- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Scattering Processes from Quantum Simulation Algorithms for Scalar Field Theories

Andrew Hardy1,*, Priyanka Mukhopadhyay2,†, M. Sohaib Alam3,4,†, Robert Konik5, Layla Hormozi6, Eleanor Rieffel3, Stuart Hadfield3,4, João Barata7, Raju Venugopalan7 et al.

Dmitri E. Kharzeev8,7 and Nathan Wiebe2,9,10,‡

  • *Contact author: andrew.hardy@mail.utoronto.ca
  • †These authors made equal contributions to this work.
  • ‡Contact author: nawiebe@cs.toronto.edu
PDF

PRX Quantum 7, 010343 – Published 3 March, 2026

DOI: https://doi.org/10.1103/3krb-wwfx

Abstract

We provide practical simulation methods for scalar field theories on a quantum computer that yield improved asymptotics as well as concrete gate estimates for the simulation and physical qubit estimates using the surface code. We achieve these improvements through two optimizations. First, we consider a finite volume approach for estimating the elements of the S-matrix. This approach is appropriate in general for 1+1D and for certain low-energy elastic collisions in higher dimensions. Second, we implement our approach using a series of different fault-tolerant simulation algorithms for Hamiltonians formulated both in the field occupation basis and field amplitude basis. Our algorithms are based on either second-order Trotterization or qubitization. The cost of Trotterization in occupation basis scales as O(λN7|Ω|3/(M5/2ϵ3/2)) where λ is the coupling strength, N is the occupation cutoff, |Ω| is the volume of the spatial lattice, M is the mass of the particles and ϵ is the uncertainty in the energy calculation used for the S-matrix determination. Qubitization in the field basis scales as O(|Ω|2(k2Λ+kM2)/ϵ), where k is the cutoff in the field and Λ is a scaled coupling constant. We find in both cases that the bounds suggest physically meaningful simulations can be performed using on the order of 4×106 physical qubits and 1012T-gates which corresponds to roughly one day on a superconducting quantum computer with surface code and a cycle time of 100 ns. This places the simulation of scalar field theory within striking distance of the gate counts for the best available chemistry simulation results.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Our paper establishes a practical roadmap for simulating particle physics on quantum computers. Classical supercomputers often struggle to calculate the complex scattering processes of quantum field theories, the fundamental framework describing how particles interact. To overcome this, we developed optimized quantum algorithms that compute these scattering matrix elements using a finite-volume approach. Scattering processes provide comprehensive information about the constituents of the theory.

We demonstrate that a physically meaningful simulation of scalar field theory is within reach, requiring computational resources comparable to those needed for breakthrough quantum chemistry calculations. By estimating that such simulations could run in roughly one day on a future fault-tolerant quantum computer, this research highlights a promising new avenue for exploring high-energy physics.

Article Text

Supplemental Material

References (171)

  1. S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996).
  2. D. W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders, Efficient quantum algorithms for simulating sparse Hamiltonians, Commun. Math. Phys. 270, 359 (2007).
  3. A. M. Childs and N. Wiebe, Hamiltonian simulation using linear combinations of unitary operations, Quantum Inf. Comput. 12, 901 (2012).
  4. A. M. Childs, R. Kothari, and R. D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision, SIAM J. Comput. 46, 1920 (2017).
  5. G. H. Low and I. L. Chuang, Hamiltonian simulation by qubitization, Quantum 3, 163 (2019).
  6. Y. Su, D. W. Berry, N. Wiebe, N. Rubin, and R. Babbush, Fault-tolerant quantum simulations of chemistry in first quantization, PRX Quantum 2, 040332 (2021).
  7. C. W. Bauer et al., Quantum simulation for high energy physics, PRX Quantum 4, 027001 (2023).
  8. J. Kogut and L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D 11, 395 (1975).
  9. S. D. Drell, H. R. Quinn, B. Svetitsky, and M. Weinstein, Quantum electrodynamics on a lattice: A Hamiltonian variational approach to the physics of the weak-coupling region, Phys. Rev. D 19, 619 (1979).
  10. S. P. Jordan, H. Krovi, K. S. M. Lee, and J. Preskill, BQP-completeness of scattering in scalar quantum field theory, Quantum 2, 44 (2018).
  11. N. Klco and M. J. Savage, Digitization of scalar fields for quantum computing, Phys. Rev. A 99, 052335 (2019).
  12. Y. Alexeev, D. Bacon, K. R. Brown, R. Calderbank, L. D. Carr, F. T. Chong, B. DeMarco, D. Englund, E. Farhi, B. Fefferman et al., Quantum computer systems for scientific discovery, PRX Quantum 2, 017001 (2021).
  13. C. W. Bauer, Z. Davoudi, A. B. Balantekin, T. Bhattacharya, M. Carena, W. A. De Jong, P. Draper, A. El-Khadra, N. Gemelke, M. Hanada et al., Quantum simulation for high-energy physics, PRX Quantum 4, 027001 (2023).
  14. N. H. Nguyen, M. C. Tran, Y. Zhu, A. M. Green, C. H. Alderete, Z. Davoudi, and N. M. Linke, Digital quantum simulation of the Schwinger model and symmetry protection with trapped ions, PRX Quantum 3, 020324 (2022).
  15. A. F. Shaw, P. Lougovski, J. R. Stryker, and N. Wiebe, Quantum algorithms for simulating the lattice Schwinger model, Quantum 4, 306 (2020).
  16. S. P. Jordan, K. S. M. Lee, and J. Preskill, Quantum algorithms for quantum field theories, Science 336, 1130 (2012).
  17. S. P. Jordan, K. S. M. Lee, and J. Preskill, Quantum algorithms for fermionic quantum field theories, arXiv:1404.7115 [hep-th, physics:quant-ph].
  18. P. G. Kevrekidis and J. Cuevas-Maraver, A Dynamical Perspective on the ϕ4 Model: Past, Present and Future, Nonlinear Systems and Complexity (Springer International Publishing, Cham, 2019), ISBN: 978-3-030-11838-9.
  19. R. P. Feynman, Photon-Hadron Interactions, Frontiers in Physics (W. A. Benjamin, Reading, Massachusetts, 2019), Vol. 35, ISBN: 978-0-805-32511-9.
  20. J. D. Bjorken and E. A. Paschos, Inelastic electron-proton and γ-proton scattering and the structure of the nucleon, Phys. Rev. 185, 1975 (1969).
  21. J. Barata, N. Mueller, A. Tarasov, and R. Venugopalan, Single-particle digitization strategy for quantum computation of a ϕ4 scalar field theory, Phys. Rev. A 103, 042410 (2021).
  22. A. C. Y. Li, A. Macridin, S. Mrenna, and P. Spentzouris, Simulating scalar field theories on quantum computers with limited resources, Phys. Rev. A 107, 032603 (2023).
  23. M. Kreshchuk, J. P. Vary, and P. J. Love, Simulating scattering of composite particles, arXiv:2310.13742 [quant-ph].
  24. M. Bagherimehrab, Y. R. Sanders, D. W. Berry, G. K. Brennen, and B. C. Sanders, Nearly optimal quantum algorithm for generating the ground state of a free quantum field theory, PRX Quantum 3, 020364 (2022).
  25. T. Farrelly and J. Streich, Discretizing quantum field theories for quantum simulation, arXiv:2002.02643 [quant-ph].
  26. N. Klco and M. J. Savage, Minimally entangled state preparation of localized wave functions on quantum computers, Phys. Rev. A 102, 012612 (2020).
  27. K. Yeter-Aydeniz, E. F. Dumitrescu, A. J. McCaskey, R. S. Bennink, R. C. Pooser, and G. Siopsis, Scalar quantum field theories as a benchmark for near-term quantum computers, Phys. Rev. A 99, 032306 (2019).
  28. Z. Davoudi, A. F. Shaw, and J. R. Stryker, General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory, Quantum 7, 1213 (2023).
  29. M. Rhodes, M. Kreshchuk, and S. Pathak, Exponential improvements in the simulation of lattice gauge theories using near-optimal techniques, PRX Quantum 5, 040347 (2024).
  30. S. Hariprakash, N. S. Modi, M. Kreshchuk, C. F. Kane, and C. W. Bauer, Strategies for simulating time evolution of Hamiltonian lattice field theories, Phys. Rev. A 111, 022419 (2025).
  31. C. W. Bauer, M. Freytsis, and B. Nachman, Simulating collider physics on quantum computers using effective field theories, Phys. Rev. Lett. 127, 212001 (2021).
  32. M. Lüscher, Two-particle states on a torus and their relation to the scattering matrix, Nucl. Phys. B 354, 531 (1991).
  33. M. T. Hansen and S. R. Sharpe, Lattice QCD and three-particle decays of resonances, Annu. Rev. Nucl. Part. Sci. 69, 65 (2019).
  34. A. Rusetsky, Three particles on the lattice, arXiv:1911.01253 [hep-lat].
  35. M. Mai, M. Döring, and A. Rusetsky, Multi-particle systems on the lattice and chiral extrapolations: A brief review, Eur. Phys. J. Spec. Top. 230, 1623 (2021).
  36. F. Romero-López, Multi-hadron interactions from lattice QCD, arXiv:2212.13793 [hep-lat].
  37. B. Gabai and X. Yin, On the S-matrix of Ising field theory in two dimensions, J. High Energy Phys. 2022, 168 (2022).
  38. Z. Bajnok and M. Lajer, Truncated Hilbert space approach to the 2d ϕ4 theory, J. High Energy Phys. 2016, 50 (2016).
  39. A. J. A. James, R. M. Konik, P. Lecheminant, N. J. Robinson, and A. M. Tsvelik, Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-Abelian bosonization to truncated spectrum methods, Rep. Prog. Phys. 81, 046002 (2018).
  40. T. Luu and M. J. Savage, Extracting scattering phase shifts in higher partial waves from lattice QCD calculations, Phys. Rev. D 83, 114508 (2011).
  41. J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards, and C. E. Thomas (for the Hadron Spectrum Collaboration), Phase shift of isospin-2 ππ scattering from lattice QCD, Phys. Rev. D 83, 071504 (2011).
  42. D. Guo, A. Alexandru, R. Molina, M. Mai, and M. Döring, Extraction of isoscalar ππ phase shifts from lattice QCD, Phys. Rev. D 98, 014507 (2018).
  43. M. Garofalo, F. Romero-López, A. Rusetsky, and C. Urbach, Testing a new method for scattering in finite volume in the ϕ4 theory, Eur. Phys. J. C 81, 1034 (2021).
  44. V. P. Yurov and Al. B. Zamolodchikov, Truncated conformal space approach to scaling Lee-Yang model, Int. J. Mod. Phys. A 05, 3221 (1990).
  45. V. P. Yurov and Al. B. Zamolodchikov, Truncated-fermionic-space approach to the critical 2D Ising model with magnetic field, Int. J. Mod. Phys. A 06, 4557 (1991).
  46. M. Hogervorst, S. Rychkov, and B. C. van Rees, Truncated conformal space approach in d dimensions: A cheap alternative to lattice field theory? Phys. Rev. D 91, 025005 (2015).
  47. T. Cohen, K. Farnsworth, R. Houtz, and M. A. Luty, Hamiltonian truncation effective theory, SciPost Phys. 13, 011 (2022).
  48. H. Chen, A. L. Fitzpatrick, and D. Karateev, Form factors and spectral densities from lightcone conformal truncation, J. High Energy Phys. 2022, 109 (2022).
  49. J. Elias-Miró and E. Hardy, Exploring Hamiltonian truncation in d=2+1, Phys. Rev. D 102, 065001 (2020).
  50. J. Liu, D. Meltzer, D. Poland, and D. Simmons-Duffin, The Lorentzian inversion formula and the spectrum of the 3d O(2) CFT, J. High Energy Phys. 2020, 115 (2020).
  51. M. Jarrell and J. E. Gubernatis, Bayesian inference and the analytic continuation of imaginary-time quantum Monte Carlo data, Phys. Rep. 269, 133 (1996).
  52. A. W. Sandvik, Stochastic method for analytic continuation of quantum Monte Carlo data, Phys. Rev. B 57, 10287 (1998).
  53. K. S. D. Beach, Identifying the maximum entropy method as a special limit of stochastic analytic continuation, arXiv:cond-mat/0403055 [cond-mat.str-el].
  54. E. Gustafson, Y. Zhu, P. Dreher, N. M. Linke, and Y. Meurice, Real-time quantum calculations of phase shifts using wave packet time delays, Phys. Rev. D 104, 054507 (2021).
  55. M. A. Carrillo, R. A. Briceño, and A. M. Sturzu, Inclusive reactions from finite Minkowski spacetime correlation functions, Phys. Rev. D 110, 054503 (2024).
  56. R. A. Briceño, M. A. Carrillo, J. V. Guerrero, M. T. Hansen, and A. M. Sturzu, Accessing scattering amplitudes using quantum computers, PoS LATTICE2021, 315 (2022).
  57. P. Mukhopadhyay, N. Wiebe, and H. T. Zhang, Synthesizing efficient circuits for Hamiltonian simulation, npj Quantum Inf. 9, 31 (2023).
  58. P. Mukhopadhyay, T. F. Stetina, and N. Wiebe, Quantum simulation of the first-quantized Pauli-Fierz Hamiltonian, PRX Quantum 5, 010345 (2024).
  59. M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer, Elucidating reaction mechanisms on quantum computers, Proc. Natl. Acad. Sci. U.S.A. 114, 7555 (2017).
  60. R. Babbush, N. Wiebe, J. McClean, J. McClain, H. Neven, and G. K.-L. Chan, Low-depth quantum simulation of materials, Phys. Rev. X 8, 011044 (2018).
  61. A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su, Toward the first quantum simulation with quantum speedup, Proc. Natl. Acad. Sci. U.S.A. 115, 9456 (2018).
  62. B. Giles and P. Selinger, Exact synthesis of multiqubit Clifford+T circuits, Phys. Rev. A 87, 032332 (2013).
  63. V. Kliuchnikov, D. Maslov, and M. Mosca, Practical approximation of single-qubit unitaries by single-qubit quantum Clifford and T circuits, IEEE Trans. Comput. 65, 161 (2016).
  64. N. J. Ross and P. Selinger, Optimal ancilla-free Clifford+T approximation of z-rotations, Quantum Inf. Comput. 16, 901 (2016).
  65. M. Mosca and P. Mukhopadhyay, A polynomial time and space heuristic algorithm for T-count, Quantum Sci. Technol. 7, 015003 (2022).
  66. V. Gheorghiu, M. Mosca, and P. Mukhopadhyay, A (quasi-)polynomial time heuristic algorithm for synthesizing T-depth optimal circuits, npj Quantum Inf. 8, 110 (2022).
  67. V. Gheorghiu, M. Mosca, and P. Mukhopadhyay, T-count and T-depth of any multi-qubit unitary, npj Quantum Inf. 8, 141 (2022).
  68. P. Mukhopadhyay, Synthesis of V-count-optimal quantum circuits for multiqubit unitaries, Phys. Rev. A 109, 052619 (2024).
  69. P. Mukhopadhyay, CS-count-optimal quantum circuits for arbitrary multi-qubit unitaries, Sci. Rep. 14, 13916 (2024).
  70. P. Mukhopadhyay, Synthesizing Toffoli-optimal quantum circuits for arbitrary multi-qubit unitaries, arXiv:2401.08950.
  71. 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).
  72. A. G. Fowler and C. Gidney, Low overhead quantum computation using lattice surgery, arXiv:1808.06709 [quant-ph].
  73. D. Litinski, Magic state distillation: Not as costly as you think, Quantum 3, 205 (2019).
  74. J. Elias-Miró, S. Rychkov, and L. G. Vitale, NLO renormalization in the Hamiltonian truncation, Phys. Rev. D 96, 065024 (2017).
  75. B. Blossier, M. D. Morte, G. v. Hippel, T. Mendes, and R. Sommer (ALPHA Collaboration), Lattice calculation of the Isgur-Wise functions τ1/2 with dynamical quarks, J. High Energy Phys. 2009, 022 (2009).
  76. R. A. Briceño, J. J. Dudek, and R. D. Young, Scattering processes and resonances from lattice QCD, Rev. Mod. Phys. 90, 025001 (2018).
  77. R. D. Mattuck and B. Johansson, Adv. Phys. 17, 509 (1968).
  78. J. Espinosa, M. Quiros, and F. Zwirner, On the phase transition in the scalar theory, Phys. Lett. B 291, 115 (1992).
  79. M. Hagan and N. Wiebe, Composite quantum simulations, Quantum 7, 1181 (2023).
  80. A. Rajput, A. Roggero, and N. Wiebe, Hybridized methods for quantum simulation in the interaction picture, Quantum 6, 780 (2022).
  81. G. H. Low and I. L. Chuang, Optimal Hamiltonian simulation by quantum signal processing, Phys. Rev. Lett. 118, 010501 (2017).
  82. G. H. Low and I. L. Chuang, Hamiltonian simulation by qubitization, Quantum 3, 163 (2019).
  83. A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (2019), pp. 193–204.
  84. M. Suzuki, General theory of fractal path integrals with applications to many-body theories and statistical physics, J. Math. Phys. 32, 400 (1991).
  85. R. D. Somma, Quantum simulations of one dimensional quantum systems, arXiv:1503.06319 [quant-ph].
  86. A. Macridin, P. Spentzouris, J. Amundson, and R. Harnik, Digital quantum computation of fermion-boson interacting systems, Phys. Rev. A 98, 042312 (2018).
  87. A. Macridin, P. Spentzouris, J. Amundson, and R. Harnik, Electron-phonon systems on a universal quantum computer, Phys. Rev. Lett. 121, 110504 (2018).
  88. E. Rinaldi, X. Han, M. Hassan, Y. Feng, F. Nori, M. McGuigan, and M. Hanada, Matrix-model simulations using quantum computing, deep learning, and lattice Monte Carlo, PRX Quantum 3, 010324 (2022).
  89. M. Hanada, J. Liu, E. Rinaldi, and M. Tezuka, Estimating truncation effects of quantum bosonic systems using sampling algorithms, Mach. Learn.: Sci. Technol. 4, 045021 (2023).
  90. Y. Akahoshi, K. Maruyama, H. Oshima, S. Sato, and K. Fujii, Partially fault-tolerant quantum computing architecture with error-corrected Clifford gates and space-time efficient analog rotations, PRX Quantum 5, 010337 (2024).
  91. E. T. Campbell, B. M. Terhal, and C. Vuillot, Roads towards fault-tolerant universal quantum computation, Nature 549, 172 (2017).
  92. 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).
  93. A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Phys. Rev. X 11, 011020 (2021).
  94. D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings, and M. Troyer, Gate-count estimates for performing quantum chemistry on small quantum computers, Phys. Rev. A 90, 022305 (2014).
  95. R. Babbush, J. McClean, D. Wecker, A. Aspuru-Guzik, and N. Wiebe, Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation, Phys. Rev. A 91, 022311 (2015).
  96. M. Li, T. Lappi, and X. Zhao, Scattering and gluon emission in a color field: A light-front Hamiltonian approach, Phys. Rev. D 104, 056014 (2021).
  97. J. Barata, X. Du, M. Li, W. Qian, and C. A. Salgado, Quantum simulation of in-medium QCD jets: Momentum broadening, gluon production, and entropy growth, Phys. Rev. D 108, 056023 (2023).
  98. G. Calajò, G. Magnifico, C. Edmunds, M. Ringbauer, S. Montangero, and P. Silvi, Digital quantum simulation of a (1+1)D SU(2) lattice gauge theory with ion qudits, PRX Quantum 5, 040309 (2024).
  99. J. Unmuth-Yockey, J. Zhang, A. Bazavov, Y. Meurice, and S.-W. Tsai, Universal features of the Abelian Polyakov loop in 1+1 dimensions, Phys. Rev. D 98, 094511 (2018).
  100. J. F. Unmuth-Yockey, Gauge-invariant rotor Hamiltonian from dual variables of 3D U(1) gauge theory, Phys. Rev. D 99, 074502 (2019).
  101. N. Klco, M. J. Savage, and J. R. Stryker, SU(2) non-Abelian gauge field theory in one dimension on digital quantum computers, Phys. Rev. D 101, 074512 (2020).
  102. R. C. Farrell, M. Illa, A. N. Ciavarella, and M. J. Savage, Scalable circuits for preparing ground states on digital quantum computers: The Schwinger model vacuum on 100 qubits, PRX Quantum 5, 020315 (2024).
  103. R. C. Farrell, M. Illa, A. N. Ciavarella, and M. J. Savage, Quantum simulations of hadron dynamics in the Schwinger model using 112 qubits, Phys. Rev. D 109, 114510 (2024).
  104. M. Illa, C. E. P. Robin, and M. J. Savage, Qu8its for quantum simulations of lattice quantum chromodynamics, Phys. Rev. D 110, 014507 (2024).
  105. A. Ciavarella, N. Klco, and M. J. Savage, Trailhead for quantum simulation of SU(3) Yang-Mills lattice gauge theory in the local multiplet basis, Phys. Rev. D 103, 094501 (2021).
  106. A. Bazavov, Y. Meurice, S.-W. Tsai, J. Unmuth-Yockey, and J. Zhang, Gauge-invariant implementation of the Abelian-Higgs model on optical lattices, Phys. Rev. D 92, 076003 (2015).
  107. J. Zhang, J. Unmuth-Yockey, J. Zeiher, A. Bazavov, S.-W. Tsai, and Y. Meurice, Quantum simulation of the universal features of the Polyakov loop, Phys. Rev. Lett. 121, 223201 (2018).
  108. A. Bazavov, S. Catterall, R. G. Jha, and J. Unmuth-Yockey, Tensor renormalization group study of the non-Abelian Higgs model in two dimensions, Phys. Rev. D 99, 114507 (2019).
  109. C. W. Bauer and D. M. Grabowska, Efficient representation for simulating U(1) gauge theories on digital quantum computers at all values of the coupling, Phys. Rev. D 107, L031503 (2023).
  110. D. M. Grabowska, C. Kane, B. Nachman, and C. W. Bauer, Overcoming exponential scaling with system size in Trotter-Suzuki implementations of constrained Hamiltonians: 2+1 U(1) lattice gauge theories, arXiv:2208.03333 [quant-ph].
  111. A. J. Buser, T. Bhattacharya, L. Cincio, and R. Gupta, State preparation and measurement in a quantum simulation of the O(3) sigma model, Phys. Rev. D 102, 114514 (2020).
  112. T. Bhattacharya, A. J. Buser, S. Chandrasekharan, R. Gupta, and H. Singh, Qubit regularization of asymptotic freedom, Phys. Rev. Lett. 126, 172001 (2021).
  113. A. H. Z. Kavaki and R. Lewis, From square plaquettes to triamond lattices for SU(2) gauge theory, Commun. Phys. 7, 208 (2024).
  114. E. M. Murairi, M. J. Cervia, H. Kumar, P. F. Bedaque, and A. Alexandru, How many quantum gates do gauge theories require?, Phys. Rev. D 106, 094504 (2022).
  115. E. Zohar, J. I. Cirac, and B. Reznik, Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices, Rep. Prog. Phys. 79, 014401 (2016).
  116. E. Zohar, J. I. Cirac, and B. Reznik, Cold-atom quantum simulator for SU(2) Yang-Mills lattice gauge theory, Phys. Rev. Lett. 110, 125304 (2013).
  117. E. Zohar, J. I. Cirac, and B. Reznik, Simulating compact quantum electrodynamics with ultracold atoms: Probing confinement and nonperturbative effects, Phys. Rev. Lett. 109, 125302 (2012).
  118. E. Zohar, J. I. Cirac, and B. Reznik, Quantum simulations of gauge theories with ultracold atoms: Local gauge invariance from angular-momentum conservation, Phys. Rev. A 88, 023617 (2013).
  119. I. Raychowdhury and J. R. Stryker, Loop, string, and hadron dynamics in SU(2) Hamiltonian lattice gauge theories, Phys. Rev. D 101, 114502 (2020).
  120. Z. Davoudi, C.-C. Hsieh, and S. V. Kadam, Scattering wave packets of hadrons in gauge theories: Preparation on a quantum computer, Quantum 8, 1520 (2024).
  121. I. Raychowdhury and J. R. Stryker, Solving Gauss’s law on digital quantum computers with loop-string-hadron digitization, Phys. Rev. Res. 2, 033039 (2020).
  122. Z. Davoudi, I. Raychowdhury, and A. Shaw, Search for efficient formulations for Hamiltonian simulation of non-Abelian lattice gauge theories, Phys. Rev. D 104, 074505 (2021).
  123. E. Mathew and I. Raychowdhury, Protecting local and global symmetries in simulating (1+1)D non-Abelian gauge theories, Phys. Rev. D 106, 054510 (2022).
  124. M. Kreshchuk, W. M. Kirby, G. Goldstein, H. Beauchemin, and P. J. Love, Quantum simulation of quantum field theory in the light-front formulation, Phys. Rev. A 105, 032418 (2022).
  125. M. Kreshchuk, S. Jia, W. M. Kirby, G. Goldstein, J. P. Vary, and P. J. Love, Simulating hadronic physics on noisy intermediate-scale quantum devices using basis light-front quantization, Phys. Rev. A 103, 062601 (2021).
  126. M. Kreshchuk, S. Jia, W. M. Kirby, G. Goldstein, J. P. Vary, and P. J. Love, Light-front field theory on current quantum computers, Entropy 23, 597 (2021).
  127. M. S. Alam, S. Hadfield, H. Lamm, and A. C. Y. Li (SQMS Collaboration), Primitive quantum gates for dihedral gauge theories, Phys. Rev. D 105, 114501 (2022).
  128. E. J. Gustafson, H. Lamm, F. Lovelace, and D. Musk, Primitive quantum gates for an SU(2) discrete subgroup: Binary tetrahedral, Phys. Rev. D 106, 114501 (2022).
  129. E. J. Gustafson, H. Lamm, and F. Lovelace, Primitive quantum gates for an SU(2) discrete subgroup: Binary octahedral, Phys. Rev. D 109, 054503 (2024).
  130. E. J. Gustafson, Y. Ji, H. Lamm, E. M. Murairi, and S. Zhu, Primitive quantum gates for an SU(3) discrete subgroup: Σ(36×3), Phys. Rev. D 110, 034515 (2024).
  131. J. Bender, E. Zohar, A. Farace, and J. I. Cirac, Digital quantum simulation of lattice gauge theories in three spatial dimensions, New J. Phys. 20, 093001 (2018).
  132. D. C. Hackett, K. Howe, C. Hughes, W. Jay, E. T. Neil, and J. N. Simone, Digitizing gauge fields: Lattice Monte Carlo results for future quantum computers, Phys. Rev. A 99, 062341 (2019).
  133. A. Alexandru, P. F. Bedaque, S. Harmalkar, H. Lamm, S. Lawrence, and N. C. Warrington (NuQS Collabration, Gluon field digitization for quantum computers, Phys. Rev. D 100, 114501 (2019).
  134. A. Yamamoto, Real-time simulation of (2+1)-dimensional lattice gauge theory on qubits, Prog. Theor. Exp. Phys. 2021, 013B06 (2021).
  135. J. F. Haase, L. Dellantonio, A. Celi, D. Paulson, A. Kan, K. Jansen, and C. A. Muschik, A resource efficient approach for quantum and classical simulations of gauge theories in particle physics, Quantum 5, 393 (2021).
  136. T. Armon, S. Ashkenazi, G. Garcia-Moreno, A. Gonzalez-Tudela, and E. Zohar, Photon-mediated stroboscopic quantum simulation of a Z2 lattice gauge theory, Phys. Rev. Lett. 127, 250501 (2021).
  137. Y. Ji, H. Lamm, and S. Zhu (NuQS Collaboration), Gluon field digitization via group space decimation for quantum computers, Phys. Rev. D 102, 114513 (2020).
  138. M. Carena, H. Lamm, Y.-Y. Li, and W. Liu, Lattice renormalization of quantum simulations, Phys. Rev. D 104, 094519 (2021).
  139. D. González-Cuadra, T. V. Zache, J. Carrasco, B. Kraus, and P. Zoller, Hardware efficient quantum simulation of non-Abelian gauge theories with qudits on Rydberg platforms, Phys. Rev. Lett. 129, 160501 (2022).
  140. K. Marshall, R. Pooser, G. Siopsis, and C. Weedbrook, Quantum simulation of quantum field theory using continuous variables, Phys. Rev. A 92, 063825 (2015).
  141. R. G. Jha, F. Ringer, G. Siopsis, and S. Thompson, Continuous-variable quantum computation of the O(3) model in 1+1 dimensions, Phys. Rev. A 109, 052412 (2024).
  142. K. Yeter-Aydeniz and G. Siopsis, Quantum computation of scattering amplitudes in scalar quantum electrodynamics, Phys. Rev. D 97, 036004 (2018).
  143. X. Deng, S. Hao, H. Guo, C. Xie, and X. Su, Continuous variable quantum optical simulation for time evolution of quantum harmonic oscillators, Sci. Rep. 6, 22914 (2016).
  144. S. Thompson and G. Siopsis, Quantum computation of phase transition in interacting scalar quantum field theory, Quantum Inf. Proc. 22, 396 (2023).
  145. E. J. Gustafson, Prospects for simulating a qudit-based model of 1+1D scalar QED, Phys. Rev. D 103, 114505 (2021).
  146. E. Gustafson, Noise improvements in quantum simulations of sQED using qutrits, arXiv:2201.04546 [quant-ph].
  147. M. Illa, C. E. P. Robin, and M. J. Savage, Quantum simulations of SO(5) many-fermion systems using qudits, Phys. Rev. C 108, 064306 (2023).
  148. F. Turro, I. A. Chernyshev, R. Bhaskar, and M. Illa, Qutrit and qubit circuits for three-flavor collective neutrino oscillations, Phys. Rev. D 111, 043038 (2025).
  149. D. M. Kurkcuoglu, M. S. Alam, J. A. Job, A. C. Y. Li, A. Macridin, G. N. Perdue, and S. Providence, Quantum simulation of ϕ4 theories in qudit systems, arXiv:2108.13357 [quant-ph].
  150. A. Vezvaee, N. Earnest-Noble, and K. Najafi, Quantum simulation of Fermi-Hubbard model based on transmon qudit interaction, arXiv:2402.01243 [quant-ph].
  151. T. Roy, Qudit-based quantum computing with SRF cavities at Fermilab (2023). https://api.semanticscholar.org/CorpusID:263296135.
  152. M. Meth, J. F. Haase, J. Zhang, C. Edmunds, L. Postler, A. Steiner, A. J. Jena, L. Dellantonio, R. Blatt, P. Zoller, T. Monz, P. Schindler, C. Muschik, and M. Ringbauer, Simulating 2D lattice gauge theories on a qudit quantum computer, Nat. Phys. 21, 570 (2025).
  153. M. Hanada, J. Liu, E. Rinaldi, and M. Tezuka, Estimating truncation effects of quantum bosonic systems using sampling algorithms, Mach. Learn. Sci. Tech. 4, 045021 (2023).
  154. See Supplemental Material at http://link.aps.org/supplemental/10.1103/3krb-wwfx for additional derivations of theorems in the main text and explicit circuit constructions.
  155. Y. He, M.-X. Luo, E. Zhang, H.-K. Wang, and X.-F. Wang, Decompositions of n-qubit Toffoli gates with linear circuit complexity, Int. J. Theor. Phys. 56, 2350 (2017).
  156. C. Gidney, Halving the cost of quantum addition, Quantum 2, 74 (2018).
  157. J. Huyghebaert and H. De Raedt, Product formula methods for time-dependent Schrodinger problems, J. Phys. A: Math. Gen. 23, 5777 (1990).
  158. 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).
  159. D. Wecker, M. B. Hastings, N. Wiebe, B. K. Clark, C. Nayak, and M. Troyer, Solving strongly correlated electron models on a quantum computer, Phys. Rev. A 92, 062318 (2015).
  160. A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su, Toward the first quantum simulation with quantum speedup, Proc. Natl. Acad. Sci. U.S.A. 115, 9456 (2018).
  161. R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Phys. Rev. X 8, 041015 (2018).
  162. Y. Nam, Y. Su, and D. Maslov, Approximate quantum Fourier transform with O(nlogn) T gates, npj Quantum Inf. 6, 1 (2020).
  163. A. M. Childs and N. Wiebe, Hamiltonian simulation using linear combinations of unitary operations, Quantum Inf. Comput. 12, 901 (2012).
  164. F. Vatan and C. Williams, Optimal quantum circuits for general two-qubit gates, Phys. Rev. A 69, 032315 (2004).
  165. A. Bocharov, M. Roetteler, and K. M. Svore, Efficient synthesis of universal repeat-until-success quantum circuits, Phys. Rev. Lett. 114, 080502 (2015).
  166. P. Selinger, Efficient Clifford+T approximation of single-qubit operators, arXiv:1212.6253 [quant-ph].
  167. 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).
  168. M. Amy, P. Azimzadeh, and M. Mosca, On the controlled-NOT complexity of controlled-NOT–phase circuits, Quantum Sci. Technol. 4, 015002 (2019).
  169. V. Gheorghiu, J. Huang, S. M. Li, M. Mosca, and P. Mukhopadhyay, Reducing the CNOT count for Clifford+T circuits on NISQ architectures, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 42, 1873 (2023).
  170. A. Kitaev and W. A. Webb, Wavefunction preparation and resampling using a quantum computer, arXiv:0801.0342 [quant-ph].
  171. D. W. Berry, A. M. Childs, and R. Kothari, Hamiltonian simulation with nearly optimal dependence on all parameters, in 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE Computer Society, Berkeley, CA, USA, 2015), pp. 792–809, ISBN: 978-1-4673-9379-6.

Outline

Information

PRX Quantum 7, 010343– Published 3 March, 2026
  • Received 9 October 2024
  • Revised 9 June 2025
  • Accepted 14 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation