Quantum algorithms and lower bounds for convex optimization

Shouvanik Chakrabarti, Andrew M. Childs, Tongyang Li, and Xiaodi Wu

Department of Computer Science, Institute for Advanced Computer Studies, and Joint Center for Quantum Information and Computer Science, University of Maryland

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Abstract

While recent work suggests that quantum computers can speed up the solution of semidefinite programs, little is known about the quantum complexity of more general convex optimization. We present a quantum algorithm that can optimize a convex function over an $n$-dimensional convex body using $\tilde{O}(n)$ queries to oracles that evaluate the objective function and determine membership in the convex body. This represents a quadratic improvement over the best-known classical algorithm. We also study limitations on the power of quantum computers for general convex optimization, showing that it requires $\tilde{\Omega}(\sqrt n)$ evaluation queries and $\Omega(\sqrt{n})$ membership queries.

Convex optimization has been a central topic in mathematics, theoretical computer science, and operations research over the last several decades. Our work, along with an independent paper by van Apeldoorn et al., gives the first quantum algorithm with provable quantum speedup for general convex optimization. On the other hand, our quantum lower bounds demonstrate that the quantum speedup for general convex optimization is at most polynomial, ruling out the possibility of an exponential speedup (such as in Shor’s factoring algorithm.

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Cited by

[1] Patrick Rebentrost and Seth Lloyd, "Quantum computational finance: quantum algorithm for portfolio optimization", arXiv:1811.03975.

[2] Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf, "Quantum SDP-Solvers: Better upper and lower bounds", arXiv:1705.01843.

[3] Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf, "Convex optimization using quantum oracles", arXiv:1809.00643.

[4] P. A. M. Casares and M. A. Martin-Delgado, "A Quantum Interior-Point Predictor-Corrector Algorithm for Linear Programming", arXiv:1902.06749.

[5] Yassine Hamoudi, Patrick Rebentrost, Ansis Rosmanis, and Miklos Santha, "Quantum and Classical Algorithms for Approximate Submodular Function Minimization", arXiv:1907.05378.

[6] Hsin-Yuan Huang, Kishor Bharti, and Patrick Rebentrost, "Near-term quantum algorithms for linear systems of equations", arXiv:1909.07344.

[7] Shouvanik Chakrabarti, Andrew M. Childs, Shih-Han Hung, Tongyang Li, Chunhao Wang, and Xiaodi Wu, "Quantum algorithm for estimating volumes of convex bodies", arXiv:1908.03903.

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