research-article

Public Access

Eliminating intermediate measurements in space-bounded Quantum computation

Authors:

Bill Fefferman,

Zachary RemscrimAuthors Info & Claims

STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

Pages 1343 - 1356

https://doi.org/10.1145/3406325.3451051

Published: 15 June 2021 Publication History

Abstract

A foundational result in the theory of quantum computation, known as the "principle of safe storage," shows that it is always possible to take a quantum circuit and produce an equivalent circuit that makes all measurements at the end of the computation. While this procedure is time efficient, meaning that it does not introduce a large overhead in the number of gates, it uses extra ancillary qubits, and so is not generally space efficient. It is quite natural to ask whether it is possible to eliminate intermediate measurements without increasing the number of ancillary qubits. We give an affirmative answer to this question by exhibiting a procedure to eliminate all intermediate measurements that is simultaneously space efficient and time efficient. In particular, this shows that the definition of a space-bounded quantum complexity class is robust to allowing or forbidding intermediate measurements. A key component of our approach, which may be of independent interest, involves showing that the well-conditioned versions of many standard linear-algebraic problems may be solved by a quantum computer in less space than seems possible by a classical computer.

References

[1]

Scott Aaronson. 2009. ON PERFECT COMPLETENESS FOR QMA. Quantum Information and Computation, 9, 1, 2009. Pages 0081–0089.

[2]

Dorit Aharonov, Alexei Kitaev, and Noam Nisan. 1998. Quantum circuits with mixed states. In Proceedings of the thirtieth annual ACM symposium on Theory of computing. Pages 20–30.

Digital Library

[3]

Dorit Aharonov and Tomer Naveh. 2002. Quantum NP-a survey. arXiv preprint quant-ph/0210077, 2002.

[4]

Eric Allender and Mitsunori Ogihara. 1996. Relationships Among PL, \#L, and the Determinant. RAIRO-Theoretical Informatics and Applications, 30, 1, 1996. Pages 1–21.

[5]

Charles H Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani. 1997. Strengths and weaknesses of quantum computing. SIAM journal on Computing, 26, 5, 1997. Pages 1510–1523.

Digital Library

[6]

Stuart J Berkowitz. 1984. On computing the determinant in small parallel time using a small number of processors. Information processing letters, 18, 3, 1984. Pages 147–150.

[7]

Dominic W Berry, Andrew M Childs, Richard Cleve, Robin Kothari, and Rolando D Somma. 2014. Exponential improvement in precision for simulating sparse Hamiltonians. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing. Pages 283–292.

Digital Library

[8]

Enric Boix-Adser\`a, Lior Eldar, and Saeed Mehraban. 2019. Approximating the Determinant of Well-Conditioned Matrices by Shallow Circuits. arXiv preprint arXiv:1912.03824, 2019.

[9]

Sergey Bravyi. 2011. Efficient algorithm for a quantum analogue of 2-SAT. Contemp. Math., 536, 2011. Pages 33–48.

[10]

Harry Buhrman, John Tromp, and Paul Vitányi. 2001. Time and space bounds for reversible simulation. In International Colloquium on Automata, Languages, and Programming. Pages 1017–1027.

[11]

Andrew M Childs. 2010. On the relationship between continuous-and discrete-time quantum walk. Communications in Mathematical Physics, 294, 2, 2010. Pages 581–603.

[12]

Stephen A Cook. 1985. A taxonomy of problems with fast parallel algorithms. Information and Control, 64, 1-3, 1985. Pages 2–22.

Digital Library

[13]

C Damm. 1991. DET=L(\#L). Fachbereich Informatik der Humboldt-Universitat zu, Berlin, 1991.

[14]

David P DiVincenzo. 2000. The physical implementation of quantum computation. Fortschritte der Physik: Progress of Physics, 48, 9-11, 2000. Pages 771–783.

[15]

Dean Doron, Amir Sarid, and Amnon Ta-Shma. 2017. On approximating the eigenvalues of stochastic matrices in probabilistic logspace. computational complexity, 26, 2, 2017. Pages 393–420.

[16]

Dean Doron and Amnon Ta-Shma. 2015. On the de-randomization of space-bounded approximate counting problems. Inform. Process. Lett., 115, 10, 2015. Pages 750–753.

Digital Library

[17]

Bill Fefferman, Hirotada Kobayashi, Cedric Yen-Yu Lin, Tomoyuki Morimae, and Harumichi Nishimura. 2016. Space-Efficient Error Reduction for Unitary Quantum Computations. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016).

[18]

Bill Fefferman and Cedric Yen-Yu Lin. 2018. A Complete Characterization of Unitary Quantum Space. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018).

[19]

Bill Fefferman and Zachary Remscrim. 2020. Eliminating intermediate measurements in space-bounded quantum computation. arXiv preprint arXiv:2006.03530, 2020.

[20]

Richard P Feynman. 1999. Simulating physics with computers. Int. J. Theor. Phys, 21, 6/7, 1999.

[21]

Uma Girish, Ran Raz, and Wei Zhan. 2020. Quantum logspace algorithm for powering matrices with bounded norm. arXiv preprint arXiv:2006.04880, 2020.

[22]

Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. 2009. Quantum algorithm for linear systems of equations. Physical review letters, 103, 15, 2009. Pages 150502.

[23]

Mark Jerrum, Alistair Sinclair, and Eric Vigoda. 2004. A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. Journal of the ACM (JACM), 51, 4, 2004. Pages 671–697.

Digital Library

[24]

Stephen P Jordan, Hirotada Kobayashi, Daniel Nagaj, and Harumichi Nishimura. 2012. Achieving perfect completeness in classical-witness quantum Merlin-Arthur proof systems. Quantum Information & Computation, 12, 5-6, 2012. Pages 461–471.

[25]

Richard Jozsa, Barbara Kraus, Akimasa Miyake, and John Watrous. 2010. Matchgate and space-bounded quantum computations are equivalent. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466, 2115, 2010. Pages 809–830.

[26]

Aleksei Yur'evich Kitaev. 1997. Quantum computations: algorithms and error correction. Uspekhi Matematicheskikh Nauk, 52, 6, 1997. Pages 53–112.

[27]

Hirotada Kobayashi, Keiji Matsumoto, and Tomoyuki Yamakami. 2003. Quantum Merlin-Arthur proof systems: Are multiple Merlins more helpful to Arthur? In International Symposium on Algorithms and Computation. Pages 189–198.

[28]

Rolf Landauer. 1961. Irreversibility and heat generation in the computing process. IBM journal of research and development, 5, 3, 1961. Pages 183–191.

[29]

Klaus-Jörn Lange, Pierre McKenzie, and Alain Tapp. 2000. Reversible space equals deterministic space. J. Comput. System Sci., 60, 2, 2000. Pages 354–367.

[30]

Seth Lloyd. 1996. Universal quantum simulators. Science, 1996. Pages 1073–1078.

[31]

Meena Mahajan and V Vinay. 1997. Determinant: Combinatorics, Algorithms, and Complexity. Chicago Journal of Theoretical Computer Science, 1997.

Digital Library

[32]

Chris Marriott and John Watrous. 2005. Quantum Arthur–Merlin games. Computational Complexity, 14, 2, 2005. Pages 122–152.

Digital Library

[33]

Dieter van Melkebeek and Thomas Watson. 2012. Time-space efficient simulations of quantum computations. Theory of Computing, 8, 1, 2012. Pages 1–51.

[34]

Daniel Nagaj, Pawel Wocjan, and Yong Zhang. 2009. Fast amplification of QMA. Quantum Information & Computation, 9, 11, 2009. Pages 1053–1068.

[35]

Michael A Nielsen and Isaac Chuang. 2002. Quantum computation and quantum information.

[36]

Noam Nisan. 1992. Pseudorandom generators for space-bounded computation. Combinatorica, 12, 4, 1992. Pages 449–461.

[37]

Simon Perdrix and Philippe Jorrand. 2006. Classically controlled quantum computation. Mathematical Structures in Computer Science, 16, 4, 2006. Pages 601–620.

Digital Library

[38]

Zachary Remscrim. 2020. The Power of a Single Qubit: Two-Way Quantum Finite Automata and the Word Problem. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs). 168, Pages 139:1–139:18.

[39]

Zachary Remscrim. 2021. Lower Bounds on the Running Time of Two-Way Quantum Finite Automata and Sublogarithmic-Space Quantum Turing Machines. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021).

[40]

Wojciech Roga, Zbigniew Puchala, Lukasz Rudnicki, and Karol Zyczkowski. 2013. Entropic trade-off relations for quantum operations. Physical Review A, 87, 3, 2013. Pages 032308.

[41]

Michael Saks. 1996. Randomization and derandomization in space-bounded computation. In Proceedings of Computational Complexity (Formerly Structure in Complexity Theory). Pages 128–149.

[42]

Michael Saks and Shiyu Zhou. 1999. BPhSPACE(s) in DSPACE(s\^3/2). Journal of computer and system sciences, 58, 2, 1999. Pages 376–403.

Digital Library

[43]

Miklos Santha and Sovanna Tan. 1998. Verifying the determinant in parallel. Computational Complexity, 7, 2, 1998. Pages 128–151.

Digital Library

[44]

Peter W Shor. 1994. Algorithms for quantum computation: Discrete logarithms and factoring. In Proceedings 35th annual symposium on foundations of computer science. Pages 124–134.

Digital Library

[45]

Peter W Shor and Stephen P Jordan. 2008. Estimating Jones polynomials is a complete problem for one clean qubit. Quantum Information & Computation, 8, 8, 2008. Pages 681–714.

[46]

Amnon Ta-Shma. 2013. Inverting well conditioned matrices in quantum logspace. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing. Pages 881–890.

Digital Library

[47]

Seinosuke Toda. 1991. Counting problems computationally equivalent to the determinant.

[48]

Leslie G Valiant. 1992. Why is Boolean complexity theory difficult. Boolean Function Complexity, 169, 1992. Pages 84–94.

[49]

V Vinay. 1991. Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits. In Proceedings of the Sixth Annual Structure in Complexity Theory Conference. Pages 270–284.

[50]

John Watrous. 1999. Space-bounded quantum complexity. J. Comput. System Sci., 59, 2, 1999. Pages 281–326.

Digital Library

[51]

John Watrous. 2001. Quantum simulations of classical random walks and undirected graph connectivity. Journal of computer and system sciences, 62, 2, 2001. Pages 376–391.

Digital Library

[52]

John Watrous. 2003. On the complexity of simulating space-bounded quantum computations. Computational Complexity, 12, 1-2, 2003. Pages 48–84.

Digital Library

[53]

John Watrous. 2009. Encyclopedia of Complexity and System Science, chapter Quantum computational complexity.

[54]

John Watrous. 2009. Zero-knowledge against quantum attacks. SIAM J. Comput., 39, 1, 2009. Pages 25–58.

Digital Library

[55]

John Watrous. 2018. The theory of quantum information. Cambridge University Press.

[56]

Stathis Zachos and Martin Furer. 1987. Probabilistic quantifiers vs. distrustful adversaries. In International Conference on Foundations of Software Technology and Theoretical Computer Science. Pages 443–455.

Cited By

Wang QZhang Z(2024)Fast Quantum Algorithms for Trace Distance EstimationIEEE Transactions on Information Theory10.1109/TIT.2023.332112170:4(2720-2733)Online publication date: Apr-2024
https://doi.org/10.1109/TIT.2023.3321121
Metger TYuen H(2023)stateQIP = statePSPACE2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00082(1349-1356)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00082
Blocki JHolman BLee S(2022)The Parallel Reversible Pebbling Game: Analyzing the Post-quantum Security of iMHFsTheory of Cryptography10.1007/978-3-031-22318-1_3(52-79)Online publication date: 7-Nov-2022
https://dl.acm.org/doi/10.1007/978-3-031-22318-1_3

Index Terms

Eliminating intermediate measurements in space-bounded Quantum computation
1. Theory of computation
  1. Models of computation
    1. Quantum computation theory
      1. Quantum complexity theory

Recommendations

Exponential separation of quantum and classical online space complexity
SPAA '06: Proceedings of the eighteenth annual ACM symposium on Parallelism in algorithms and architectures

The main objective of quantum computation is to exploit the natural parallelism of quantum mechanics to solve problems using less computational resources than classical computers. Although quantum algorithms realizing an exponential time speed-up over ...
Teleportation-based quantum computation, extended Temperley---Lieb diagrammatical approach and Yang---Baxter equation

This paper focuses on the study of topological features in teleportation-based quantum computation and aims at presenting a detailed review on teleportation-based quantum computation (Gottesman and Chuang in Nature 402: 390, 1999). In the extended ...
Construction of quantum gates for concatenated Greenberger---Horne---Zeilinger-type logic qubit

Concatenated Greenberger---Horne---Zeilinger (C-GHZ) state is a kind of logic qubit which is robust in noisy environment. In this paper, we encode the C-GHZ state as the logic qubit and design two kinds of quantum gates for such logic qubit. The first ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

June 2021

1797 pages

ISBN:9781450380539

DOI:10.1145/3406325

General Chair:
Samir Khuller
Northwestern University, USA
,
Program Chair:
Virginia Vassilevska Williams
Massachusetts Institute of Technology, USA

Copyright © 2021 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Sponsors

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 15 June 2021

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

AFOSR
NSF (National Science Foundation)

Conference

STOC '21

Sponsor:

SIGACT

STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing

June 21 - 25, 2021

Virtual, Italy

Acceptance Rates

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

3
Total Citations
View Citations
260
Total Downloads

Downloads (Last 12 months)107
Downloads (Last 6 weeks)19

Reflects downloads up to 22 Sep 2024

Other Metrics

View Author Metrics

Citations

Cited By

Wang QZhang Z(2024)Fast Quantum Algorithms for Trace Distance EstimationIEEE Transactions on Information Theory10.1109/TIT.2023.332112170:4(2720-2733)Online publication date: Apr-2024
https://doi.org/10.1109/TIT.2023.3321121
Metger TYuen H(2023)stateQIP = statePSPACE2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00082(1349-1356)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00082
Blocki JHolman BLee S(2022)The Parallel Reversible Pebbling Game: Analyzing the Post-quantum Security of iMHFsTheory of Cryptography10.1007/978-3-031-22318-1_3(52-79)Online publication date: 7-Nov-2022
https://dl.acm.org/doi/10.1007/978-3-031-22318-1_3

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Table of Contents