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Self-Correcting Gottesman-Kitaev-Preskill Qubit and Gates in a Driven-Dissipative Circuit

Frederik Nathan1,2,*, Liam O’Brien2, Kyungjoo Noh3, Matthew H. Matheny3, Arne L. Grimsmo3, Liang Jiang3,4, and Gil Refael2,3

  • *Contact author: frederik.nathan@nbi.ku.dk
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PRX Quantum 6, 030352 – Published 17 September, 2025

DOI: https://doi.org/10.1103/ykqb-m52z

Abstract

We show that a self-correcting Gottesman-Kitaev-Preskill (GKP) qubit can be realized with a high-impedance LC circuit coupled to a resistor and a Josephson junction via a controllable switch. When activating the switch in a particular stepwise pattern, the resonator relaxes into a subspace of GKP states that encode a protected qubit. Under continued operation, the resistor dissipatively error corrects the qubit against bit flips and decoherence by absorbing noise-induced entropy. We show that this leads to an exponential enhancement of the coherence time (T1 and T2), even in the presence of extrinsic noise, imperfect control, and device-parameter variations. We show that the qubit supports exponentially robust single-qubit Clifford gates, implemented via appropriate control of the switch, and readout and/or initialization via supercurrent measurement. The self-correcting properties of the qubit allow it to operate at approximately 1-K temperatures and resonator Q factors down to approximately 1000 for realistic parameters, and make it amenable to parallel control through global control signals. We discuss how the effects of quasiparticle poisoning—potentially, though not necessarily, a limiting factor—might be mitigated. We finally demonstrate that a related device supports a self-correcting magic T gate.

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Popular Summary

Noise is a formidable challenge for quantum computers: if any data qubit is affected by noise, all processed information is lost. Standard approaches, such as surface codes, solve the problem by hiding the processed information in configurations of multiple physical qubits that are monitored and corrected before fluctuations accumulate to affect the hidden information. These standard approaches are challenging to scale up, because they require a large number of qubits to protect quantum information, leading to significant engineering overhead in system size and complexity. In addition, measurement, signal transmission, and data processing must be fast to ensure that readout, decoding, and feedback can compete with the rate of decoherence.

Here, we discover a new approach to quantum error correction that bypasses these challenges in the form of a self-correcting qubit. The qubit consists of a superconducting LC resonator coupled to a resistor and Josephson junction through a controllable switch. The logical “0” and “1” states correspond to even and odd parities of the resonator flux, relative to the flux quantum h/2e. When activating the switch in a particular stepwise pattern, the resistor absorbs noise-induced fluctuations without affecting the encoded information. The result is a physical qubit the lifetime of which scales exponentially with the system parameters and can exceed 106 lifetimes of an unprotected resonator state without any need for physical-qubit overhead, readout, or feedback.

Our work introduces a new paradigm for quantum error correction that leverages dissipation, thereby bypassing the main scalability challenges of existing approaches. Since the scheme is based on existing superconducting circuit technology, this could provide a promising new route to scalable quantum computing.

Article Text

References (103)

  1. P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493 (1995).
  2. A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54, 1098 (1996).
  3. A. M. Steane, Error correcting codes in quantum theory, Phys. Rev. Lett. 77, 793 (1996).
  4. E. Knill, R. Laflamme, and W. H. Zurek, Resilient quantum computation, Science 279, 342 (1998).
  5. R. Raussendorf and J. Harrington, Fault-tolerant quantum computation with high threshold in two dimensions, Phys. Rev. Lett. 98, 190504 (2007).
  6. 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).
  7. R. Acharya et al., Google Quantum AI, Suppressing quantum errors by scaling a surface code logical qubit, Nature 614, 676 (2023).
  8. B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015).
  9. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  10. S. J. Pauka, K. Das, R. Kalra, A. Moini, Y. Yang, M. Trainer, A. Bousquet, C. Cantaloube, N. Dick, G. C. Gardner, M. J. Manfra, and D. J. Reilly, A cryogenic CMOS chip for generating control signals for multiple qubits, Nat. Electron. 4, 64 (2021).
  11. S. Bravyi and B. Terhal, A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes, New J. Phys. 11, 043029 (2009).
  12. H. Bombin, R. W. Chhajlany, M. Horodecki, and M. A. Martin-Delgado, Self-correcting quantum computers, New J. Phys. 15, 055023 (2013), publisher: IOP Publishing.
  13. J. P. Barnes and W. S. Warren, Automatic quantum error correction, Phys. Rev. Lett. 85, 856 (2000).
  14. B. Kraus, H. P. Büchler, S. Diehl, A. Kantian, A. Micheli, and P. Zoller, Preparation of entangled states by quantum Markov processes, Phys. Rev. A 78, 042307 (2008).
  15. F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Quantum computation and quantum-state engineering driven by dissipation, Nat. Phys. 5, 633 (2009).
  16. Z. Leghtas, G. Kirchmair, B. Vlastakis, R. J. Schoelkopf, M. H. Devoret, and M. Mirrahimi, Hardware-efficient autonomous quantum memory protection, Phys. Rev. Lett. 111, 120501 (2013).
  17. F. Reiter, A. S. Sørensen, P. Zoller, and C. A. Muschik, Dissipative quantum error correction and application to quantum sensing with trapped ions, Nat. Commun. 8, 1822 (2017).
  18. S. Touzard, A. Grimm, Z. Leghtas, S. O. Mundhada, P. Reinhold, C. Axline, M. Reagor, K. Chou, J. Blumoff, K. M. Sliwa, S. Shankar, L. Frunzio, R. J. Schoelkopf, M. Mirrahimi, and M. H. Devoret, Coherent oscillations inside a quantum manifold stabilized by dissipation, Phys. Rev. X 8, 021005 (2018).
  19. 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).
  20. D. R. Pérez and E. Kapit, Improved autonomous error correction using variable dissipation in small logical qubit architectures, Quantum Sci. Technol. 6, 015006 (2020), publisher: IOP Publishing.
  21. P. M. Harrington, E. J. Mueller, and K. W. Murch, Engineered dissipation for quantum information science, Nat. Rev. Phys. 4, 660 (2022).
  22. 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), number: 7820 Publisher: Nature Publishing Group.
  23. A. Gyenis, A. Di Paolo, J. Koch, A. Blais, A. A. Houck, and D. I. Schuster, Moving beyond the transmon: Noise-protected superconducting quantum circuits, PRX Quantum 2, 030101 (2021).
  24. B. 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).
  25. L.-A. Sellem, A. Sarlette, Z. Leghtas, M. Mirrahimi, P. Rouchon, and P. Campagne-Ibarcq, Dissipative protection of a GKP qubit in a high-impedance superconducting circuit driven by a microwave frequency comb, Phys. Rev. X 15, 011011 (2025).
  26. D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001).
  27. J. Conrad, A. G. Burchards, and S. T. Flammia, Lattices, gates, and curves: GKP codes as a Rosetta stone, ArXiv:2407.03270.
  28. Note that our protocol does not directly generate a magic H state by projecting the LC oscillator vacuum state into the code subspace, as proposed in Ref. [103], since the nontrivial evolution toward the code subspace generated during by our stabilization protocol is inequivalent to a projection.
  29. P. Manset, J. Palomo, A. Schmitt, K. Gerashchenko, R. Rousseau, H. Patange, P. Abgrall, E. Flurin, S. Deléglise, T. Jacqmin, and L. Balembois, Hyperinductance based on stacked Josephson junctions, ArXiv:2505.02764.
  30. E. Afshari and A. Hajimiri, Nonlinear transmission lines for pulse shaping in silicon, IEEE J. Solid-State Circuits 40, 744 (2005), conference Name: IEEE Journal of Solid-State Circuits.
  31. 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), number: 7821 Publisher: Nature Publishing Group.
  32. 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).
  33. 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).
  34. 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, F. Hopfmueller, M. Tremblay, N. E. Frattini, J. C. Lemyre, and P. St-Jean, Autonomous quantum error correction of Gottesman-Kitaev-Preskill states, Phys. Rev. Lett. 132, 150607 (2024).
  35. X. C. Kolesnikow, R. W. Bomantara, A. C. Doherty, and A. L. Grimsmo, Gottesman-Kitaev-Preskill state preparation using periodic driving, Phys. Rev. Lett. 132, 130605 (2024).
  36. J. Conrad, Twirling and Hamiltonian engineering via dynamical decoupling for Gottesman-Kitaev-Preskill quantum computing, Phys. Rev. A 103, 022404 (2021).
  37. M. Rymarz, S. Bosco, A. Ciani, and D. P. DiVincenzo, Hardware-encoding grid states in a nonreciprocal superconducting circuit, Phys. Rev. X 11, 011032 (2021).
  38. The operator Ξ(φ/φ0) may conveniently be computed as ∑ncne−πinφ/φ0, where the {cn} are the Fourier coefficients of Ξ. Note that cn is only nonzero for odd n due to the antisymmetry Ξ(x)=−Ξ(x+1).
  39. G. Pantaleoni, B. Q. Baragiola, and N. C. Menicucci, Modular bosonic subsystem codes, Phys. Rev. Lett. 125, 040501 (2020).
  40. Y. Ouyang and E. T. Campbell, Trade-offs on number and phase shift resilience in bosonic quantum codes, IEEE Trans. Inf. Theory 67, 6644 (2021).
  41. M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, Dynamically protected cat-qubits: A new paradigm for universal quantum computation, New J. Phys. 16, 045014 (2014), publisher: IOP Publishing.
  42. 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).
  43. M. V. Larsen et al., Integrated photonic source of Gottesman-Kitaev-Preskill qubits, Nature 642, 587 (2025), publisher: Nature Publishing Group.
  44. M. Holthaus and M. E. Flatté, Subharmonic generation in quantum systems, Phys. Lett. A 187, 151 (1994).
  45. K. Sacha, Modeling spontaneous breaking of time-translation symmetry, Phys. Rev. A 91, 033617 (2015).
  46. V. Khemani, A. Lazarides, R. Moessner, and S. L. Sondhi, Phase structure of driven quantum systems, Phys. Rev. Lett. 116, 250401 (2016).
  47. D. V. Else, B. Bauer, and C. Nayak, Floquet time crystals, Phys. Rev. Lett. 117, 090402 (2016).
  48. Y. Zhang, J. Gosner, S. M. Girvin, J. Ankerhold, and M. I. Dykman, Time-translation-symmetry breaking in a driven oscillator: From the quantum coherent to the incoherent regime, Phys. Rev. A 96, 052124 (2017).
  49. Z. Gong, R. Hamazaki, and M. Ueda, Discrete time-crystalline order in cavity and circuit QED systems, Phys. Rev. Lett. 120, 040404 (2018).
  50. N. Lörch, Y. Zhang, C. Bruder, and M. I. Dykman, Quantum state preparation for coupled period tripling oscillators, Phys. Rev. Res. 1, 023023 (2019).
  51. F. Nathan, G. Refael, M. S. Rudner, and I. Martin, Quantum frequency locking and downconversion in a driven qubit-cavity system, Phys. Rev. Res. 2, 043411 (2020).
  52. H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, United Kingdom, 2002).
  53. C. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, Berlin, 2004).
  54. F. Nathan and M. S. Rudner, Universal Lindblad equation for open quantum systems, Phys. Rev. B 102, 115109 (2020).
  55. J. Ankerhold, H. Grabert, and G.-L. Ingold, Dissipative quantum systems with a potential barrier: General theory and the parabolic barrier, Phys. Rev. E 51, 4267 (1995).
  56. Specifically, Hrf is defined such that ∂tV†ρV=−i[Hrf,V†ρV].
  57. This result can, e.g., be seen by noting that φ¯ is a φ0-periodic function of φ and thus can be written as φ¯=∑ncne−2πinφ/φ0 for some coefficients {cn}. Since e−2πinφ/φ0 translates q by 2en, we see that e2πinφ/φ0,S2=0, and hence also φ¯,S2=0.
  58. This may, e.g., be defined such that ρ(t) has its φ¯ support confined in the interval |φ¯|≤Δφ(t), up to a residual weight given by some exponentially small error tolerance δp≪1 which we specify. Due to the Gaussian envelope of the probability support in each well, Δφ(t) can be significantly smaller than φ0/2 even when we pick δp exponentially small.
  59. More precisely, with the definition of Δφ in note [58], the evolution of ρ~(t) is unaffected up to (exponentially suppressed) δp corrections if we replace φ¯ with the time-dependent operator φ¯c(t) in Eq. (17), where φ¯(t)=φ¯ for |φ¯|≤Δφ(t) and φ¯c(t)=Δφ(t) for |φ¯|≥Δφ(t).
  60. For instance, with the trunctation scheme in note [59], it would relax to cλφ0, where c is the O(1) cutoff scale introduced there.
  61. Specifically, Sij=δizδjz+ϵijz, where ϵijk is the Levi-Civita symbol.
  62. To see this, note that for z∈Z and |δφ|≤φ0/2, σ¯x|zφ0+δφ⟩=|−(z−1)φ0+δφ⟩ and σ¯z|zφ0+δφ⟩=(−1)z|zφ0+δφ⟩. Noting that the states |zφ0+δφ⟩ form a complete orthormal basis, the above relations imply that σ¯x2=σ¯z2=1 and σ¯xσ¯z=−σ¯zσ¯x, implying that the anticommutator relations hold.
  63. This T gate is the Hermitian conjugate of the usual definition of the T gate, which would assign a e−iπ/8 phase to logical |0⟩ states relative to logical |1⟩ states. The gate is exponentially protected by the same mechanism as the S gate for the configuration discussed in the previous sections.
  64. S. Danilin, A. V. Lebedev, A. Vepsäläinen, G. B. Lesovik, G. Blatter, and G. S. Paraoanu, Quantum-enhanced magnetometry by phase estimation algorithms with a single artificial atom, npj Quantum Inf. 4, 29 (2018).
  65. R. P. Budoyo, K. Kakuyanagi, H. Toida, Y. Matsuzaki, and S. Saito, Electron spin resonance with up to 20 spin sensitivity measured using a superconducting flux qubit, Appl. Phys. Lett. 116, 194001 (2020).
  66. The initial state consisted of well ground states with a Gaussian envelope and had wave function Ψs(φ) for s=0,1, where Ψs(φ)=∑ke−8π2λ02k2e−(φ−[2k+s]φ0)/2λ02φ02 (up to a normalization).
  67. A. Isar, W. Scheid, and A. Sandulescu, Quasiprobability distributions for open quantum systems within the Lindblad theory, J. Math. Phys. 32, 2128 (1991).
  68. The inverse scaling with fLC arises due to the parametrization we use for charge noise: the coupling between system and charge noise is proportional to C−2 while C∝fLC−1, due to Eq. (5) fixing the resonator impedance. The charge-noise-induced diffusion rate thus scales with fLC2 for fixed γq, while the tolerance for the diffusion rate scales with fLC−1. As a result, smaller fLC leads to more stability with our parametrization.
  69. G. Kiršanskas, M. Franckié, and A. Wacker, Phenomenological position and energy resolving Lindblad approach to quantum kinetics, Phys. Rev. B 97, 035432 (2018).
  70. F. Nathan, Ph.D. Thesis, Niels Bohr Institute, University of Copenhagen, 2018.
  71. F. Nathan, I. Martin, and G. Refael, Topological frequency conversion in a driven dissipative quantum cavity, Phys. Rev. B 99, 094311 (2019).
  72. D. Davidović, Geometric-arithmetic master equation in large and fast open quantum systems, J. Phys. A: Math. Theor. 55, 455301 (2022).
  73. F. Nathan and M. S. Rudner, Quantifying the accuracy of steady states obtained from the universal Lindblad equation, Phys. Rev. B 109, 205140 (2024).
  74. Here, we have used that ε0=4πλ02EJ.
  75. J. Dalibard, Y. Castin, and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Phys. Rev. Lett. 68, 580 (1992).
  76. H. Carmichael, An Open Systems Approach to Quantum Optics (Springer Berlin, Heidelberg, 1993).
  77. Specifically, we initialize the system in a state with wave function Ψ0(φ)=∑n=−4545∑k=035cnkψk(φ−nφ0), with ψk(φ) the kth harmonic oscillator eigenstate with vacuum fluctuation width λ0φ0 (see Fig. 4), and {cnk} random complex numbers with zero mean and identical variances.
  78. Specifically, the system has been initialized in a superposition of Gaussian wave functions with a Gaussian envelope, with the wave function Ψ(φ)=cos(θ2)Ψ0(φ)+sin(θ2)eiφΨ1(φ), where Ψs(φ)=N∑ke−(2k+s)2π2λ02/4e−(φ−[2k+s]φ0)2/2λ02φ02, with N a normalization constant and (θ,ϕ)=(1.93,1.62) chosen randomly on the unit sphere. The state above is well confined in the code subspace and realizes the logical state cos(θ2)|0⟩+sin(θ2)eiφ|1⟩. We also note that the results do not rely on the particular details of the initialization, except for the initial expectation values of the logical operators. After a few cycles of the protocol, the resonator has reached a thermal ensemble of generalized GKP states with no memory of the initialization, except for the initially encoded logical information.
  79. Due to the long qubit lifetime in this regime, the level of decay that we can resolve on the simulated timescale is very small and thus subject to significant statistical fluctuations between SSE trajectories. To overcome this, we show time-coarse-grained data, obtained by time-averaging the Bloch-vector length over 400 driving periods. We also use the time-coarse-grained data for our fit. We have confirmed that the lifetime fit varies little with the chosen window length, while the quality of the linear fit increases with n.
  80. We speculate that the near-stationary interval arises from a small effective Lieb-Robinson velocity in phase space, which implies that it takes a long time for exponential tails of the phase-space support of the state to spread to the domain boundaries of the logical operators.
  81. The values of log||⟨σ⟩||2 shown in Fig. 7 are very small and hence the logarithm can be well approximated by its first-order Taylor series over this range of values. Thus, log||⟨σ⟩||2¯ being linear over this range is not conclusive evidence that the Bloch-vector length decays exponentially in time; nonetheless, the inverse slope still defines a meaningful lifetime for the qubit.
  82. I. V. Pechenezhskiy, R. A. Mencia, L. B. Nguyen, Y.-H. Lin, and V. E. Manucharyan, The superconducting quasicharge qubit, Nature 585, 368 (2020).
  83. M. Peruzzo, A. Trioni, F. Hassani, M. Zemlicka, and J. M. Fink, Surpassing the resistance quantum with a geometric superinductor, Phys. Rev. Appl. 14, 044055 (2020).
  84. W. C. Smith, M. Villiers, A. Marquet, J. Palomo, M. R. Delbecq, T. Kontos, P. Campagne-Ibarcq, B. Douçot, and Z. Leghtas, Magnifying quantum phase fluctuations with Cooper-pair pairing, Phys. Rev. X 12, 021002 (2022).
  85. A. Anferov, F. Wan, S. P. Harvey, J. Simon, and D. I. Schuster, Millimeter-wave superconducting qubit, PRX Quantum 6, 020336 (2025).
  86. L. Casparis, M. R. Connolly, M. Kjaergaard, N. J. Pearson, A. Kringhøj, T. W. Larsen, F. Kuemmeth, T. Wang, C. Thomas, S. Gronin, G. C. Gardner, M. J. Manfra, C. M. Marcus, and K. D. Petersson, Superconducting gatemon qubit based on a proximitized two-dimensional electron gas, Nat. Nanotechnol. 13, 915 (2018), number: 10 Publisher: Nature Publishing Group.
  87. M. Samizadeh Nikoo, A. Jafari, N. Perera, M. Zhu, G. Santoruvo, and E. Matioli, Nanoplasma-enabled picosecond switches for ultrafast electronics, Nature 579, 534 (2020), publisher: Nature Publishing Group.
  88. A. Kemppinen, A. J. Manninen, M. Möttönen, J. J. Vartiainen, J. T. Peltonen, and J. P. Pekola, Suppression of the critical current of a balanced superconducting quantum interference device, Appl. Phys. Lett. 92, 052110 (2008).
  89. 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), publisher: Nature Publishing Group.
  90. R. Lescanne, M. Villiers, T. Peronnin, A. Sarlette, M. Delbecq, B. Huard, T. Kontos, M. Mirrahimi, and Z. Leghtas, Exponential suppression of bit-flips in a qubit encoded in an oscillator, Nat. Phys. 16, 509 (2020), publisher: Nature Publishing Group.
  91. A. Vanselow, B. Beauseigneur, L. Lattier, M. Villiers, A. Denis, P. Morfin, Z. Leghtas, and P. Campagne-Ibarcq, Dissipating quartets of excitations in a superconducting circuit, ArXiv:2501.05960.
  92. M. Geier and F. Nathan, Self-correcting GKP qubit in a superconducting circuit with an oscillating voltage bias, ArXiv:2412.03650.
  93. L. Glazman and G. Catelani, Bogoliubov quasiparticles in superconducting qubits, SciPost Physics Lecture Notes, 31 (2021).
  94. R.-P. Riwar, A. Hosseinkhani, L. D. Burkhart, Y. Y. Gao, R. J. Schoelkopf, L. I. Glazman, and G. Catelani, Normal-metal quasiparticle traps for superconducting qubits, Phys. Rev. B 94, 104516 (2016), publisher: American Physical Society.
  95. Google Quantum AI and Collaborators, Quantum error correction below the surface code threshold, Nature 638, 920 (2025).
  96. Microsoft Azure Quantum, M. Aghaee et al., Interferometric single-shot parity measurement in InAs-Al hybrid devices, Nature 638, 651 (2025).
  97. For example, setting w0=6κ leads to an error of order erfc(6)∼10−17.
  98. This holds since ‖X‖tr=‖X†‖tr, and ‖UX‖tr=‖X‖tr for any unitary U.
  99. The choice k=k0 is chosen because it results in simple expressions for the bounds that we obtain. We emphasize that sharper bounds can possibly be obtained for different values of k.
  100. To see that the right-hand side is Hermitian, note that the 2e periodicity of fα′(q/e) implies [φ¯,fα′(q/e)]=0.
  101. To see this, note that ‖AB‖tr=Tr[UAB] for some unitary U that generates the polar decomposition of AB. The result follows using |Tr(AB)|≤‖A‖‖B‖tr.
  102. Specifically, ‖Oρ‖tr=Tr[UOρ] for some unitary U that generates the polar decompositon of Oρ. We identify Tr[UOρ]=(ρ,ρUO)HS, where (x,y)HS=Tr[x†y] is the Hilbert-Schmidt inner product. The result follows when using the Cauchy-Schwartz inequality, (x,y)HS≤(x,x)HS(y,y)HS.
  103. 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).

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PRX Quantum 6, 030352– Published 17 September, 2025
  • Received 18 June 2024
  • Revised 10 April 2025
  • Accepted 9 July 2025
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