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Probing the universality of topological defect formation in a quantum annealer: Kibble-Zurek mechanism and beyond

Yuki Bando1,*, Yuki Susa2,†, Hiroki Oshiyama3, Naokazu Shibata3, Masayuki Ohzeki4,1,5, Fernando Javier Gómez-Ruiz6, Daniel A. Lidar7,8, Sei Suzuki9, Adolfo del Campo6,10,11 et al.

Hidetoshi Nishimori1,4,12

  • 1Institute of Innovative Research, Tokyo Institute of Technology, Nagatsuta-cho, Midori-ku, Yokohama 226-8503, Japan
  • 2Institute of Innovative Research, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8550, Japan
  • 3Department of Physics, Tohoku University, Sendai 980-8578, Japan
  • 4Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan
  • 5Sigma-i Co. Ltd., Konan, Minato-ku, Tokyo 108-0075, Japan
  • 6Donostia International Physics Center, E-20018 San Sebastián, Spain
  • 7Departments of Electrical and Computer Engineering, Chemistry, and Physics, University of Southern California, Los Angeles, California 90089, USA
  • 8Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA
  • 9Department of Liberal Arts, Saitama Medical University, Moroyama, Saitama 350-0495, Japan
  • 10IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao, Spain
  • 11Department of Physics, University of Massachusetts Boston, 100 Morrissey Boulevard, Boston, Massachusetts 02125, USA
  • 12RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), Wako, Saitama 351-0198, Japan

  • *y-bando@qa.iir.titech.ac.jp
  • †Present address: System Platform Research Laboratories, NEC Corporation, Kawasaki 211-8666, Japan.
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Phys. Rev. Research 2, 033369 – Published 8 September, 2020

DOI: https://doi.org/10.1103/PhysRevResearch.2.033369

Abstract

The number of topological defects created in a system driven through a quantum phase transition exhibits a power-law scaling with the driving time. This universal scaling law is the key prediction of the Kibble-Zurek mechanism (KZM), and testing it using a hardware-based quantum simulator is a coveted goal of quantum information science. Here we provide such a test using quantum annealing. Specifically, we report on extensive experimental tests of topological defect formation via the one-dimensional transverse-field Ising model on two different D-Wave quantum annealing devices. We find that the quantum simulator results can indeed be explained by the KZM for open-system quantum dynamics with phase-flip errors, with certain quantitative deviations from the theory likely caused by factors such as random control errors and transient effects. In addition, we probe physics beyond the KZM by identifying signatures of universality in the distribution and cumulants of the number of kinks and their decay, and again find agreement with the quantum simulator results. This implies that the theoretical predictions of the generalized KZM theory, which assumes isolation from the environment, applies beyond its original scope to an open system. We support this result by extensive numerical computations. To check whether an alternative, classical interpretation of these results is possible, we used the spin-vector Monte Carlo model, a candidate classical description of the D-Wave device. We find that the degree of agreement with the experimental data from the D-Wave annealing devices is better for the KZM, a quantum theory, than for the classical spin-vector Monte Carlo model, thus favoring a quantum description of the device. Our work provides an experimental test of quantum critical dynamics in an open quantum system, and paves the way to new directions in quantum simulation experiments.

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References (95)

  1. T. Kadowaki and H. Nishimori, Quantum annealing in the transverse Ising model, Phys. Rev. E 58, 5355 (1998).
  2. A. Das and B. K. Chakrabarti, Colloquium: Quantum annealing and analog quantum computation, Rev. Mod. Phys. 80, 1061 (2008).
  3. T. Albash and D. A. Lidar, Adiabatic quantum computation, Rev. Mod. Phys. 90, 015002 (2018).
  4. P. Hauke, H. G. Katzgraber, W. Lechner, H. Nishimori, and W. D. Oliver, Perspectives of quantum annealing: Methods and implementations, Rep. Prog. Phys. 83, 054401 (2020).
  5. S. Boixo, T. F. Ronnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, Evidence for quantum annealing with more than one hundred qubits, Nat. Phys. 10, 218 (2014).
  6. T. F. Rønnow, Z. Wang, J. Job, S. Boixo, S. V. Isakov, D. Wecker, J. M. Martinis, D. A. Lidar, and M. Troyer, Defining and detecting quantum speedup, Science 345, 420 (2014).
  7. I. Hen, J. Job, T. Albash, T. F. Ronnow, M. Troyer, and D. A. Lidar, Probing for quantum speedup in spin-glass problems with planted solutions, Phys. Rev. A 92, 042325 (2015).
  8. J. King, S. Yarkoni, J. Raymond, I. Ozfidan, A. D. King, M. M. Nevisi, J. P. Hilton, and C. C. McGeoch, Quantum Annealing amid Local Ruggedness and Global Frustration, J. Phys. Soc. Jpn. 88, 061007 (2019).
  9. J. King, S. Yarkoni, M. M. Nevisi, J. P. Hilton, and C. C. McGeoch, Benchmarking a quantum annealing processor with the time-to-target metric, arXiv:1508.05087.
  10. V. S. Denchev, S. Boixo, S. V. Isakov, N. Ding, R. Babbush, V. Smelyanskiy, J. Martinis, and H. Neven, What is the Computational Value of Finite-Range Tunneling?, Phys. Rev. X 6, 031015 (2016).
  11. S. Mandrà, Z. Zhu, W. Wang, A. Perdomo-Ortiz, and H. G. Katzgraber, Strengths and weaknesses of weak-strong cluster problems: A detailed overview of state-of-the-art classical heuristics versus quantum approaches, Phys. Rev. A 94, 022337 (2016).
  12. W. Vinci and D. A. Lidar, Optimally Stopped Optimization, Phys. Rev. Appl. 6, 054016 (2016).
  13. T. Albash and D. A. Lidar, Demonstration of a Scaling Advantage for a Quantum Annealer over Simulated Annealing, Phys. Rev. X 8, 031016 (2018).
  14. S. Mandrà and H. G. Katzgraber, A deceptive step towards quantum speedup detection, Quantum Sci. Technol. 3, 04LT01 (2018).
  15. A. Pearson, A. Mishra, I. Hen, and D. A. Lidar, Analog errors in quantum annealing: Doom and hope, npj Quant. Inf. 5, 107 (2019).
  16. A. D. King, J. Carrasquilla, J. Raymond, I. Ozfidan, E. Andriyash, A. Berkley, M. Reis, T. Lanting, R. Harris, F. Altomare, K. Boothby, P. I. Bunyk, C. Enderud, A. Fréchette, E. Hoskinson, N. Ladizinsky, T. Oh, G. Poulin-Lamarre, C. Rich, Y. Sato, A. Y. Smirnov, L. J. Swenson, M. H. Volkmann, J. Whittaker, J. Yao, E. Ladizinsky, M. W. Johnson, J. Hilton, and M. H. Amin, Observation of topological phenomena in a programmable lattice of 1,800 qubits, Nature (London) 560, 456 (2018).
  17. A. D. King, J. Raymond, T. Lanting, S. V. Isakov, M. Mohseni, G. Poulin-Lamarre, S. Ejtemaee, W. Bernoudy, I. Ozfidan, A. Y. Smirnov, M. Reis, F. Altomare, M. Babcock, C. Baron, A. J. Berkley, K. Boothby, P. I. Bunyk, H. Christiani, C. Enderud, B. Evert, R. Harris, E. Hoskinson, S. Huang, K. Jooya, A. Khodabandelou, N. Ladizinsky, R. Li, P. A. Lott, A. J. R. MacDonald, D. Marsden, G. Marsden, T. Medina, R. Molavi, R. Neufeld, M. Norouzpour, T. Oh, I. Pavlov, I. Perminov, T. Prescott, C. Rich, Y. Sato, B. Sheldan, G. Sterling, L. J. Swenson, N. Tsai, M. H. Volkmann, J. D. Whittaker, W. Wilkinson, J. Yao, H. Neven, J. P. Hilton, E. Ladizinsky, M. W. Johnson, and M. H. Amin, Scaling advantage in quantum simulation of geometrically frustrated magnets, arXiv:1911.03446.
  18. R. Harris, Y. Sato, A. J. Berkley, M. Reis, F. Altomare, M. H. Amin, K. Boothby, P. Bunyk, C. Deng, C. Enderud, S. Huang, E. Hoskinson, M. W. Johnson, E. Ladizinsky, N. Ladizinsky, T. Lanting, R. Li, T. Medina, R. Molavi, R. Neufeld, T. Oh, I. Pavlov, I. Perminov, G. Poulin-Lamarre, C. Rich, A. Smirnov, L. Swenson, N. Tsai, M. Volkmann, J. Whittaker, and J. Yao, Phase transitions in a programmable quantum spin glass simulator, Science 361, 162 (2018).
  19. R. Islam, C. Senko, W. C. Campbell, S. Korenblit, J. Smith, A. Lee, E. E. Edwards, C. C. J. Wang, J. K. Freericks, and C. Monroe, Emergence and frustration of magnetism with variable-range interactions in a quantum simulator, Science 340, 583 (2013).
  20. J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, Many-body localization in a quantum simulator with programmable random disorder, Nat. Phys. 12, 907 (2016).
  21. J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z. X. Gong, and C. Monroe, Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator, Nature (London) 551, 601 (2017).
  22. T. W. Kibble, Topology of cosmic domains and strings, J. Phys. A: Math. Gen. 9, 1387 (1976).
  23. W. H. Zurek, Cosmological experiments in superfluid helium? Nature (London) 317, 505 (1985).
  24. We use both terms interchangeably.
  25. A. Polkovnikov, Universal adiabatic dynamics in the vicinity of a quantum critical point, Phys. Rev. B 72, 161201(R) (2005).
  26. B. Damski, The Simplest Quantum Model Supporting the Kibble-Zurek Mechanism of Topological Defect Production: Landau-Zener Transitions from a New Perspective, Phys. Rev. Lett. 95, 035701 (2005).
  27. J. Dziarmaga, Dynamics of a Quantum Phase Transition: Exact Solution of the Quantum Ising Model, Phys. Rev. Lett. 95, 245701 (2005).
  28. W. H. Zurek, U. Dorner, and P. Zoller, Dynamics of a Quantum Phase Transition, Phys. Rev. Lett. 95, 105701 (2005).
  29. A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83, 863 (2011).
  30. A. del Campo and W. H. Zurek, Universality of phase transition dynamics: Topological defects from symmetry breaking, Int. J. Mod. Phys. A 29, 1430018 (2014).
  31. M. Uhlmann, R. Schützhold, and U. R. Fischer, Vortex Quantum Creation and Winding Number Scaling in a Quenched Spinor Bose Gas, Phys. Rev. Lett. 99, 120407 (2007).
  32. M. Uhlmann, R. Schützhold, and U. R. Fischer, O(N) symmetry-breaking quantum quench: Topological defects versus quasiparticles, Phys. Rev. D 81, 025017 (2010).
  33. X.-Y. Xu, Y.-J. Han, K. Sun, J.-S. Xu, J.-S. Tang, C.-F. Li, and G.-C. Guo, Quantum Simulation of Landau-Zener Model Dynamics Supporting the Kibble-Zurek Mechanism, Phys. Rev. Lett. 112, 035701 (2014).
  34. J.-M. Cui, Y.-F. Huang, Z. Wang, D.-Y. Cao, J. Wang, W.-M. Lv, L. Luo, A. del Campo, Y.-J. Han, C.-F. Li, and G.-C. Guo, Experimental trapped-ion quantum simulation of the Kibble-Zurek dynamics in momentum space, Sci. Rep. 6, 33381 (2016).
  35. M. Gong, X. Wen, G. Sun, D.-W. Zhang, D. Lan, Y. Zhou, Y. Fan, Y. Liu, X. Tan, H. Yu, Y. Yu, S.-L. Zhu, S. Han, and P. Wu, Simulating the Kibble-Zurek mechanism of the Ising model with a superconducting qubit system, Sci. Rep. 6, 22667 (2016).
  36. J.-M. Cui, F. J. Gómez-Ruiz, Y.-F. Huang, C.-F. Li, G.-C. Guo, and A. del Campo, Experimentally testing quantum critical dynamics beyond the Kibble-Zurek mechanism, Commun. Phys. 3, 44 (2020).
  37. H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Probing many-body dynamics on a 51-atom quantum simulator, Nature (London) 551, 579 (2017).
  38. D. Sen, K. Sengupta, and S. Mondal, Defect Production in Nonlinear Quench across a Quantum Critical Point, Phys. Rev. Lett. 101, 016806 (2008).
  39. R. Barankov and A. Polkovnikov, Optimal Nonlinear Passage Through a Quantum Critical Point, Phys. Rev. Lett. 101, 076801 (2008).
  40. J. Dziarmaga, Dynamics of a quantum phase transition in the random Ising model: Logarithmic dependence of the defect density on the transition rate, Phys. Rev. B 74, 064416 (2006).
  41. T. Caneva, R. Fazio, and G. E. Santoro, Adiabatic quantum dynamics of a random Ising chain across its quantum critical point, Phys. Rev. B 76, 144427 (2007).
  42. W. H. Zurek and U. Dorner, Phase transition in space: How far does a symmetry bend before it breaks? Phil. Trans. R. Soc. A 366, 2953 (2008).
  43. J. Dziarmaga and M. M. Rams, Dynamics of an inhomogeneous quantum phase transition, New J. Phys. 12, 055007 (2010).
  44. A. del Campo, T. W. B. Kibble, and W. H. Zurek, Causality and non-equilibrium second-order phase transitions in inhomogeneous systems, J. Phys.: Condens. Matter 25, 404210 (2013).
  45. F. J. Gómez-Ruiz and A. del Campo, Universal Dynamics of Inhomogeneous Quantum Phase Transitions: Suppressing Defect Formation, Phys. Rev. Lett. 122, 080604 (2019).
  46. D. Patanè, A. Silva, L. Amico, R. Fazio, and G. E. Santoro, Adiabatic Dynamics in Open Quantum Critical Many-Body Systems, Phys. Rev. Lett. 101, 175701 (2008).
  47. P. Nalbach, S. Vishveshwara, and A. A. Clerk, Quantum Kibble-Zurek physics in the presence of spatially correlated dissipation, Phys. Rev. B 92, 014306 (2015).
  48. A. Dutta, A. Rahmani, and A. del Campo, Anti-Kibble-Zurek Behavior in Crossing the Quantum Critical Point of a Thermally Isolated System Driven by a Noisy Control Field, Phys. Rev. Lett. 117, 080402 (2016).
  49. R. Puebla, A. Smirne, S. F. Huelga, and M. B. Plenio, Universal Anti-Kibble-Zurek Scaling in Fully Connected Systems, Phys. Rev. Lett. 124, 230602 (2020).
  50. B. Gardas, J. Dziarmaga, W. H. Zurek, and M. Zwolak, Defects in quantum computers, Sci. Rep. 8, 4539 (2018).
  51. P. Weinberg, M. Tylutki, J. M. Rönkkö, J. Westerholm, J. A. Åström, P. Manninen, P. Törmä, and A. W. Sandvik, Scaling and Diabatic effects in Quantum Annealing with a D-Wave Device, Phys. Rev. Lett. 124, 090502 (2020).
  52. A. del Campo, Universal Statistics of Topological Defects Formed in a Quantum Phase Transition, Phys. Rev. Lett. 121, 200601 (2018).
  53. V. Choi, Minor-embedding in adiabatic quantum computation: I. the parameter setting problem, Quant. Inf. Proc. 7, 193 (2008).
  54. K. L. Pudenz, T. Albash, and D. A. Lidar, Error-corrected quantum annealing with hundreds of qubits, Nat. Commun. 5, 3243 (2014).
  55. A. Mishra, T. Albash, and D. A. Lidar, Performance of two different quantum annealing correction codes, Quantum Inf. Process. 15, 609 (2016).
  56. A. Mishra, T. Albash, and D. A. Lidar, Finite temperature quantum annealing solving exponentially small gap problem with non-monotonic success probability, Nat. Commun. 9, 2917 (2018).
  57. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
  58. H. Nishimori and G. Ortiz, Elements of Phase Transitions and Critical Phenomena (Oxford University Press, Oxford, UK, 2011).
  59. B. Damski and W. H. Zurek, Adiabatic-impulse approximation for avoided level crossings: From phase-transition dynamics to Landau-Zener evolutions and back again, Phys. Rev. A 73, 063405 (2006).
  60. L. Cincio, J. Dziarmaga, M. M. Rams, and W. H. Zurek, Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum ising model, Phys. Rev. A 75, 052321 (2007).
  61. QPU-Specific Physical Properties: dw_2000q_5, D-Wave Systems Inc., 3033 Beta Ave Burnaby, BC V5G 4M9 Canada (2019).
  62. A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Dynamics of the dissipative two-state system, Rev. Mod. Phys. 59, 1 (1987).
  63. T. Lanting, M. H. S. Amin, M. W. Johnson, F. Altomare, A. J. Berkley, S. Gildert, R. Harris, J. Johansson, P. Bunyk, E. Ladizinsky, E. Tolkacheva, and D. V. Averin, Probing high-frequency noise with macroscopic resonant tunneling, Phys. Rev. B 83, 180502(R) (2011).
  64. T. Albash, S. Boixo, D. A. Lidar, and P. Zanardi, Quantum adiabatic Markovian master equations, New J. Phys. 14, 123016 (2012).
  65. S. Boixo, T. Albash, F. M. Spedalieri, N. Chancellor, and D. A. Lidar, Experimental signature of programmable quantum annealing, Nat. Commun. 4, 2067 (2013).
  66. T. Albash, W. Vinci, A. Mishra, P. A. Warburton, and D. A. Lidar, Consistency tests of classical and quantum models for a quantum annealer, Phys. Rev. A 91, 042314 (2015).
  67. T. Albash, T. F. Rønnow, M. Troyer, and D. A. Lidar, Reexamining classical and quantum models for the D-Wave One processor, Eur. Phys. J. Spec. Top. 224, 111 (2015).
  68. T. Albash, I. Hen, F. M. Spedalieri, and D. A. Lidar, Reexamination of the evidence for entanglement in a quantum annealer, Phys. Rev. A 92, 062328 (2015).
  69. M. H. Amin, Searching for quantum speedup in quasistatic quantum annealers, Phys. Rev. A 92, 052323 (2015).
  70. T. Albash and D. A. Lidar, Decoherence in adiabatic quantum computation, Phys. Rev. A 91, 062320 (2015).
  71. S. Boixo, V. N. Smelyanskiy, A. Shabani, S. V. Isakov, M. Dykman, V. S. Denchev, M. H. Amin, A. Y. Smirnov, M. Mohseni, and H. Neven, Computational multiqubit tunneling in programmable quantum annealers, Nat. Commun. 7, 10327 (2016).
  72. H. Munoz-Bauza, H. Chen, and D. Lidar, A double-slit proposal for quantum annealing, npj Quant. Inf. 5, 51 (2019).
  73. P. Werner, K. Völker, M. Troyer, and S. Chakravarty, Phase Diagram and Critical Exponents of a Dissipative Ising Spin Chain in a Transverse Magnetic Field, Phys. Rev. Lett. 94, 047201 (2005).
  74. S. Pankov, S. Florens, A. Georges, G. Kotliar, and S. Sachdev, Non-Fermi-liquid behavior from two-dimensional antiferromagnetic fluctuations: A renormalization-group and large-N analysis, Phy. Rev. B 69, 054426 (2004).
  75. S. Sachdev, P. Werner, and M. Troyer, Universal Conductance of Nanowires Near the Superconductor-Metal Quantum Transition, Phys. Rev. Lett. 92, 237003 (2004).
  76. S. Suzuki, H. Oshiyama, and N. Shibata, Quantum annealing of pure and random ising chains coupled to a bosonic environment, J. Phys. Soc. Jpn. 88, 061003 (2019).
  77. H. Oshiyama, S. Suzuki, and N. Shibata, Kibble-Zurek mechanism in a dissipative transverse ising chain, arXiv:2005.05621.
  78. D. A. Huse and D. S. Fisher, Residual Energies after Slow Cooling of Disordered Systems, Phys. Rev. Lett. 57, 2203 (1986).
  79. F. J. Gómez-Ruiz, J. J. Mayo, and A. del Campo, Full Counting Statistics of Topological Defects after Crossing a Phase Transition, Phys. Rev. Lett. 124, 240602 (2020).
  80. S. W. Shin, G. Smith, J. A. Smolin, and U. Vazirani, How “Quantum” is the D-Wave machine? arXiv:1401.7087.
  81. A. D. King, E. Hoskinson, T. Lanting, E. Andriyash, and M. H. Amin, Degeneracy, degree, and heavy tails in quantum annealing, Phys. Rev. A 93, 052320 (2016).
  82. Y. Susa, Y. Yamashiro, M. Yamamoto, I. Hen, D. A. Lidar, and H. Nishimori, Quantum annealing of the p-spin model under inhomogeneous transverse field driving, Phys. Rev. A 98, 042326 (2018).
  83. Y. Yamashiro, M. Ohkuwa, H. Nishimori, and D. A. Lidar, Dynamics of reverse annealing for the fully connected p-spin model, Phys. Rev. A 100, 052321 (2019).
  84. M. H. Amin, E. Andriyash, J. Rolfe, B. Kulchytskyy, and R. Melko, Quantum Boltzmann Machine, Phys. Rev. X 8, 021050 (2018).
  85. M. Benedetti, J. Realpe-Gómez, R. Biswas, and A. Perdomo-Ortiz, Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models, Phys. Rev. X 7, 041052 (2017).
  86. R. Y. Li, T. Albash, and D. A. Lidar, Limitations of error corrected quantum annealing in improving the performance of Boltzmann machines, Quantum Sci. Technol. 5, 045010 (2020).
  87. J. Marshall, E. G. Rieffel, and I. Hen, Thermalization, Freeze-out, and Noise: Deciphering Experimental Quantum Annealers, Phys. Rev. Appl. 8, 064025 (2017).
  88. W. Vinci and D. A. Lidar, Scalable effective-temperature reduction for quantum annealers via nested quantum annealing correction, Phys. Rev. A 97, 022308 (2018).
  89. L. C. Venuti, T. Albash, D. A. Lidar, and P. Zanardi, Adiabaticity in open quantum systems, Phys. Rev. A 93, 032118 (2016).
  90. J. A. Smolin and G. Smith, Classical signature of quantum annealing, Front. Phys. 2, 52 (2014).
  91. S. W. Shin, G. Smith, J. A. Smolin, and U. Vazirani, Comment on “Distinguishing Classical and Quantum Models for the D-Wave Device”, arXiv:1404.6499.
  92. L. Wang, T. F. Rønnow, S. Boixo, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, Comment on: ‘Classical signature of quantum annealing', arXiv:1305.5837.
  93. T. Lanting, A. J. Przybysz, A. Y. Smirnov, F. M. Spedalieri, M. H. Amin, A. J. Berkley, R. Harris, F. Altomare, S. Boixo, P. Bunyk, N. Dickson, C. Enderud, J. P. Hilton, E. Hoskinson, M. W. Johnson, E. Ladizinsky, N. Ladizinsky, R. Neufeld, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, S. Uchaikin, A. B. Wilson, and G. Rose, Entanglement in a Quantum Annealing Processor, Phys. Rev. X 4, 021041 (2014).
  94. J. Job and D. Lidar, Test-driving 1000 qubits, Quantum Sci. Technol. 3, 030501 (2018).
  95. A. Caldeira and A. Leggett, Quantum tunneling in a dissipative system, Ann. Phys. 149, 374 (1983).

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Phys. Rev. Research 2, 033369– Published 8 September, 2020
  • Received 12 February 2020
  • Accepted 10 August 2020
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