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 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 Ω~(n) evaluation queries and Ω(n) membership queries.

Highlights

  • Convex optimization has been a central topic in the study of mathematical optimization, theoretical computer science, and operations research over the last several decades

  • The fastest known classical algorithm for general convex optimization solves an instance of dimension n using O(n2) queries to oracles for the convex body and the objective function, and runs in time O(n3) [21]

  • semidefinite programs (SDPs) are specified by positive semidefinite matrices, and their solution is related to well-understood tasks in quantum computation such as solving linear systems (e.g., [9, 13]) and Gibbs sampling (e.g., [2, 6])

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Summary

Introduction

Convex optimization has been a central topic in the study of mathematical optimization, theoretical computer science, and operations research over the last several decades. The fastest known classical algorithm for general convex optimization solves an instance of dimension n using O(n2) queries to oracles for the convex body and the objective function, and runs in time O(n3) [21].1. It relies on a reduction from optimization to separation that makes O(n) separation oracle calls and runs in time O(n3) [22] It is unclear whether the query complexity of O(n2) is optimal for all possible classical algorithms, it is the best possible result using the above framework. SDPs are specified by positive semidefinite matrices, and their solution is related to well-understood tasks in quantum computation such as solving linear systems (e.g., [9, 13]) and Gibbs sampling (e.g., [2, 6]).

Convex optimization
Contributions
Upper bound
Lower bound
Open questions
Oracle definitions
Evaluation to subgradient
Smooth functions
Start with n b-bit registers set to 0 and Hadamard transform each to obtain
Extension to non-smooth functions
Membership to separation
Separation to optimization
Membership queries
Evaluation queries
Discretization: simulating perfectly precise queries by low-precision queries
Smoothed hypercube
Classical gradient computation
Mollified functions
B Proof details for upper bound
Convexity of max-norm optimization
Correctness of Line 1 and Line 2
Correctness of Line 3

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