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Quantum Logspace Algorithm for Powering Matrices with Bounded Norm

Authors Uma Girish, Ran Raz, Wei Zhan

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Author Details

Uma Girish
  • Department of Computer Science, Princeton University, NJ, USA
Ran Raz
  • Department of Computer Science, Princeton University, NJ, USA
Wei Zhan
  • Department of Computer Science, Princeton University, NJ, USA

Funding

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.

Acknowledgements

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.

Cite As Get BibTex

Uma Girish, Ran Raz, and Wei Zhan. Quantum Logspace Algorithm for Powering Matrices with Bounded Norm. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 73:1-73:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.73

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • BQL
  • Matrix Powering
  • Quantum Circuit
  • Reversible Computation

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