Quantum SDP-Solvers: Better Upper and Lower Bounds

Brandão and Svore recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension n of the problem and the number m of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure.We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimizations problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with mn when m is approximately n, which is the same as classical.

[1]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[2]  R. Cleve,et al.  Quantum algorithms revisited , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[3]  Sanjeev Arora,et al.  A combinatorial, primal-dual approach to semidefinite programs , 2007, STOC '07.

[4]  I. Chuang,et al.  Optimal Hamiltonian Simulation by Quantum Signal Processing. , 2016, Physical review letters.

[5]  Krysta Marie Svore,et al.  Quantum Speed-ups for Semidefinite Programming , 2016, ArXiv.

[6]  James Renegar,et al.  "Efficient" Subgradient Methods for General Convex Optimization , 2016, SIAM J. Optim..

[7]  Ronald de Wolf,et al.  Convex optimization using quantum oracles , 2018, Quantum.

[8]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[9]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[10]  Guy Kindler,et al.  Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[11]  Sanjeev Arora,et al.  The Multiplicative Weights Update Method: a Meta-Algorithm and Applications , 2012, Theory Comput..

[12]  I. Chuang,et al.  Hamiltonian Simulation by Qubitization , 2016, Quantum.

[13]  Andris Ambainis,et al.  Quantum walk algorithm for element distinctness , 2003, 45th Annual IEEE Symposium on Foundations of Computer Science.

[14]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[15]  Ben Reichardt,et al.  Span Programs and Quantum Query Complexity: The General Adversary Bound Is Nearly Tight for Every Boolean Function , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[16]  András Gilyén,et al.  Improvements in Quantum SDP-Solving with Applications , 2018, ICALP.

[17]  Krysta Marie Svore,et al.  Quantum Speed-Ups for Solving Semidefinite Programs , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[18]  D. Poulin,et al.  Preparing ground States of quantum many-body systems on a quantum computer. , 2008, Physical review letters.

[19]  Mehdi Mhalla,et al.  Quantum Query Complexity of Some Graph Problems , 2004, SIAM J. Comput..

[20]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[21]  Aleksandrs Belovs,et al.  Applications of Adversary Method in Quantum Query Algorithms , 2014, 1402.3858.

[22]  Gunnar Rätsch,et al.  Matrix Exponentiated Gradient Updates for On-line Learning and Bregman Projection , 2004, J. Mach. Learn. Res..

[23]  Yin Tat Lee,et al.  A Faster Cutting Plane Method and its Implications for Combinatorial and Convex Optimization , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[24]  Nathan Wiebe,et al.  Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics , 2018, STOC.

[25]  Andrew M. Childs,et al.  Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[26]  James Renegar,et al.  Accelerated first-order methods for hyperbolic programming , 2015, Mathematical Programming.

[27]  Andrew M. Childs,et al.  Simulating Hamiltonian dynamics with a truncated Taylor series. , 2014, Physical review letters.

[28]  Ryan O'Donnell,et al.  Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs? , 2007, SIAM J. Comput..

[29]  Nathan Wiebe,et al.  Hamiltonian simulation using linear combinations of unitary operations , 2012, Quantum Inf. Comput..

[30]  Shelby Kimmel.  Quantum Adversary (Upper) Bound , 2013, Chic. J. Theor. Comput. Sci..

[31]  Shouvanik Chakrabarti,et al.  Quantum algorithms and lower bounds for convex optimization , 2018, Quantum.

[32]  A. Harrow,et al.  Quantum algorithm for linear systems of equations. , 2008, Physical review letters.

[33]  Aram W. Harrow,et al.  Quantum algorithm for solving linear systems of equations , 2010 .

[34]  Joran van Apeldoorn,et al.  Quantum algorithms for zero-sum games , 2019, 1904.03180.

[35]  Iordanis Kerenidis,et al.  A Quantum Interior Point Method for LPs and SDPs , 2018, ACM Transactions on Quantum Computing.

[36]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[37]  Rolando D. Somma,et al.  Quantum algorithms for Gibbs sampling and hitting-time estimation , 2016, Quantum Inf. Comput..

[38]  Gilles Brassard,et al.  Tight bounds on quantum searching , 1996, quant-ph/9605034.

[39]  Troy Lee,et al.  Quantum Query Complexity of State Conversion , 2010, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[40]  D. Poulin,et al.  Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. , 2009, Physical review letters.

[41]  Troy Lee,et al.  Negative weights make adversaries stronger , 2007, STOC '07.

[42]  Mehdi Mhalla,et al.  Quantum Query Complexity of Some Graph Problems , 2006, SIAM J. Comput..

[43]  T. Koopmans,et al.  Activity Analysis of Production and Allocation. , 1952 .

[44]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[45]  Manfred K. Warmuth,et al.  Online variance minimization , 2011, Machine Learning.

[46]  Xiaodi Wu,et al.  Quantum SDP Solvers: Large Speed-Ups, Optimality, and Applications to Quantum Learning , 2017, ICALP.

[47]  Ronald de Wolf,et al.  Quantum SDP-Solvers: Better Upper and Lower Bounds , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[48]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[49]  Lov K. Grover,et al.  Creating superpositions that correspond to efficiently integrable probability distributions , 2002, quant-ph/0208112.

[50]  Andrew M. Childs,et al.  Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision , 2015, SIAM J. Comput..

[51]  Christoph Dürr,et al.  A Quantum Algorithm for Finding the Minimum , 1996, ArXiv.

[52]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[53]  Ronald de Wolf,et al.  Quantum lower bounds by polynomials , 2001, JACM.

[54]  Andrew M. Childs,et al.  Quantum linear systems algorithm with exponentially improved dependence on precision , 2015 .

[55]  G. Brassard,et al.  Quantum Amplitude Amplification and Estimation , 2000, quant-ph/0005055.