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Toward the nonequilibrium thermodynamic analog of complexity and the Jarzynski identity. (English) Zbl 1522.81300

Summary: The Jarzynski identity can describe small-scale nonequilibrium systems through stochastic thermodynamics. The identity considers fluctuating trajectories in a phase space. The complexity geometry frames the discussions on quantum computational complexity using the method of Riemannian geometry, which builds a bridge between optimal quantum circuits and classical geodesics in the space of unitary operators. Complexity geometry enables the application of the methods of classical physics to deal with pure quantum problems. By combining the two frameworks, i.e., the Jarzynski identity and complexity geometry, we derived a complexity analog of the Jarzynski identity using the complexity geometry. We considered a set of geodesics in the space of unitary operators instead of the trajectories in a phase space. The obtained complexity version of the Jarzynski identity strengthened the evidence for the existence of a well-defined resource theory of uncomplexity and presented an extensive discussion on the second law of complexity. Furthermore, analogous to the thermodynamic fluctuation-dissipation theorem, we proposed a version of the fluctuation-dissipation theorem for the complexity. Although this study does not focus on holographic fluctuations, we found that the results are surprisingly suitable for capturing their information. The results obtained using nonequilibrium methods may contribute to understand the nature of the complexity and study the features of the holographic fluctuations.

MSC:

81T28 Thermal quantum field theory
80M60 Stochastic analysis in thermodynamics and heat transfer
81P68 Quantum computation
81T35 Correspondence, duality, holography (AdS/CFT, gauge/gravity, etc.)
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[1] J. A. Wheeler, Information, Physics, Quantum: The Search for Links, the proceedings of The 1988 Workshop on Complexity, Entropy, and the Physics of Information, Westview Press, Santa Fe, New Mexico, Boulder, CO, U.S.A. (1990).
[2] Maldacena, JM, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys., 38, 1113 (1999) · Zbl 0969.81047 · doi:10.1023/A:1026654312961
[3] Gubser, SS; Klebanov, IR; Polyakov, AM, Gauge theory correlators from noncritical string theory, Phys. Lett. B, 428, 105 (1998) · Zbl 1355.81126 · doi:10.1016/S0370-2693(98)00377-3
[4] Witten, E., Anti-de Sitter space and holography, Adv. Theor. Math. Phys., 2, 253 (1998) · Zbl 0914.53048 · doi:10.4310/ATMP.1998.v2.n2.a2
[5] Ryu, S.; Takayanagi, T., Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett., 96 (2006) · Zbl 1228.83110 · doi:10.1103/PhysRevLett.96.181602
[6] L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys.64 (2016) 24 [Addendum ibid.64 (2016) 44] [arXiv:1403.5695] [INSPIRE]. · Zbl 1429.81019
[7] Stanford, D.; Susskind, L., Complexity and Shock Wave Geometries, Phys. Rev. D, 90 (2014) · doi:10.1103/PhysRevD.90.126007
[8] Susskind, L., Entanglement is not enough, Fortsch. Phys., 64, 49 (2016) · Zbl 1429.81021 · doi:10.1002/prop.201500095
[9] Brown, AR; Roberts, DA; Susskind, L.; Swingle, B.; Zhao, Y., Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett., 116 (2016) · doi:10.1103/PhysRevLett.116.191301
[10] Brown, AR; Roberts, DA; Susskind, L.; Swingle, B.; Zhao, Y., Complexity, action, and black holes, Phys. Rev. D, 93 (2016) · doi:10.1103/PhysRevD.93.086006
[11] Chemissany, W.; Osborne, TJ, Holographic fluctuations and the principle of minimal complexity, JHEP, 12, 055 (2016) · Zbl 1390.83044 · doi:10.1007/JHEP12(2016)055
[12] M. A. Nielsen, A geometric approach to quantum circuit lower bounds, quant-ph/0502070. · Zbl 1152.81788
[13] M. A. Nielsen, M. R. Dowling, M. Gu and A. C. Doherty, Quantum Computation as Geometry, Science311 (2006) 1133. · Zbl 1226.81049
[14] Nielsen, MA; Dowling, MR; Gu, M.; Doherty, AC, Optimal control, geometry, and quantum computing, Phys. Rev. A, 73 (2006) · doi:10.1103/PhysRevA.73.062323
[15] M. R. Dowling and M. A. Nielsen, The geometry of quantum computation, quant-ph/0701004. · Zbl 1158.81313
[16] Brown, AR; Susskind, L., Complexity geometry of a single qubit, Phys. Rev. D, 100 (2019) · doi:10.1103/PhysRevD.100.046020
[17] Auzzi, R.; Baiguera, S.; De Luca, GB; Legramandi, A.; Naredelli, G.; Zenoni, N., Geometry of quantum complexity, Phys. Rev. D, 103 (2018) · doi:10.1103/PhysRevD.103.106021
[18] Brown, AR; Susskind, L., Second law of quantum complexity, Phys. Rev. D, 97 (2018) · doi:10.1103/PhysRevD.97.086015
[19] W. Sun and X.-H. Ge, Complexity growth rate, grand potential and partition function, arXiv:1912.00153 [INSPIRE].
[20] Ge, X-H; Wang, B., Quantum computational complexity, Einstein’s equations and accelerated expansion of the Universe, JCAP, 02, 047 (2018) · Zbl 1527.83137 · doi:10.1088/1475-7516/2018/02/047
[21] Caputa, P.; Magan, JM, Quantum Computation as Gravity, Phys. Rev. Lett., 122 (2019) · doi:10.1103/PhysRevLett.122.231302
[22] Jarzynski, C., Nonequilibrium Equality for Free Energy Differences, Phys. Rev. Lett., 78, 2690 (1997) · doi:10.1103/PhysRevLett.78.2690
[23] C. Jarzynski, Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach, Phys. Rev. E56 (1997) 5018 [cond-mat/9707325].
[24] C. Jarzynski, Microscopic analysis of Clausius-Duhem processes, J. Stat. Phys.96 (1999) 415 [cond-mat/9802249]. · Zbl 0964.82018
[25] G. E. Crooks, Nonequilibrium Measurements of Free Energy Differences for Microscopically Reversible Markovian Systems, J. Stat. Phys.90 (1998) 1481. · Zbl 0946.82029
[26] G. E. Crooks, Path-ensemble averages in systems driven far from equilibrium, Phys. Rev. E61 (2000) 2361 [cond-mat/9908420].
[27] G. Hummer and A. Szabo, Free energy reconstruction from nonequilibrium single-molecule pulling experiments, Proc. Nat. Acad. Sci.98 (2001) 3658.
[28] Van den Broeck, C.; Esposito, M., Ensemble and Trajectory Thermodynamics: A Brief Introduction, Physica A, 418, 6 (2015) · doi:10.1016/j.physa.2014.04.035
[29] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path integrals, Higher Education Press, Beijing, P.R. China (2015). · Zbl 0176.54902
[30] M. Kac, On distributions of certain Wiener functionals, Trans. Amer. Math. Soc.65 (1949) 1. · Zbl 0032.03501
[31] B. Øksendal, Stochastic differential equations, Springer, Heidelberg, Germany (2000).
[32] Feynman, RP, Space-time approach to nonrelativistic quantum mechanics, Rev. Mod. Phys., 20, 367 (1948) · Zbl 1371.81126 · doi:10.1103/RevModPhys.20.367
[33] Minic, D.; Pleimling, M., The Jarzynski Identity and the AdS/CFT Duality, Phys. Lett. B, 700, 277 (2011) · doi:10.1016/j.physletb.2011.05.021
[34] Yunger Halpern, N., Jarzynski-like equality for the out-of-time-ordered correlator, Phys. Rev. A, 95 (2017) · doi:10.1103/PhysRevA.95.012120
[35] N. Y. Halpern, A. J. P. Garner, O. C. O. Dahlsten and V. Vedral, Maximum one-shot dissipated work form Rényi divergences, Phys. Rev. E97 (2018) 052135.
[36] S. S. Chern, W. H. Chen and K. S. Lam, Lectures on Differential Geometry, World Scientific, New Jersey, U.S.A. (1999) [DOI]. · Zbl 0940.53001
[37] Maldacena, J.; Stanford, D., Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D, 94 (2016) · doi:10.1103/PhysRevD.94.106002
[38] Maldacena, J.; Stanford, D.; Yang, Z., Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP, 2016, 12C104 (2016) · Zbl 1361.81112
[39] Witten, E., An SYK-Like Model Without Disorder, J. Phys. A, 52 (2019) · Zbl 1509.81564 · doi:10.1088/1751-8121/ab3752
[40] Gu, Y.; Qi, X-L; Stanford, D., Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, JHEP, 05, 125 (2017) · Zbl 1380.81325 · doi:10.1007/JHEP05(2017)125
[41] Cottrell, W.; Freivogel, B.; Hofman, DM; Lokhande, SF, How to Build the Thermofield Double State, JHEP, 02, 058 (2019) · Zbl 1411.81174 · doi:10.1007/JHEP02(2019)058
[42] Chapman, S., Complexity and entanglement for thermofield double states, SciPost Phys., 6, 034 (2019) · doi:10.21468/SciPostPhys.6.3.034
[43] Spengler, C.; Huber, M.; Hiesmayr, BC, Composite parameterization and Haar measure for all unitary and special unitary groups, J. Math. Phys., 53 (2012) · Zbl 1273.28012 · doi:10.1063/1.3672064
[44] Brown, AR; Susskind, L.; Zhao, Y., Quantum Complexity and Negative Curvature, Phys. Rev. D, 95 (2017) · doi:10.1103/PhysRevD.95.045010
[45] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer New York, NY, U.S.A. (1989) [DOI].
[46] Lashkari, N.; Stanford, D.; Hastings, M.; Osborne, T.; Hayden, P., Towards the Fast Scrambling Conjecture, JHEP, 04, 022 (2013) · Zbl 1342.81374 · doi:10.1007/JHEP04(2013)022
[47] C. G. Gray, Principle of least action, Scholarpedia4 (2009) 8291.
[48] R. Zwanzig, Nonequilibrium statistical mechanics, Oxford University Press, New York, U.S.A. (2000). · Zbl 1267.82001
[49] K. Jacobs, Stochastic Processes for Physicists: Understanding Noisy System, Cambridge University Press, Cambridge, U.K. (2010) [DOI]. · Zbl 1206.82001
[50] Z. Schuss, Theory and Applications of Stochastic Processes: An Analytical Approach, Springer, Heidelberg, Germany (2010) [DOI]. · Zbl 1202.60005
[51] B. E. Baaquie, Path Integrals and Hamiltonians: Principles and Methods, Cambridge University Press, Cambridge, U.K. (2014). · Zbl 1329.81001
[52] D. J. Toms, The Schwinger action principle and the Feynman path integral for quantum mechanics in curved space, hep-th/0411233 [INSPIRE].
[53] Van Vleck, JH, The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics, Proc. Nat. Acad. Sci., 14, 178 (1928) · JFM 54.0976.01 · doi:10.1073/pnas.14.2.178
[54] Morette, C., On the definition and approximation of Feynman’s path integrals, Phys. Rev., 81, 848 (1951) · Zbl 0042.45506 · doi:10.1103/PhysRev.81.848
[55] B. S. DeWitt, Dynamical Theory of Groups and Fields, Gordan and Breach, New York, U.S.A. (1964). · Zbl 0148.46102
[56] H. S. Ruse, Taylor’s Theorem in the Tensor Calculus, Proc. Lond. Math. Soc.32 (1931) 87. · JFM 57.0980.01
[57] R. Livi and P. Politi, Nonequilibrium Statistical Physics: A Modern Perspective, Cambridge University Press, Cambridge, U.K. (2017) [DOI]. · Zbl 1451.82001
[58] R. P. Feynman, Statistical Mechanics: A Set of Lectures, CRC Press, Los Angeles, U.S.A. (2017). · Zbl 0997.82500
[59] Bernamonti, A.; Galli, F.; Hernandez, J.; Myers, RC; Ruan, S-M; Simón, J., First Law of Holographic Complexity, Phys. Rev. Lett., 123 (2019) · doi:10.1103/PhysRevLett.123.081601
[60] Bernamonti, A.; Galli, F.; Hernandez, J.; Myers, RC; Ruan, S-M; Simón, J., Aspects of The First Law of Complexity, J. Phys. A, 53, 29 (2020) · Zbl 1518.81021 · doi:10.1088/1751-8121/ab8e66
[61] Gour, G.; Müller, MP; Narasimhachar, V.; Spekkens, RW; Halpern, NY, The resource theory of informational nonequilibrium in thermodynamics, Phys. Rept., 583, 1 (2015) · Zbl 1357.81040 · doi:10.1016/j.physrep.2015.04.003
[62] F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes and R. W. Spekkens, Resource Theory of Quantum States out of Thermal Equilibrium, Phys. Rev. Lett.111 (2013) 250404.
[63] V. Veitch, S. A. H. Mousavian, D. Gottesman and J. Emerson, The resource theory of stablizer computation, New J. Phys.16 (2014) 013009. · Zbl 1451.81184
[64] E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys.91 (2019) 025001.
[65] M. Horodecki, P. Horodecki and J. Oppenheim, Reversible transformations from pure to mixed states and the unique measure of information, Phys. Rev. A67 (2003) 062104. · Zbl 1267.81098
[66] M. Horodecki et al., Local Information as a Resource in Distributed Quantum Systems, Phys. Rev. Lett.90 (2003) 100402. · Zbl 1267.81088
[67] L. Susskind, Three Lectures on Complexity and Black Holes, SpringerBriefs in Physics, Springer, Cham, Germany (2018) [DOI] [arXiv:1810.11563] [INSPIRE]. · Zbl 1435.83004
[68] N. Y. Halpern, N. B. T. Kothakonda, J. Haferkamp, A. Munson, J. Eisert and P. Faist, Resource theory of quantum uncomplexity, arXiv:2110.11371 [INSPIRE].
[69] L. Susskind and Y. Zhao, Switchbacks and the Bridge to Nowhere, arXiv:1408.2823 [INSPIRE].
[70] S. Vinjanampathy and J. Anders, Quantum Thermodynamics, Contemp. Phys.57 (2016) 545 [arXiv:1508.06099].
[71] C. Jarzynski, Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale, Ann. Rev. Condens. Matter Phys.2 (2011) 329.
[72] Callen, HB; Welton, TA, Irreversibility and generalized noise, Phys. Rev., 83, 34 (1951) · Zbl 0044.41201 · doi:10.1103/PhysRev.83.34
[73] J. Hermans, Simple Analysis of Noise and Hysteresis in (Slow-Growth) Free Energy Simulations, J. Phys. Chem.95 (1991) 9029.
[74] Camilo, G.; Teixeira, D., Complexity and Floquet dynamics: Nonequilibrium Ising phase transitions, Phys. Rev. B, 102 (2020) · doi:10.1103/PhysRevB.102.174304
[75] Lieb, EH; Schultz, T.; Mattis, D., Two soluble models of an antiferromagnetic chain, Annals Phys., 16, 407 (1961) · Zbl 0129.46401 · doi:10.1016/0003-4916(61)90115-4
[76] E. Barouch, B. M. McCoy and M. Dresden, Statistical Mechanics of the XY Model. I, Phys. Rev. A2 (1970) 1075.
[77] M. S. Kalyan, G. A. Prasad, V. S. S. Sastry and K. P. N. Murthy, A Note on Non-equilibrium Work Fluctuations and Equilibrium Free Energies, J. Phys. A390 (2011) 1240 [arXiv:1011.4413].
[78] M. Esposito and C. V. den Broeck, Three detailed fluctuation theorems, Phys. Rev. Lett.104 (2010) 090601.
[79] Jefferson, R.; Myers, RC, Circuit complexity in quantum field theory, JHEP, 10, 107 (2017) · Zbl 1383.81233 · doi:10.1007/JHEP10(2017)107
[80] Yang, R-Q; An, Y-S; Niu, C.; Zhang, C-Y; Kim, K-Y, What kind of “complexity” is dual to holographic complexity?, Eur. Phys. J. C, 82, 262 (2022) · doi:10.1140/epjc/s10052-022-10151-0
[81] L. Bassman, K. Klymko, N. M. Tubman and W. A. de Jong, Computing Free Energies with Fluctuation Relations on Quantum Computers, arXiv:2103.09846.
[82] G. S. Chirikjian, Stochastic Models, Information theory, and Lie Groups, Springer, Heidelberg, Germany (2000).
[83] Schwinger, JS, The Theory of quantized fields. 1, Phys. Rev., 82, 914 (1951) · Zbl 0043.42202 · doi:10.1103/PhysRev.82.914
[84] Schwinger, JS, The Theory of quantized fields. 2, Phys. Rev., 91, 713 (1953) · Zbl 0057.43401 · doi:10.1103/PhysRev.91.713
[85] Qiuping A. Wang, Maximum path information and the principle of least action for chaotic system, Chaos Solitons Fractals23 (2005) 1253 [cond-mat/0405373]. · Zbl 1086.37518
[86] https://brilliant.org/wiki/ergodic-markov-chains/.
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