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 optimization 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 $mapprox n$, which is the same as classical.
Semidefinite programs (SDPs) are an important tool in convex optimization tasks and approximation algorithms. They allow to optimize a linear function over positive semidefinite matrices, subject to linear constraints on those matrices, and they are solvable in polynomial time on a classical computer. Brand~ao and Svore recently gave a quantum algorithm for solving semidefinite programs that (in some regimes) is faster than the best-possible classical algorithm. In this paper we improve on their algorithm in several ways, in particular we obtain a 4-th root improvement in the running time with respect to the required precision. We also show strong limits for this particular approach to quantum SDP-solvers, for instance for combinatorial optimization problems that have a lot of symmetry, and we prove some general limitations for quantum SDP-solvers.
 Sanjeev Arora, Elad Hazan, and Satyen Kale. The multiplicative weights update method: a meta-algorithm and applications. toc, 8(6):121–164, 2012.
 Sanjeev Arora and Satyen Kale. A combinatorial, primal-dual approach to semidefinite programs. jacm, 63(2):12:1–12:35, 2016. Earlier version in STOC’07.
 Andris Ambainis. Quantum walk algorithm for element distinctness. siamjc, 37(1):210–239, 2007. Earlier version in FOCS’04. arXiv: quant-ph/0311001?.
 Joran van Apeldoorn and András Gilyén. Improvements in quantum SDP-solving with applications. In icalp46th, pages 99:1–99:15, 2019. arXiv: 1804.05058?.
 Joran van Apeldoorn and András Gilyén. Quantum algorithms for zero-sum games. arXiv: 1904.03180?, 2019.
 Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf. Quantum SDP-solvers: Better upper and lower bounds. In focs58th, pages 403–414, 2017. arXiv: 1705.01843?.
 Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf. Convex optimization using quantum oracles. quantum, 4:220, 2020. arXiv: 1809.00643?.
 Noga Alon and Joel H. Spencer. The Probabilistic Method. Wiley-Interscience, third edition, 2008.
 Michel Boyer, Gilles Brassard, Peter Høyer, and Alain Tapp. Tight bounds on quantum searching. Fortschritte der Physik, 46(4–5):493–505, 1998. arXiv: quant-ph/9605034?.
 Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. Simulating Hamiltonian dynamics with a truncated Taylor series. prl, 114(9):090502, 2015. arXiv: 1412.4687?.
 Dominic W. Berry, Andrew M. Childs, and Robin Kothari. Hamiltonian simulation with nearly optimal dependence on all parameters. In focs56th, pages 792–809, 2015. arXiv: 1501.01715?.
 Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. Quantum amplitude amplification and estimation. In Quantum Computation and Quantum Information: A Millennium Volume, volume 305 of Contemporary Mathematics Series, pages 53–74. AMS, 2002. arXiv: quant-ph/0005055?.
 Fernando G. S. L. Brandão, Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M. Svore, and Xiaodi Wu. Quantum SDP solvers: Large speed-ups, optimality, and applications to quantum learning. In icalp46th, pages 27:1–27:14, 2019. arXiv: 1710.02581?.
 Fernando G. S. L. Brandão and Krysta M. Svore. Quantum speed-ups for solving semidefinite programs. In focs58th, pages 415–426, 2017. arXiv: 1609.05537?.
 Shouvanik Chakrabarti, Andrew M. Childs, Tongyang Li, and Xiaodi Wu. Quantum algorithms and lower bounds for convex optimization. quantum, 4:221, 2020. arXiv: 1809.01731?.
 Richard Cleve, Artur Ekert, Chiara Macchiavello, and Michele Mosca. Quantum algorithms revisited. rspa, 454(1969):339–354, 1998. arXiv: quant-ph/9708016?.
 Andrew M. Childs, Robin Kothari, and Rolando D. Somma. Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. siamjc, 46(6):1920–1950, 2017. arXiv: 1511.02306?.
 Anirban Narayan Chowdhury and Rolando D. Somma. Quantum algorithms for Gibbs sampling and hitting-time estimation. qic, 17(1&2):41–64, 2017. arXiv: 1603.02940?.
 Andrew M. Childs and Nathan Wiebe. Hamiltonian simulation using linear combinations of unitary operations. qic, 12(11&12):901–924, 2012. arXiv: 1202.5822?.
 George B. Dantzig. Maximization of a linear function of variables subject to linear inequalities. In Activity Analysis of Production and Allocation, Cowles Commission Monograph No. 13, pages 339–347. John Wiley & Sons Inc., New York, N. Y., 1951.
 Christoph Dürr and Peter Høyer. A quantum algorithm for finding the minimum. arXiv: quant-ph/9607014?, 1996.
 Christoph Dürr, Mark Heiligman, Peter Høyer, and Mehdi Mhalla. Quantum query complexity of some graph problems. siamjc, 35(6):1310–1328, 2006. Earlier version in ICALP’04. arXiv: quant-ph/0401091?.
 Martin Grötschel, László Lovász, and Alexander Schrijver. The ellipsoid method and its consequences in combinatorial optimization. comb, 1(2):169–197, 1981.
 Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric Algorithms and Combinatorial Optimization. Springer, 1988.
 Lov Grover and Terry Rudolph. Creating superpositions that correspond to efficiently integrable probability distributions. arXiv: quant-ph/0208112?, 2002.
 Lov K. Grover. A fast quantum mechanical algorithm for database search. In stoc28th, pages 212–219, 1996. arXiv: quant-ph/9605043?.
 András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In stoc51st, pages 193–204, 2019. arXiv: 1806.01838?.
 Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. jacm, 42(6):1115–1145, 1995. Earlier version in STOC’94.
 Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. prl, 103(15):150502, 2009. arXiv: 0811.3171?.
 Shelby Kimmel. Quantum adversary (upper) bound. cjtcs, 2013:4:1–14, 2013. Earlier version in ICALP’12. arXiv: 1101.0797?.
 Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O’Donnell. Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? siamjc, 37(1):319–357, 2007. Earlier version in FOCS’04.
 Iordanis Kerenidis and Anupam Prakash. A quantum interior point method for LPs and SDPs. arXiv: 1808.09266?, 2018.
 Guang Hao Low and Isaac L. Chuang. Optimal Hamiltonian simulation by quantum signal processing. prl, 118(1):010501, 2017. arXiv: 1606.02685?.
 Guang Hao Low and Isaac L. Chuang. Hamiltonian simulation by qubitization. quantum, 3:163, 2019. arXiv: 1610.06546?.
 Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong. A faster cutting plane method and its implications for combinatorial and convex optimization. In focs56th, pages 1049–1065, 2015. arXiv: 1508.04874?.
 Michael A. Nielsen and Isaac L. Chuang. Quantum computation and quantum information. Cambridge University Press, 2000.
 Y. Nesterov and A. Nemirovski. Interior-point polynomial algorithms in convex programming, volume 13 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), 1994.
 David Poulin and Pawel Wocjan. Preparing ground states of quantum many-body systems on a quantum computer. prl, 102(13):130503, 2009. arXiv: 0809.2705?.
 David Poulin and Pawel Wocjan. Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. prl, 103(22):220502, 2009. arXiv: 0905.2199?.
 James Renegar. “Efficient” subgradient methods for general convex optimization. siamjc, 26(4):2649–2676, 2016. arXiv: 1605.08712?.
 James Renegar. Accelerated first-order methods for hyperbolic programming. Mathematical Programming, 173(1):1–35, 2019. arXiv: 1512.07569?.
 Alexander Schrijver. Theory of Linear and Integer Programming. John Wiley & Sons, Inc., New York, NY, USA, 1986.
 Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. siamjc, 26(5):1484–1509, 1997. Earlier version in FOCS’94. arXiv: quant-ph/9508027?.
 Koji Tsuda, Gunnar Rätsch, and Manfred K. Warmuth. Matrix exponentiated gradient updates for on-line learning and Bregman projection. Journal of Machine Learning Research, 6:995–1018, 2005. Earlier version in NIPS’04.
 Manfred K. Warmuth and Dima Kuzmin. Online variance minimization. Machine Learning, 87(1):1–32, 2012. Earlier version in COLT’06.
 András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe, “Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics”, arXiv:1806.01838.
 Carlo Ciliberto, Mark Herbster, Alessandro Davide Ialongo, Massimiliano Pontil, Andrea Rocchetto, Simone Severini, and Leonard Wossnig, “Quantum machine learning: a classical perspective”, Proceedings of the Royal Society of London Series A 474 2209, 20170551 (2018).
 Patrick Rebentrost and Seth Lloyd, “Quantum computational finance: quantum algorithm for portfolio optimization”, arXiv:1811.03975.
 Joran van Apeldoorn and András Gilyén, “Quantum algorithms for zero-sum games”, arXiv:1904.03180.
 András Gilyén, Srinivasan Arunachalam, and Nathan Wiebe, “Optimizing quantum optimization algorithms via faster quantum gradient computation”, arXiv:1711.00465.
 Patrick Rebentrost, Maria Schuld, Leonard Wossnig, Francesco Petruccione, and Seth Lloyd, “Quantum gradient descent and Newton’s method for constrained polynomial optimization”, arXiv:1612.01789.
 Fernando G. S. L. Brandão, Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M. Svore, and Xiaodi Wu, “Quantum SDP Solvers: Large Speed-ups, Optimality, and Applications to Quantum Learning”, arXiv:1710.02581.
 Joran van Apeldoorn and András Gilyén, “Improvements in Quantum SDP-Solving with Applications”, arXiv:1804.05058.
 Nai-Hui Chia, Tongyang Li, Han-Hsuan Lin, and Chunhao Wang, “Quantum-inspired classical sublinear-time algorithm for solving low-rank semidefinite programming via sampling approaches”, arXiv:1901.03254.
 Simon Apers, “Quantum Walk Sampling by Growing Seed Sets”, arXiv:1904.11446.
 Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf, “Convex optimization using quantum oracles”, arXiv:1809.00643.
 Shouvanik Chakrabarti, Andrew M. Childs, Tongyang Li, and Xiaodi Wu, “Quantum algorithms and lower bounds for convex optimization”, arXiv:1809.01731.
 Yimin Ge, Jordi Tura, and J. Ignacio Cirac, “Faster ground state preparation and high-precision ground energy estimation with fewer qubits”, arXiv:1712.03193.
 Ivan Bardet and Cambyse Rouzé, “Hypercontractivity and logarithmic Sobolev Inequality for non-primitive quantum Markov semigroups and estimation of decoherence rates”, arXiv:1803.05379.
 Ashley Montanaro, “Quantum speedup of branch-and-bound algorithms”, arXiv:1906.10375.
 Johannes Bausch, Sathyawageeswar Subramanian, and Stephen Piddock, “A Quantum Search Decoder for Natural Language Processing”, arXiv:1909.05023.
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