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Quantum speedups of some general-purpose numerical optimisation algorithms

Ashley Montanaro, Noah Linden, Cezar-Mihail Alexandru, Ella Bridgett-Tomkinson, Joseph Paul MacManus, Hannah Morris

Published Sep 3, 2020• 14 citations
Publisher

Abstract

Abstract We give quantum speedups of several general-purpose numerical optimisation methods for minimising a function <?CDATA $f:{\mathbb{R}}^{n}\to \mathbb{R}$?> f : R n → R . First, we show that many techniques for global optimisation under a Lipschitz constraint can be accelerated near-quadratically. Second, we show that backtracking line search, an ingredient in quasi-Newton optimisation algorithms, can be accelerated up to quadratically. Third, we show that a component of the Nelder–Mead algorithm can be accelerated by up to a multiplicative factor of <?CDATA $O\left(\sqrt{n}\right)$?> O ( n ) . Fourth, we show that a quantum gradient computation algorithm of Gilyén et al can be used to approximately compute gradients in the framework of stochastic gradient descent. In each case, our results are based on applying existing quantum algorithms to accelerate specific components of the classical algorithms, rather than developing new quantum techniques.

DOI
10.1088/2058-9565/abb003

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Abstract We give quantum speedups of several general-purpose numerical optimisation methods for minimising a function <?CDATA $f:{\mathbb{R}}^{n}\to \mathbb{R}$?> f : R n → R . First, we show that many techniques for global optimisation under a Lipschitz constraint can be accelerated near-quadratically. Second, we show that backtracking line search, an ingredient in quasi-Newton optimisation algorithms, can be accelerated up to quadratically. Third, we show that a component of the Nelder–Mead algorithm can be accelerated by up to a multiplicative factor of <?CDATA $O\left(\sqrt{n}\right)$?> O ( n ) . Fourth, we show that a quantum gradient computation algorithm of Gilyén et al can be used to approximately compute gradients in the framework of stochastic gradient descent. In each case, our results are based on applying existing quantum algorithms to accelerate specific components of the classical algorithms, rather than developing new quantum techniques.
Abstract We give quantum speedups of several general-purpose numerical optimisation methods for minimising a function f : R n → R . First, we show that many techniques for global optimisation under a Lipschitz constraint can be accelerated near-quadratically. Second, we show that backtracking line search, an ingredient in quasi-Newton optimisation algorithms, can be accelerated up to quadratically. Third, we show that a component of the Nelder–Mead algorithm can be accelerated by up to a multiplicative factor of O ( n ) . Fourth, we show that a quantum gradient computation algorithm of Gilyén et al can be used to approximately compute gradients in the framework of stochastic gradient descent. In each case, our results are based on applying existing quantum algorithms to accelerate specific components of the classical algorithms, rather than developing new quantum techniques.
We consider the problem of minimizing a continuous function given quantum access to a stochastic gradient oracle. We provide two new methods for the special case of minimizing a Lipschitz convex function. Each method obtains a dimension versus accuracy trade-off which is provably unachievable classically and we prove that one method is asymptotically optimal in low-dimensional settings. Additionally, we provide quantum algorithms for computing a critical point of a smooth non-convex function at rates not known to be achievable classically. To obtain these results we build upon the quantum multivariate mean estimation result of Cornelissen et al. 2022 and provide a general quantum-variance reduction technique of independent interest.
We study the first-order convex optimization problem, where we have black-box access to a (not necessarily smooth) function f:Rn→Rf:\mathbb{R}^n \to \mathbb{R}f:Rn→R and its (sub)gradient. Our goal is to find an εεε-approximate minimum of fff starting from a point that is distance at most RRR from the true minimum. If fff is GGG-Lipschitz, then the classic gradient descent algorithm solves this problem with O((GR/ε)2)O((GR/ε)^{2})O((GR/ε)2) queries. Importantly, the number of queries is independent of the dimension nnn and gradient descent is optimal in this regard: No deterministic or randomized algorithm can achieve better complexity that is still independent of the dimension nnn. In this paper we reprove the randomized lower bound of Ω((GR/ε)2)Ω((GR/ε)^{2})Ω((GR/ε)2) using a simpler argument than previous lower bounds. We then show that although the function family used in the lower bound is hard for randomized algorithms, it can be solved using O(GR/ε)O(GR/ε)O(GR/ε) quantum queries. We then show an improved lower bound against quantum algorithms using a different set of instances and establish our main result that in general even quantum algorithms need Ω((GR/ε)2)Ω((GR/ε)^2)Ω((GR/ε)2) queries to solve the problem. Hence there is no quantum speedup over gradient descent for black-box first-order convex optimization without further assumptions on the function family.
Monte Carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically. One important example is the use in statistical physics of rapidly mixing Markov chains to approximately compute partition functions. In this work, we describe a quantum algorithm which can accelerate Monte Carlo methods in a very general setting. The algorithm estimates the expected output value of an arbitrary randomized or quantum subroutine with bounded variance, achieving a near-quadratic speedup over the best possible classical algorithm. Combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions, which use multiple-stage Markov chain Monte Carlo techniques. The quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently.
Quantum algorithms can deliver asymptotic speedups over their classical counterparts. However, there are few cases where a substantial quantum speedup has been worked out in detail for reasonably-sized problems, when compared with the best classical algorithms and taking into account realistic hardware parameters and overheads for fault-tolerance. All known examples of such speedups correspond to problems related to simulation of quantum systems and cryptography. Here we apply general-purpose quantum algorithms for solving constraint satisfaction problems to two families of prototypical NP-complete problems: boolean satisfiability and graph colouring. We consider two quantum approaches: Grover's algorithm and a quantum algorithm for accelerating backtracking algorithms. We compare the performance of optimised versions of these algorithms, when applied to random problem instances, against leading classical algorithms. Even when considering only problem instances that can be solved within one day, we find that there are potentially large quantum speedups available. In the most optimistic parameter regime we consider, this could be a factor of over 105 relative to a classical desktop computer; in the least optimistic regime, the speedup is reduced to a factor of over 103 . However, the number of physical qubits used is extremely large, and improved fault-tolerance methods will likely be needed to make these results practical. In particular, the quantum advantage disappears if one includes the cost of the classical processing power required to perform decoding of the surface code using current techniques.
Gradient-based numerical methods are ubiquitous in optimization techniques frequently applied in industry to solve practical problems. Often times, evaluating the objective function is a complicated process, so estimating the gradient of a function with as few function evaluations as possible is a natural problem. We investigate whether quantum computers can perform ℓ∞\ell^{\infty}ℓ∞-approximate gradient estimation of multivariate functions f:Rd→Rf : \mathbb{R}^d \to \mathbb{R}f:Rd→R with fewer function evaluations than classically. Following previous work by Jordan [Jor05] and Gilyén et al. [GAW18], we prove that one can calculate an ℓ∞\ell^{\infty}ℓ∞-approximation of the gradient of fff with a query complexity that scales sublinearly with ddd under weaker smoothness conditions than previously considered. Furthermore, for a particular subset of smoothness conditions, we prove a new lower bound on the query complexity of the gradient estimation problem, proving essential optimality of Gilyén et al.'s gradient estimation algorithm in a broader range of parameter values, and affirming the validity of their conjecture [GAW18]. Moreover, we improve their lower bound qualitatively by showing that their algorithm is also optimal for functions that satisfy the imposed smoothness conditions globally instead of locally. Finally, we introduce new ideas to prove lower bounds on the query complexity of the ℓp\ell^pℓp-approximate gradient estimation problem where p∈[1,∞)p \in [1,\infty)p∈[1,∞), and prove that lifting Gilyén et al.'s algorithm to this domain in the canonical manner is essentially optimal.