Quantum Logspace Algorithm for Powering Matrices with Bounded Norm
We give a quantum logspace algorithm for powering contraction matrices, that is, matrices with spectral norm at most 1. The algorithm gets as an input an arbitrary n× n contraction matrix A, and a parameter T ≤ poly(n) and outputs the entries of A^T, up to (arbitrary) polynomially small additive error. The algorithm applies only unitary operators, without intermediate measurements. We show various implications and applications of this result:
First, we use this algorithm to show that the class of quantum logspace algorithms with only quantum memory and with intermediate measurements is equivalent to the class of quantum logspace algorithms with only quantum memory without intermediate measurements. This shows that the deferred-measurement principle, a fundamental principle of quantum computing, applies also for quantum logspace algorithms (without classical memory). More generally, we give a quantum algorithm with space O(S + log T) that takes as an input the description of a quantum algorithm with quantum space S and time T, with intermediate measurements (without classical memory), and simulates it unitarily with polynomially small error, without intermediate measurements.
Since unitary transformations are reversible (while measurements are irreversible) an interesting aspect of this result is that it shows that any quantum logspace algorithm (without classical memory) can be simulated by a reversible quantum logspace algorithm. This proves a quantum analogue of the result of Lange, McKenzie and Tapp that deterministic logspace is equal to reversible logspace [Lange et al., 2000].
Finally, we use our results to show non-trivial classical simulations of quantum logspace learning algorithms.
BQL
Matrix Powering
Quantum Circuit
Reversible Computation
Theory of computation~Quantum complexity theory
73:1-73:20
Track A: Algorithms, Complexity and Games
Research supported by the Simons Collaboration on Algorithms and Geometry, by a Simons Investigator Award and by the National Science Foundation grant No. CCF-1714779.
https://eccc.weizmann.ac.il/report/2020/087/
We would like to thank Dieter van Melkebeek and Subhayan Roy Moulik for very helpful suggestions and comments on a previous version of this work. We also thank the anonymous reviewers for their thorough feedback.
Uma
Girish
Uma Girish
Department of Computer Science, Princeton University, NJ, USA
Ran
Raz
Ran Raz
Department of Computer Science, Princeton University, NJ, USA
Wei
Zhan
Wei Zhan
Department of Computer Science, Princeton University, NJ, USA
10.4230/LIPIcs.ICALP.2021.73
Paul Beame, Shayan Oveis Gharan, and Xin Yang. Time-space tradeoffs for learning finite functions from random evaluations, with applications to polynomials. In Conference On Learning Theory (COLT 2018), pages 843-856, 2018.
Charles Bennett. Notes on landauer’s principle, reversible computation, and maxwell’s demon. Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 34:501-510, September 2003. URL: https://doi.org/10.1016/S1355-2198(03)00039-X.
https://doi.org/10.1016/S1355-2198(03)00039-X
Jin-Yi Cai, Venkatesan T Chakaravarthy, and Dieter van Melkebeek. Time-space tradeoff in derandomizing probabilistic logspace. Theory of Computing Systems, 39(1):189-208, 2006.
Shantanav Chakraborty, András Gilyén, and Stacey Jeffery. The power of block-encoded matrix powers: Improved regression techniques via faster hamiltonian simulation. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2019.
Bill Fefferman, Hirotada Kobayashi, Cedric Yen-Yu Lin, Tomoyuki Morimae, and Harumichi Nishimura. Space-efficient error reduction for unitary quantum computations. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016), volume 55. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.14.
https://doi.org/10.4230/LIPIcs.ICALP.2016.14
Bill Fefferman and Cedric Yen-Yu Lin. A complete characterization of unitary quantum space. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018.
Bill Fefferman and Zachary Remscrim. Eliminating intermediate measurements in space-bounded quantum computation, 2020. URL: http://arxiv.org/abs/2006.03530.
http://arxiv.org/abs/2006.03530
Sumegha Garg, Ran Raz, and Avishay Tal. Extractor-based time-space lower bounds for learning. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2018), pages 990-1002, 2018.
András Gilyén. Quantum singular value transformation & its algorithmic applications. ILLC Dissertation Series, 2019.
András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 193-204, 2019.
Ofer Grossman and Yang P Liu. Reproducibility and pseudo-determinism in log-space. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 606-620. SIAM, 2019.
Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. Physical review letters, 103(15):150502, 2009.
Gillat Kol, Ran Raz, and Avishay Tal. Time-space hardness of learning sparse parities. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2017), pages 1067-1080, 2017.
R. Landauer. Irreversibility and heat generation in the computing process. IBM Journal of Research and Development, 5(3):183-191, 1961.
Klaus-Jörn Lange, Pierre McKenzie, and Alain Tapp. Reversible space equals deterministic space. Journal of Computer and System Sciences, 60(2):354-367, 2000.
Chris Marriott and John Watrous. Quantum arthur-merlin games. Computational Complexity, 14(2):122-152, 2005.
Dieter van Melkebeek and Thomas Watson. Time-space efficient simulations of quantum computations. Theory of Computing, 8(1):1-51, 2012.
Dana Moshkovitz and Michal Moshkovitz. Entropy samplers and strong generic lower bounds for space bounded learning. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018.
Michael A Nielsen and Isaac L Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2010.
David Perez-Garcia, Michael M Wolf, Denes Petz, and Mary Beth Ruskai. Contractivity of positive and trace-preserving maps under l p norms. Journal of Mathematical Physics, 47(8):083506, 2006.
Ran Raz. A time-space lower bound for a large class of learning problems. In 58th IEEE Annual Symposium on Foundations of Computer Science (FOCS 2017), pages 732-742, 2017.
Ran Raz. Fast learning requires good memory: A time-space lower bound for parity learning. Journal of the ACM (JACM), 66(1):1-18, 2018.
Michael Saks and Shiyu Zhou. BPHSPACE(s) ⊆ DSPACE(s^3/2). Journal of computer and system sciences, 58(2):376-403, 1999.
Ohad Shamir. Fundamental limits of online and distributed algorithms for statistical learning and estimation. In Advances in Neural Information Processing Systems 27 (NIPS 2014), pages 163-171, 2014.
Yaoyun Shi. Both toffoli and controlled-not need little help to do universal quantum computing. Quantum Info. Comput., 3(1):84–92, 2003.
Jacob Steinhardt, Gregory Valiant, and Stefan Wager. Memory, communication, and statistical queries. In Conference on Learning Theory (COLT 2016), pages 1490-1516, 2016.
Amnon Ta-Shma. Inverting well conditioned matrices in quantum logspace. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC 2013), pages 881-890, 2013.
John Watrous. Space-bounded quantum complexity. Journal of Computer and System Sciences, 59(2):281-326, 1999.
Fuzhen Zhang. Matrix theory: basic results and techniques. Springer Science & Business Media, 2011.
Uma Girish, Ran Raz, and Wei Zhan
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