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  • Perspective
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Quantum simulation of fundamental particles and forces

Abstract

Key static and dynamic properties of matter — from creation in the Big Bang to evolution into subatomic and astrophysical environments — arise from the underlying fundamental quantum fields of the standard model and their effective descriptions. However, the simulation of these properties lies beyond the capabilities of classical computation alone. Advances in quantum technologies have improved control over quantum entanglement and coherence to the point at which robust simulations of quantum fields are anticipated in the foreseeable future. In this Perspective article, we discuss the emerging area of quantum simulations of standard-model physics, outlining the challenges and opportunities for progress in the context of nuclear and high-energy physics.

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Fig. 1: From the standard model to the simulation of dynamics in extreme environments.
Fig. 2: On the path to quantum simulations of standard-model physics.

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References

  1. Glashow, S. Partial symmetries of weak interactions. Nucl. Phys. 22, 579–588 (1961).

    Article  Google Scholar 

  2. Higgs, P. W. Broken symmetries and the masses of gauge bosons. Phys. Rev. Lett. 13, 508–509 (1964).

    Article  ADS  MathSciNet  Google Scholar 

  3. Weinberg, S. A model of leptons. Phys. Rev. Lett. 19, 1264–1266 (1967).

    Article  ADS  Google Scholar 

  4. Salam, A. Weak and electromagnetic interactions. Conf. Proc. C 680519, 367–377 (1968).

    Google Scholar 

  5. Politzer, H. Reliable perturbative results for strong interactions? Phys. Rev. Lett. 30, 1346–1349 (1973).

    Article  ADS  Google Scholar 

  6. Gross, D. J. & Wilczek, F. Ultraviolet behavior of nonabelian gauge theories. Phys. Rev. Lett. 30, 1343–1346 (1973).

    Article  ADS  Google Scholar 

  7. Aoki, Y. et al. FLAG review 2021. Eur. Phys. J. C 82, 869 (2022).

    Article  ADS  Google Scholar 

  8. Davoudi, Z. et al. Report of the Snowmass 2021 Topical group on lattice gauge theory. Preprint at https://arxiv.org/abs/2209.10758 (2022).

  9. Kronfeld, A. S. et al. Lattice QCD and particle physics. Preprint at https://arxiv.org/abs/2207.07641 (2022).

  10. Davoudi, Z. et al. Nuclear matrix elements from lattice QCD for electroweak and beyond-standard-model processes. Phys. Rept. 900, 1–74 (2021).

    Article  ADS  MATH  Google Scholar 

  11. Nagata, K. Finite-density lattice QCD and sign problem: current status and open problems. Prog. Part. Nucl. Phys. 127, 103991 (2022).

    Article  Google Scholar 

  12. Bazavov, A., Karsch, F., Mukherjee, S. & Petreczky, P. Hot-dense lattice QCD: USQCD whitepaper 2018. Eur. Phys. J. A 55, 194 (2019).

    Article  ADS  Google Scholar 

  13. Alexandru, A., Basar, G., Bedaque, P. F. & Warrington, N. C. Complex paths around the sign problem. Rev. Mod. Phys. 94, 015006 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  14. Troyer, M. & Wiese, U.-J. Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations. Phys. Rev. Lett. 94, 170201 (2005).

    Article  ADS  Google Scholar 

  15. Nuclear Physics and Quantum Information Science: Report by the NSAC QIS Subcommittee. Tech. Rep. NSF & DOE Office of Science. https://science.osti.gov/-/media/np/pdf/Reports/NSAC_QIS_Report.pdf (2019).

  16. Bauer, C. W. et al. Quantum simulation for high energy physics. PRX Quantum 4, 027001 (2023).

    Article  ADS  Google Scholar 

  17. Catterall, S. et al. Report of the Snowmass 2021 theory frontier topical group on quantum information science. Preprint at https://arxiv.org/abs/2209.14839 (2022).

  18. Humble, T. S., Perdue, G. N. & Savage, M. J. Snowmass Computational Frontier: Topical Group Report on quantum computing. Preprint at https://arxiv.org/abs/2209.06786 (2022).

  19. Beck, D. et al. Quantum information science and technology for nuclear physics. Input into U.S. Long-Range Planning, 2023. Preprint at https://arxiv.org/abs/2303.00113 (2023).

  20. Manin, Y. Computable and Uncomputable (Sovetskoye Radio, 1980).

  21. Benioff, P. The computer as a physical system: a microscopic quantum mechanical hamiltonian model of computers as represented by turing machines. J. Stat. Phys. 22, 563–591 (1980).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982).

    Article  MathSciNet  Google Scholar 

  23. Feynman, R. P. Quantum mechanical computers. Found. Phys. 16, 507–531 (1986).

    Article  ADS  MathSciNet  Google Scholar 

  24. Landauer, R. Information is physical. Phys. Today 44, 23–29 (1991).

    Article  ADS  Google Scholar 

  25. Bennett, C. H. Logical reversibility of computation. IBM J. Res. Dev. 17, 525–532 (1973).

    Article  MATH  MathSciNet  Google Scholar 

  26. Landauer, R. Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5, 183–191 (1961).

    Article  MATH  MathSciNet  Google Scholar 

  27. Fredkin, E. & Toffoli, T. Conservative logic. Int. J. Theor. Phys. 21, 219–253 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  28. Bañuls, M. C. et al. Simulating lattice gauge theories within quantum technologies. Eur. Phys. J. D 74, 165 (2020).

    Article  ADS  Google Scholar 

  29. Kasper, V., Juzeliunas, G., Lewenstein, M., Jendrzejewski, F. & Zohar, E. From the Jaynes–Cummings model to non-Abelian gauge theories: a guided tour for the quantum engineer. New J. Phys. 22, 103027 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  30. Aidelsburger, M. et al. Cold atoms meet lattice gauge theory. Phil. Trans. Roy. Soc. Lond. A 380, 20210064 (2021).

    ADS  Google Scholar 

  31. Klco, N., Roggero, A. & Savage, M. J. Standard model physics and the digital quantum revolution: thoughts about the interface. Rept. Prog. Phys. 85, 064301 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  32. Mil, A. et al. A scalable realization of local U(1) gauge invariance in cold atomic mixtures. Science 367, 1128–1130 (2020).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. Yang, B. et al. Observation of gauge invariance in a 71-site Bose–Hubbard quantum simulator. Nature 587, 392–396 (2020).

    Article  ADS  Google Scholar 

  34. Zhou, Z.-Y. et al. Thermalization dynamics of a gauge theory on a quantum simulator. Science 377, abl6277 (2022).

    Article  MathSciNet  Google Scholar 

  35. Schweizer, C. et al. Floquet approach to \({\mathbb{Z}}\)2 lattice gauge theories with ultracold atoms in optical lattices. Nat. Phys. 15, 1168–1173 (2019).

  36. Görg, F. et al. Realization of density-dependent Peierls phases to engineer quantized gauge fields coupled to ultracold matter. Nat. Phys. 15, 1161–1167 (2019).

    Article  Google Scholar 

  37. Klco, N. & Savage, M. J. Entanglement spheres and a UV-IR connection in effective field theories. Phys. Rev. Lett. 127, 211602 (2021).

    Article  ADS  Google Scholar 

  38. Kaplan, D. B. A method for simulating chiral fermions on the lattice. Phys. Lett. B 288, 342–347 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  39. Kaplan, D. B. Chiral fermions on the lattice. Nucl. Phys. B Proc. Suppl. 30, 597–600 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  40. Narayanan, R. & Neuberger, H. Infinitely many regulator fields for chiral fermions. Phys. Lett. B 302, 62–69 (1993).

    Article  ADS  Google Scholar 

  41. Narayanan, R. & Neuberger, H. Chiral fermions on the lattice. Phys. Rev. Lett. 71, 3251 (1993).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  42. Shamir, Y. Chiral fermions from lattice boundaries. Nucl. Phys. B 406, 90–106 (1993).

    Article  ADS  Google Scholar 

  43. Augusiak, R., Cucchietti, F. & Lewenstein, M. Many-body physics from a quantum information perspective. In Modern Theories of Many-Particle Systems in Condensed Matter Physics 245–294 (Springer, 2012).

  44. Zeng, B., Chen, X., Zhou, D.-L. & Wen, X.-G. Quantum information meets quantum matter — from quantum entanglement to topological phase in many-body systems. Preprint at https://doi.org/10.48550/arXiv.1508.02595 (2015).

  45. Savary, L. & Balents, L. Quantum spin liquids: a review. Rep. Prog. Phys. 80, 016502 (2017).

    Article  ADS  Google Scholar 

  46. Sachdev, S. Topological order and emergent gauge fields and Fermi surface reconstruction. Rept. Prog. Phys. 82, 014001 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  47. White, S. R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863–2866 (1992).

    Article  ADS  Google Scholar 

  48. Rommer, S. & Ostlund, S. Class of ansatz wave functions for one-dimensional spin systems and their relation to the density matrix renormalization group. Phys. Rev. B 55, 2164–2181 (1997).

    Article  ADS  Google Scholar 

  49. Vidal, G. Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91, 147902 (2003).

    Article  ADS  Google Scholar 

  50. Vidal, G. Efficient simulation of one-dimensional quantum many-body systems. Phys. Rev. Lett. 93, 040502 (2004).

    Article  ADS  Google Scholar 

  51. Pichler, T., Dalmonte, M., Rico, E., Zoller, P. & Montangero, S. Real-time dynamics in U(1) lattice gauge theories with tensor networks. Phys. Rev. X 6, 011023 (2016).

    Google Scholar 

  52. Kühn, S., Zohar, E., Cirac, J. I. & Bañuls, M. C. Non-Abelian string breaking phenomena with matrix product states. JHEP 07, 130 (2015).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  53. Montangero, S. & Evenson. Introduction to Tensor Network Methods (Springer, 2018).

  54. Tilloy, A. & Cirac, J. I. Continuous tensor network states for quantum fields. Phys. Rev. X 9, 021040 (2019).

    Google Scholar 

  55. Silvi, P. et al. The tensor networks anthology: simulation techniques for many-body quantum lattice systems. SciPost Physics Lecture Notes https://doi.org/10.21468/SciPostPhysLectNotes.8 (2019).

  56. Bañuls, M. C., Cichy, K., Cirac, J. I., Jansen, K. & Kühn, S. Tensor networks and their use for lattice gauge theories. Proc. Sci. https://doi.org/10.22323/1.334.0022 (2018).

    Article  Google Scholar 

  57. Bañuls, M. C. & Cichy, K. Review on novel methods for lattice gauge theories. Rept. Prog. Phys. 83, 024401 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  58. Banuls, M. C., Heller, M. P., Jansen, K., Knaute, J. & Svensson, V. From spin chains to real-time thermal field theory using tensor networks. Phys. Rev. Res. 2, 033301 (2020).

    Article  Google Scholar 

  59. Emonts, P., Bañuls, M. C., Cirac, I. & Zohar, E. Variational Monte Carlo simulation with tensor networks of a pure \({{\mathbb{Z}}}_{3}\) gauge theory in (2+1)d. Phys. Rev. D 102, 074501 (2020).

  60. Meurice, Y., Sakai, R. & Unmuth-Yockey, J. Tensor lattice field theory for renormalization and quantum computing. Rev. Mod. Phys. 94, 025005 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  61. Meurice, Y. et al. Tensor networks for high energy physics: contribution to Snowmass 2021. Preprint at https://arxiv.org/abs/2203.04902 (2022).

  62. Bañuls, M. C. Tensor network algorithms: a route map. Annu. Rev. Condens. Matter Phys. 14, 173–191 (2023).

    Article  ADS  MathSciNet  Google Scholar 

  63. Milsted, A., Liu, J., Preskill, J. & Vidal, G. Collisions of false-vacuum bubble walls in a quantum spin chain. PRX Quantum 3, 020316 (2022).

    Article  ADS  Google Scholar 

  64. Klco, N. & Savage, M. J. Systematically localizable operators for quantum simulations of quantum field theories. Phys. Rev. A 102, 012619 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  65. Klco, N. & Savage, M. J. Fixed-point quantum circuits for quantum field theories. Phys. Rev. A 102, 052422 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  66. Ho, C. M. & Hsu, S. D. H. Entanglement and fast quantum thermalization in heavy ion collisions. Mod. Phys. Lett. A 31, 1650110 (2016).

    Article  ADS  Google Scholar 

  67. Kharzeev, D. E. & Levin, E. M. Deep inelastic scattering as a probe of entanglement. Phys. Rev. D 95, 114008 (2017).

    Article  ADS  Google Scholar 

  68. Baker, O. K. & Kharzeev, D. E. Thermal radiation and entanglement in proton–proton collisions at energies available at the CERN Large Hadron Collider. Phys. Rev. D 98, 054007 (2018).

    Article  ADS  Google Scholar 

  69. Cervera-Lierta, A., Latorre, J. I., Rojo, J. & Rottoli, L. Maximal entanglement in high energy physics. SciPost Phys. 3, 036 (2017).

    Article  ADS  Google Scholar 

  70. Beane, S. R., Kaplan, D. B., Klco, N. & Savage, M. J. Entanglement suppression and emergent symmetries of strong interactions. Phys. Rev. Lett. 122, 102001 (2019).

    Article  ADS  Google Scholar 

  71. Beane, S. R. & Ehlers, P. Chiral symmetry breaking entanglement and the nucleon spin decomposition. Mod. Phys. Lett. A 35, 2050048 (2019).

    Article  ADS  MATH  Google Scholar 

  72. Tu, Z., Kharzeev, D. E. & Ullrich, T. Einstein–Podolsky–Rosen paradox and quantum entanglement at subnucleonic scales. Phys. Rev. Lett. 124, 062001 (2020).

    Article  ADS  Google Scholar 

  73. Beane, S. R. & Farrell, R. C. Geometry and entanglement in the scattering matrix. Ann. Phys. 433, 168581 (2021).

    Article  MATH  MathSciNet  Google Scholar 

  74. Beane, S. R., Farrell, R. C. & Varma, M. Entanglement minimization in hadronic scattering with pions. Int. J. Mod. Phys. A 36, 2150205 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  75. Kharzeev, D. E. & Levin, E. Deep inelastic scattering as a probe of entanglement: confronting experimental data. Phys. Rev. D 104, L031503 (2021).

    Article  ADS  Google Scholar 

  76. Robin, C., Savage, M. J. & Pillet, N. Entanglement rearrangement in self-consistent nuclear structure calculations. Phys. Rev. C 103, 034325 (2021).

    Article  ADS  Google Scholar 

  77. Low, I. & Mehen, T. Symmetry from entanglement suppression. Phys. Rev. D 104, 074014 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  78. Gong, W., Parida, G., Tu, Z. & Venugopalan, R. Measurement of Bell-type inequalities and quantum entanglement from Λ-hyperon spin correlations at high energy colliders. Phys. Rev. D 106, L031501 (2022).

    Article  ADS  Google Scholar 

  79. Roggero, A. Entanglement and many-body effects in collective neutrino oscillations. Phys. Rev. D 104, 103016 (2021).

    Article  ADS  Google Scholar 

  80. Johnson, C. W. & Gorton, O. C. Proton–neutron entanglement in the nuclear shell model. Preprint at https://arxiv.org/abs/2210.14338 (2022).

  81. Reeh, H. & Schlieder, S. Bemerkungen zur unitäräquivalenz von lorentzinvarianten feldern. Il Nuovo Cimento (1955–1965) 22, 1051–1068 (1961).

    Article  ADS  MATH  Google Scholar 

  82. Summers, S. J. & Werner, R. The vacuum violates Bell’s inequalities. Phys. Lett. A 110, 257–259 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  83. Summers, S. J. & Werner, R. Bell’s inequalities and quantum field theory. I. General setting. J. Math. Phys. 28, 2440–2447 (1987).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  84. Summers, S. J. & Werner, R. Bell’s inequalities and quantum field theory. II. Bell’s inequalities are maximally violated in the vacuum. J. Math. Phys. 28, 2448–2456 (1987).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  85. Valentini, A. Non-local correlations in quantum electrodynamics. Phys. Lett. A 153, 321 – 325 (1991).

    Article  Google Scholar 

  86. Srednicki, M. Entropy and area. Phys. Rev. Lett. 71, 666–669 (1993).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  87. Holzhey, C., Larsen, F. & Wilczek, F. Geometric and renormalized entropy in conformal field theory. Nucl. Phys. B 424, 443–467 (1994).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  88. Halvorson, H. & Clifton, R. Generic Bell correlation between arbitrary local algebras in quantum field theory. J. Math. Phys. 41, 1711–1717 (2000).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  89. Audenaert, K., Eisert, J., Plenio, M. B. & Werner, R. F. Entanglement properties of the harmonic chain. Phys. Rev. A 66, 042327 (2002).

    Article  ADS  Google Scholar 

  90. Reznik, B. Entanglement from the vacuum. Found. Phys. 33, 167–176 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  91. Reznik, B., Retzker, A. & Silman, J. Violating Bell’s inequalities in the vacuum. Phys. Rev. A 71, 042104 (2005).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  92. Calabrese, P. & Cardy, J. L. Entanglement entropy and quantum field theory. J. Stat. Mech. 0406, P06002 (2004).

    MATH  Google Scholar 

  93. Botero, A. & Reznik, B. Spatial structures and localization of vacuum entanglement in the linear harmonic chain. Phys. Rev. A. 70, 052329 (2004).

    Article  ADS  Google Scholar 

  94. Retzker, A., Cirac, J. I. & Reznik, B. Detecting vacuum entanglement in a linear ion trap. Phys. Rev. Lett. Phys. Rev. Lett. 94, 050504 (2005).

    Article  ADS  Google Scholar 

  95. Kofler, J., Vedral, V., Kim, M. S. & Brukner, Č. Entanglement between collective operators in a linear harmonic chain. Phys. Rev. A 73, 052107 (2006).

    Article  ADS  Google Scholar 

  96. Ryu, S. & Takayanagi, T. Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett. 96, 181602 (2006).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  97. Marcovitch, S., Retzker, A., Plenio, M. & Reznik, B. Critical and noncritical long-range entanglement in Klein–Gordon fields. Phys. Rev. A 80, 012325 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  98. Calabrese, P., Cardy, J. & Tonni, E. Entanglement entropy of two disjoint intervals in conformal field theory. J. Stat. Mech. 0911, P11001 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  99. Calabrese, P. & Cardy, J. Entanglement entropy and conformal field theory. J. Phys. A 42, 504005 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  100. Casini, H. & Huerta, M. Entanglement entropy in free quantum field theory. J. Phys. A 42, 504007 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  101. Zych, M., Costa, F., Kofler, J. & Brukner, C. Entanglement between smeared field operators in the Klein–Gordon vacuum. Phys. Rev. D 81, 125019 (2010).

    Article  ADS  Google Scholar 

  102. Calabrese, P., Cardy, J. & Tonni, E. Entanglement negativity in extended systems: a field theoretical approach. J. Stat. Mech. 1302, P02008 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  103. Calabrese, P., Cardy, J. & Tonni, E. Entanglement negativity in quantum field theory. Phys. Rev. Lett. 109, 130502 (2012).

    Article  ADS  Google Scholar 

  104. Ghosh, S., Soni, R. M. & Trivedi, S. P. On the entanglement entropy for gauge theories. J. High Energy Phys. 09, 069 (2015).

    Article  MATH  MathSciNet  Google Scholar 

  105. Soni, R. M. & Trivedi, S. P. Aspects of entanglement entropy for gauge theories. J. High Energy Phys. 01, 136 (2016).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  106. Dalmonte, M., Vermersch, B. & Zoller, P. Quantum simulation and spectroscopy of entanglement Hamiltonians. Nat. Phys. 14, 827–831 (2018).

    Article  Google Scholar 

  107. Witten, E. APS medal for exceptional achievement in research: invited article on entanglement properties of quantum field theory. Rev. Mod. Phys. 90, 045003 (2018).

    Article  ADS  Google Scholar 

  108. Mendes-Santos, T., Giudici, G., Dalmonte, M. & Rajabpour, M. A. Entanglement Hamiltonian of quantum critical chains and conformal field theories. Phys. Rev. B 100, 155122 (2019).

    Article  ADS  Google Scholar 

  109. Di Giulio, G. & Tonni, E. On entanglement hamiltonians of an interval in massless harmonic chains. J. Stat. Mech. 2003, 033102 (2020).

    Article  MATH  MathSciNet  Google Scholar 

  110. Klco, N. & Savage, M. J. Geometric quantum information structure in quantum fields and their lattice simulation. Phys. Rev. D 103, 065007 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  111. Kokail, C., van Bijnen, R., Elben, A., Vermersch, B. & Zoller, P. Entanglement Hamiltonian tomography in quantum simulation. Nat. Phys. 17, 936–942 (2021).

    Article  Google Scholar 

  112. Roy, A., Pollmann, F. & Saleur, H. Entanglement Hamiltonian of the 1+1-dimensional free, compactified boson conformal field theory. J. Stat. Mech. 2008, 083104 (2020).

    Article  MATH  MathSciNet  Google Scholar 

  113. Klco, N., Beck, D. H. & Savage, M. J. Entanglement structures in quantum field theories: negativity cores and bound entanglement in the vacuum. Phys. Rev. A 107, 012415 (2023).

    Article  ADS  MathSciNet  Google Scholar 

  114. Mueller, N., Zache, T. V. & Ott, R. Thermalization of gauge theories from their entanglement spectrum. Phys. Rev. Lett. 129, 011601 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  115. Dalmonte, M., Eisler, V., Falconi, M. & Vermersch, B. Entanglement Hamiltonians: from field theory, to lattice models and experiments. Ann. Phys. 534, 2200064 (2022).

    Article  Google Scholar 

  116. Kogut, J. B. & Susskind, L. Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D 11, 395–408 (1975).

    Article  ADS  Google Scholar 

  117. Banks, T., Susskind, L. & Kogut, J. B. Strong coupling calculations of lattice gauge theories: (1+1)-dimensional exercises. Phys. Rev. D 13, 1043 (1976).

    Article  ADS  Google Scholar 

  118. Klco, N. & Savage, M. J. Digitization of scalar fields for quantum computing. Phys. Rev. A 99, 052335 (2019).

    Article  ADS  Google Scholar 

  119. Briceño, R. A., Guerrero, J. V., Hansen, M. T. & Sturzu, A. M. Role of boundary conditions in quantum computations of scattering observables. Phys. Rev. D 103, 014506 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  120. Carena, M., Lamm, H., Li, Y.-Y. & Liu, W. Lattice renormalization of quantum simulations. Phys. Rev. D 104, 094519 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  121. Ciavarella, A. N. & Chernyshev, I. A. Preparation of the SU(3) lattice Yang–Mills vacuum with variational quantum methods. Phys. Rev. D 105, 074504 (2022).

    Article  ADS  Google Scholar 

  122. Clemente, G., Crippa, A. & Jansen, K. Strategies for the determination of the running coupling of (2 + 1)-dimensional QED with quantum computing. Phys. Rev. D 106, 114511 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  123. Farrell, R. C. et al. Preparations for quantum simulations of quantum chromodynamics in 1 + 1 dimensions. I. Axial gauge. Phys. Rev. D 107, 054512 (2023).

    Article  ADS  Google Scholar 

  124. Brower, R., Chandrasekharan, S. & Wiese, U. J. QCD as a quantum link model. Phys. Rev. D 60, 094502 (1999).

    Article  ADS  Google Scholar 

  125. Byrnes, T. & Yamamoto, Y. Simulating lattice gauge theories on a quantum computer. Phys. Rev. A 73, 022328 (2006).

    Article  ADS  Google Scholar 

  126. Anishetty, R., Mathur, M. & Raychowdhury, I. Prepotential formulation of SU(3) lattice gauge theory. J. Phys. A 43, 035403 (2010).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  127. Zohar, E. & Burrello, M. Formulation of lattice gauge theories for quantum simulations. Phys. Rev. D 91, 054506 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  128. Bañuls, M. C., Cichy, K., Cirac, J. I., Jansen, K. & Kühn, S. Efficient basis formulation for 1+1 dimensional SU(2) lattice gauge theory: spectral calculations with matrix product states. Phys. Rev. X 7, 041046 (2017).

    Google Scholar 

  129. Kaplan, D. B. & Stryker, J. R. Gauss’s law, duality, and the hamiltonian formulation of u(1) lattice gauge theory. Phys. Rev. D 102, 094515 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  130. Zohar, E. & Cirac, J. I. Removing staggered fermionic matter in U(N) and SU(N) lattice gauge theories. Phys. Rev. D 99, 114511 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  131. Raychowdhury, I. & Stryker, J. R. Loop string and hadron dynamics in SU(2) Hamiltonian lattice gauge theories. Phys. Rev. D 101, 114502 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  132. Alexandru, A. et al. Gluon field digitization for quantum computers. Phys. Rev. D 100, 114501 (2019).

    Article  ADS  Google Scholar 

  133. Klco, N., Stryker, J. R. & Savage, M. J. SU(2) non-Abelian gauge field theory in one dimension on digital quantum computers. Phys. Rev. D 101, 074512 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  134. Singh, H. & Chandrasekharan, S. Qubit regularization of the O(3) sigma model. Phys. Rev. D 100, 054505 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  135. Davoudi, Z., Raychowdhury, I. & Shaw, A. Search for efficient formulations for Hamiltonian simulation of non-Abelian lattice gauge theories. Phys. Rev. D 104, 074505 (2021).

    Article  ADS  Google Scholar 

  136. Haase, J. F. et al. A resource efficient approach for quantum and classical simulations of gauge theories in particle physics. Quantum 5, 393 (2021).

    Article  Google Scholar 

  137. Ji, Y., Lamm, H. & Zhu, S. Gluon field digitization via group space decimation for quantum computers. Phys. Rev. D 102, 114513 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  138. Kreshchuk, M., Kirby, W. M., Goldstein, G., Beauchemin, H. & Love, P. J. Quantum simulation of quantum field theory in the light-front formulation. Phys. Rev. A 105, 032418 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  139. Buser, A. J., Gharibyan, H., Hanada, M., Honda, M. & Liu, J. Quantum simulation of gauge theory via orbifold lattice. J. High Energy Phys. 09, 034 (2021).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  140. Ciavarella, A., Klco, N. & Savage, M. J. Trailhead for quantum simulation of SU(3) Yang–Mills lattice gauge theory in the local multiplet basis. Phys. Rev. D 103, 094501 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  141. Bauer, C. W. & Grabowska, D. M. Efficient representation for simulating U(1) gauge theories on digital quantum computers at all values of the coupling. Preprint at https://arxiv.org/abs/2111.08015 (2021).

  142. Ciavarella, A., Klco, N. & Savage, M. J. Some conceptual aspects of operator design for quantum simulations of non-Abelian lattice gauge theories. Preprint at https://arxiv.org/abs/2203.11988 (2022).

  143. Kane, C., Grabowska, D. M., Nachman, B. & Bauer, C. W. Efficient quantum implementation of 2+1 U(1) lattice gauge theories with Gauss law constraints. Preprint at https://arxiv.org/abs/2211.10497 (2022).

  144. Jordan, S. P., Lee, K. S. M. & Preskill, J. Quantum computation of scattering in scalar quantum field theories. Quant. Inf. Comput. 14, 1014–1080 (2014).

    MathSciNet  Google Scholar 

  145. Kempe, J. & Regev, O. 3-Local Hamiltonian is QMA-complete. Quantum Inf. Comput. 3, 258–264 (2003).

    MATH  MathSciNet  Google Scholar 

  146. Kempe, J., Kitaev, A. & Regev, O. The complexity of the local Hamiltonian problem. SIAM J. Comput. 35, quant–ph/0406180 (2004).

    MathSciNet  Google Scholar 

  147. Oliveira, R. & Terhal, B. M. The complexity of quantum spin systems on a two-dimensional square lattice. Quantum Inf. Comput. 8, quant–ph/0504050 (2005).

    MathSciNet  Google Scholar 

  148. Calabrese, P. & Cardy, J. L. Time-dependence of correlation functions following a quantum quench. Phys. Rev. Lett. 96, 136801 (2006).

    Article  ADS  Google Scholar 

  149. Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).

    Article  ADS  Google Scholar 

  150. McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New J. Phys. 18, 023023 (2016).

    Article  ADS  MATH  Google Scholar 

  151. Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).

    Article  ADS  Google Scholar 

  152. Grimsley, H. R., Economou, S. E., Barnes, E. & Mayhall, N. J. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nat. Commun. 10, 3007 (2019).

    Article  ADS  Google Scholar 

  153. Tang, H. L. et al. Qubit-ADAPT-VQE: an adaptive algorithm for constructing hardware-efficient Ansätze on a quantum processor. PRX Quantum 2, 020310 (2021).

    Article  ADS  Google Scholar 

  154. Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nat. Phys. 16, 205–210 (2020).

    Article  Google Scholar 

  155. McArdle, S. et al. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Inf. 5, 75 (2019).

    Article  ADS  Google Scholar 

  156. Yeter-Aydeniz, K., Pooser, R. C. & Siopsis, G. Practical quantum computation of chemical and nuclear energy levels using quantum imaginary time evolution and Lanczos algorithms. npj Quantum Inf. 6, 63 (2020).

    Article  ADS  Google Scholar 

  157. Liu, J. & Xin, Y. Quantum simulation of quantum field theories as quantum chemistry. J. High Energy Phys. 12, 011 (2020).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  158. de Jong, W. A. et al. Quantum simulation of nonequilibrium dynamics and thermalization in the Schwinger model. Phys. Rev. D 106, 054508 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  159. Czajka, A. M., Kang, Z.-B., Ma, H. & Zhao, F. Quantum simulation of chiral phase transitions. J. High Energy Phys. 08, 209 (2022).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  160. Davoudi, Z., Mueller, N. & Powers, C. Toward quantum computing phase diagrams of gauge theories with thermal pure quantum states. Preprint at https://arxiv.org/abs/2208.13112 (2022).

  161. Aaronson, S. Shadow tomography of quantum states. In STOC 2018 https://doi.org/10.1145/3188745.3188802 (Association for Computing Machinery, 2018).

  162. Huang, H.-Y., Kueng, R. & Preskill, J. Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16, 1050–1057 (2020).

    Article  Google Scholar 

  163. Elben, A. et al. The randomized measurement toolbox. Nat. Rev. Phys. 5, 9–24 (2023).

    Article  Google Scholar 

  164. Barata, J. A., Mueller, N., Tarasov, A. & Venugopalan, R. Single-particle digitization strategy for quantum computation of a ϕ4 scalar field theory. Phys. Rev. A 103, 042410 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  165. Shaw, A. F., Lougovski, P., Stryker, J. R. & Wiebe, N. Quantum algorithms for simulating the lattice Schwinger model. Quantum 4, 306 (2020).

    Article  Google Scholar 

  166. Kan, A. & Nam, Y. Lattice quantum chromodynamics and electrodynamics on a universal quantum computer. Preprint at https://arxiv.org/abs/2107.12769 (2021).

  167. Lamm, H., Lawrence, S. & Yamauchi, Y. General methods for digital quantum simulation of gauge theories. Phys. Rev. D 100, 034518 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  168. Paulson, D. et al. Towards simulating 2D effects in lattice gauge theories on a quantum computer. PRX Quantum 2, 030334 (2021).

    Article  ADS  Google Scholar 

  169. Stryker, J. R. Shearing approach to gauge invariant Trotterization. Preprint at https://arxiv.org/abs/2105.11548 (2021).

  170. Davoudi, Z., Shaw, A. F. & Stryker, J. R. General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory. Preprint at https://arxiv.org/abs/2212.14030 (2022).

  171. Martinez, E. A. et al. Real-time dynamics of lattice gauge theories with a few-qubit quantum computer. Nature 534, 516–519 (2016).

    Article  ADS  Google Scholar 

  172. Stryker, J. R. Oracles for Gauss’s law on digital quantum computers. Phys. Rev. A 99, 042301 (2019).

    Article  ADS  Google Scholar 

  173. Raychowdhury, I. & Stryker, J. R. Solving Gauss’s Law on digital quantum computers with Loop-String-Hadron digitization. Phys. Rev. Res. 2, 033039 (2020).

    Article  Google Scholar 

  174. Stannigel, K. et al. Constrained dynamics via the Zeno effect in quantum simulation: implementing non-Abelian lattice gauge theories with cold atoms. Phys. Rev. Lett. 112, 120406 (2014).

    Article  ADS  Google Scholar 

  175. Kasper, V., Zache, T. V., Jendrzejewski, F., Lewenstein, M. & Zohar, E. Non-Abelian gauge invariance from dynamical decoupling. Preprint at https://arxiv.org/abs/2012.08620 (2020).

  176. Halimeh, J. C., Lang, H., Mildenberger, J., Jiang, Z. & Hauke, P. Gauge-symmetry protection using single-body terms. PRX Quantum 2, 040311 (2021).

    Article  ADS  Google Scholar 

  177. Tran, M. C., Su, Y., Carney, D. & Taylor, J. M. Faster digital quantum simulation by symmetry protection. PRX Quantum 2, 010323 (2021).

    Article  Google Scholar 

  178. Lamm, H., Lawrence, S. & Yamauchi, Y. Suppressing coherent gauge drift in quantum simulations. Preprint at https://arxiv.org/abs/2005.12688 (2020).

  179. Nguyen, N. H. et al. Digital quantum simulation of the Schwinger model and symmetry protection with trapped ions. PRX Quantum 3, 020324 (2022).

    Article  ADS  Google Scholar 

  180. Zohar, E., Cirac, J. I. & Reznik, B. Cold-atom quantum simulator for SU(2) Yang–Mills lattice gauge theory. Phys. Rev. Lett. 110, 125304 (2013).

    Article  ADS  Google Scholar 

  181. Banerjee, D. et al. Atomic quantum simulation of dynamical gauge fields coupled to fermionic matter: from string breaking to evolution after a quench. Phys. Rev. Lett. 109, 175302 (2012).

    Article  ADS  Google Scholar 

  182. Tagliacozzo, L., Celi, A., Orland, P. & Lewenstein, M. Simulations of non-Abelian gauge theories with optical lattices. Nat. Commun. 4, 2615 (2013).

    Article  ADS  Google Scholar 

  183. Zohar, E., Cirac, J. I. & Reznik, B. Quantum simulations of gauge theories with ultracold atoms: local gauge invariance from angular momentum conservation. Phys. Rev. A 88, 023617 (2013).

    Article  ADS  Google Scholar 

  184. Hauke, P., Marcos, D., Dalmonte, M. & Zoller, P. Quantum simulation of a lattice Schwinger model in a chain of trapped ions. Phys. Rev. X 3, 041018 (2013).

    Google Scholar 

  185. Kühn, S., Cirac, J. I. & Bañuls, M.-C. Quantum simulation of the Schwinger model: a study of feasibility. Phys. Rev. A 90, 042305 (2014).

    Article  ADS  Google Scholar 

  186. Kasper, V., Hebenstreit, F., Oberthaler, M. & Berges, J. Schwinger pair production with ultracold atoms. Phys. Lett. B 760, 742–746 (2016).

    Article  ADS  MATH  Google Scholar 

  187. Zohar, E., Cirac, J. I. & Reznik, B. Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices. Rept. Prog. Phys. 79, 014401 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  188. Mezzacapo, A. et al. Non-Abelian SU(2) lattice gauge theories in superconducting circuits. Phys. Rev. Lett. 115, 240502 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  189. Bazavov, A., Meurice, Y., Tsai, S.-W., Unmuth-Yockey, J. & Zhang, J. Gauge-invariant implementation of the Abelian Higgs model on optical lattices. Phys. Rev. D 92, 076003 (2015).

    Article  ADS  Google Scholar 

  190. Yang, D. et al. Analog quantum simulation of (1+1)-dimensional lattice QED with trapped ions. Phys. Rev. A 94, 052321 (2016).

    Article  ADS  Google Scholar 

  191. González-Cuadra, D., Zohar, E. & Cirac, J. I. Quantum simulation of the Abelian-Higgs lattice gauge theory with ultracold atoms. New J. Phys. 19, 063038 (2017).

    Article  ADS  MATH  Google Scholar 

  192. Davoudi, Z. et al. Towards analog quantum simulations of lattice gauge theories with trapped ions. Phys. Rev. Res. 2, 023015 (2020).

    Article  Google Scholar 

  193. Surace, F. M. et al. Lattice gauge theories and string dynamics in Rydberg atom quantum simulators. Phys. Rev. X 10, 021041 (2020).

    Google Scholar 

  194. Luo, D. et al. Framework for simulating gauge theories with dipolar spin systems. Phys. Rev. A 102, 032617 (2020).

    Article  ADS  Google Scholar 

  195. Ott, R., Zache, T. V., Jendrzejewski, F. & Berges, J. Scalable cold-atom quantum simulator for two-dimensional QED. Phys. Rev. Lett. 127, 130504 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  196. Dasgupta, R. & Raychowdhury, I. Cold-atom quantum simulator for string and hadron dynamics in non-Abelian lattice gauge theory. Phys. Rev. A 105, 023322 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  197. Andrade, B. et al. Engineering an effective three-spin Hamiltonian in trapped-ion systems for applications in quantum simulation. Quantum Sci. Technol. 7, 034001 (2022).

    Article  ADS  Google Scholar 

  198. Osborne, J., McCulloch, I. P., Yang, B., Hauke, P. & Halimeh, J. C. Large-scale 2 + 1D U(1) gauge theory with dynamical matter in a cold-atom quantum simulator. Preprint at https://arxiv.org/abs/2211.01380 (2022).

  199. Zohar, E., Farace, A., Reznik, B. & Cirac, J. I. Digital quantum simulation of Z2 lattice gauge theories with dynamical fermionic matter. Phys. Rev. Lett. 118, 070501 (2017).

    Article  ADS  Google Scholar 

  200. Zohar, E., Farace, A., Reznik, B. & Cirac, J. I. Digital lattice gauge theories. Phys. Rev. A 95, 023604 (2017).

    Article  ADS  Google Scholar 

  201. Bender, J., Zohar, E., Farace, A. & Cirac, J. I. Digital quantum simulation of lattice gauge theories in three spatial dimensions. New J. Phys. 20, 093001 (2018).

    Article  ADS  Google Scholar 

  202. Davoudi, Z., Linke, N. M. & Pagano, G. Toward simulating quantum field theories with controlled phonon-ion dynamics: a hybrid analog-digital approach. Phys. Rev. Res. 3, 043072 (2021).

    Article  Google Scholar 

  203. Zhang, X. et al. Experimental quantum simulation of fermion–antifermion scattering via boson exchange in a trapped ion. Nat. Commun. 9, 195 (2018).

    Article  ADS  Google Scholar 

  204. González-Cuadra, D., Zache, T. V., Carrasco, J., Kraus, B. & Zoller, P. Hardware efficient quantum simulation of non-Abelian gauge theories with qudits on Rydberg platforms. Phys. Rev. Lett. 129, 160501 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  205. Bennett, C. H., DiVincenzo, D. P., Smolin, J. A. & Wootters, W. K. Mixed state entanglement and quantum error correction. Phys. Rev. A 54, 3824–3851 (1996).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  206. Dankert, C., Cleve, R., Emerson, J. & Livine, E. Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A 80, 012304 (2009).

    Article  ADS  Google Scholar 

  207. Dür, W., Hein, M., Cirac, J. I. & Briegel, H. J. Standard forms of noisy quantum operations via depolarization. Phys. Rev. A. 72, 052326 (2005).

    Article  ADS  Google Scholar 

  208. Emerson, J. et al. Symmetrized characterization of noisy quantum processes. Science 317, 1893 (2007).

    Article  ADS  Google Scholar 

  209. Temme, K., Bravyi, S. & Gambetta, J. M. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett. 119, 180509 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  210. Li, Y. & Benjamin, S. C. Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X 7, 021050 (2017).

    Google Scholar 

  211. Endo, S., Benjamin, S. C. & Li, Y. Practical quantum error mitigation for near-future applications. Phys. Rev. X 8, 031027 (2018).

    Google Scholar 

  212. Kandala, A. et al. Error mitigation extends the computational reach of a noisy quantum processor. Nature 567, 491–495 (2019).

    Article  ADS  Google Scholar 

  213. He, A., Nachman, B., de Jong, W. A. & Bauer, C. W. Zero-noise extrapolation for quantum-gate error mitigation with identity insertions. Phys. Rev. A 102, 012426 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  214. Viola, L., Knill, E. & Lloyd, S. Dynamical decoupling of open quantum systems. Phys. Rev. Lett. 82, 2417 (1999).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  215. Souza, A. M., Álvarez, G. A. & Suter, D. Robust dynamical decoupling. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 370, 4748–4769 (2012).

    Article  ADS  Google Scholar 

  216. Suter, D. & Álvarez, G. A. Colloquium: protecting quantum information against environmental noise. Rev. Mod. Phys. 88, 041001 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  217. Rahman, S. A., Lewis, R., Mendicelli, E. & Powell, S. Self-mitigating Trotter circuits for SU(2) lattice gauge theory on a quantum computer. Phys. Rev. D 106, 074502 (2022).

    Article  ADS  Google Scholar 

  218. Urbanek, M. et al. Mitigating depolarizing noise on quantum computers with noise-estimation circuits. Phys. Rev. Lett. 127, 270502 (2021).

    Article  MathSciNet  Google Scholar 

  219. Zhang, B. et al. Hidden inverses: coherent error cancellation at the circuit level. Phys. Rev. Appl. 17, 034074 (2022).

    Article  ADS  Google Scholar 

  220. Leyton-Ortega, V., Majumder, S. & Pooser, R. C. Quantum error mitigation by hidden inverses protocol in superconducting quantum devices. Preprint at https://arxiv.org/abs/2204.12407 (2022).

  221. Klco, N. & Savage, M. J. Hierarchical qubit maps and hierarchically implemented quantum error correction. Phys. Rev. A 104, 062425 (2021).

    Article  ADS  Google Scholar 

  222. Rajput, A., Roggero, A. & Wiebe, N. Quantum error correction with gauge symmetries. Preprint at https://arxiv.org/abs/2112.05186 (2021).

  223. Roggero, A., Li, A. C. Y., Carlson, J., Gupta, R. & Perdue, G. N. Quantum computing for neutrino-nucleus scattering. Phys. Rev. D 101, 074038 (2020).

    Article  ADS  Google Scholar 

  224. Holland, E. T. et al. Optimal control for the quantum simulation of nuclear dynamics. Phys. Rev. A 101, 062307 (2020).

    Article  ADS  Google Scholar 

  225. Roggero, A., Gu, C., Baroni, A. & Papenbrock, T. Preparation of excited states for nuclear dynamics on a quantum computer. Phys. Rev. C 102, 064624 (2020).

    Article  ADS  Google Scholar 

  226. Stetcu, I., Baroni, A. & Carlson, J. Variational approaches to constructing the many-body nuclear ground state for quantum computing. Phys. Rev. C 105, 064308 (2022).

    Article  ADS  Google Scholar 

  227. Choi, K., Lee, D., Bonitati, J., Qian, Z. & Watkins, J. Rodeo algorithm for quantum computing. Phys. Rev. Lett. 127, 040505 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  228. Baroni, A. et al. Nuclear two point correlation functions on a quantum computer. Phys. Rev. D 105, 074503 (2022).

    Article  ADS  Google Scholar 

  229. Turro, F. et al. Imaginary-time propagation on a quantum chip. Phys. Rev. A 105, 022440 (2022).

    Article  ADS  Google Scholar 

  230. Faba, J., Martin, V. & Robledo, L. Correlation energy and quantum correlations in a solvable model. Phys. Rev. A 104, 032428 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  231. Kruppa, A. T., Kovács, J., Salamon, P., Legeza, O. & Zaránd, G. Entanglement and seniority. Phys. Rev. C 106, 024303 (2022).

    Article  ADS  Google Scholar 

  232. Klco, N. et al. Quantum-classical computation of Schwinger model dynamics using quantum computers. Phys. Rev. A 98, 032331 (2018).

    Article  ADS  Google Scholar 

  233. Kokail, C. et al. Self-verifying variational quantum simulation of lattice models. Nature 569, 355–360 (2019).

    Article  ADS  Google Scholar 

  234. Atas, Y. Y. et al. SU(2) hadrons on a quantum computer via a variational approach. Nat. Commun. 12, 6499 (2021).

    Article  ADS  Google Scholar 

  235. Farrell, R. C. et al. Preparations for quantum simulations of quantum chromodynamics in 1+1 dimensions. II. Single-baryon β-decay in real time. Phys. Rev. D 107, 054513 (2023).

    Article  ADS  Google Scholar 

  236. A Rahman, S., Lewis, R., Mendicelli, E. & Powell, S. SU(2) lattice gauge theory on a quantum annealer. Phys. Rev. D 104, 034501 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  237. Bauer, C. W., Fleming, S., Pirjol, D. & Stewart, I. W. An effective field theory for collinear and soft gluons: heavy to light decays. Phys. Rev. D 63, 114020 (2001).

    Article  ADS  Google Scholar 

  238. Bauer, C. W., Freytsis, M. & Nachman, B. Simulating collider physics on quantum computers using effective field theories. Phys. Rev. Lett. 127, 212001 (2021).

    Article  ADS  Google Scholar 

  239. Bepari, K., Malik, S., Spannowsky, M. & Williams, S. Towards a quantum computing algorithm for helicity amplitudes and parton showers. Phys. Rev. D 103, 076020 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  240. Bepari, K., Malik, S., Spannowsky, M. & Williams, S. Quantum walk approach to simulating parton showers. Phys. Rev. D 106, 056002 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  241. Lamm, H., Lawrence, S. & Yamauchi, Y. Parton physics on a quantum computer. Phys. Rev. Res. 2, 013272 (2020).

    Article  Google Scholar 

  242. Echevarria, M. G., Egusquiza, I. L., Rico, E. & Schnell, G. Quantum simulation of light-front parton correlators. Phys. Rev. D 104, 014512 (2021).

    Article  ADS  Google Scholar 

  243. Li, T. et al. Partonic collinear structure by quantum computing. Phys. Rev. D 105, L111502 (2022).

    Article  ADS  Google Scholar 

  244. Mueller, N., Tarasov, A. & Venugopalan, R. Deeply inelastic scattering structure functions on a hybrid quantum computer. Phys. Rev. D 102, 016007 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  245. Pérez-Salinas, A., Cruz-Martinez, J., Alhajri, A. A. & Carrazza, S. Determining the proton content with a quantum computer. Phys. Rev. D 103, 034027 (2021).

    Article  ADS  Google Scholar 

  246. Qian, W., Basili, R., Pal, S., Luecke, G. & Vary, J. P. Solving hadron structures using the basis light-front quantization approach on quantum computers. Phys. Rev. Research 4, 043193 (2022).

    Article  ADS  Google Scholar 

  247. Pedernales, J. S., Candia, R. D., Egusquiza, I. L., Casanova, J. & Solano, E. Efficient quantum algorithm for computing n-time correlation functions. Phys. Rev. Lett. 113, 1401.2430 (2014).

    Article  ADS  Google Scholar 

  248. Berges, J., Heller, M. P., Mazeliauskas, A. & Venugopalan, R. QCD thermalization: ab initio approaches and interdisciplinary connections. Rev. Mod. Phys. 93, 035003 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  249. Lovato, A. et al. Long range plan: dense matter theory for heavy-ion collisions and neutron stars. Preprint at https://arxiv.org/abs/2211.02224 (2022).

  250. Kaufman, A. M. et al. Quantum thermalization through entanglement in an isolated many-body system. Science 353, 794–800 (2016).

    Article  ADS  Google Scholar 

  251. Geraedts, S. D., Nandkishore, R. & Regnault, N. Many-body localization and thermalization: insights from the entanglement spectrum. Phys. Rev. B 93, 174202 (2016).

    Article  ADS  Google Scholar 

  252. Turner, C. J., Michailidis, A. A., Abanin, D. A., Serbyn, M. & Papić, Z. Weak ergodicity breaking from quantum many-body scars. Nat. Phys. 14, 745–749 (2018).

    Article  Google Scholar 

  253. Serbyn, M., Abanin, D. A. & Papić, Z. Quantum many-body scars and weak breaking of ergodicity. Nat. Phys. 17, 675–685 (2021).

    Article  Google Scholar 

  254. Heyl, M. Dynamical quantum phase transitions: a review. Rep. Prog. Phys. 81, 054001 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  255. Heyl, M., Polkovnikov, A. & Kehrein, S. Dynamical quantum phase transitions in the transverse-field Ising model. Phys. Rev. Lett. 110, 135704 (2013).

    Article  ADS  Google Scholar 

  256. Zhang, J. et al. Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator. Nature 551, 601–604 (2017).

    Article  ADS  Google Scholar 

  257. Guo, X.-Y. et al. Observation of a dynamical quantum phase transition by a superconducting qubit simulation. Phys. Rev. Appl. 11, 044080 (2019).

    Article  ADS  Google Scholar 

  258. Aramthottil, A. S. et al. Scar states in deconfined Z2 lattice gauge theories. Phys. Rev. B 106, L041101 (2022).

    Article  ADS  Google Scholar 

  259. Desaules, J.-Y. et al. Prominent quantum many-body scars in a truncated Schwinger model. Phys. Rev. B 107, 205112 (2023).

    Article  ADS  Google Scholar 

  260. Halimeh, J. C., Barbiero, L., Hauke, P., Grusdt, F. & Bohrdt, A. Robust quantum many-body scars in lattice gauge theories. Quantum 7, 1004 (2023).

    Article  Google Scholar 

  261. Zache, T. V. et al. Dynamical topological transitions in the massive Schwinger model with a θ term. Phys. Rev. Lett. 122, 050403 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  262. Mueller, N. et al. Quantum computation of dynamical quantum phase transitions and entanglement tomography in a lattice gauge theory. Preprint at https://arxiv.org/abs/2210.03089 (2022).

  263. Van Damme, M., Zache, T. V., Banerjee, D., Hauke, P. & Halimeh, J. C. Dynamical quantum phase transitions in spin-SU(1) quantum link models. Phys. Rev. B 106, 245110 (2022).

    Article  ADS  Google Scholar 

  264. Van Damme, M., Desaules, J.-Y., Papić, Z. & Halimeh, J. C. The anatomy of dynamical quantum phase transitions. Preprint at https://arxiv.org/abs/2210.02453 (2022).

  265. Jensen, R. B., Pedersen, S. P. & Zinner, N. T. Dynamical quantum phase transitions in a noisy lattice gauge theory. Phys. Rev. B 105, 224309 (2022).

    Article  ADS  Google Scholar 

  266. Pantaleone, J. T. Dirac neutrinos in dense matter. Phys. Rev. D 46, 510–523 (1992).

    Article  ADS  Google Scholar 

  267. Pantaleone, J. T. Neutrino oscillations at high densities. Phys. Lett. B 287, 128–132 (1992).

    Article  ADS  Google Scholar 

  268. Friedland, A. & Lunardini, C. Do many particle neutrino interactions cause a novel coherent effect? J. High Energy Phys.10, 043 (2003).

    Article  ADS  Google Scholar 

  269. Bell, N. F., Rawlinson, A. A. & Sawyer, R. F. Speedup through entanglement: many body effects in neutrino processes. Phys. Lett. B 573, 86–93 (2003).

    Article  ADS  Google Scholar 

  270. Sawyer, R. F. ‘Classical’ instabilities and ‘quantum’ speed-up in the evolution of neutrino clouds. Preprint at https://arxiv.org/abs/hep-ph/0408265 (2004).

  271. Rrapaj, E. Exact solution of multiangle quantum many-body collective neutrino-flavor oscillations. Phys. Rev. C 101, 065805 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  272. Cervia, M. J., Patwardhan, A. V., Balantekin, A. B., Coppersmith, S. N. & Johnson, C. W. Entanglement and collective flavor oscillations in a dense neutrino gas. Phys. Rev. D 100, 083001 (2019).

    Article  ADS  Google Scholar 

  273. Martin, J. D., Roggero, A., Duan, H., Carlson, J. & Cirigliano, V. Classical and quantum evolution in a simple coherent neutrino problem. Phys. Rev. D 105, 083020 (2022).

    Article  ADS  Google Scholar 

  274. Roggero, A. Dynamical phase transitions in models of collective neutrino oscillations. Phys. Rev. D 104, 123023 (2021).

    Article  ADS  Google Scholar 

  275. Roggero, A., Rrapaj, E. & Xiong, Z. Entanglement and correlations in fast collective neutrino flavor oscillations. Phys. Rev. D 106, 043022 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  276. Amitrano, V. et al. Trapped-ion quantum simulation of collective neutrino oscillations. Phys. Rev. D 107, 023007 (2023).

    Article  ADS  Google Scholar 

  277. Illa, M. & Savage, M. J. Multi-neutrino entanglement and correlations in dense neutrino systems. Phys. Rev. Lett. 130, 221003 (2023).

    Article  ADS  Google Scholar 

  278. Hall, B., Roggero, A., Baroni, A. & Carlson, J. Simulation of collective neutrino oscillations on a quantum computer. Phys. Rev. D 104, 063009 (2021).

    Article  ADS  Google Scholar 

  279. Yeter-Aydeniz, K., Bangar, S., Siopsis, G. & Pooser, R. C. Collective neutrino oscillations on a quantum computer. Quant. Inf. Proc. 21, 84 (2022).

    Article  MATH  MathSciNet  Google Scholar 

  280. Illa, M. & Savage, M. J. Basic elements for simulations of standard-model physics with quantum annealers: multigrid and clock states. Phys. Rev. A 106, 052605 (2022).

    Article  ADS  Google Scholar 

  281. Preskill, J. Quantum computing and the entanglement frontier. Preprint at https://arxiv.org/abs/1203.5813 (2012).

  282. Daley, A. J. et al. Practical quantum advantage in quantum simulation. Nature 607, 667–676 (2022).

    Article  ADS  Google Scholar 

  283. Alexeev, Y. et al. Quantum computer systems for scientific discovery. PRX Quantum 2, 017001 (2021).

    Article  Google Scholar 

  284. Bernard, C. et al. Panel discussion on chiral extrapolation of physical observables. Nucl. Phys. B Proc. Suppl. 119, 170–184 (2003).

    Article  ADS  MATH  Google Scholar 

  285. Christ, N. H. In Encyclopedia of Parallel Computing (ed Padua, D.) 1668–1677 (Springer, 2011); https://doi.org/10.1007/978-0-387-09766-4_304

  286. Davis, E., Bentsen, G. & Schleier-Smith, M. Approaching the Heisenberg limit without single-particle detection. Phys. Rev. Lett. 116, 053601 (2016).

    Article  ADS  Google Scholar 

  287. Zhou, S., Zhang, M., Preskill, J. & Jiang, L. Achieving the Heisenberg limit in quantum metrology using quantum error correction. Nat. Commun. 9, 78 (2018).

    Article  ADS  Google Scholar 

  288. Ahmed, Z. et al. Quantum sensing for high energy physics. Preprint at https://arxiv.org/abs/1803.11306 (2018).

  289. Zhuang, Q., Preskill, J. & Jiang, L. Distributed quantum sensing enhanced by continuous-variable error correction. New J. Phys. 22, 022001 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  290. Guo, X. et al. Distributed quantum sensing in a continuous-variable entangled network. Nat. Phys. 16, 281–284 (2020).

    Article  Google Scholar 

  291. Kaubruegger, R., Vasilyev, D. V., Schulte, M., Hammerer, K. & Zoller, P. Quantum variational optimization of Ramsey interferometry and atomic clocks. Phys. Rev. X 11, 041045 (2021).

    Google Scholar 

  292. Marciniak, C. D. et al. Optimal metrology with programmable quantum sensors. Nature 603, 604–609 (2022).

    Article  ADS  Google Scholar 

  293. Xia, Y., Li, W., Zhuang, Q. & Zhang, Z. Quantum-enhanced data classification with a variational entangled sensor network. Phys. Rev. X 11, 021047 (2021).

    Google Scholar 

  294. Hernández-Gómez, S. et al. Optimal control of a quantum sensor: a fast algorithm based on an analytic solution. Preprint at https://arxiv.org/abs/2112.14998 (2021).

  295. Alderete, C. H. et al. Inference-based quantum sensing. Phys. Rev. Lett. 129, 190501 (2022).

    Article  MathSciNet  Google Scholar 

  296. Brady, A. J. et al. Entangled sensor-networks for dark-matter searches. PRX Quantum 3, 030333 (2022).

    Article  ADS  Google Scholar 

  297. Altman, E. et al. Quantum simulators: architectures and opportunities. PRX Quantum 2, 017003 (2021).

    Article  Google Scholar 

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Acknowledgements

The authors thank all of our collaborators and co-workers (represented in the references and far beyond) for shaping our perspectives of the evolving field of quantum simulation for nuclear-physics and high-energy physics. C.W.B. acknowledges support by the Director, Office of Science, Office of High Energy Physics of the US Department of Energy under the Contract No. DE-AC02-05CH11231, in particular through Quantum Information Science Enabled Discovery (QuantISED) for High Energy Physics (KA2401032). Z.D. acknowledges support by the US DOE’s Office of Science Early Career Award, under award no. DE-SC0020271, National Science Foundation Quantum Leap Challenge Institute for Robust Quantum Simulation (https://rqs.umd.edu) under grant OMA-2120757, Maryland Center for Fundamental Physics (https://mcfp.physics.umd.edu) at the University of Maryland and the DOE’s Office of Science, Office of Advanced Scientific Computing Research, Quantum Computing Application Teams program, under fieldwork proposal number ERKJ347, and the Accelerated Research in Quantum Computing program under award DESC0020312. M.J.S. acknowledges support by the US Department of Energy, Office of Science, Office of Nuclear Physics, InQubator for Quantum Simulation (IQuS) (https://iqus.uw.edu) under Award Number DOE (NP) Award DE-SC0020970, the Department of Physics (https://phys.washington.edu) and the College of Arts and Sciences (https://www.artsci.washington.edu) at the University of Washington.

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Bauer, C.W., Davoudi, Z., Klco, N. et al. Quantum simulation of fundamental particles and forces. Nat Rev Phys 5, 420–432 (2023). https://doi.org/10.1038/s42254-023-00599-8

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