Abstract
The “quantum complexity” of a unitary operator measures the difficulty of its construction from a set of elementary quantum gates. While the notion of quantum complexity was first introduced as a quantum generalization of the classical computational complexity, it has since been argued to hold a fundamental significance in its own right, as a physical quantity analogous to the thermodynamic entropy. In this paper, we present a unified perspective on various notions of quantum complexity, viewed as functions on the space of unitary operators. One striking feature of these functions is that they can exhibit non-smooth and even fractal behaviour. We use ideas from Diophantine approximation theory and sub-Riemannian geometry to rigorously quantify this lack of smoothness. Implications for the physical meaning of quantum complexity are discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M.A. Nielsen and I. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge U.K. (2002).
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 and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018) 086015 [arXiv:1701.01107] [INSPIRE].
L. Susskind, Three Lectures on Complexity and Black Holes, in SpringerBriefs in Physics, Springer, Cham Switzerland (2018) [arXiv:1810.11563] [INSPIRE].
L. Susskind, Black Holes at Exp-time, arXiv:2006.01280 [INSPIRE].
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 and L. Susskind, Complexity geometry of a single qubit, Phys. Rev. D 100 (2019) 046020 [arXiv:1903.12621] [INSPIRE].
M.A. Nielsen, A geometric approach to quantum circuit lower bounds, quant-ph/0502070.
N. Khaneja, S.J. Glaser and R. Brockett, Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer, Phys. Rev. A 65 (2002) 032301 [Erratum ibid. A 68 (2003) 049903] [Erratum ibid. A 71 (2005) 039906].
E. Le Donne, Lecture notes on sub-Riemannian geometry, (2010) and online at https://sites.google.com/site/enricoledonne/.
M. Gromov, Carnot-Carathéodory spaces seen from within, in Sub-Riemannian Geometry, A. Bellaïche and J.-J. Risler eds., Birkhäuser Basel, Basel Switzerland (1996), pp. 79–323.
S. Lloyd, Almost Any Quantum Logic Gate is Universal, Phys. Rev. Lett. 75 (1995) 346 [INSPIRE].
M.A. Nielsen, M.R. Dowling, M. Gu and A.C. Doherty, Quantum Computation as Geometry, Science 311 (2006) 1133.
H.W. Lin, Cayley graphs and complexity geometry, JHEP 02 (2019) 063 [arXiv:1808.06620] [INSPIRE].
C.M. Dawson and M.A. Nielsen, The Solovay-Kitaev algorithm, Quantum Info. Comput. 6 (2006) 81 [quant-ph/0505030].
A.W. Harrow, B. Recht and I.L. Chuang, Efficient discrete approximations of quantum gates, J. Math. Phys. 43 (2002) 4445.
A. Gamburd, D. Jakobson and P. Sarnak, Spectra of elements in the group ring of SU(2), J. Eur. Math. Soc. 1 (1999) 51.
J. Bourgain and A. Gamburd, On the spectral gap for finitely-generated subgroups of SU(2), Invent. Math. 171 (2008) 83.
J. Bourgain and A. Gamburd, A Spectral Gap Theorem in SU(d), J. Eur. Math. Soc. 14 (2012) 1455.
W.M. Schmidt, Diophantine Approximation, in Lect. Notes Math. 785, Springer-Verlag (2009).
J. Pöschel, A lecture on the classical KAM theorem, in Proceedings of Symposia in Pure Mathematics 69, American Mathematical Society, Providence RI U.S.A. (2001), pp. 707–732 [arXiv:0908.2234].
S. Aubry and G. André, Analyticity breaking and Anderson localization in incommensurate lattices, in Group theoretical methods in physics, proceedings of the VIII International Colloquium on Group-theoretical Methods in Physics, Kiryat Anavim, Israel, 25–29 March 1979, Annals of the Israel Physical Society 3, A. Hilger and Israel Physical Society (1980), pp. 133–164.
M. Takahashi, Thermodynamics of one-dimensional solvable models, Cambridge University Press, Cambridge U.K. (2005).
O. Parzanchevski and P. Sarnak, Super-Golden-Gates for PU(2), Adv. Math. 327 (2018) 869.
E. Breuillard and T. Gelander, On dense free subgroups of Lie groups, J. Algebra 261 (2003) 448.
S. Świerczkowski, A class of free rotation groups, Indagat. Math. 5 (1994) 221.
E. Breuillard and A. Lubotzky, Expansion in simple groups, arXiv:1807.03879.
T. Prosen, Open XXZ Spin Chain: Nonequilibrium Steady State and a Strict Bound on Ballistic Transport, Phys. Rev. Lett. 106 (2011) 217206.
J. Bausch and S. Piddock, The complexity of translationally invariant low-dimensional spin lattices in 3D editors-pick, J. Math. Phys. 58 (2017) 111901.
E. Breuillard and T. Gelander, Uniform independence in linear groups, Invent. Math. 173 (2008) 225.
K. Shizume, T. Nakajima, R. Nakayama and Y. Takahashi, Quantum computational Riemannian and sub-Riemannian geodesics, Prog. Theor. Phys. 127 (2012) 997 [INSPIRE] and online pdf version at https://academic.oup.com/ptp/article-pdf/127/6/997/5437884/ 127-6-997.pdf.
T. Frankel, The geometry of physics: an introduction, Cambridge University Press, Cambridge U.K. (2011).
A. Bellaïche, The tangent space in sub-Riemannian geometry, in Sub-Riemannian Geometry, A. Bellaïche and J.-J. Risler eds., Birkhäuser Basel, Basel Switzerland (1996), pp. 1–78.
M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. IHÉS 53 (1981) 53.
P. Pansu, Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergod. Theory Dyn. Syst. 3 (1983) 415.
J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972) 250.
M.R. Dowling and M.A. Nielsen, The Geometry of Quantum Computation, Quant. Inf. Comput. 8 (2008) 861 [quant-ph/0701004]
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2106.08324
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Bulchandani, V.B., Sondhi, S.L. How smooth is quantum complexity?. J. High Energ. Phys. 2021, 230 (2021). https://doi.org/10.1007/JHEP10(2021)230
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2021)230