. Srinivas, H. M. Knaack, W. Ge, A. C. Wilson, D. J. Wineland, D. Leibfried, J. J. Bollinger, D. T. C. Allcock, and D. H. Slichter, Quantum amplification of boson-mediated interactions, Nat. Phys. 17, 898 (2021).
V. So, M. D. Suganthi, A. Menon, M. Zhu, R. Zhuravel, H. Pu, P. G. Wolynes, J. N. Onuchic, and G. Pagano, Trapped-ion quantum simulation of electron transfer models with tunable dissipation, Sci. Adv. 10, eads8011 (2024).
M. Mielenz, H. Kalis, M. Wittemer, F. Hakelberg, U. Warring, R. Schmied, M. Blain, P. Maunz, D. L. Moehring, D. Leibfried, et al., Arrays of individually controlled ions suitable for two-dimensional quantum simulations, Nat. Commun. 7, 11839 (2016).
S. Jain, J. Alonso, M. Grau, and J. P. Home, Scalable arrays of micro-Penning traps for quantum computing and simulation, Phys. Rev. X 10, 031027 (2020).
S. Jain, T. Sägesser, P. Hrmo, C. Torkzaban, M. Stadler, R. Oswald, C. Axline, A. Bautista-Salvador, C. Ospelkaus, D. Kienzler, et al., Penning micro-trap for quantum computing, Nature 627, 510 (2024).
F. Hakelberg, P. Kiefer, M. Wittemer, U. Warring, and T. Schaetz, Interference in a prototype of a two-dimensional ion trap array quantum simulator, Phys. Rev. Lett. 123, 100504 (2019).
R. C. Sterling, H. Rattanasonti, S. Weidt, K. Lake, P. Srinivasan, S. Webster, M. Kraft, and W. K. Hensinger, Fabrication and operation of a two-dimensional ion-trap lattice on a high-voltage microchip, Nat. Commun. 5, 3637 (2014).
D. Palani, F. Hasse, P. Kiefer, F. Boeckling, J.-P. Schroeder, U. Warring, and T. Schaetz, High-fidelity transport of trapped-ion qubits in a multilayer array, Phys. Rev. A 107, L050601 (2023).
T. Olsacher, L. Postler, P. Schindler, T. Monz, P. Zoller, and L. M. Sieberer, Scalable and parallel tweezer gates for quantum computing with long ion strings, PRX Quantum 1, 020316 (2020).
D. Schwerdt, L. Peleg, Y. Shapira, N. Priel, Y. Florshaim, A. Gross, A. Zalic, G. Afek, N. Akerman, A. Stern, A. B. Kish, and R. Ozeri, Scalable architecture for trapped-ion quantum computing using rf traps and dynamic optical potentials, Phys. Rev. X 14, 041017 (2024).
A. Schuckert, O. Katz, L. Feng, E. Crane, A. De, M. Hafezi, A. V. Gorshkov, and C. Monroe, Observation of a finite-energy phase transition in a one-dimensional quantum simulator, Nat. Phys. 21, 374 (2025).
M. K. Joshi, F. Kranzl, A. Schuckert, I. Lovas, C. Maier, R. Blatt, M. Knap, and C. F. Roos, Observing emergent hydrodynamics in a long-range quantum magnet, Science 376, 720 (2022).
W. Chen, J. Gan, J.-N. Zhang, D. Matuskevich, and K. Kim, Quantum computation and simulation with vibrational modes of trapped ions, Chin. Phys. B 30, 060311 (2021).
W. Chen, Y. Lu, S. Zhang, K. Zhang, G. Huang, M. Qiao, X. Su, J. Zhang, J.-N. Zhang, L. Banchi, M. S. Kim, and K. Kim, Scalable and programmable phononic network with trapped ions, Nat. Phys. 19, 877 (2023).
S. Saner, O. Băzăvan, D. J. Webb, G. Araneda, D. M. Lucas, C. J. Ballance, and R. Srinivas, Generating arbitrary superpositions of nonclassical quantum harmonic oscillator states, ArXiv:2409.03482.
K. Toyoda, R. Hiji, A. Noguchi, and S. Urabe, Hong-Ou-Mandel interference of two phonons in trapped ions, Nature 527, 74 (2015).
C.-H. Nguyen, K.-W. Tseng, G. Maslennikov, H. Gan, and D. Matsukevich, Experimental swap test of infinite dimensional quantum states, ArXiv:2103.10219.
J. Y. Araz, M. Grau, J. Montgomery, and F. Ringer, Hybrid quantum simulations with qubits and qumodes on trapped-ion platforms, Phys. Rev. A 112, 012620 (2025).
C. Flühmann, T. L. Nguyen, M. Marinelli, V. Negnevitsky, K. Mehta, and J. P. Home, Encoding a qubit in a trapped-ion mechanical oscillator, Nature 566, 513 (2019).
B. de Neeve, T.-L. Nguyen, T. Behrle, and J. P. Home, Error correction of a logical grid state qubit by dissipative pumping, Nat. Phys. 18, 296 (2022).
C. Flühmann and J. P. Home, Direct characteristic-function tomography of quantum states of the trapped-ion motional oscillator, Phys. Rev. Lett. 125, 043602 (2020).
V. G. Matsos, C. H. Valahu, T. Navickas, A. D. Rao, M. J. Millican, X. C. Kolesnikow, M. J. Biercuk, and T. R. Tan, Robust and deterministic preparation of bosonic logical states in a trapped ion, Phys. Rev. Lett. 133, 050602 (2024).
M. Kang, H. Nuomin, S. N. Chowdhury, J. L. Yuly, K. Sun, J. Whitlow, J. Valdiviezo, Z. Zhang, P. Zhang, D. N. Beratan, and K. R. Brown, Seeking a quantum advantage with trapped-ion quantum simulations of condensed-phase chemical dynamics, Nat. Rev. Chem. 8, 340 (2024).
C. H. Valahu, V. C. Olaya-Agudelo, R. J. MacDonell, T. Navickas, A. D. Rao, M. J. Millican, J. B. Pérez-Sánchez, J. Yuen-Zhou, M. J. Biercuk, C. Hempel, T. R. Tan, and I. Kassal, Direct observation of geometric-phase interference in dynamics around a conical intersection, Nat. Chem. 15, 1503 (2023).
T. Nguyen, D. Lyakh, R. C. Pooser, T. S. Humble, T. Proctor, and M. Sarovar, Quantum circuit transformations with a multi-level intermediate representation compiler, ArXiv:2112.10677.
Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, Diffraction-limited optics for single-atom manipulation, Phys. Rev. A 75, 013406 (2007).
A. M. Kaufman, B. J. Lester, and C. A. Regal, Cooling a single atom in an optical tweezer to its quantum ground state, Phys. Rev. X 2, 041014 (2012).
A. L. Shaw, P. Scholl, R. Finkelstein, R. B.-S. Tsai, J. Choi, and M. Endres, Erasure cooling, control, and hyperentanglement of motion in optical tweezers, Science 388, 845 (2025).
A. M. Kaufman, B. J. Lester, C. M. Reynolds, M. L. Wall, M. Foss-Feig, K. R. A. Hazzard, A. M. Rey, and C. A. Regal, Two-particle quantum interference in tunnel-coupled optical tweezers, Science 345, 306 (2014).
A. W. Young, S. Geller, W. J. Eckner, N. Schine, S. Glancy, E. Knill, and A. M. Kaufman, An atomic boson sampler, Nature 629, 311 (2024).
L. H. Bohnmann, D. F. Locher, J. Zeiher, and M. Müller, Bosonic quantum error correction with neutral atoms in optical dipole traps, Phys. Rev. A 111, 022432 (2025).
P. T. Grochowski, H. Pichler, C. A. Regal, and O. Romero-Isart, Quantum control of continuous systems via nonharmonic potential modulation, Quantum 9, 1824 (2025).
A. V. Aho, M. S. Lam, R. Sethi, and J. D. Ullman, Compilers: Principles, Techniques, & Tools (Pearson Education India, Boston, 2007), 2nd ed.
S. Khatri and M. M. Wilde, Principles of quantum communication theory: A modern approach, ArXiv:2011.04672.
P. Murali, D. M. Debroy, K. R. Brown, and M. Martonosi, Architecting noisy intermediate-scale trapped ion quantum computers, in Proceedings of the ACM/IEEE 47th Annual International Symposium on Computer Architecture, ISCA ’20 (IEEE Press, 2020), p. 529.
M. Oskin, F. T. Chong, I. L. Chuang, and J. Kubiatowicz, Building quantum wires: the long and the short of it, SIGARCH Comput. Archit. News 31, 374 (2003).
S. Xu, C. T. Hann, B. Foxman, S. M. Girvin, and Y. Ding, in Proceedings of the 56th Annual IEEE/ACM International Symposium on Microarchitecture, MICRO ’23 (ACM, New York, 2023), p. 526.
V. Giovannetti, S. Lloyd, and L. Maccone, Quantum random access memory, Phys. Rev. Lett. 100, 160501 (2008).
V. Giovannetti, S. Lloyd, and L. Maccone, Architectures for a quantum random access memory, Phys. Rev. A 78, 052310 (2008).
S. Jaques and A. G. Rattew, QRAM: A survey and critique, Quantum 9, 1922 (2025).
C. T. Hann, G. Lee, S. M. Girvin, and L. Jiang, Resilience of quantum random access memory to generic noise, PRX Quantum 2, 020311 (2021).
F.-Y. Hong, Y. Xiang, Z.-Y. Zhu, L.-Z. Jiang, and L.-N. Wu, Robust quantum random access memory, Phys. Rev. A 86, 010306(R) (2012).
J. Liu, C. T. Hann, and L. Jiang, Data centers with quantum random access memory and quantum networks, Phys. Rev. A 108, 032610 (2023).
O. D. Matteo, V. Gheorghiu, and M. Mosca, Fault-tolerant resource estimation of quantum random-access memories, IEEE Trans. Quantum Eng. 1, 1 (2020).
D. K. Weiss, S. Puri, and S. M. Girvin, Qram architectures using superconducting cavities, PRX Quantum 5, 020312 (2024).
K. Phalak, A. Chatterjee, and S. Ghosh, Quantum random access memory for dummies, Sensors 23, 17 (2023).
H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, Measurement-based quantum computation, Nat. Phys. 5, 19 (2009).
J. Preskill, Plug-in quantum software, Nature 402, 357 (1999).
S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005).
qiskit website, https://www.ibm.com/quantum/qiskit, 2025.
H. Buhrman, M. Folkertsma, B. Loff, and N. M. P. Neumann, State preparation by shallow circuits using feed forward, Quantum 8, 1552 (2024).
N. M. P. Neumann, Ph.D. thesis, University of Amsterdam, https://hdl.handle.net/11245.1/8b878351-22bb-4fe9-bb5d-591a41d8a027, 2025.
P. Pham and K. M. Svore, A 2D nearest-neighbor quantum architecture for factoring in polylogarithmic depth, Quantum Inf. Comput. 13, 937 (2013).
D. D. Thaker, T. S. Metodi, A. W. Cross, I. L. Chuang, and F. T. Chong, in Proceedings of the 33rd Annual International Symposium on Computer Architecture, ISCA ’06 (IEEE Computer Society, Washington, DC, 2006), p. 378.
J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Coupling superconducting qubits via a cavity bus, Nature 449, 443 (2007).
C. R. Monroe, R. J. Schoelkopf, and M. D. Lukin, Quantum connections, Sci. Am. 314, 50 (2016).
P. O. Schmidt, T. Rosenband, C. Langer, W. M. Itano, J. C. Bergquist, and D. J. Wineland, Spectroscopy using quantum logic, Science 309, 749 (2005).
I. Strandberg, A. M. Eriksson, B. Royer, M. Kervinen, and S. Gasparinetti, Digital homodyne and heterodyne detection for stationary bosonic modes, Phys. Rev. Lett. 133, 063601 (2024).
B. R. Johnson, M. D. Reed, A. A. Houck, D. I. Schuster, L. S. Bishop, E. Ginossar, J. M. Gambetta, L. DiCarlo, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Quantum non-demolition detection of single microwave photons in a circuit, Nat. Phys. 6, 663 (2010).
C. S. Wang, J. C. Curtis, B. J. Lester, Y. Zhang, Y. Y. Gao, J. Freeze, V. S. Batista, P. H. Vaccaro, I. L. Chuang, L. Frunzio, L. Jiang, S. M. Girvin, and R. J. Schoelkopf, Efficient multiphoton sampling of molecular vibronic spectra on a superconducting bosonic processor, Phys. Rev. X 10, 021060 (2020).
C. T. Hann, S. S. Elder, C. S. Wang, K. Chou, R. J. Schoelkopf, and L. Jiang, Robust readout of bosonic qubits in the dispersive coupling regime, Phys. Rev. A 98, 022305 (2018).
A. V. Dixit, S. Chakram, K. He, A. Agrawal, R. K. Naik, D. I. Schuster, and A. Chou, Searching for dark matter with a superconducting qubit, Phys. Rev. Lett. 126, 141302 (2021).
B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Deterministically encoding quantum information using 100-photon Schrödinger cat states, Science 342, 607 (2013).
B. Yurke, S. L. McCall, and J. R. Klauder, SU(2) and SU(1,1) interferometers, Phys. Rev. A 33, 4033 (1986).
K. M. Backes, et al., A quantum enhanced search for dark matter axions, Nature 590, 238 (2021).
C. Oh, S. Chen, Y. Wong, S. Zhou, H.-Y. Huang, J. A. H. Nielsen, Z.-H. Liu, J. S. Neergaard-Nielsen, U. L. Andersen, L. Jiang, and J. Preskill, Entanglement-enabled advantage for learning a bosonic random displacement channel, Phys. Rev. Lett. 133, 230604 (2024).
S. Aaronson and A. Arkhipov, in Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, USA, 2011), p. 333.
Y.-H. Deng, et al., Gaussian boson sampling with pseudo-photon-number-resolving detectors and quantum computational advantage, Phys. Rev. Lett. 131, 150601 (2023).
M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, Experimental boson sampling, Nat. Photonics 7, 540 (2013).
J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, et al., Boson sampling on a photonic chip, Science 339, 798 (2013).
G. S. Thekkadath, S. Sempere-Llagostera, B. A. Bell, R. B. Patel, M. S. Kim, and I. A. Walmsley, Experimental demonstration of Gaussian boson sampling with displacement, PRX Quantum 3, 020336 (2022).
H.-S. Zhong, L.-C. Peng, Y. Li, Y. Hu, W. Li, J. Qin, D. Wu, W. Zhang, H. Li, L. Zhang, et al., Experimental Gaussian boson sampling, Sci. Bull. 64, 511 (2019).
J. Huh, G. G. Guerreschi, B. Peropadre, J. R. McClean, and A. Aspuru-Guzik, Boson sampling for molecular vibronic spectra, Nat. Photonics 9, 615 (2015).
C. Oh, M. Liu, Y. Alexeev, B. Fefferman, and L. Jiang, Classical algorithm for simulating experimental Gaussian boson sampling, Nat. Phys. 20, 1461 (2024).
H. Qi, D. J. Brod, N. Quesada, and R. García-Patrón, Regimes of classical simulability for noisy Gaussian boson sampling, Phys. Rev. Lett. 124, 100502 (2020).
R. García-Patrón, J. J. Renema, and V. Shchesnovich, Simulating boson sampling in lossy architectures, Quantum 3, 169 (2019).
R. Babbush, D. W. Berry, R. Kothari, R. D. Somma, and N. Wiebe, Exponential quantum speedup in simulating coupled classical oscillators, Phys. Rev. X 13, 041041 (2023).
N. Heurtel, A. Fyrillas, G. d. Gliniasty, R. Le Bihan, S. Malherbe, M. Pailhas, E. Bertasi, B. Bourdoncle, P.-E. Emeriau, R. Mezher, L. Music, N. Belabas, B. Valiron, P. Senellart, S. Mansfield, and J. Senellart, perceval: A software platform for discrete variable photonic quantum computing, Quantum 7, 931 (2023).
C. Liu, M. Wang, S. A. Stein, Y. Ding, and A. Li, Quantum memory: A missing piece in quantum computing units, ArXiv:2309.14432.
S. Stein, S. Sussman, T. Tomesh, C. Guinn, E. Tureci, S. F. Lin, W. Tang, J. Ang, S. Chakram, A. Li, M. Martonosi, F. Chong, A. A. Houck, I. L. Chuang, and M. Demarco, in Proceedings of the 56th Annual IEEE/ACM International Symposium on Microarchitecture, MICRO ’23 (Association for Computing Machinery, New York, 2023), p. 539.
S. Arunachalam, V. Gheorghiu, T. Jochym-O’Connor, M. Mosca, and P. V. Srinivasan, On the robustness of bucket brigade quantum ram, New J. Phys. 17, 123010 (2015).
N. Guo, Y. Huang, T. Mai, S. Patil, C. Cao, M. Seok, S. Sethumadhavan, and Y. Tsividis, Energy-efficient hybrid analog/digital approximate computation in continuous time, IEEE J. Solid-State Circuits 51, 1514 (2016).
Y. Huang, N. Guo, M. Seok, Y. Tsividis, K. Mandli, and S. Sethumadhavan, in Proceedings of the 50th Annual IEEE/ACM International Symposium on Microarchitecture, MICRO ’17 (Association for Computing Machinery, New York, 2017), p. 665.
S. M. Girvin, Introduction to quantum error correction and fault tolerance, SciPost Phys. Lect. Notes, 70 (2023).
J. M. Arrazola, T. R. Bromley, J. Izaac, C. R. Myers, K. Brádler, and N. Killoran, Machine learning method for state preparation and gate synthesis on photonic quantum computers, Quantum Sci. Technol. 4, 024004 (2019).
M. Krenn, J. S. Kottmann, N. Tischler, and A. Aspuru-Guzik, Conceptual understanding through efficient automated design of quantum optical experiments, Phys. Rev. X 11, 031044 (2021).
M. Krenn, M. Erhard, and A. Zeilinger, Computer-inspired quantum experiments, Nat. Rev. Phys. 2, 649 (2020).
A. Cervera-Lierta, M. Krenn, and A. Aspuru-Guzik, Design of quantum optical experiments with logic artificial intelligence, Quantum 6, 836 (2022).
R. Nichols, L. Mineh, J. Rubio, J. C. F. Matthews, and P. A. Knott, Designing quantum experiments with a genetic algorithm, Quantum Sci. Technol. 4, 045012 (2019).
T. Fösel, S. Krastanov, F. Marquardt, and L. Jiang, Efficient cavity control with snap gates, ArXiv:2004.14256.
U. Chabaud and M. Walschaers, Resources for bosonic quantum computational advantage, Phys. Rev. Lett. 130, 090602 (2023).
U. Chabaud, M. Joseph, S. Mehraban, and A. Motamedi, Bosonic quantum computational complexity, ArXiv:2410.04274.
P. Reinhold, S. Rosenblum, W.-L. Ma, L. Frunzio, L. Jiang, and R. J. Schoelkopf, Error-corrected gates on an encoded qubit, Nat. Phys. 16, 822 (2020).
F. T. Chong, D. Franklin, and M. Martonosi, Programming languages and compiler design for realistic quantum hardware, Nature 549, 180 (2017).
B. Heim, M. Soeken, S. Marshall, C. Granade, M. Roetteler, A. Geller, M. Troyer, and K. Svore, Quantum programming languages, Nat. Rev. Phys. 2, 709 (2020).
A. JavadiAbhari, S. Patil, D. Kudrow, J. Heckey, A. Lvov, F. T. Chong, and M. Martonosi, ScaffCC, Parallel Comput. 45, 2 (2015).
Cirq Developers, Cirq, 2024, https://doi.org/10.5281/zenodo.11398048.
K. Svore, A. Geller, M. Troyer, J. Azariah, C. Granade, B. Heim, V. Kliuchnikov, M. Mykhailova, A. Paz, and M. Roetteler, in Proceedings of the Real World Domain Specific Languages Workshop 2018, RWDSL ’18 (Association for Computing Machinery, New York, 2018).
T. Häner, D. S. Steiger, K. Svore, and M. Troyer, A software methodology for compiling quantum programs, Quantum Sci. Technol. 3, 020501 (2018).
A. Cross, A. Javadi-Abhari, T. Alexander, N. De Beaudrap, L. S. Bishop, S. Heidel, C. A. Ryan, P. Sivarajah, J. Smolin, J. M. Gambetta, et al., OpenQASM 3: A broader and deeper quantum assembly language, ACM Trans. Quantum Comput. 3, 1 (2022).
Quantum Intermediate Representation, 2020, https://devblogs.microsoft.com/qsharp/introducing-quantum-intermediate-representation-qir/.
Quantum Intermediate Representation Alliance, 2022, https://github.com/qir-alliance.
R. S. Smith, M. J. Curtis, and W. J. Zeng, A practical quantum instruction set architecture, ArXiv:1608.03355.
Superconducting microwave cavity modes have weak amplitude damping (photon loss) but very little intrinsic dephasing. The mechanical modes in ion traps tend to suffer from dephasing but have little amplitude damping.
We conservatively define memory break-even as the logical qubit lifetime exceeding the lifetime of the best quantum component among the physical qubits comprising the logical qubit.
L. Hu, Y. Ma, W. Cai, X. Mu, Y. Xu, W. Wang, Y. Wu, H. Wang, Y. P. Song, C.-L. Zou, S. M. Girvin, L.-M. Duan, and L. Sun, Quantum error correction and universal gate set operation on a binomial bosonic logical qubit, Nat. Phys. 15, 503 (2019).
P. Jordan and E. Wigner, Über das Paulische Äquivalenzverbot, Z. Phys. 47, 631 (1928).
G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Quantum algorithms for fermionic simulations, Phys. Rev. A 64, 022319 (2001).
G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Erratum: Quantum algorithms for fermionic simulations [Phys. Rev. A 64, 022319 2001)], Phys. Rev. A 65, 029902(E) (2002).
J. T. Seeley, M. J. Richard, and P. J. Love, The Bravyi-Kitaev transformation for quantum computation of electronic structure, J. Chem. Phys. 137, 224109 (2012).
V. c. v. Havlíček, M. Troyer, and J. D. Whitfield, Operator locality in the quantum simulation of fermionic models, Phys. Rev. A 95, 032332 (2017).
N. P. Sawaya and J. Huh, Quantum algorithm for calculating molecular vibronic spectra, J. Chem. Phys. Lett. 10, 3586 (2019).
A. F. Shaw, P. Lougovski, J. R. Stryker, and N. Wiebe, Quantum algorithms for simulating the lattice Schwinger model, Quantum 4, 306 (2020).
M. Soeken, M. Roetteler, N. Wiebe, and G. De Micheli, in Proceedings of the 54th Annual Design Automation Conference 2017, DAC ’17 (Association for Computing Machinery, New York, 2017).
M. K. Bhaskar, S. Hadfield, A. Papageorgiou, and I. Petras, Quantum algorithms and circuits for scientific computing, Quant. Inf. Comput. 16, 197 (2016).
C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, Simulating the vibrational quantum dynamics of molecules using photonics, Nature 557, 660 (2018).
Y. Shen, Y. Lu, K. Zhang, J. Zhang, S. Zhang, J. Huh, and K. Kim, Quantum optical emulation of molecular vibronic spectroscopy using a trapped-ion device, Chem. Sci. 9, 836 (2018).
L. Hu, Y.-C. Ma, Y. Xu, W.-T. Wang, Y.-W. Ma, K. Liu, H.-Y. Wang, Y.-P. Song, M.-H. Yung, and L.-Y. Sun, Simulation of molecular spectroscopy with circuit quantum electrodynamics, Sci. Bull. 63, 293 (2018).
C. S. Wang, N. E. Frattini, B. J. Chapman, S. Puri, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, Observation of wave-packet branching through an engineered conical intersection, Phys. Rev. X 13, 011008 (2023).
L. G. Lutterbach and L. Davidovich, Method for direct measurement of the Wigner function in cavity QED and ion traps, Phys. Rev. Lett. 78, 2547 (1997).
P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, Direct measurement of the Wigner function of a one-photon Fock state in a cavity, Phys. Rev. Lett. 89, 200402 (2002).
M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. O’Connell, D. Sank, J. Wenner, J. M. Martinis, et al., Synthesizing arbitrary quantum states in a superconducting resonator, Nature 459, 546 (2009).
S. Haroche and J.-M. Raimond, Exploring the Quantum (Oxford University Press, Oxford, 2006).
D. F. Walls and G. J. Milburn, Quantum Optics (Springer Science & Business Media, London, 2007).
B. Vlastakis, A. Petrenko, N. Ofek, L. Sun, Z. Leghtas, K. Sliwa, Y. Liu, M. Hatridge, J. Blumoff, L. Frunzio, M. Mirrahimi, L. Jiang, M. H. Devoret, and R. J. Schoelkopf, Characterizing entanglement of an artificial atom and a cavity cat state with Bell’s inequality, Nat. Commun. 6, 8970 (2015).
Azure quantum documentation, https://learn.microsoft.com/en-us/azure/quantum/, 2025.
H. C. P. Kendell, G. Ferranti, and C. A. Weidner, Deterministic generation of highly squeezed GKP states in ultracold atoms, APL Quantum 1, 026109 (2024).
A. Krasnok, P. Dhakal, A. Fedorov, P. Frigola, M. Kelly, and S. Kutsaev, Superconducting microwave cavities and qubits for quantum information systems, Appl. Phys. Rev. 11, 011302 (2024).
P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, A quantum engineer’s guide to superconducting qubits, Appl. Phys. Rev. 6, 021318 (2019).
S. M. Girvin, in Circuit QED: Superconducting Qubits Coupled to Microwave Photons, in Proceedings of the 2011 Les Houches Summer School on Quantum Machines, edited by M. H. Devoret, R. Schoelkopf, B. Huard, and C. Leticia (Oxford University Press, Oxford, 2014).
S. L. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77, 513 (2005).
S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, Efficient classical simulation of continuous variable quantum information processes, Phys. Rev. Lett. 88, 097904 (2002).
A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy, S. L. Braunstein, and G. J. Milburn, Schrödinger cats and their power for quantum information processing, J. Opt. B: Quantum Semiclassical Opt. 6, S828 (2004).
T. Aoki, G. Takahashi, T. Kajiya, J.-i. Yoshikawa, S. L. Braunstein, P. van Loock, and A. Furusawa, Quantum error correction beyond qubits, Nat. Phys. 5, 541 (2009).
C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012).
N. Liu, J. Thompson, C. Weedbrook, S. Lloyd, V. Vedral, M. Gu, and K. Modi, Power of one qumode for quantum computation, Phys. Rev. A 93, 052304 (2016).
V. V. Albert, C. Shu, S. Krastanov, C. Shen, R.-B. Liu, Z.-B. Yang, R. J. Schoelkopf, M. Mirrahimi, M. H. Devoret, and L. Jiang, Holonomic quantum control with continuous variable systems, Phys. Rev. Lett. 116, 140502 (2016).
D. D. Stancil and G. T. Byrd, Principles of Superconducting Quantum Computers (John Wiley & Sons, Hoboken, New Jersey, 2022).
P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Linear optical quantum computing with photonic qubits, Rev. Mod. Phys. 79, 135 (2007).
E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature 409, 46 (2001).
J. E. Bourassa, R. N. Alexander, M. Vasmer, A. Patil, I. Tzitrin, T. Matsuura, D. Su, B. Q. Baragiola, S. Guha, G. Dauphinais, K. K. Sabapathy, N. C. Menicucci, and I. Dhand, Blueprint for a scalable photonic fault-tolerant quantum computer, Quantum 5, 392 (2021).
S. Bartolucci, P. Birchall, H. Bombín, H. Cable, C. Dawson, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pant, F. Pastawski, T. Rudolph, and C. Sparrow, Fusion-based quantum computation, Nat. Commun. 14, 912 (2023).
K. Sahay, J. Claes, and S. Puri, Tailoring fusion-based error correction for high thresholds to biased fusion failures, Phys. Rev. Lett. 131, 120604 (2023).
S. Takeda and A. Furusawa, Toward large-scale fault-tolerant universal photonic quantum computing, APL Photonics 4, 060902 (2019).
J. L. O’Brien, Optical quantum computing, Science 318, 1567 (2007).
J. D. Teoh, P. Winkel, H. K. Babla, B. J. Chapman, J. Claes, S. J. de Graaf, J. W. O. Garmon, W. D. Kalfus, Y. Lu, A. Maiti, K. Sahay, N. Thakur, T. Tsunoda, S. H. Xue, L. Frunzio, S. M. Girvin, S. Puri, and R. J. Schoelkopf, Dual-rail encoding with superconducting cavities, PNAS 120, e2221736120 (2023).
K. S. Chou, et al., A superconducting dual-rail cavity qubit with erasure-detected logical measurements, Nat. Phys. 20, 1454 (2024).
T. Tsunoda, J. D. Teoh, W. D. Kalfus, S. J. de Graaf, B. J. Chapman, J. C. Curtis, N. Thakur, S. M. Girvin, and R. J. Schoelkopf, Error-detectable bosonic entangling gates with a noisy ancilla, PRX Quantum 4, 020354 (2023).
S. J. de Graaf, S. H. Xue, B. J. Chapman, J. D. Teoh, T. Tsunoda, P. Winkel, J. W. Garmon, K. M. Chang, L. Frunzio, S. Puri, et al., A mid-circuit erasure check on a dual-rail cavity qubit using the joint-photon number-splitting regime of circuit QED, npj Quantum Inf. 11, 1 (2025).
B. J. Chapman, S. J. de Graaf, S. H. Xue, Y. Zhang, J. Teoh, J. C. Curtis, T. Tsunoda, A. Eickbusch, A. P. Read, A. Koottandavida, S. O. Mundhada, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, High-on-off-ratio beam-splitter interaction for gates on bosonically encoded qubits, PRX Quantum 4, 020355 (2023).
D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Resolving photon number states in a superconducting circuit, Nature 445, 515 (2007).
S. S. Elder, C. S. Wang, P. Reinhold, C. T. Hann, K. S. Chou, B. J. Lester, S. Rosenblum, L. Frunzio, L. Jiang, and R. J. Schoelkopf, High-fidelity measurement of qubits encoded in multilevel superconducting circuits, Phys. Rev. X 10, 011001 (2020).
L. Sun, A. Petrenko, Z. Leghtas, B. Vlastakis, G. Kirchmair, K. M. Sliwa, A. Narla, M. Hatridge, S. Shankar, J. Blumoff, L. Frunzio, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Tracking photon jumps with repeated quantum non-demolition parity measurements, Nature 511, 444 (2014).
I. L. Chuang and Y. Yamamoto, Simple quantum computer, Phys. Rev. A 52, 3489 (1995).
A. Koottandavida, I. Tsioutsios, A. Kargioti, C. R. Smith, V. R. Joshi, W. Dai, J. D. Teoh, J. C. Curtis, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, Erasure detection of a dual-rail qubit encoded in a double-post superconducting cavity, Phys. Rev. Lett. 132, 180601 (2024).
A. Kubica, A. Haim, Y. Vaknin, H. Levine, F. Brandão, and A. Retzker, Erasure qubits: Overcoming the $T_{1}$ limit in superconducting circuits, Phys. Rev. X 13, 041022 (2023).
H. Levine, A. Haim, J. S. Hung, N. Alidoust, M. Kalaee, L. DeLorenzo, E. A. Wollack, P. A. Arriola, A. Khalajhedayati, Y. Vaknin, et al., Demonstrating a long-coherence dual-rail erasure qubit using tunable transmons, Phys. Rev. X 14, 011051 (2024).
Y. Wu, S. Kolkowitz, S. Puri, and J. D. Thompson, Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays, Nat. Commun. 13, 4657 (2022).
M. Kang, W. C. Campbell, and K. R. Brown, Quantum error correction with metastable states of trapped ions using erasure conversion, PRX Quantum 4, 020358 (2023).
K. Sahay, J. Jin, J. Claes, J. D. Thompson, and S. Puri, High-threshold codes for neutral-atom qubits with biased erasure errors, Phys. Rev. X 13, 041013 (2023).
S. Ma, G. Liu, P. Peng, B. Zhang, S. Jandura, J. Claes, A. P. Burgers, G. Pupillo, S. Puri, and J. D. Thompson, High-fidelity gates and mid-circuit erasure conversion in an atomic qubit, Nature 622, 279 (2023).
The standard physics notation agrees that $| ↑ ⟩ = | e ⟩$ is the excited state and $| ↓ ⟩ = | g ⟩$ is the ground state. However, in traditional quantum physics the excited states are often labeled as $| e ⟩ = | 1 ⟩$ and $| g ⟩ = | 0 ⟩$ , in contrast to the conventions of quantum information.
A. Macridin, A. C. Y. Li, and P. Spentzouris, Qumode transfer between continuous and discrete variable devices by near-deterministic teleportation protocols, Phys. Rev. A 109, 032419 (2024).
A. Z. Goldberg, A. B. Klimov, G. Leuchs, and L. L. Sanchez-Soto, Covariant operator bases for continuous variables, Quantum 8, 1363 (2024).
A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Introduction to quantum noise, measurement, and amplification, Rev. Mod. Phys. 82, 1155 (2010).
E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749 (1932).
Y.-S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics: Group Theoretical Approach (World Scientific, 1991), Vol. 40.
Here, we are using the standard definitions of position and momentum that have Poisson bracket equal to unity rather than choosing the value of one-half that would correspond in the quantum case to the choice of Wigner units.
U. Chabaud, Ph.D. thesis, Sorbonne Université, https://doi.org/10.48550/arXiv.2102.05227, 2021.
U. Chabaud, D. Markham, and F. Grosshans, Stellar representation of non-Gaussian quantum states, Phys. Rev. Lett. 124, 063605 (2020).
U. Chabaud and S. Mehraban, Holomorphic representation of quantum computations, Quantum 6, 831 (2022).
S. M. Girvin and T. Jach, Formalism for the quantum Hall effect: Hilbert space of analytic functions, Phys. Rev. B 29, 5617 (1984).
J. B. Conway, Functions of One Complex Variable II (Springer Science & Business Media, 2012), Vol. 159.
Uniquely since the overall normalization of the state is fixed and the global phase is irrelevant.
M. Walschaers, Non-Gaussian quantum states and where to find them, PRX Quantum 2, 030204 (2021).
A. Aralov, Émilie Gillet, V. Nguyen, A. Cosentino, M. Walschaers, and M. Frigerio, Photon catalysis for general multimode multi-photon quantum state preparation, ArXiv:2507.19397.
F. Arzani, R. I. Booth, and U. Chabaud, Can effective descriptions of bosonic systems be considered complete? ArXiv:2501.13857.
Y. Tong, V. V. Albert, J. R. McClean, J. Preskill, and Y. Su, Provably accurate simulation of gauge theories and bosonic systems, Quantum 6, 816 (2022).
R. W. Heeres, B. Vlastakis, E. Holland, S. Krastanov, V. V. Albert, L. Frunzio, L. Jiang, and R. J. Schoelkopf, Cavity state manipulation using photon-number selective phase gates, Phys. Rev. Lett. 115, 137002 (2015).
M. Kudra, M. Kervinen, I. Strandberg, S. Ahmed, M. Scigliuzzo, A. Osman, D. P. Lozano, M. O. Tholén, R. Borgani, D. B. Haviland, G. Ferrini, J. Bylander, A. F. Kockum, F. Quijandría, P. Delsing, and S. Gasparinetti, Robust preparation of Wigner-negative states with optimized snap-displacement sequences, PRX Quantum 3, 030301 (2022).
H.-Y. Lo, D. Kienzler, L. de Clercq, M. Marinelli, V. Negnevitsky, B. C. Keitch, and J. P. Home, Spin-motion entanglement and state diagnosis with squeezed oscillator wavepackets, Nature 521, 336 (2015).
D. Kienzler, H.-Y. Lo, B. Keitch, L. de Clercq, F. Leupold, F. Lindenfelser, M. Marinelli, V. Negnevitsky, and J. P. Home, Quantum harmonic oscillator state synthesis by reservoir engineering, Science 347, 53 (2015).
D. Gottesman, in Group22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, edited by S. P. Corney, R. Delbourgo, and P. D. Jarvis (International Press, Cambridge, MA, 1999), p. 32.
S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A 70, 052328 (2004).
A. Mari and J. Eisert, Positive Wigner functions render classical simulation of quantum computation efficient, Phys. Rev. Lett. 109, 230503 (2012).
V. Veitch, N. Wiebe, C. Ferrie, and J. Emerson, Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation, New J. Phys. 15, 013037 (2013).
G. H. Low and I. L. Chuang, Optimal Hamiltonian simulation by quantum signal processing, Phys. Rev. Lett. 118, 010501 (2017).
Z. M. Rossi and I. L. Chuang, Multivariable quantum signal processing (M-QSP): Prophecies of the two-headed oracle, Quantum 6, 811 (2022).
J. M. Martyn, Y. Liu, Z. E. Chin, and I. L. Chuang, Efficient fully-coherent quantum signal processing algorithms for real-time dynamics simulation, J. Chem. Phys. 158, 024106 (2023).
Z. M. Rossi, V. M. Bastidas, W. J. Munro, and I. L. Chuang, Quantum signal processing with continuous variables, ArXiv:2304.14383.
J. Sinanan-Singh, G. L. Mintzer, I. L. Chuang, and Y. Liu, Single-shot quantum signal processing interferometry, Quantum 8, 1427 (2024).
C. Shen, K. Noh, V. V. Albert, S. Krastanov, M. H. Devoret, R. J. Schoelkopf, S. M. Girvin, and L. Jiang, Quantum channel construction with circuit quantum electrodynamics, Phys. Rev. B 95, 134501 (2017).
B. Royer, S. Singh, and S. M. Girvin, Encoding qubits in multimode grid states, PRX Quantum 3, 010335 (2022).
J. Schwinger, Unitary transformations and the action principle, PNAS 46, 883 (1960).
D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001).
C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University Press, 2004).
R. Dassonneville, R. Assouly, T. Peronnin, A. A. Clerk, A. Bienfait, and B. Huard, Dissipative stabilization of squeezing beyond 3 dB in a microwave mode, PRX Quantum 2, 020323 (2021).
Y. Zhang, B. J. Lester, Y. Y. Gao, L. Jiang, R. J. Schoelkopf, and S. M. Girvin, Engineering bilinear mode coupling in circuit QED: Theory and experiment, Phys. Rev. A 99, 012314 (2019).
T.-C. Chien, O. Lanes, C. Liu, X. Cao, P. Lu, S. Motz, G. Liu, D. Pekker, and M. Hatridge, Multiparametric amplification and qubit measurement with a Kerr-free Josephson ring modulator, Phys. Rev. A 101, 042336 (2020).
J.-I. Yoshikawa, Y. Miwa, A. Huck, U. L. Andersen, P. van Loock, and A. Furusawa, Demonstration of a quantum nondemolition SUM gate, Phys. Rev. Lett. 101, 250501 (2008).
D. Marković, S. Jezouin, Q. Ficheux, S. Fedortchenko, S. Felicetti, T. Coudreau, P. Milman, Z. Leghtas, and B. Huard, Demonstration of an effective ultrastrong coupling between two oscillators, Phys. Rev. Lett. 121, 040505 (2018).
S. L. Braunstein and R. I. McLachlan, Generalized squeezing, Phys. Rev. A 35, 1659 (1987).
C. W. Sandbo Chang, C. Sabín, P. Forn-Díaz, F. Quijandría, A. M. Vadiraj, I. Nsanzineza, G. Johansson, and C. M. Wilson, Observation of three-photon spontaneous parametric down-conversion in a superconducting parametric cavity, Phys. Rev. X 10, 011011 (2020).
A. M. Eriksson, T. Sépulcre, M. Kervinen, T. Hillmann, M. Kudra, S. Dupouy, Y. Lu, M. Khanahmadi, J. Yang, C. Castillo-Moreno, P. Delsing, and S. Gasparinetti, Universal control of a bosonic mode via drive-activated native cubic interactions, Nat. Commun. 15, 2512 (2024).
D. Zhang, D. Barral, Y. Zhang, M. Xiao, and K. Bencheikh, Genuine tripartite non-Gaussian entanglement, Phys. Rev. Lett. 130, 093602 (2023).
A. Agustí, C. W. Sandbo Chang, F. Quijandría, G. Johansson, C. M. Wilson, and C. Sabín, Tripartite genuine non-Gaussian entanglement in three-mode spontaneous parametric down-conversion, Phys. Rev. Lett. 125, 020502 (2020).
K. Marshall, R. Pooser, G. Siopsis, and C. Weedbrook, Repeat-until-success cubic phase gate for universal continuous-variable quantum computation, Phys. Rev. A 91, 032321 (2015).
Y. Zheng, O. Hahn, P. Stadler, P. Holmvall, F. Quijandría, A. Ferraro, and G. Ferrini, Gaussian conversion protocols for cubic phase state generation, PRX Quantum 2, 010327 (2021).
G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Observation of quantum state collapse and revival due to the single-photon Kerr effect, Nature 495, 205 (2013).
H.-K. Lau and M. B. Plenio, Universal quantum computing with arbitrary continuous-variable encoding, Phys. Rev. Lett. 117, 100501 (2016).
Y. Y. Gao, B. J. Lester, K. Chou, L. Frunzio, M. H. Devoret, L. Jiang, S. M. Girvin, and R. J. Schoelkopf, Entanglement of bosonic modes through an engineered exchange interaction, Nature 566, 509 (2019).
D. C. McKay, C. J. Wood, S. Sheldon, J. M. Chow, and J. M. Gambetta, Efficient $z$ gates for quantum computing, Phys. Rev. A 96, 022330 (2017).
S. L. Braunstein, Squeezing as an irreducible resource, Phys. Rev. A 71, 055801 (2005).
G. Cariolaro and G. Pierobon, Bloch-Messiah reduction of Gaussian unitaries by Takagi factorization, Phys. Rev. A 94, 062109 (2016).
A. L. Grimsmo and S. Puri, Quantum error correction with the Gottesman-Kitaev-Preskill code, PRX Quantum 2, 020101 (2021).
M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, Amplification and squeezing of quantum noise with a tunable Josephson metamaterial, Nat. Phys. 4, 929 (2008).
X. Pan, J. Schwinger, N.-N. Huang, P. Song, W. Chua, F. Hanamura, A. Joshi, F. Valadares, R. Filip, and Y. Y. Gao, Protecting the quantum interference of cat states by phase-space compression, Phys. Rev. X 13, 021004 (2023).
C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff, Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at microwave frequencies, Nat. Phys. 9, 345 (2013).
Y. Y. Gao, B. J. Lester, Y. Zhang, C. Wang, S. Rosenblum, L. Frunzio, L. Jiang, S. M. Girvin, and R. J. Schoelkopf, Programmable interference between two microwave quantum memories, Phys. Rev. X 8, 021073 (2018).
E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, Generating entangled microwave radiation over two transmission lines, Phys. Rev. Lett. 109, 183901 (2012).
G. Andersson, S. W. Jolin, M. Scigliuzzo, R. Borgani, M. O. Tholén, J. C. Rivera Hernández, V. Shumeiko, D. B. Haviland, and P. Delsing, Squeezing and multimode entanglement of surface acoustic wave phonons, PRX Quantum 3, 010312 (2022).
C. Eichler, D. Bozyigit, C. Lang, M. Baur, L. Steffen, J. M. Fink, S. Filipp, and A. Wallraff, Observation of two-mode squeezing in the microwave frequency domain, Phys. Rev. Lett. 107, 113601 (2011).
A. Roy and M. Devoret, Introduction to parametric amplification of quantum signals with Josephson circuits, C. R. Phys. 17, 740 (2016).
M. Esposito, A. Ranadive, L. Planat, S. Leger, D. Fraudet, V. Jouanny, O. Buisson, W. Guichard, C. Naud, J. Aumentado, F. Lecocq, and N. Roch, Observation of two-mode squeezing in a traveling wave parametric amplifier, Phys. Rev. Lett. 128, 153603 (2022).
I. Tzitrin, J. E. Bourassa, N. C. Menicucci, and K. K. Sabapathy, Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes, Phys. Rev. A 101, 032315 (2020).
S. D. Bartlett and B. C. Sanders, Requirement for quantum computation, J. Mod. Opt. 50, 2331 (2003).
S. Singh, PhD Thesis, Yale University, 2025.
G. C. Wick, The evaluation of the collision matrix, Phys. Rev. 80, 268 (1950).
M. Howard, J. Wallman, V. Veitch, and J. Emerson, Contextuality supplies the “magic” for quantum computation, Nature 510, 351 (2014).
H. Pashayan, J. J. Wallman, and S. D. Bartlett, Estimating outcome probabilities of quantum circuits using quasiprobabilities, Phys. Rev. Lett. 115, 070501 (2015).
J. R. Seddon and E. T. Campbell, Quantifying magic for multi-qubit operations, Proc. R. Soc. A 475, 20190251 (2019).
S. Bravyi, D. Browne, P. Calpin, E. Campbell, D. Gosset, and M. Howard, Simulation of quantum circuits by low-rank stabilizer decompositions, Quantum 3, 181 (2019).
T. Haug and M. S. Kim, Scalable measures of magic resource for quantum computers, PRX Quantum 4, 010301 (2023).
K. Bu, W. Gu, and A. Jaffe, Quantum entropy and central limit theorem, PNAS 120, e2304589120 (2023).
E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Randomized benchmarking of quantum gates, Phys. Rev. A 77, 012307 (2008).
J. Niset, J. Fiurášek, and N. J. Cerf, No-go theorem for Gaussian quantum error correction, Phys. Rev. Lett. 102, 120501 (2009).
C.-H. Wang, K. Noh, J. Lebreuilly, S. M. Girvin, and L. Jiang, Photon-number-dependent Hamiltonian engineering for cavities, Phys. Rev. Appl. 15, 044026 (2021).
K. Jacobs, Engineering quantum states of a nanoresonator via a simple auxiliary system, Phys. Rev. Lett. 99, 117203 (2007).
B. Royer, S. Singh, and S. M. Girvin, Stabilization of finite-energy Gottesman-Kitaev-Preskill states, Phys. Rev. Lett. 125, 260509 (2020).
Z. Leghtas, G. Kirchmair, B. Vlastakis, M. H. Devoret, R. J. Schoelkopf, and M. Mirrahimi, Deterministic protocol for mapping a qubit to coherent state superpositions in a cavity, Phys. Rev. A 87, 042315 (2013).
P. C. Haljan, K.-A. Brickman, L. Deslauriers, P. J. Lee, and C. Monroe, Spin-dependent forces on trapped ions for phase-stable quantum gates and entangled states of spin and motion, Phys. Rev. Lett. 94, 153602 (2005).
R. W. Heeres, P. Reinhold, N. Ofek, L. Frunzio, L. Jiang, M. H. Devoret, and R. J. Schoelkopf, Implementing a universal gate set on a logical qubit encoded in an oscillator, Nat. Commun. 8, 94 (2017).
J. Job, Efficient, direct compilation of SU(N) operations into snap & Displacement gates, ArXiv:2307.11900.
K. Park, P. Marek, and R. Filip, Efficient quantum simulation of nonlinear interactions using snap and Rabi gates, Quantum Sci. Technol. 9, 025004 (2024).
P.-T. Fong and H.-K. Lau, Engineering non-Gaussian bosonic gates through quantum signal processing, ArXiv:2508.20261.
N. E. Frattini, U. Vool, S. Shankar, A. Narla, K. M. Sliwa, and M. H. Devoret, 3-wave mixing Josephson dipole element, Appl. Phys. Lett. 110, 222603 (2017).
T. Hillmann, F. Quijandría, G. Johansson, A. Ferraro, S. Gasparinetti, and G. Ferrini, Universal gate set for continuous-variable quantum computation with microwave circuits, Phys. Rev. Lett. 125, 160501 (2020).
S. Sefi and P. van Loock, How to decompose arbitrary continuous-variable quantum operations, Phys. Rev. Lett. 107, 170501 (2011).
T. W. Borneman, C. E. Granade, and D. G. Cory, Parallel information transfer in a multinode quantum information processor, Phys. Rev. Lett. 108, 140502 (2012).
A. M. Childs and N. Wiebe, Product formulas for exponentials of commutators, J. Math. Phys. 54, 062202 (2013).
Y.-A. Chen, A. M. Childs, M. Hafezi, Z. Jiang, H. Kim, and Y. Xu, Efficient product formulas for commutators and applications to quantum simulation, Phys. Rev. Res. 4, 013191 (2022).
A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Theory of Trotter error with commutator scaling, Phys. Rev. X 11, 011020 (2021).
C. Kang, M. B. Soley, E. Crane, S. M. Girvin, and N. Wiebe, Leveraging Hamiltonian simulation techniques to compile operations on bosonic devices, J. Phys. A: Math. Theor. 58, 175301 (2025).
C.-F. Chen, H.-Y. Huang, R. Kueng, and J. A. Tropp, Concentration for random product formulas, PRX Quantum 2, 040305 (2021).
A. A. Diringer, E. Blumenthal, A. Grinberg, L. Jiang, and S. Hacohen-Gourgy, Conditional-not displacement: Fast multioscillator control with a single qubit, Phys. Rev. X 14, 011055 (2024).
M. Um, J. Zhang, D. Lv, Y. Lu, S. An, J.-N. Zhang, H. Nha, M. S. Kim, and K. Kim, Phonon arithmetic in a trapped ion system, Nat. Commun. 7, 11410 (2016).
S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.-Q. Yin, H. T. Quan, and K. Kim, Experimental test of the quantum Jarzynski equality with a trapped-ion system, Nat. Phys. 11, 193 (2015).
F. Beaudoin, M. P. da Silva, Z. Dutton, and A. Blais, First-order sidebands in circuit QED using qubit frequency modulation, Phys. Rev. A 86, 022305 (2012).
P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, and A. Wallraff, Using sideband transitions for two-qubit operations in superconducting circuits, Phys. Rev. B 79, 180511(R) (2009).
A. Z. Goldberg and A. M. Steinberg, Transcoherent states: Optical states for maximal generation of atomic coherence, PRX Quantum 1, 020306 (2020).
A. Z. Goldberg, A. M. Steinberg, and K. Heshami, Beyond transcoherent states: Field states for effecting optimal coherent rotations on single or multiple qubits, Quantum 7, 963 (2023).
A. B. Özgüler and D. Venturelli, Numerical gate synthesis for quantum heuristics on bosonic quantum processors, Front. Phys. 10, 900612 (2022).
N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, Universal quantum computation with continuous-variable cluster states, Phys. Rev. Lett. 97, 110501 (2006).
M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, Quantum computing with continuous-variable clusters, Phys. Rev. A 79, 062318 (2009).
S. Ghose and B. C. Sanders, Non-Gaussian ancilla states for continuous variable quantum computation via Gaussian maps, J. Mod. Opt. 54, 855 (2007).
B. Q. Baragiola, G. Pantaleoni, R. N. Alexander, A. Karanjai, and N. C. Menicucci, All-Gaussian universality and fault tolerance with the Gottesman-Kitaev-Preskill code, Phys. Rev. Lett. 123, 200502 (2019).
C. Calcluth, N. Reichel, A. Ferraro, and G. Ferrini, Sufficient condition for universal quantum computation using bosonic circuits, PRX Quantum 5, 020337 (2024).
A. Tranter, P. J. Love, F. Mintert, and P. V. Coveney, A comparison of the Bravyi-Kitaev and Jordan-Wigner transformations for the quantum simulation of quantum chemistry, J. Chem. Theory Comput. 14, 5617 (2018).
E. Jaynes and F. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc. IEEE 51, 89 (1963).
H. A. Gersch and G. C. Knollman, Quantum cell model for bosons, Phys. Rev. 129, 959 (1963).
W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42, 1698 (1979).
I. I. Rabi, Space quantization in a gyrating magnetic field, Phys. Rev. 51, 652 (1937).
K. Hepp and E. H. Lieb, On the superradiant phase transition for molecules in a quantized radiation field: The Dicke maser model, Ann. Phys. 76, 360 (1973).
T. Holstein, Studies of polaron motion: Part II. The “small” polaron, Ann. Phys. 8, 343 (1959).
E. A. Arsenault, A. J. Schile, D. T. Limmer, and G. R. Fleming, Vibronic coupling in energy transfer dynamics and two-dimensional electronic-vibrational spectra, J. Chem. Phys. 155, 054201 (2021).
C. Schweizer, F. Grusdt, M. Berngruber, L. Barbiero, E. Demler, N. Goldman, I. Bloch, and M. Aidelsburger, Floquet approach to $Z_{2}$ lattice gauge theories with ultracold atoms in optical lattices, Nat. Phys. 15, 1168 (2019).
J. Murugan, D. Stanford, and E. Witten, More on supersymmetric and 2D analogs of the SYK model, J. High Energy Phys. 2017, 146 (2017).
W. Fu and S. Sachdev, Numerical study of fermion and boson models with infinite-range random interactions, Phys. Rev. B 94, 035135 (2016).
I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59, 799 (1987).
A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory (Courier Corporation, 2012).
D. P. Arovas, E. Berg, S. A. Kivelson, and S. Raghu, The Hubbard model, Annu. Rev. Condens. Matter Phys. 13, 239 (2022).
S. Lloyd, Coherent quantum feedback, Phys. Rev. A 62, 022108 (2000).
R. J. Nelson, Y. Weinstein, D. Cory, and S. Lloyd, Experimental demonstration of fully coherent quantum feedback, Phys. Rev. Lett. 85, 3045 (2000).
J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, et al., The variational quantum eigensolver: A review of methods and best practices, Phys. Rep. 986, 1 (2022).
M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, et al., Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
K. Temme, S. Bravyi, and J. M. Gambetta, Error mitigation for short-depth quantum circuits, Phys. Rev. Lett. 119, 180509 (2017).
Y. Suzuki, S. Endo, K. Fujii, and Y. Tokunaga, Quantum error mitigation as a universal error reduction technique: Applications from the NISQ to the fault-tolerant quantum computing eras, PRX Quantum 3, 010345 (2022).
S. Endo, S. C. Benjamin, and Y. Li, Practical quantum error mitigation for near-future applications, Phys. Rev. X 8, 031027 (2018).
Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y. Li, J. R. McClean, and T. E. O’Brien, Quantum error mitigation, Rev. Mod. Phys. 95, 045005 (2023).
T. Steckmann, D. Luo, Y.-X. Wang, S. R. Muleady, A. Seif, C. Monroe, M. J. Gullans, A. V. Gorshkov, O. Katz, and A. Schuckert, Error mitigation of shot-to-shot fluctuations in analog quantum simulators, ArXiv:2506.16509.
D. Su, R. Israel, K. Sharma, H. Qi, I. Dhand, and K. Brádler, Error mitigation on a near-term quantum photonic device, Quantum 5, 452 (2021).
J. Ang, et al., arquin: Architectures for multinode superconducting quantum computers, ACM Trans. Quantum Comput. 5, 3 (2024).
S. P. Kumar and I. Dhand, Optimal modular architectures for universal linear optics, ArXiv:2001.02012.
A. J. Friedman, C. Yin, Y. Hong, and A. Lucas, Locality and error correction in quantum dynamics with measurement, ArXiv:2206.09929.
L. Piroli, G. Styliaris, and J. I. Cirac, Quantum circuits assisted by local operations and classical communication: Transformations and phases of matter, Phys. Rev. Lett. 127, 220503 (2021).
N. Tantivasadakarn, A. Vishwanath, and R. Verresen, Hierarchy of topological order from finite-depth unitaries, measurement, and feedforward, PRX Quantum 4, 020339 (2023).
K. C. Smith, E. Crane, N. Wiebe, and S. M. Girvin, Deterministic constant-depth preparation of the AKLT state on a quantum processor using fusion measurements, PRX Quantum 4, 020315 (2023).
K. C. Smith, A. Khan, B. K. Clark, S. M. Girvin, and T.-C. Wei, Constant-depth preparation of matrix product states with adaptive quantum circuits, PRX Quantum 5, 030344 (2024).
S. H. Choe and R. König, How to fault-tolerantly realize any quantum circuit with local operations, PRX Quantum 6, 010357 (2025).
M. Arienzo, D. Grinko, M. Kliesch, and M. Heinrich, Bosonic randomized benchmarking with passive transformations, PRX Quantum 6, 020305 (2025).
M. Kliesch and I. Roth, Theory of quantum system certification, PRX Quantum 2, 010201 (2021).
J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, Quantum certification and benchmarking, Nat. Rev. Phys. 2, 382 (2020).
I. Šupić and J. Bowles, Self-testing of quantum systems: A review, Quantum 4, 337 (2020).
U. Chabaud, T. Douce, F. Grosshans, E. Kashefi, and D. Markham, in 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020), Leibniz International Proceedings in Informatics (LIPIcs), edited by S. T. Flammia (Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2020), Vol. 158, p. 3:1.
U. Chabaud, F. Grosshans, E. Kashefi, and D. Markham, Efficient verification of boson sampling, Quantum 5, 578 (2021).
C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, A Schrödinger cat living in two boxes, Science 352, 1087 (2016).
H. Jeon, J. Kang, W. Choi, K. Kim, J. You, and T. Kim, Multimode bosonic state tomography with single-shot joint parity measurement of a trapped ion, ArXiv:2506.12628.
D. Gottesman, Stabilizer codes and quantum error correction, ArXiv:quant-ph/9705052.
E. Magesan, J. M. Gambetta, B. R. Johnson, C. A. Ryan, J. M. Chow, S. T. Merkel, M. P. da Silva, G. A. Keefe, M. B. Rothwell, T. A. Ohki, M. B. Ketchen, and M. Steffen, Efficient measurement of quantum gate error by interleaved randomized benchmarking, Phys. Rev. Lett. 109, 080505 (2012).
R. Harper and S. T. Flammia, Estimating the fidelity of T gates using standard interleaved randomized benchmarking, Quantum Sci. Technol. 2, 015008 (2017).
J. T. Iosue, K. Sharma, M. J. Gullans, and V. V. Albert, Continuous-variable quantum state designs: Theory and applications, Phys. Rev. X 14, 011013 (2024).
J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, and A. G. White, Ancilla-assisted quantum process tomography, Phys. Rev. Lett. 90, 193601 (2003).
U. Chabaud, G. Roeland, M. Walschaers, F. Grosshans, V. Parigi, D. Markham, and N. Treps, Certification of non-Gaussian states with operational measurements, PRX Quantum 2, 020333 (2021).
S. Becker, N. Datta, L. Lami, and C. Rouze, Classical shadow tomography for continuous variables quantum systems, IEEE Trans. Inf. Theory 70, 3427 (2024).
S. Gandhari, V. V. Albert, T. Gerrits, J. M. Taylor, and M. J. Gullans, Precision bounds on continuous-variable state tomography using classical shadows, PRX Quantum 5, 010346 (2024).
K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, Direct measurement of the Wigner function by photon counting, Phys. Rev. A 60, 674 (1999).
M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10, 285 (1975).
A. Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep. Math. Phys. 3, 275 (1972).
N. Johnston and D. W. Kribs, Quantum gate fidelity in terms of Choi matrices, J. Phys. A: Math. Theor. 44, 495303 (2011).
J. M. Chow, J. M. Gambetta, A. D. Córcoles, S. T. Merkel, J. A. Smolin, C. Rigetti, S. Poletto, G. A. Keefe, M. B. Rothwell, J. R. Rozen, M. B. Ketchen, and M. Steffen, Universal quantum gate set approaching fault-tolerant thresholds with superconducting qubits, Phys. Rev. Lett. 109, 060501 (2012).
E. Nielsen, J. K. Gamble, K. Rudinger, T. Scholten, K. Young, and R. Blume-Kohout, Gate set tomography, Quantum 5, 557 (2021).
In practice this is a poor approximation since low-frequency dephasing noise often has a $1 / f$ -like spectrum.
A. Joshi, K. Noh, and Y. Y. Gao, Quantum information processing with bosonic qubits in circuit QED, Quantum Sci. Technol. 6, 033001 (2021).
W. Cai, Y. Ma, W. Wang, C.-L. Zou, and L. Sun, Bosonic quantum error correction codes in superconducting quantum circuits, Fundam. Res. 1, 50 (2021).
M. Mazyar, L. Zaki, V. A. Victor, T. Steven, J. S. Robert, J. Liang, and H. D. Michel, Dynamically protected cat-qubits: A new paradigm for universal quantum computation, New J. Phys. 16, 045014 (2014).
V. V. Albert, K. Noh, K. Duivenvoorden, D. J. Young, R. T. Brierley, P. Reinhold, C. Vuillot, L. Li, C. Shen, S. M. Girvin, B. M. Terhal, and L. Jiang, Performance and structure of single-mode bosonic codes, Phys. Rev. A 97, 032346 (2018).
A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Quantum computing with rotation-symmetric bosonic codes, Phys. Rev. X 10, 011058 (2020).
M. H. Michael, M. Silveri, R. T. Brierley, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, New class of quantum error-correcting codes for a bosonic mode, Phys. Rev. X 6, 031006 (2016).
For a discussion of why one cannot achieve hardware scalability by replacing a large number of physical qubits by a single oscillator mode, see Ref. [733].
D. Lachance-Quirion, M.-A. Lemonde, J. O. Simoneau, L. St-Jean, P. Lemieux, S. Turcotte, W. Wright, A. Lacroix, J. Fréchette-Viens, R. Shillito, et al., Autonomous quantum error correction of Gottesman-Kitaev-Preskill states, Phys. Rev. Lett. 132, 150607 (2024).
S. Rosenblum, P. Reinhold, M. Mirrahimi, L. Jiang, L. Frunzio, and R. J. Schoelkopf, Fault-tolerant detection of a quantum error, Science 361, 266 (2018).
O. Vy, X. Wang, and K. Jacobs, Error-transparent evolution: The ability of multi-body interactions to bypass decoherence, New J. Phys. 15, 053002 (2013).
E. Kapit, Error-transparent quantum gates for small logical qubit architectures, Phys. Rev. Lett. 120, 050503 (2018).
W.-L. Ma, M. Zhang, Y. Wong, K. Noh, S. Rosenblum, P. Reinhold, R. J. Schoelkopf, and L. Jiang, Path-independent quantum gates with noisy ancilla, Phys. Rev. Lett. 125, 110503 (2020).
W.-L. Ma, S.-S. Li, and L. Jiang, Algebraic structure of path-independent quantum control, Phys. Rev. Res. 4, 023102 (2022).
E. Knill, R. Laflamme, R. Martinez, and C. Negrevergne, Benchmarking quantum computers: The five-qubit error correcting code, Phys. Rev. Lett. 86, 5811 (2001).
D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y. Yamamoto, Approximate quantum error correction can lead to better codes, Phys. Rev. A 56, 2567 (1997).
C. Gidney and M. Ekerå, How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits, Quantum 5, 433 (2021).
E. Gouzien, D. Ruiz, F.-M. Le Régent, J. Guillaud, and N. Sangouard, Performance analysis of a repetition cat code architecture: Computing 256-bit elliptic curve logarithm in 9 hours with 126 133 cat qubits, Phys. Rev. Lett. 131, 040602 (2023).
N. Raveendran, N. Rengaswamy, F. Rozpędek, A. Raina, L. Jiang, and B. Vasić, Finite rate QLDPC-GKP coding scheme that surpasses the CSS Hamming bound, Quantum 6, 767 (2022).
I. L. Chuang, D. W. Leung, and Y. Yamamoto, Bosonic quantum codes for amplitude damping, Phys. Rev. A 56, 1114 (1997).
P. T. Cochrane, G. J. Milburn, and W. J. Munro, Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping, Phys. Rev. A 59, 2631 (1999).
K. Noh, V. V. Albert, and L. Jiang, Quantum capacity bounds of Gaussian thermal loss channels and achievable rates with Gottesman-Kitaev-Preskill codes, IEEE Trans. Inf. Theory 65, 2563 (2019).
Q. Xu, P. Zeng, D. Xu, and L. Jiang, Fault-tolerant operation of bosonic qubits with discrete-variable ancillae, Phys. Rev. X 14, 031016 (2024).
M.-J. Liang, T. Chen, and Z.-Y. Xue, Error-mitigated geometric quantum control over an oscillator, Phys. Rev. Appl. 23, 024033 (2025).
D. S. Schlegel, F. Minganti, and V. Savona, Quantum error correction using squeezed Schrödinger cat states, Phys. Rev. A 106, 022431 (2022).
P. David Rodríguez and K. Eliot, Improved autonomous error correction using variable dissipation in small logical qubit architectures, Quantum Sci. Technol. 6, 015006 (2020).
J. Lebreuilly, K. Noh, C.-H. Wang, S. M. Girvin, and L. Jiang, Autonomous quantum error correction and quantum computation, ArXiv:2103.05007.
Z. Li, T. Roy, D. Rodríguez Pérez, K.-H. Lee, E. Kapit, and D. I. Schuster, Autonomous error correction of a single logical qubit using two transmons, Nat. Commun. 15, 1681 (2024).
Z. Li, T. Roy, D. R. Pérez, D. I. Schuster, and E. Kapit, Hardware-efficient autonomous error correction with linear couplers in superconducting circuits, Phys. Rev. Res. 6, 013171 (2024).
Q. Xu, G. Zheng, Y.-X. Wang, P. Zoller, A. A. Clerk, and L. Jiang, Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits, npj Quantum Inf. 9, 78 (2023).
S. Touzard, A. Grimm, Z. Leghtas, S. O. Mundhada, P. Reinhold, C. Axline, M. Reagor, K. Chou, J. Blumoff, K. M. Sliwa, et al., Coherent oscillations inside a quantum manifold stabilized by dissipation, Phys. Rev. X 8, 021005 (2018).
C. Berdou, et al., One hundred second bit-flip time in a two-photon dissipative oscillator, PRX Quantum 4, 020350 (2023).
U. Réglade, A. Bocquet, R. Gautier, J. Cohen, A. Marquet, E. Albertinale, N. Pankratova, M. Hallén, F. Rautschke, L.-A. Sellem, P. Rouchon, A. Sarlette, M. Mirrahimi, P. Campagne-Ibarcq, R. Lescanne, S. Jezouin, and Z. Leghtas, Quantum control of a cat qubit with bit-flip times exceeding ten seconds, Nature 629, 778 (2024).
S. Puri, S. Boutin, and A. Blais, Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving, npj Quantum Inf. 3, 18 (2017).
S. Puri, L. St-Jean, J. A. Gross, A. Grimm, N. E. Frattini, P. S. Iyer, A. Krishna, S. Touzard, L. Jiang, A. Blais, S. T. Flammia, and S. M. Girvin, Bias-preserving gates with stabilized cat qubits, Sci. Adv. 6, eaay5901 (2020).
A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. H. Devoret, Stabilization and operation of a Kerr-cat qubit, Nature 584, 205 (2020).
H. Putterman, J. Iverson, Q. Xu, L. Jiang, O. Painter, F. G. S. L. Brandão, and K. Noh, Stabilizing a bosonic qubit using colored dissipation, Phys. Rev. Lett. 128, 110502 (2022).
N. E. Frattini, R. G. Cortiñas, J. Venkatraman, X. Xiao, Q. Su, C. U. Lei, B. J. Chapman, V. R. Joshi, S. Girvin, R. J. Schoelkopf, S. Puri, and M. H. Devoret, Observation of pairwise level degeneracies and the quantum regime of the Arrhenius law in a double-well parametric oscillator, Phys. Rev. X 14, 031040 (2024).
J. Venkatraman, R. G. Cortinas, N. E. Frattini, X. Xiao, and M. H. Devoret, Quantum interference of tunneling paths under a double-well barrier, ArXiv:2211.04605.
V. E. Manucharyan, J. Koch, L. I. Glazman, and M. H. Devoret, Fluxonium: Single Cooper-pair circuit free of charge offsets, Science 326, 113 (2009).
A. Somoroff, Q. Ficheux, R. A. Mencia, H. Xiong, R. Kuzmin, and V. E. Manucharyan, Millisecond coherence in a superconducting qubit, Phys. Rev. Lett. 130, 267001 (2023).
V. V. Albert, S. O. Mundhada, A. Grimm, S. Touzard, M. H. Devoret, and L. Jiang, Pair-cat codes: Autonomous error-correction with low-order nonlinearity, Quantum Sci. Technol. 4, 035007 (2019).
J. Harrington and J. Preskill, Achievable rates for the Gaussian quantum channel, Phys. Rev. A 64, 062301 (2001).
J. Conrad, J. Eisert, and F. Arzani, Gottesman-Kitaev-Preskill codes: A lattice perspective, Quantum 6, 648 (2022).
S. P. Jain, J. T. Iosue, A. Barg, and V. V. Albert, Quantum spherical codes, Nat. Phys. 20, 1300 (2024).
J. Conrad, J. Eisert, and J.-P. Seifert, Good Gottesman-Kitaev-Preskill codes from the NTRU cryptosystem, Quantum 8, 1398 (2024).
M. Lin, C. Chamberland, and K. Noh, Closest lattice point decoding for multimode Gottesman-Kitaev-Preskill codes, PRX Quantum 4, 040334 (2023).
J. Schwinger, On angular momentum, Technical Report (1952), https://doi.org/10.2172/4389568..
M. Mathur, I. Raychowdhury, and R. Anishetty, SU(N) irreducible Schwinger bosons, J. Math. Phys. 51, 093504 (2010).
A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, J. Ye, P. Zoller, E. Demler, M. D. Lukin, and A. M. Rey, Two-orbital SU(N) magnetism with ultracold alkaline-earth atoms, Nat. Phys. 6, 289 (2010).
S. Chaudhury, S. Merkel, T. Herr, A. Silberfarb, I. H. Deutsch, and P. S. Jessen, Quantum control of the hyperfine spin of a $Cs$ atom ensemble, Phys. Rev. Lett. 99, 163002 (2007).
C. Vuillot, H. Asasi, Y. Wang, L. P. Pryadko, and B. M. Terhal, Quantum error correction with the toric Gottesman-Kitaev-Preskill code, Phys. Rev. A 99, 032344 (2019).
K. Noh, S. M. Girvin, and L. Jiang, Encoding an oscillator into many oscillators, Phys. Rev. Lett. 125, 080503 (2020).
J. Wu and Q. Zhuang, Continuous-variable error correction for general Gaussian noises, Phys. Rev. Appl. 15, 034073 (2021).
J. Wu, A. J. Brady, and Q. Zhuang, Optimal encoding of oscillators into more oscillators, Quantum 7, 1082 (2023).
L. Hänggli and R. König, Oscillator-to-oscillator codes do not have a threshold, IEEE Trans. Inf. Theory 68, 1068 (2022).
A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012).
K. Noh and C. Chamberland, Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code, Phys. Rev. A 101, 012316 (2020).
K. Noh, C. Chamberland, and F. G. S. L. Brandão, Low-overhead fault-tolerant quantum error correction with the surface-GKP code, PRX Quantum 3, 010315 (2022).
B. M. Terhal, J. Conrad, and C. Vuillot, Towards scalable bosonic quantum error correction, Quantum Sci. Technol. 5, 043001 (2020).
A. S. Darmawan, B. J. Brown, A. L. Grimsmo, D. K. Tuckett, and S. Puri, Practical quantum error correction with the XZZX code and Kerr-cat qubits, PRX Quantum 2, 030345 (2021).
S. Singh, A. S. Darmawan, B. J. Brown, and S. Puri, High-fidelity magic-state preparation with a biased-noise architecture, Phys. Rev. A 105, 052410 (2022).
C. Chamberland, K. Noh, P. Arrangoiz-Arriola, E. T. Campbell, C. T. Hann, J. Iverson, H. Putterman, T. C. Bohdanowicz, S. T. Flammia, A. Keller, G. Refael, J. Preskill, L. Jiang, A. H. Safavi-Naeini, O. Painter, and F. G. S. L. Brandão, Building a fault-tolerant quantum computer using concatenated cat codes, PRX Quantum 3, 010329 (2022).
J. Guillaud and M. Mirrahimi, Repetition cat qubits for fault-tolerant quantum computation, Phys. Rev. X 9, 041053 (2019).
H. Putterman, et al., Hardware-efficient quantum error correction via concatenated bosonic qubits, Nature 638, 927 (2025).
D. Ruiz, J. Guillaud, A. Leverrier, M. Mirrahimi, and C. Vuillot, LDPC-cat codes for low-overhead quantum computing in 2D, Nat. Commun. 16, 1040 (2025).
K. Fukui, A. Tomita, and A. Okamoto, Analog quantum error correction with encoding a qubit into an oscillator, Phys. Rev. Lett. 119, 180507 (2017).
Y. Xu, Y. Wang, E.-J. Kuo, and V. V. Albert, Qubit-oscillator concatenated codes: Decoding formalism and code comparison, PRX Quantum 4, 020342 (2023).
S. Singh, B. Royer, and S. M. Girvin, Towards non-Abelian quantum signal processing: Efficient control of hybrid continuous- and discrete-variable architectures, ArXiv:2504.19992.
J. Hastrup, K. Park, J. B. Brask, R. Filip, and U. L. Andersen, Measurement-free preparation of grid states, npj Quantum Inf. 7, 17 (2021).
C. K. Law and J. H. Eberly, Arbitrary control of a quantum electromagnetic field, Phys. Rev. Lett. 76, 1055 (1996).
Disentangling the ancilla from cat states parallel to position axis in phase space using non-Abelian QSP entangles the ancilla with cat states parallel to the momentum axis.
This is admissible since it can be tracked in software with a deterministic Pauli frame change after each stabilization step.
X. Zhou, D. W. Leung, and I. L. Chuang, Methodology for quantum logic gate construction, Phys. Rev. A 62, 052316 (2000).
The sBs circuit rotates the ancilla state entangled with ${| P_{+} ⟩}_{L}$ ( ${| P_{-} ⟩}_{L}$ ) by $4 π$ ( $2 π$ ) Thus, the qubit is disentangled with an additional local phase of $π$ on the logical state. Hence, this operation applies a deterministic logical Pauli $P$ operation along with the logical single-qubit rotation. See Ref. [478] for details.
I. Rojkov, P. M. Röggla, M. Wagener, M. Fontboté-Schmidt, S. Welte, J. Home, and F. Reiter, Two-qubit operations for finite-energy Gottesman-Kitaev-Preskill encodings, Phys. Rev. Lett. 133, 100601 (2024).
S. Chakraborty, Implementing any linear combination of unitaries on intermediate-term quantum computers, Quantum 8, 1496 (2024).
C. Yuan, C. McNally, and M. Carbin, twist: Sound reasoning for purity and entanglement in quantum programs, Proc. ACM Program. Lang. 6, 1 (2022).
S. Becker, N. Datta, L. Lami, and C. Rouzé, Energy-constrained discrimination of unitaries, quantum speed limits, and a Gaussian Solovay-Kitaev theorem, Phys. Rev. Lett. 126, 190504 (2021).
M. Gschwendtner and A. Winter, Infinite-dimensional programmable quantum processors, PRX Quantum 2, 030308 (2021).
K. Sharma, S. Khatri, M. Cerezo, and P. J. Coles, Noise resilience of variational quantum compiling, New J. Phys. 22, 043006 (2020).
H. Wang, Y. Ding, J. Gu, Y. Lin, D. Z. Pan, F. T. Chong, and S. Han, in IEEE International Symposium on High-Performance Computer Architecture, HPCA ’22 (2022), p. 692.
P. Murali, J. M. Baker, A. Javadi-Abhari, F. T. Chong, and M. Martonosi, in Proceedings of the Twenty-Fourth International Conference on Architectural Support for Programming Languages and Operating Systems, ASPLOS ’19 (Association for Computing Machinery, New York, 2019), p. 1015.
G. H. Low, T. J. Yoder, and I. L. Chuang, Methodology of resonant equiangular composite quantum gates, Phys. Rev. X 6, 041067 (2016).
D. J. Singh and L. Nordstrom, Planewaves, Pseudopotentials, and the LAPW Method (Springer Science & Business Media, 2006).
G. J. Milburn, Simulating nonlinear spin models in an ion trap, ArXiv:quant-ph/9908037.
Using vector identities, it can also be shown that $W_{η_{2}, η_{1}} = \frac{1}{4} [\hat{I} (1 + η_{1} η_{2} \hat{c} \cdot \hat{b}) + η_{2} \hat{c} \cdot \vec{σ} + η_{1} \hat{b} \cdot \vec{σ} + i η_{1} η_{2} \hat{c} \times \hat{b} \cdot \vec{σ}]$ .
B. Dias and R. König, Classical simulation of non-Gaussian bosonic circuits, Phys. Rev. A 110, 042402 (2024).
A. W. Cross and J. M. Gambetta, Optimized pulse shapes for a resonator-induced phase gate, Phys. Rev. A 91, 032325 (2015).
H. Paik, A. Mezzacapo, M. Sandberg, D. T. McClure, B. Abdo, A. D. Corcoles, O. Dial, D. F. Bogorin, B. L. T. Plourde, M. Steffen, A. W. Cross, J. M. Gambetta, and J. M. Chow, Experimental demonstration of a resonator-induced phase gate in a multiqubit circuit-QED system, Phys. Rev. Lett. 117, 250502 (2016).
S. Puri and A. Blais, High-fidelity resonator-induced phase gate with single-mode squeezing, Phys. Rev. Lett. 116, 180501 (2016).
S. Sheldon, E. Magesan, J. M. Chow, and J. M. Gambetta, Procedure for systematically tuning up cross-talk in the cross-resonance gate, Phys. Rev. A 93, 060302(R) (2016).
A. M. Childs and N. Wiebe, Hamiltonian simulation using linear combinations of unitary operations, Quantum Inf. Comput. 12, 901 (2012).
J. Han, W. Cai, L. Hu, X. Mu, Y. Ma, Y. Xu, W. Wang, H. Wang, Y. P. Song, C.-L. Zou, and L. Sun, Experimental simulation of open quantum system dynamics via Trotterization, Phys. Rev. Lett. 127, 020504 (2021).
M. Suzuki, General theory of fractal path integrals with applications to many-body theories and statistical physics, J. Math. Phys. 32, 400 (1991).
J. Hastrup, K. Park, R. Filip, and U. L. Andersen, Unconditional preparation of squeezed vacuum from Rabi interactions, Phys. Rev. Lett. 126, 153602 (2021).
M. Ayyash, X. Xu, S. Ashhab, and M. Mariantoni, Driven multiphoton qubit-resonator interactions, Phys. Rev. A 110, 053711 (2024).
M. Ayyash, Multimode qubit-conditional operations via generalized cross-resonance, ArXiv:2503.15941.
M. K. Hope, J. Lidal, and F. Massel, Preparation of conditionally-squeezed states in qubit-oscillator systems, ArXiv:2504.01664.
A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, A variational eigenvalue solver on a photonic quantum processor, Nat. Commun. 5, 5213 (2014).
E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, ArXiv:1411.4028.
D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Simulating Hamiltonian dynamics with a truncated Taylor series, Phys. Rev. Lett. 114, 090502 (2015).
G. H. Low and I. L. Chuang, Hamiltonian simulation by uniform spectral amplification, ArXiv:1707.05391.
L. Bell, Y. Wang, K. C. Smith, Y. Liu, E. Dumitrescu, and S. M. Girvin, Co-designing eigen- and singular-value transformation oracles: From algorithmic applications to hardware compilation, ArXiv:2502.16029.
A. Gilyén, Ph.D. thesis, Universiteit van Amsterdam, https://hdl.handle.net/11245.1/20e9733e-6014-402d-afa9-20f3cc4a0568, 2019.
N. Stamatopoulos and W. J. Zeng, Derivative pricing using quantum signal processing, Quantum 8, 1322 (2024).
S. Wimperis, Broadband, narrowband, and passband composite pulses for use in advanced NMR experiments, J. Magn. Reson., Ser. A 109, 221 (1994).
Y. Dong, L. Lin, and Y. Tong, Ground-state preparation and energy estimation on early fault-tolerant quantum computers via quantum eigenvalue transformation of unitary matrices, PRX Quantum 3, 040305 (2022).
R. Chao, D. Ding, A. Gilyen, C. Huang, and M. Szegedy, Finding angles for quantum signal processing with machine precision, ArXiv:2003.02831.
Z. M. Rossi, J. L. Ceroni, and I. L. Chuang, Modular quantum signal processing in many variables, Quantum 9, 1776 (2025).
B. Németh, B. Kövér, B. Kulcsár, R. B. Miklósi, and A. Gilyén, On variants of multivariate quantum signal processing and their characterizations, ArXiv:2312.09072.
H. Le Jeannic, A. Cavaillès, K. Huang, R. Filip, and J. Laurat, Slowing quantum decoherence by squeezing in phase space, Phys. Rev. Lett. 120, 073603 (2018).
T. Hillmann and F. Quijandría, Quantum error correction with dissipatively stabilized squeezed-cat qubits, Phys. Rev. A 107, 032423 (2023).
S. Kan, M. Palma, Z. Du, S. A. Stein, C. Liu, J. Chen, A. Li, and Y. Mao, in 2024 IEEE International Conference on Quantum Computing and Engineering, QCE ’24 (2024), Vol. 1, p. 1558.
V. V. Sivak, A. Eickbusch, H. Liu, B. Royer, I. Tsioutsios, and M. H. Devoret, Model-free quantum control with reinforcement learning, Phys. Rev. X 12, 011059 (2022).
T. Schulte-Herbrüggen, A. Spörl, N. Khaneja, and S. J. Glaser, Optimal control for generating quantum gates in open dissipative systems, J. Phys. B: At. Mol. Opt. Phys. 44, 154013 (2011).
H. Ball, M. J. Biercuk, A. R. R. Carvalho, J. Chen, M. Hush, L. A. D. Castro, L. Li, P. J. Liebermann, H. J. Slatyer, C. Edmunds, V. Frey, C. Hempel, and A. Milne, Software tools for quantum control: Improving quantum computer performance through noise and error suppression, Quantum Sci. Technol. 6, 044011 (2021).
D. P. Kingma and J. Ba, in 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7–9, 2015, Conference Track Proceedings, edited by Y. Bengio and Y. LeCun (2015).
We note through private communication that the fidelities presented here for points at circuit depth of $N = 2$ correct a minor simulation error that was later found for points at $N = 2$ in Fig. 2(a) of Ref. [60]. To the best of our knowledge, at $N = 2$ the results here represent the best known numerically optimized protocols.
B. Zhang and Q. Zhuang, Energy-dependent barren plateau in bosonic variational quantum circuits, Quantum Sci. Technol. 10, 015009 (2024).
R. Porotti, V. Peano, and F. Marquardt, Gradient-ascent pulse engineering with feedback, PRX Quantum 4, 030305 (2023).
S. Khatri, R. LaRose, A. Poremba, L. Cincio, A. T. Sornborger, and P. J. Coles, Quantum-assisted quantum compiling, Quantum 3, 140 (2019).
M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, Cost function dependent barren plateaus in shallow parametrized quantum circuits, Nat. Commun. 12, 1791 (2021).
A. K. Pati and S. L. Braunstein, in Quantum Information with Continuous Variables, edited by S. L. Braunstein and A. K. Pati (Springer Netherlands, Dordrecht, 2003), p. 31.
A. K. Pati, S. L. Braunstein, and S. Lloyd, Quantum searching with continuous variables, ArXiv:quant-ph/0002082.
S. J. Lomonaco Jr and L. H. Kauffman, A continuous variable Shor algorithm, ArXiv:quant-ph/0210141.
S.-C. Wang, Y. Chi, L. Yu, Z. Sun, Q.-P. Su, and C.-P. Yang, Implementing a quantum search algorithm with nonorthogonal states, Phys. Rev. A 103, 032413 (2021).
A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. O’Brien, and T. C. Ralph, Boson sampling from a Gaussian state, Phys. Rev. Lett. 113, 100502 (2014).
C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, Gaussian boson sampling, Phys. Rev. Lett. 119, 170501 (2017).
H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, et al., Quantum computational advantage using photons, Science 370, 1460 (2020).
L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins, et al., Quantum computational advantage with a programmable photonic processor, Nature 606, 75 (2022).
Y. Liu, J. M. Martyn, J. Sinanan-Singh, K. C. Smith, S. M. Girvin, and I. L. Chuang, Toward mixed analog-digital quantum signal processing: Quantum AD/DA conversion and the Fourier transform, IEEE Trans. Signal Process. 73, 3641 (2025).
Here, we are taking $| x, Δ ⟩_{B}$ to be an exact position eigenstate; we will remedy this assumption and evaluate performance on $| x, Δ ⟩_{B}$ (which is not an exact position eigenstate) shortly.
J. Hastrup, K. Park, J. B. Brask, R. Filip, and U. L. Andersen, Universal unitary transfer of continuous-variable quantum states into a few qubits, Phys. Rev. Lett. 128, 110503 (2022).
As mentioned in Ref. [543], this exact state is unphysical because it has infinite energy; however, it can be well approximated by a squeezed vacuum.
P. Shor, in Proceedings 35th Annual Symposium on Foundations of Computer Science, FOCS ’94 (1994), p. 124.
A. Gilyén, S. Arunachalam, and N. Wiebe, in Proceedings of the 2019 Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’19 (2019), p. 1425.
National Science Technology Council, National quantum initiative supplement to the presidents FY 2022 budget (2022), https://www.quantum.gov/wp-content/uploads/2021/12/NQI-Annual-Report-FY2022.pdf.
European Quantum Flagship, European quantum flagship strategic research agenda (2020), https://qt.eu//app/uploads/2020/04/Strategic_Research-_Agenda_d_FINAL.pdf.
C. W. Bauer, et al., Quantum simulation for high-energy physics, PRX Quantum 4, 027001 (2023).
Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. Johnson, M. Kieferová, I. D. Kivlichan, T. Menke, B. Peropadre, N. P. D. Sawaya, S. Sim, L. Veis, and A. Aspuru-Guzik, Quantum chemistry in the age of quantum computing, Chem. Rev. 119, 10856 (2019).
I. M. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Rev. Mod. Phys. 86, 153 (2014).
H.-D. Meyer, U. Manthe, and L. Cederbaum, The multi-configurational time-dependent Hartree approach, Chem. Phys. Lett. 165, 73 (1990).
M. Marques and E. Gross, Time-dependent density functional theory, Annu. Rev. Phys. Chem. 55, 427 (2004).
A. Polkovnikov, Phase space representation of quantum dynamics, Ann. Phys. 325, 1790 (2010).
R. J. Bartlett, Many-body perturbation theory and coupled cluster theory for electron correlation in molecules, Annu. Rev. Phys. Chem. 32, 359 (1981).
R. J. Bartlett and M. Musiał, Coupled-cluster theory in quantum chemistry, Rev. Mod. Phys. 79, 291 (2007).
M. C. Bañuls, Tensor network algorithms: A route map, Annu. Rev. Condens. Matter Phys. 14, 173 (2023).
W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Quantum Monte Carlo simulations of solids, Rev. Mod. Phys. 73, 33 (2001).
J. Danzer, C. Gattringer, L. Liptak, and M. Marinkovic, A study of the sign problem for lattice QCD with chemical potential, Phys. Lett. B 682, 240 (2009).
L. Bongiovanni, Numerical methods for the sign problem in lattice field theory, ArXiv:1603.06458.
R. P. Feynman, Quantum mechanical computers, Opt. News 11, 11 (1985).
It is important to distinguish between two cases: (a) simulation of the time evolution of a quantum system with a specified Hamiltonian and a specified starting state; and (b) finding the ground state (or other eigenstate) of a specified Hamiltonian. For most physical systems, the former is in the complexity class BQP and the latter is in QMA which is a quantum analog of NP and is believed to be exponentially more challenging than BQP [1].
E. Altman, et al., Quantum simulators: Architectures and opportunities, PRX Quantum 2, 017003 (2021).
A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Practical quantum advantage in quantum simulation, Nature 607, 667 (2022).
Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, Digital quantum simulation of spin models with circuit quantum electrodynamics, Phys. Rev. X 5, 021027 (2015).
Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. Van Den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, et al., Evidence for the utility of quantum computing before fault tolerance, Nature 618, 500 (2023).
R. Barends, L. Lamata, J. Kelly, L. García-Álvarez, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, et al., Digital quantum simulation of fermionic models with a superconducting circuit, Nat. Commun. 6, 7654 (2015).
T. V. Zache, D. González-Cuadra, and P. Zoller, Fermion-qudit quantum processors for simulating lattice gauge theories with matter, Quantum 7, 1140 (2023).
J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner, Quantum simulation of antiferromagnetic spin chains in an optical lattice, Nature 472, 307 (2011).
C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys. 93, 025001 (2021).
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature 415, 39 (2002).
A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A cold-atom Fermi-Hubbard antiferromagnet, Nature 545, 462 (2017).
X. Zhang, K. Zhang, Y. Shen, S. Zhang, J.-N. Zhang, M.-H. Yung, J. Casanova, J. S. Pedernales, L. Lamata, E. Solano, and K. Kim, Experimental quantum simulation of fermion-antifermion scattering via boson exchange in a trapped ion, Nat. Commun. 9, 195 (2018).
A. Mil, T. V. Zache, A. Hegde, A. Xia, R. P. Bhatt, M. K. Oberthaler, P. Hauke, J. Berges, and F. Jendrzejewski, A scalable realization of local U(1) gauge invariance in cold atomic mixtures, Science 367, 1128 (2020).
W. Domcke and D. R. Yarkony, Role of conical intersections in molecular spectroscopy and photoinduced chemical dynamics, Annu. Rev. Phys. Chem. 63, 325 (2012).
S. P. Jordan, K. S. Lee, and J. Preskill, Quantum algorithms for quantum field theories, Science 336, 1130 (2012).
J.-P. Bureik, G. Hercé, M. Allemand, A. Tenart, T. Roscilde, and D. Clément, Suppression of Bogoliubov momentum pairing and emergence of non-Gaussian correlations in ultracold interacting Bose gases, Nat. Phys. 21, 57 (2025).
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, 110404 (2006).
M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96, 110405 (2006).
S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996).
G. H. Low, V. Kliuchnikov, and N. Wiebe, Well-conditioned multiproduct Hamiltonian simulation, ArXiv:1907.11679.
A. M. Childs, Universal computation by quantum walk, Phys. Rev. Lett. 102, 180501 (2009).
D. W. Berry and A. M. Childs, Black-box Hamiltonian simulation and unitary implementation, Quantum Inf. Comput. 12, 29 (2012).
D. W. Berry, A. M. Childs, and R. Kothari, in 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS) (2015).
G. H. Low and I. L. Chuang, Hamiltonian simulation by qubitization, Quantum 3, 163 (2019).
E. Campbell, Random compiler for fast Hamiltonian simulation, Phys. Rev. Lett. 123, 070503 (2019).
P. K. Faehrmann, M. Steudtner, R. Kueng, M. Kieferová, and J. Eisert, Randomizing multi-product formulas for Hamiltonian simulation, Quantum 6, 806 (2022).
A. Y. Kitaev, Quantum measurements and the Abelian stabilizer problem, ArXiv:quant-ph/9511026.
S. Bravyi, A. Chowdhury, D. Gosset, and P. Wocjan, Quantum Hamiltonian complexity in thermal equilibrium, Nat. Phys. 18, 1367 (2022).
S. Lu, M. C. Bañuls, and J. I. Cirac, Algorithms for quantum simulation at finite energies, PRX Quantum 2, 020321 (2021).
A. Schuckert, A. Bohrdt, E. Crane, and M. Knap, Probing finite-temperature observables in quantum simulators of spin systems with short-time dynamics, Phys. Rev. B 107, L140410 (2023).
Y. Zhang, S. Jahanbani, A. Riswadkar, S. Shankar, and A. C. Potter, Sequential quantum simulation of spin chains with a single circuit QED device, Phys. Rev. A 109, 022606 (2024).
S. Morita and H. Nishimori, Mathematical foundation of quantum annealing, J. Math. Phys. 49, 125210 (2008).
S. Lee, J. Lee, H. Zhai, Y. Tong, A. M. Dalzell, A. Kumar, P. Helms, J. Gray, Z.-H. Cui, W. Liu, M. Kastoryano, R. Babbush, J. Preskill, D. R. Reichman, E. T. Campbell, E. F. Valeev, L. Lin, and G. K.-L. Chan, Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistry, Nat. Commun. 14, 1952 (2023).
F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Symmetry protection of topological phases in one-dimensional quantum spin systems, Phys. Rev. B 85, 075125 (2012).
A. Taylor, G. Bressanini, H. Kwon, and M. S. Kim, Quantum error cancellation in photonic systems: Undoing photon losses, Phys. Rev. A 110, 022622 (2024).
H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
C. Eichler, D. Bozyigit, C. Lang, L. Steffen, J. Fink, and A. Wallraff, Experimental state tomography of itinerant single microwave photons, Phys. Rev. Lett. 106, 220503 (2011).
D. Braak, Integrability of the Rabi model, Phys. Rev. Lett. 107, 100401 (2011).
A. F. Kockum, A. Miranowicz, S. D. Liberato, S. Savasta, and F. Nori, Ultrastrong coupling between light and matter, Nat. Rev. Phys. 1, 19 (2019).
P. Forn-Díaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Ultrastrong coupling regimes of light-matter interaction, Rev. Mod. Phys. 91, 025005 (2019).
S. Aaronson and A. Arkhipov, The computational complexity of linear optics, Theory Comput. 9, 143 (2013).
S. M. Girvin, in Topological Aspects of Low-dimensional Systems: Proceedings, Les Houches Summer School of Theoretical Physics, Session 69: Les Houches, France, July 7–31, 1998, Les Houches—École d’Été de Physique Theorique, edited by A. Comtet, T. Jolicœur, S. Ouvry, and F. David (Springer, Berlin, 1999), Vol. 69.
E. Kapit, Quantum simulation architecture for lattice bosons in arbitrary, tunable, external gauge fields, Phys. Rev. A 87, 062336 (2013).
M. Hafezi, P. Adhikari, and J. M. Taylor, Engineering three-body interaction and Pfaffian states in circuit QED systems, Phys. Rev. B 90, 060503(R) (2014).
C. Owens, A. LaChapelle, B. Saxberg, B. M. Anderson, R. Ma, J. Simon, and D. I. Schuster, Quarter-flux Hofstadter lattice in a qubit-compatible microwave cavity array, Phys. Rev. A 97, 013818 (2018).
J. C. Owens, M. G. Panetta, B. Saxberg, G. Roberts, S. Chakram, R. Ma, A. Vrajitoarea, J. Simon, and D. I. Schuster, Chiral cavity quantum electrodynamics, Nat. Phys. 18, 1048 (2022).
P. D. Kurilovich, V. D. Kurilovich, J. Lebreuilly, and S. M. Girvin, Stabilizing the Laughlin state of light: Dynamics of hole fractionalization, SciPost Phys. 13, 107 (2022).
M. Aidelsburger, Artificial Gauge Fields with Ultracold Atoms in Optical Lattices (Springer International Publishing, 2016).
J. Léonard, S. Kim, J. Kwan, P. Segura, F. Grusdt, C. Repellin, N. Goldman, and M. Greiner, Realization of a fractional quantum Hall state with ultracold atoms, Nature 619, 495 (2023).
L. W. Clark, N. Schine, C. Baum, N. Jia, and J. Simon, Observation of Laughlin states made of light, Nature 582, 41 (2020).
C. Wang, F.-M. Liu, M.-C. Chen, H. Chen, X.-H. Zhao, C. Ying, Z.-X. Shang, J.-W. Wang, Y.-H. Huo, C.-Z. Peng, et al., Realization of fractional quantum Hall state with interacting photons, Science 384, 579 (2024).
M. C. Tran, D. K. Mark, W. W. Ho, and S. Choi, Measuring arbitrary physical properties in analog quantum simulation, Phys. Rev. X 13, 011049 (2023).
A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. Choi, V. Khemani, J. Léonard, and M. Greiner, Probing entanglement in a many-body–localized system, Science 364, 256 (2019).
R. N. Palmer and D. Jaksch, High-field fractional quantum Hall effect in optical lattices, Phys. Rev. Lett. 96, 180407 (2006).
S. G. Tabrizi, A. V. Arbuznikov, and M. Kaupp, Construction of giant-spin Hamiltonians from many-spin Hamiltonians by third-order perturbation theory and application to an $Fe$ $_{3}$ $Cr$ single-molecule magnet, Chem. Eur. J. 22, 6853 (2016).
J. Schwinger, in Quantum Mechanics (Springer, 2001), p. 149.
U. Wiese, Ultracold quantum gases and lattice systems: Quantum simulation of lattice gauge theories, Ann. Phys. 525, 777 (2013).
S. Roy, A. Senanian, C. S. Wang, O. C. Wetherbee, L. Zhang, B. Cole, C. P. Larson, E. Yelton, K. Arora, P. L. McMahon, B. L. T. Plourde, B. Royer, and V. Fatemi, Synthetic high angular momentum spin dynamics in a microwave oscillator, Phys. Rev. X 15, 021009 (2025).
E. Champion, Z. Wang, R. W. Parker, and M. S. Blok, Efficient control of a transmon qudit using effective spin-7/2 rotations, Phys. Rev. X 15, 021096 (2025).
P. Giorda, P. Zanardi, and S. Lloyd, Universal quantum control in irreducible state-space sectors: Application to bosonic and spin-boson systems, Phys. Rev. A 68, 062320 (2003).
B. E. Anderson, H. Sosa-Martinez, C. A. Riofrío, I. H. Deutsch, and P. S. Jessen, Accurate and robust unitary transformations of a high-dimensional quantum system, Phys. Rev. Lett. 114, 240401 (2015).
R. G. Newton and B.-L. Young, Measurability of the spin density matrix, Ann. Phys. 49, 393 (1968).
M. A. Perlin, D. Barberena, and A. M. Rey, Spin qudit tomography and state reconstruction error, Phys. Rev. A 104, 062413 (2021).
S. M. Girvin and K. Yang, Modern Condensed Matter Physics (Cambridge University Press, 2019).
R. F. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M. O. Ellis, and R. W. Chantrell, Atomistic spin model simulations of magnetic nanomaterials, J. Phys.: Condens. Matter 26, 103202 (2014).
J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, PNAS 79, 2554 (1982).
M. P. Fisher, Quantum cognition: The possibility of processing with nuclear spins in the brain, Ann. Phys. 362, 593 (2015).
A. Winter and H. Rieger, Quantum phase transition and correlations in the multi-spin-boson model, Phys. Rev. B 90, 224401 (2014).
S. Schmidt and G. Blatter, Strong coupling theory for the Jaynes-Cummings-Hubbard model, Phys. Rev. Lett. 103, 086403 (2009).
K. C. Smith, A. Bhattacharya, and D. J. Masiello, Exact k-body representation of the Jaynes-Cummings interaction in the dressed basis: Insight into many-body phenomena with light, Phys. Rev. A 104, 013707 (2021).
S. Kundu and N. Makri, Intramolecular vibrations in excitation energy transfer: Insights from real-time path integral calculations, Annu. Rev. Phys. Chem. 73, 349 (2022).
M. R. Wasielewski, M. D. Forbes, N. L. Frank, K. Kowalski, G. D. Scholes, J. Yuen-Zhou, M. A. Baldo, D. E. Freedman, R. H. Goldsmith, T. Goodson III, et al., Exploiting chemistry and molecular systems for quantum information science, Nat. Rev. Chem. 4, 490 (2020).
M. Kebric, L. Barbiero, C. Reinmoser, U. Schollwöck, and F. Grusdt, Confinement and Mott transitions of dynamical charges in one-dimensional lattice gauge theories, Phys. Rev. Lett. 127, 167203 (2021).
C. Ratti, Lattice QCD and heavy ion collisions: a review of recent progress, Rep. Prog. Phys. 81, 084301 (2018).
J. A. Flores-Livas, L. Boeri, A. Sanna, G. Profeta, R. Arita, and M. Eremets, A perspective on conventional high-temperature superconductors at high pressure: Methods and materials, Phys. Rep. 856, 1 (2020).
R. Dutta, et al., Simulating chemistry on bosonic quantum devices, J. Chem. Theory Comput. 20, 6426 (2024).
R. Dutta, N. P. Vu, C. Xu, D. G. A. Cabral, N. Lyu, A. V. Soudackov, X. Dan, H. Li, C. Wang, and V. S. Batista, Simulating electronic structure on bosonic quantum computers, J. Chem. Theory Comput. 21, 2281 (2025).
F. Verstraete and J. I. Cirac, Mapping local Hamiltonians of fermions to local Hamiltonians of spins, J. Stat. Mech: Theory Exp. 2005, P09012 (2005).
C. Derby, J. Klassen, J. Bausch, and T. Cubitt, Compact fermion to qubit mappings, Phys. Rev. B 104, 035118 (2021).
I. D. Kivlichan, J. McClean, N. Wiebe, C. Gidney, A. Aspuru-Guzik, GarnetKin-Lic Chan, and R. Babbush, Quantum simulation of electronic structure with linear depth and connectivity, Phys. Rev. Lett. 120, 110501 (2018).
M. G. Algaba, P. V. Sriluckshmy, M. Leib, and F. Šimkovic IV, Low-depth simulations of fermionic systems on square-grid quantum hardware, Quantum 8, 1327 (2024).
Y. Aharonov, L. Davidovich, and N. Zagury, Quantum random walks, Phys. Rev. A 48, 1687 (1993).
E. Flurin, V. V. Ramasesh, S. Hacohen-Gourgy, L. S. Martin, N. Y. Yao, and I. Siddiqi, Observing topological invariants using quantum walks in superconducting circuits, Phys. Rev. X 7, 031023 (2017).
S. P. Jordan, K. S. Lee, and J. Preskill, Quantum computation of scattering in scalar quantum field theories, Quantum Inf. Comput. 14, 1014 (2014).
F. Gray, Pulse code communication, United States Patent Number 2,632,058 (1953), https://patentimages.storage.googleapis.com/a3/d7/f2/0343f5f2c0cf50/US2632058.pdf.
N. P. D. Sawaya, T. Menke, T. H. Kyaw, S. Johri, A. Aspuru-Guzik, and G. G. Guerreschi, Resource-efficient digital quantum simulation of $d$ -level systems for photonic, vibrational, and spin- $s$ Hamiltonians, npj Quantum Inf. 6, 49 (2020).
Note, however, that $L$ cannot be made too large because the maximum energy stored in the oscillator (and thus its decay rate) grows exponentially with $L$ . See Ref. [733] for a general discussion of why the Hilbert space of quantum computers must be built up from tensor products of multiple subsystems.
D. Janković, J.-G. Hartmann, M. Ruben, and P.-A. Hervieux, Noisy qudit vs multiple qubits: Conditions on gate efficiency for enhancing fidelity, npj Quantum Inf. 10, 59 (2024).
S. Rosenblum, Y. Y. Gao, P. Reinhold, C. Wang, C. J. Axline, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, A cnot gate between multiphoton qubits encoded in two cavities, Nat. Commun. 9, 652 (2018).
https://en.wikipedia.org/wiki/Gray_code.
X. Li, G. Yang, C. M. Torres, D. Zheng, and K. L. Wang, A class of efficient quantum incrementer gates for quantum circuit synthesis, Int. J. Mod. Phys. B 28, 1350191 (2014).
A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Elementary gates for quantum computation, Phys. Rev. A 52, 3457 (1995).
T. Häner, M. Roetteler, and K. M. Svore, Optimizing quantum circuits for arithmetic, ArXiv:1805.12445.
N. Wiebe, D. Berry, P. Høyer, and B. C. Sanders, Higher order decompositions of ordered operator exponentials, J. Phys. A: Math. Theor. 43, 065203 (2010).
A. M. Krol and Z. Al-Ars, Beyond quantum Shannon: Circuit construction for general $n$ -qubit gates based on block ZXZ-decomposition, ArXiv:2403.13692.
C. F. Kane, S. Hariprakash, N. S. Modi, M. Kreshchuk, and C. W. Bauer, Block encoding bosons by signal processing, Quantum 9, 1747 (2025).
M. Hanada, S. Matsuura, E. Mendicelli, and E. Rinaldi, Exponential improvement in quantum simulations of bosons, ArXiv:2505.02553.
D. A. Patterson and J. L. Hennessy, Computer Organization and Design RISC-V Edition: The Hardware Software Interface (Morgan Kaufmann Publishers Inc., San Francisco, California, 2017), 1st ed.
D. R. Weinberg, C. J. Gagliardi, J. F. Hull, C. F. Murphy, C. A. Kent, B. C. Westlake, A. Paul, D. H. Ess, D. G. McCafferty, and T. J. Meyer, Proton-coupled electron transfer, Chem. Rev. 112, 4016 (2012).
S. Hammes-Schiffer, Proton-coupled electron transfer: Moving together and charging forward, JACS 137, 8860 (2015).
J. Bloch, A. Cavalleri, V. Galitski, M. Hafezi, and A. Rubio, Strongly correlated electron-photon systems, Nature 606, 41 (2022).
R. F. Ribeiro, L. A. Martínez-Martínez, M. Du, J. Campos-Gonzalez-Angulo, and J. Yuen-Zhou, Polariton chemistry: Controlling molecular dynamics with optical cavities, Chem. Sci. 9, 6325 (2018).
A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (3rd Ed.), (Pearson, Upper Saddle River, Boston, Columbus, San Francisco, New York, Indianapolis, London, Toronto, Sydney, Singapore. Tokyo, Montreal, Dubai, Madrid, Hong Kong, Mexico City, Munich, Paris, Amsterdam, Cape Town, 2009).
C. M. Dawson and M. A. Nielsen, The Solovay-Kitaev algorithm, Quantum Inf. Comput. 6, 81 (2006).
V. Kliuchnikov, D. Maslov, and M. Mosca, Asymptotically optimal approximation of single qubit unitaries by Clifford and $T$ circuits using a constant number of ancillary qubits, Phys. Rev. Lett. 110, 190502 (2013).
J. Haah, Product decomposition of periodic functions in quantum signal processing, Quantum 3, 190 (2019).
E. Gouzien and N. Sangouard, Factoring 2048-bit RSA integers in 177 days with 13 436 qubits and a multimode memory, Phys. Rev. Lett. 127, 140503 (2021).
C. Duckering, J. M. Baker, D. I. Schuster, and F. T. Chong, in 53rd Annual IEEE/ACM International Symposium on Microarchitecture, MICRO ’20 (Association for Computing Machinery, New York, 2020), p. 173.
I. Pietikäinen, O. Černotík, A. Eickbusch, A. Maiti, J. W. O. Garmon, R. Filip, and S. M. Girvin, Strategies and trade-offs for controllability and memory time of ultra-high-quality microwave cavities in circuit QED, PRX Quantum 5, 040307 (2024).
A. Krishna and J.-P. Tillich, Towards low overhead magic state distillation, Phys. Rev. Lett. 123, 070507 (2019).
E. T. Campbell, H. Anwar, and D. E. Browne, Magic-state distillation in all prime dimensions using quantum Reed-Muller codes, Phys. Rev. X 2, 041021 (2012).
Y. Yamamoto, K. Aihara, T. Leleu, K.-i. Kawarabayashi, S. Kako, M. Fejer, K. Inoue, and H. Takesue, Coherent Ising machines—optical neural networks operating at the quantum limit, npj Quantum Inf. 3, 49 (2017).
T. Wang, L. Wu, P. Nobel, and J. Roychowdhury, Solving combinatorial optimisation problems using oscillator based Ising machines, Nat. Comput. 20, 287 (2021).
M. Bonnin, F. L. Traversa, and F. Bonani, in Advances in Artificial Intelligence-based Technologies: Selected Papers in Honour of Professor Nikolaos G. Bourbakis—Vol. 1, edited by M. Virvou, G. A. Tsihrintzis, L. H. Tsoukalas, and L. C. Jain (Springer International Publishing, Cham, 2022), p. 179.
B. Coecke and R. Duncan, Interacting quantum observables: Categorical algebra and diagrammatics, New J. Phys. 13, 043016 (2011).
J. van de Wetering, ZX-calculus for the working quantum computer scientist, ArXiv:2012.13966.
H. Nagayoshi, W. Asavanant, R. Ide, K. Fukui, A. Sakaguchi, J. ichi Yoshikawa, N. C. Menicucci, and A. Furusawa, ZX graphical calculus for continuous-variable quantum processes, Phys. Rev. Res. 7, 033141 (2025).
C. Chamberland and M. E. Beverland, Flag fault-tolerant error correction with arbitrary distance codes, Quantum 2, 53 (2018).
E. van den Berg, S. Bravyi, J. M. Gambetta, P. Jurcevic, D. Maslov, and K. Temme, Single-shot error mitigation by coherent Pauli checks, Phys. Rev. Res. 5, 033193 (2023).
Y. Liu, S. Singh, K. C. Smith, E. Crane, J. M. Martyn, A. Eickbusch, A. Schuckert, R. D. Li, J. Sinanan-Singh, M. B. Soley, et al., Hybrid Oscillator-Qubit Quantum Processors: Instruction Set Architectures, Abstract Machine Models, and Applications [GitHub], 2025, https://github.com/shraggy/ISA-simulations.
J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design derived from the Cooper pair box, Phys. Rev. A 76, 042319 (2007).
J. Gambetta, A. Blais, D. I. Schuster, A. Wallraff, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Qubit-photon interactions in a cavity: Measurement-induced dephasing and number splitting, Phys. Rev. A 74, 042318 (2006).
Y. Shi, P. Gokhale, P. Murali, J. M. Baker, C. Duckering, Y. Ding, N. C. Brown, C. Chamberland, A. Javadi-Abhari, A. W. Cross, et al., Resource-efficient quantum computing by breaking abstractions, Proc. IEEE 108, 1353 (2020).
C. Arenz, D. I. Bondar, D. Burgarth, C. Cormick, and H. Rabitz, Amplification of quadratic Hamiltonians, Quantum 4, 271 (2020).
S. C. Burd, R. Srinivas, J. J. Bollinger, A. C. Wilson, D. J. Wineland, D. Leibfried, D. H. Slichter, and D. T. C. Allcock, Quantum amplification of mechanical oscillator motion, Science 364, 1163 (2019).
Y. Shapira, R. Shaniv, T. Manovitz, N. Akerman, L. Peleg, L. Gazit, R. Ozeri, and A. Stern, Theory of robust multiqubit nonadiabatic gates for trapped ions, Phys. Rev. A 101, 032330 (2020).
N. A. Gershenfeld and I. L. Chuang, Bulk spin-resonance quantum computation, Science 275, 350 (1997).
U. Warring, C. Ospelkaus, Y. Colombe, R. Jördens, D. Leibfried, and D. J. Wineland, Individual-ion addressing with microwave field gradients, Phys. Rev. Lett. 110, 173002 (2013).
S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuletić, and M. D. Lukin, High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622, 268 (2023).
M. O. Brown, S. R. Muleady, W. J. Dworschack, R. J. Lewis-Swan, A. M. Rey, O. Romero-Isart, and C. A. Regal, Time-of-flight quantum tomography of an atom in an optical tweezer, Nat. Phys. 19, 569 (2023).
Y. Wang, A. Kumar, T.-Y. Wu, and D. S. Weiss, Single-qubit gates based on targeted phase shifts in a 3D neutral atom array, Science 352, 1562 (2016).
L. F. Buchmann, K. Mølmer, and D. Petrosyan, Creation and transfer of nonclassical states of motion using Rydberg dressing of atoms in a lattice, Phys. Rev. A 95, 013403 (2017).
K. Eckert, J. Mompart, X. X. Yi, J. Schliemann, D. Bruß, G. Birkl, and M. Lewenstein, Quantum computing in optical microtraps based on the motional states of neutral atoms, Phys. Rev. A 66, 042317 (2002).
L. Lami and M. M. Wilde, Exact solution for the quantum and private capacities of bosonic dephasing channels, Nat. Photonics 17, 525 (2023).
M. Brownnutt, M. Kumph, P. Rabl, and R. Blatt, Ion-trap measurements of electric-field noise near surfaces, Rev. Mod. Phys. 87, 1419 (2015).
Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, Heating of trapped ions from the quantum ground state, Phys. Rev. A 61, 063418 (2000).
C. Noel, M. Berlin-Udi, C. Matthiesen, J. Yu, Y. Zhou, V. Lordi, and H. Häffner, Electric-field noise from thermally activated fluctuators in a surface ion trap, Phys. Rev. A 99, 063427 (2019).
R. Dubessy, T. Coudreau, and L. Guidoni, Electric field noise above surfaces: A model for heating-rate scaling law in ion traps, Phys. Rev. A 80, 031402(R) (2009).
G. H. Low, P. F. Herskind, and I. L. Chuang, Finite-geometry models of electric field noise from patch potentials in ion traps, Phys. Rev. A 84, 053425 (2011).
M. Kumph, C. Henkel, P. Rabl, M. Brownnutt, and R. Blatt, Electric-field noise above a thin dielectric layer on metal electrodes, New J. Phys. 18, 023020 (2016).
M. Teller, D. A. Fioretto, P. C. Holz, P. Schindler, V. Messerer, K. Schüppert, Y. Zou, R. Blatt, J. Chiaverini, J. Sage, and T. E. Northup, Heating of a trapped ion induced by dielectric materials, Phys. Rev. Lett. 126, 230505 (2021).
A. Safavi-Naini, P. Rabl, P. F. Weck, and H. R. Sadeghpour, Microscopic model of electric-field-noise heating in ion traps, Phys. Rev. A 84, 023412 (2011).
B. L. Foulon, K. G. Ray, C.-E. Kim, Y. Liu, B. M. Rubenstein, and V. Lordi, $1 / ω$ electric-field noise in surface ion traps from correlated adsorbate dynamics, Phys. Rev. A 105, 013107 (2022).
K. G. Ray, B. M. Rubenstein, W. Gu, and V. Lordi, van der Waals-corrected density functional study of electric field noise heating in ion traps caused by electrode surface adsorbates, New J. Phys. 21, 053043 (2019).
E. Kim, A. Safavi-Naini, D. A. Hite, K. S. McKay, D. P. Pappas, P. F. Weck, and H. R. Sadeghpour, Electric-field noise from carbon-adatom diffusion on a $Au$ (110) surface: First-principles calculations and experiments, Phys. Rev. A 95, 033407 (2017).
C. D. Bruzewicz, J. M. Sage, and J. Chiaverini, Measurement of ion motional heating rates over a range of trap frequencies and temperatures, Phys. Rev. A 91, 041402(R) (2015).
A. Pagano, D. Jaschke, W. Weiss, and S. Montangero, Optimal control transport of neutral atoms in optical tweezers at finite temperature, Phys. Rev. Res. 6, 033282 (2024).
R. A. Bertlmann and P. Krammer, Bloch vectors for qudits, J. Phys. A: Math. Theor. 41, 235303 (2008).
F. Albertini and D. D’Alessandro, Notions of controllability for bilinear multilevel quantum systems, IEEE Trans. Autom. Control 48, 1399 (2003).
D. Dong and I. Petersen, Quantum control theory and applications: A survey, IET Control Theory Appl. 4, 2651 (2010).
M. Suzuki, Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems, Commun. Math. Phys. 51, 183 (1976).
D. Edward Bruschi, A. Xuereb, and R. Zeier, Deciding finiteness of bosonic dynamics with tunable interactions, J. Phys. A: Math. Theor. 58, 025204 (2024).
F. A. Mele, A. A. Mele, L. Bittel, J. Eisert, V. Giovannetti, L. Lami, L. Leone, and S. F. E. Oliviero, Learning quantum states of continuous variable systems, ArXiv:2405.01431.
C. H. Valahu, T. Navickas, M. J. Biercuk, and T. R. Tan, Benchmarking bosonic modes for quantum information with randomized displacements, PRX Quantum 5, 040337 (2024).
A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Approaching unit visibility for control of a superconducting qubit with dispersive readout, Phys. Rev. Lett. 95, 060501 (2005).
T. Walter, P. Kurpiers, S. Gasparinetti, P. Magnard, A. Potočnik, Y. Salathé, M. Pechal, M. Mondal, M. Oppliger, C. Eichler, and A. Wallraff, Rapid high-fidelity single-shot dispersive readout of superconducting qubits, Phys. Rev. Appl. 7, 054020 (2017).
P. D. Kurilovich, T. Connolly, C. G. L. Bøttcher, D. K. Weiss, S. Hazra, V. R. Joshi, A. Z. Ding, H. Nho, S. Diamond, V. D. Kurilovich, W. Dai, V. Fatemi, L. Frunzio, L. I. Glazman, and M. H. Devoret, High-frequency readout free from transmon multi-excitation resonances, ArXiv:2501.09161.
K. Vogel and H. Risken, Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase, Phys. Rev. A 40, 2847 (1989).
M. Ahmad, S. Qamar, and M. S. Zubairy, Quantum-state tomography using phase-sensitive amplification, Phys. Rev. A 62, 043814 (2000).
M. Ahmad, S. Qamar, and M. S. Zubairy, Reconstruction of a multimode entangled state using a two-photon phase-sensitive linear amplifier, Phys. Rev. A 67, 043815 (2003).
A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, Quantum homodyne tomography of a two-photon Fock state, Phys. Rev. Lett. 96, 213601 (2006).
K. He, M. Yuan, Y. Wong, S. Chakram, A. Seif, L. Jiang, and D. I. Schuster, Efficient multimode Wigner tomography, Nat. Commun. 15, 4138 (2024).
C. Shen, R. W. Heeres, P. Reinhold, L. Jiang, Y.-K. Liu, R. J. Schoelkopf, and L. Jiang, Optimized tomography of continuous variable systems using excitation counting, Phys. Rev. A 94, 052327 (2016).
T. Krisnanda, C. Y. Fontaine, A. Copetudo, P. Song, K. X. Lee, N.-N. Huang, F. Valadares, T. C. H. Liew, and Y. Y. Gao, Demonstrating efficient and robust bosonic state reconstruction via optimized excitation counting, PRX Quantum 6, 010303 (2025).
S. T. Flammia, D. Gross, Y.-K. Liu, and J. Eisert, Quantum tomography via compressed sensing: Error bounds, sample complexity and efficient estimators, New J. Phys. 14, 095022 (2012).
K. R. Brown, A. W. Harrow, and I. L. Chuang, Arbitrarily accurate composite pulse sequences, Phys. Rev. A 70, 052318 (2004).
K. R. Brown, A. W. Harrow, and I. L. Chuang, Erratum: Arbitrarily accurate composite pulse sequences [Phys. Rev. A 70, 052318 (2004)], Phys. Rev. A 72, 039905(E) (2005).
W. Gong and S. Choi, Robust and parallel control of many qubits, ArXiv:2312.08426.
L. Ying, Stable factorization for phase factors of quantum signal processing, Quantum 6, 842 (2022).
Y. Dong, X. Meng, K. B. Whaley, and L. Lin, Efficient phase-factor evaluation in quantum signal processing, Phys. Rev. A 103, 042419 (2021).
J. Wang, Y. Dong, and L. Lin, On the energy landscape of symmetric quantum signal processing, Quantum 6, 850 (2022).
R. Blume-Kohout, C. M. Caves, and I. H. Deutsch, Climbing mount scalable: Physical resource requirements for a scalable quantum computer, Found. Phys. 32, 1641 (2002).

PRX Quantum 7, 010201– Published 7 January, 2026

Vol. 7, Iss. 1 — January - March 2026

Received 28 August 2024
Revised 15 August 2025

DOI: https://doi.org/10.1103/4rf7-9tfx

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Hybrid Oscillator-Qubit Quantum Processors: Instruction Set Architectures, Abstract Machine Models, and Applications

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