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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Cited by

[1] Shouvanik Chakrabarti, Andrew M. Childs, Tongyang Li, and Xiaodi Wu, "Quantum algorithms and lower bounds for convex optimization", arXiv:1809.01731, Quantum 4, 221 (2020).

[2] Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf, "Quantum SDP-Solvers: Better upper and lower bounds", arXiv:1705.01843, Quantum 4, 230 (2020).

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

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

[5] Aram W. Harrow, "Small quantum computers and large classical data sets", arXiv:2004.00026.

[6] Joran van Apeldoorn and Sander Gribling, "Simon's problem for linear functions", arXiv:1810.12030.

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

[8] 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.

[9] Yassine Hamoudi, Maharshi Ray, Patrick Rebentrost, Miklos Santha, Xin Wang, and Siyi Yang, "Quantum algorithms for hedging and the Sparsitron", arXiv:2002.06003.

[10] Ammar Daskin, "The quantum version of the shifted power method and its application in quadratic binary optimization", arXiv:1809.01378.

[11] Cezar-Mihail Alexandru, Ella Bridgett-Tomkinson, Noah Linden, Joseph MacManus, Ashley Montanaro, and Hannah Morris, "Quantum speedups of some general-purpose numerical optimisation algorithms", Quantum Science and Technology 5 4, 045014 (2020).

[12] Chenyi Zhang, Jiaqi Leng, and Tongyang Li, "Quantum Algorithms for Escaping from Saddle Points", arXiv:2007.10253.

The above citations are from Crossref's cited-by service (last updated successfully 2020-10-28 00:09:44) and SAO/NASA ADS (last updated successfully 2020-10-28 00:09:45). The list may be incomplete as not all publishers provide suitable and complete citation data.

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