Convex optimization using quantum oracles

Joran van Apeldoorn1, András Gilyén1, Sander Gribling1, and Ronald de Wolf1,2

1QuSoft, CWI, Amsterdam, the Netherlands
2University of Amsterdam, Amsterdam, the Netherlands

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We study to what extent quantum algorithms can speed up solving convex optimization problems. Following the classical literature we assume access to a convex set via various oracles, and we examine the efficiency of reductions between the different oracles. In particular, we show how a separation oracle can be implemented using $\tilde{O}(1)$ quantum queries to a membership oracle, which is an exponential quantum speed-up over the $\Omega(n)$ membership queries that are needed classically. We show that a quantum computer can very efficiently compute an approximate subgradient of a convex Lipschitz function. Combining this with a simplification of recent classical work of Lee, Sidford, and Vempala gives our efficient separation oracle. This in turn implies, via a known algorithm, that $\tilde{O}(n)$ quantum queries to a membership oracle suffice to implement an optimization oracle (the best known classical upper bound on the number of membership queries is quadratic). We also prove several lower bounds: $\Omega(\sqrt{n})$ quantum separation (or membership) queries are needed for optimization if the algorithm knows an interior point of the convex set, and $\Omega(n)$ quantum separation queries are needed if it does not.

Optimizing a function subject to various constraints is an important task, including practical problems like scheduling, energy minimization, learning neural networks, etc. In many cases the set K of points that satisfy the constraints is "convex", meaning that the line between any two points of K also lies in K. There can be different types of access to K: in some cases we can efficiently determine whether a given point lies in K ("membership queries"), in some cases we can efficiently find a hyperplane separating K from a given point outside of K, and in some cases we can efficiently optimize any well-behaved function over K. We study quantum algorithms that efficiently convert between different such types of access to K. Our work, along with an independent Quantum paper by Chakrabarti et al., gives a quantum algorithm that finds a separating hyperplane based on very few membership queries. This in turn leads to a quadratic quantum improvement in the number of membership queries needed for optimization, compared to the best known classical algorithm. Interestingly, our speed-up is based on the Fourier transform (Jordan's algorithm for computing gradients) rather than Grover search. We also prove that quantum algorithms can speed up the general problem of convex optimization only polynomially.

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