research-article

A Quantum Interior Point Method for LPs and SDPs

Authors:

Iordanis Kerenidis,

Anupam PrakashAuthors Info & Claims

ACM Transactions on Quantum Computing, Volume 1, Issue 1

Article No.: 5, Pages 1 - 32

https://doi.org/10.1145/3406306

Published: 02 October 2020 Publication History

Abstract

We present a quantum interior point method (IPM) for semi-definite programs that has a worst-case running time of Õ(n^2.5 / ξ² μ κ ³ log(1/ϵ)). The algorithm outputs a pair of matrices (S,Y) that have objective value within ϵ of the optimal and satisfy the constraints approximately to error xi. The parameter mu is at most √2n while kappa is an upper bound on the condition number of the intermediate solution matrices arising in the classical IPM. For the case where κ ≪ n^5/6, our method provides a significant polynomial speedup over the best-known classical semi-definite program solvers that have a worst-case running time of Õ(n⁶). For linear programs, our algorithm has a running time of Õ(n^1.5 / ξ² μ κ ³ log (1/ϵ)) with the same guarantees and with parameter μ < √2n. Our technical contributions include an efficient quantum procedure for solving the Newton linear systems arising in the classical IPMs, an efficient pure state tomography algorithm, and an analysis of the IPM where the linear systems are solved approximately. Our results pave the way for the development of quantum algorithms with significant polynomial speedups for applications in optimization and machine learning.

References

[1]

Dana Angluin and Leslie G. Valiant. 1977. Fast probabilistic algorithms for Hamiltonian circuits and matchings. In Proceedings of the 9th Annual ACM Symposium on Theory of Computing. ACM, New York, NY, 30--41.

[2]

Joran van Apeldoorn and András Gilyén. 2019. Improvements in quantum SDP-solving with applications. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP’19).

[3]

Joran van Apeldoorn and András Gilyén. 2019. Quantum algorithms for zero-sum games. arXiv:1904.03180.

[4]

Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf. 2017. Quantum SDP-solvers: Better upper and lower bounds. In Proceedings of the 2017 58th Annual Symposium on Foundations of Computer Science (FOCS’17). IEEE, Los Alamitos, CA, 403--414.

[5]

Sanjeev Arora, Elad Hazan, and Satyen Kale. 2005. Fast algorithms for approximate semidefinite programming using the multiplicative weights update method. In Proceedings of the 2005 46th Annual Symposium on Foundations of Computer Science (FOCS’05). IEEE, Los Alamitos, CA, 339--348.

[6]

Sanjeev Arora, Elad Hazan, and Satyen Kale. 2012. The multiplicative weights update method: A meta-algorithm and applications. Theory of Computing 8, 1 (2012), 121--164.

[7]

Aharon Ben-Tal and Arkadi Nemirovski. 2001. Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. SIAM.

[8]

Fernando G. S. L. Brandão, Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M. Svore, and Xiaodi Wu. 2019. Quantum SDP solvers: Large speed-ups, optimality, and applications to quantum learning. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP’19).

[9]

Fernando G. S. L. Brandão and Krysta M. Svore. 2017. Quantum speed-ups for solving semidefinite programs. In Proceedings of the 2017 58th Annual Symposium on Foundations of Computer Science (FOCS’17). IEEE, Los Alamitos, CA, 415--426.

[10]

Sébastien Bubeck. 2015. Convex optimization: Algorithms and complexity. Foundations and Trends in Machine Learning 8, 3--4 (2015), 231--357.

Digital Library

[11]

Joo-Siong Chai and Kim-Chuan Toh. 2007. Preconditioning and iterative solution of symmetric indefinite linear systems arising from interior point methods for linear programming. Computational Optimization and Applications 36, 2--3 (2007), 221--247.

[12]

Shantanav Chakraborty, András Gilyén, and Stacey Jeffery. 2019. The power of block-encoded matrix powers: Improved regression techniques via faster Hamiltonian simulation. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP’19).

[13]

Nai-Hui Chia, Tongyang Li, Han-Hsuan Lin, and Chunhao Wang. 2019. Quantum-inspired classical sublinear-time algorithm for solving low-rank semidefinite programming via sampling approaches. arXiv:1901.03254.

[14]

Michael B. Cohen, Yin Tat Lee, and Zhao Song. 2019. Solving linear programs in the current matrix multiplication time. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing. ACM, New York, NY, 938--942.

Digital Library

[15]

Hilary Dollar. 2005. Iterative Linear Algebra for Constrained Optimization. Ph.D. Dissertation. Keble College, University of Oxford.

[16]

H. Sue Dollar, Nicholas I. M. Gould, Wil H. A. Schilders, and Andy J. Wathen. 2007. Using constraint preconditioners with regularized saddle-point problems. Computational Optimization and Applications 36, 2--3 (2007), 249--270.

[17]

Dan Garber and Elad Hazan. 2011. Approximating semidefinite programs in sublinear time. In Advances in Neural Information Processing Systems. 1080--1088.

[18]

András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. 2019. 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. ACM, New York, NY, 193--204.

Digital Library

[19]

Michel X. Goemans and David P. Williamson. 1995. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42, 6 (1995), 1115--1145.

Digital Library

[20]

David Gross, Yi-Kai Liu, Steven T. Flammia, Stephen Becker, and Jens Eisert. 2010. Quantum state tomography via compressed sensing. Physical Review Letters 105, 15 (2010), 150401.

[21]

Narendra Karmarkar. 1984. A new polynomial-time algorithm for linear programming. In Proceedings of the 16th Annual ACM Symposium on Theory of Computing. ACM, New York, NY, 302--311.

[22]

Iordanis Kerenidis and Anupam Prakash. 2017. Quantum recommendation systems. In Proceedings of the 8th Innovations in Theoretical Computer Science Conference.

[23]

Iordanis Kerenidis and Anupam Prakash. 2020. Quantum gradient descent for linear systems and least squares. Physical Review A 101, 2 (2020), 022316.

[24]

Iordanis Kerenidis, Anupam Prakash, and Daniel Szilagyi. 2019. Quantum algorithms for second-order cone programming and support vector machines. arXiv:1908.06720.

[25]

Michael Keyl. 2006. Quantum state estimation and large deviations. Reviews in Mathematical Physics 18, 01 (2006), 19--60.

[26]

Leonid G. Khachiyan. 1980. Polynomial algorithms in linear programming. USSR Computational Mathematics and Mathematical Physics 20, 1 (1980), 53--72.

[27]

Yin Tat Lee and Aaron Sidford. 2015. Efficient inverse maintenance and faster algorithms for linear programming. In Proceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS’15). IEEE, Los Alamitos, CA, 230--249.

Digital Library

[28]

Yin Tat Lee, Aaron Sidford, and Sam Chiu-Wai Wong. 2015. A faster cutting plane method and its implications for combinatorial and convex optimization. In Proceedings of the 2015 56th Annual Symposium on Foundations of Computer Science (FOCS’15). IEEE, Los Alamitos, CA, 1049--1065.

Digital Library

[29]

Yurii Nesterov and Arkadi Nemirovski. 1994. Interior-Point Polynomial Algorithms in Convex Programming. Vol. 13. SIAM.

[30]

Ryan O’Donnell and John Wright. 2016. Efficient quantum tomography. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing. ACM, New York, NY, 899--912.

[31]

Pravin M. Vaidya. 1989. Speeding-up linear programming using fast matrix multiplication. In Proceedings of the 1989 30th Annual Symposium on Foundations of Computer Science (FOCS’89). IEEE, Los Alamitos, CA, 332--337.

[32]

Pravin M. Vaidya. 1990. An algorithm for linear programming which requires O((m + n) n² + (m + n)^1.5n)L) arithmetic operations. Mathematical Programming 47, 1--3 (1990), 175--201.

Cited By

Mohammadisiahroudi MWu ZAugustino BCarr ATerlaky T(2025)Improvements to Quantum Interior Point Method for Linear OptimizationACM Transactions on Quantum Computing10.1145/37022446:1(1-24)Online publication date: 14-Jan-2025
https://dl.acm.org/doi/10.1145/3702244
Amani FKargarian A(2025)Optimal power flow solution via noise-resilient quantum interior-point methodsElectric Power Systems Research10.1016/j.epsr.2024.111216240(111216)Online publication date: Mar-2025
https://doi.org/10.1016/j.epsr.2024.111216
Ajagekar AYou F(2025)Decarbonization of Building Operations with Adaptive Quantum Computing-Based Model Predictive ControlEngineering10.1016/j.eng.2025.02.002Online publication date: Feb-2025
https://doi.org/10.1016/j.eng.2025.02.002
Show More Cited By

Index Terms

A Quantum Interior Point Method for LPs and SDPs
1. Mathematics of computing
  1. Mathematical analysis
    1. Mathematical optimization
      1. Continuous optimization
2. Theory of computation
  1. Models of computation
    1. Quantum computation theory

Recommendations

Extending the QCR method to general mixed-integer programs

Let (MQP) be a general mixed integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. In this paper, we present a convex reformulation of (MQP), i.e. we reformulate (MQP) into an equivalent program, with ...
Branching on hyperplane methods for mixed integer linear and convex programming using adjoint lattices

We present branching-on-hyperplane methods for solving mixed integer linear and mixed integer convex programs. In particular, we formulate the problem of finding a good branching hyperplane using a novel concept of adjoint lattice. We also reformulate ...
An interior point method for solving systems of linear equations and inequalities

A simple interior point method is proposed for solving a system of linear equations subject to nonnegativity constraints. The direction of update is defined by projection of the current solution on a linear manifold defined by the equations. ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Quantum Computing

ACM Transactions on Quantum Computing Volume 1, Issue 1

December 2020

139 pages

EISSN:2643-6817

DOI:10.1145/3427922

Editors:
Travis S. Humble
Oak Ridge National Laboratory, USA
,
Mingsheng Ying
University of Technology Sydney, Australia

Issue’s Table of Contents

Copyright © 2020 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].

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 02 October 2020

Accepted: 01 June 2020

Revised: 01 May 2020

Received: 01 December 2019

Published in TQC Volume 1, Issue 1

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

QuantERA, QuantAlgo, and ANR QuBIC

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

63
Total Citations
View Citations
542
Total Downloads

Downloads (Last 12 months)84
Downloads (Last 6 weeks)4

Reflects downloads up to 08 Mar 2025

Other Metrics

View Author Metrics

Citations

Cited By

Mohammadisiahroudi MWu ZAugustino BCarr ATerlaky T(2025)Improvements to Quantum Interior Point Method for Linear OptimizationACM Transactions on Quantum Computing10.1145/37022446:1(1-24)Online publication date: 14-Jan-2025
https://dl.acm.org/doi/10.1145/3702244
Amani FKargarian A(2025)Optimal power flow solution via noise-resilient quantum interior-point methodsElectric Power Systems Research10.1016/j.epsr.2024.111216240(111216)Online publication date: Mar-2025
https://doi.org/10.1016/j.epsr.2024.111216
Ajagekar AYou F(2025)Decarbonization of Building Operations with Adaptive Quantum Computing-Based Model Predictive ControlEngineering10.1016/j.eng.2025.02.002Online publication date: Feb-2025
https://doi.org/10.1016/j.eng.2025.02.002
Bellante AFioravanti TCarminati MZanero SLuongo A(2025)Evaluating the potential of quantum machine learning in cybersecurity: A case-study on PCA-based intrusion detection systemsComputers & Security10.1016/j.cose.2025.104341(104341)Online publication date: Feb-2025
https://doi.org/10.1016/j.cose.2025.104341
Mohammadisiahroudi MAugustino BSampourmahani PTerlaky T(2025)Quantum computing inspired iterative refinement for semidefinite optimizationMathematical Programming10.1007/s10107-024-02183-zOnline publication date: 11-Jan-2025
https://doi.org/10.1007/s10107-024-02183-z
Ciacco AGuerriero FMacrina G(2025)Review of quantum algorithms for medicine, finance and logisticsSoft Computing10.1007/s00500-025-10540-zOnline publication date: 25-Feb-2025
https://doi.org/10.1007/s00500-025-10540-z
Chen ZLu YWang HLiu YLi T(2025)Quantum Langevin Dynamics for OptimizationCommunications in Mathematical Physics10.1007/s00220-025-05234-4406:3Online publication date: 17-Feb-2025
https://doi.org/10.1007/s00220-025-05234-4
Zhang YZhang CFang CWang LLi TSalakhutdinov RKolter ZHeller KWeller AOliver NScarlett JBerkenkamp F(2024)Quantum algorithms and lower bounds for finite-sum optimizationProceedings of the 41st International Conference on Machine Learning10.5555/3692070.3694562(60244-60270)Online publication date: 21-Jul-2024
https://dl.acm.org/doi/10.5555/3692070.3694562
He JLiu CLiu XLi LLui JSalakhutdinov RKolter ZHeller KWeller AOliver NScarlett JBerkenkamp F(2024)Quantum algorithm for online exp-concave optimizationProceedings of the 41st International Conference on Machine Learning10.5555/3692070.3692789(17946-17971)Online publication date: 21-Jul-2024
https://dl.acm.org/doi/10.5555/3692070.3692789
Zhang ZWang QYing M(2024)Parallel Quantum Algorithm for Hamiltonian SimulationQuantum10.22331/q-2024-01-15-12288(1228)Online publication date: 15-Jan-2024
https://doi.org/10.22331/q-2024-01-15-1228
Show More Cited By

View Options

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 Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

HTML Format

View this article in HTML Format.

Figures

Tables

Media

View Issue’s Table of Contents