Abstract
It is conjectured that in the geometric formulation of quantum computing, one can study quantum complexity through classical entropy of statistical ensembles established non-relativistically in the group manifold of unitary operators. The kinetic and positional decompositions of statistical entropy are conjectured to correspond to the Kolmogorov complexity and computational complexity, respectively, of corresponding quantum circuits. In this paper, we claim that by applying the virial theorem to the group manifold, one can derive a generic relation between Kolmogorov complexity and computational complexity in the thermal equilibrium.
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References
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].
A.R. Brown, L. Susskind and Y. Zhao, Quantum Complexity and Negative Curvature, Phys. Rev. D 95 (2017) 045010 [arXiv:1608.02612] [INSPIRE].
S. Chapman, H. Marrochio and R.C. Myers, Complexity of Formation in Holography, JHEP 01 (2017) 062 [arXiv:1610.08063] [INSPIRE].
D. Carmi, R.C. Myers and P. Rath, Comments on Holographic Complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].
R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].
D. Carmi, S. Chapman, H. Marrochio, R.C. Myers and S. Sugishita, On the Time Dependence of Holographic Complexity, JHEP 11 (2017) 188 [arXiv:1709.10184] [INSPIRE].
Z. Fu, A. Maloney, D. Marolf, H. Maxfield and Z. Wang, Holographic complexity is nonlocal, JHEP 02 (2018) 072 [arXiv:1801.01137] [INSPIRE].
A. Bernamonti, F. Galli, R.C. Myers and J. Oppenheim, Holographic second laws of black hole thermodynamics, JHEP 07 (2018) 111 [arXiv:1803.03633] [INSPIRE].
L. Hackl and R.C. Myers, Circuit complexity for free fermions, JHEP 07 (2018) 139 [arXiv:1803.10638] [INSPIRE].
S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part I, JHEP 06 (2018) 046 [arXiv:1804.07410] [INSPIRE].
S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part II, JHEP 06 (2018) 114 [arXiv:1805.07262] [INSPIRE].
M. Nielsen, A geometric approach to quantum circuit lower bounds, quant-ph/0502070.
M. Dowling and M. Nielsen, The geometry of quantum computation, quant-ph/0701004.
A.R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018) 086015 [arXiv:1701.01107] [INSPIRE].
L. Susskind, Dear Qubitzers, GR=QM, arXiv:1708.03040 [INSPIRE].
L. Susskind, Black Holes and Complexity Classes, arXiv:1802.02175 [INSPIRE].
S. Chandrasekhar, The Post-Newtonian Equations of Hydrodynamics in General Relativity., Astrophys. J. 142 (1965) 1488 [INSPIRE].
S. Bonazzola, The Virial Theorem in General Relativity, Astrophys. J. 182 (1973) 335.
J. Katz and A. Ori, Localisation of field energy, Class. Quant. Grav. 7 (1990) 787.
E. Gourgoulhon and S. Bonazzola, A formulation of the virial theorem in general relativity, Class. Quant. Grav. 11 (1994) 443.
S. Bonazzola and E. Gourgoulhon, Noncircular axisymmetric stationary spacetimes, Phys. Rev. D 48 (1993) 2635.
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, University of California, Santa Barbara, U.S.A., 7 April 2015.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, University of California, Santa Barbara, U.S.A., 27 May 2015.
A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, talk at KITP, University of California, Santa Barbara, U.S.A., 12 February 2015.
A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, talk given at The Fundamental Physics Prize Symposium, 10 November 2014.
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016)106002 [arXiv:1604.07818] [INSPIRE].
D.A. Roberts and B. Yoshida, Chaos and complexity by design, JHEP 04 (2017) 121 [arXiv:1610.04903] [INSPIRE].
J. Cotler, N. Hunter-Jones, J. Liu and B. Yoshida, Chaos, Complexity and Random Matrices, JHEP 11 (2017) 048 [arXiv:1706.05400] [INSPIRE].
N. Hunter-Jones and J. Liu, Chaos and random matrices in supersymmetric SYK, JHEP 05 (2018)202 [arXiv:1710.08184] [INSPIRE].
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Bao, N., Liu, J. Quantum complexity and the virial theorem. J. High Energ. Phys. 2018, 144 (2018). https://doi.org/10.1007/JHEP08(2018)144
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DOI: https://doi.org/10.1007/JHEP08(2018)144