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arXiv:2210.06539 · neural information processing systems

Quantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constants

Andrew M. Childs, Tongyang Li, Jin-Peng Liu, Chunhao Wang, Ruizhe Zhang

Published Jan 1, 2022• 0 citations
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Abstract

Given a convex function $f\colon\mathbb{R}^{d}\to\mathbb{R}$, the problem of sampling from a distribution $\propto e^{-f(x)}$ is called log-concave sampling. This task has wide applications in machine learning, physics, statistics, etc. In this work, we develop quantum algorithms for sampling log-concave distributions and for estimating their normalizing constants $\int_{\mathbb{R}^d}e^{-f(x)}\mathrm{d} x$. First, we use underdamped Langevin diffusion to develop quantum algorithms that match the query complexity (in terms of the condition number $κ$ and dimension $d$) of analogous classical algorithms that use gradient (first-order) queries, even though the quantum algorithms use only evaluation (zeroth-order) queries. For estimating normalizing constants, these algorithms also achieve quadratic speedup in the multiplicative error $ε$. Second, we develop quantum Metropolis-adjusted Langevin algorithms with query complexity $\widetilde{O}(κ^{1/2}d)$ and $\widetilde{O}(κ^{1/2}d^{3/2}/ε)$ for log-concave sampling and normalizing constant estimation, respectively, achieving polynomial speedups in $κ,d,ε$ over the best known classical algorithms by exploiting quantum analogs of the Monte Carlo method and quantum walks. We also prove a $1/ε^{1-o(1)}$ quantum lower bound for estimating normalizing constants, implying near-optimality of our quantum algorithms in $ε$.

DOI
10.52202/068431-1686

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We establish a systematic framework of unbiased quantum sampling and estimation protocols for the classical Gibbs expectation. This framework generalizes existing approaches to the partition function estimation and has broader applications in various fields. We consider sampling and estimation for a wide class of non-log-concave distributions, particularly heavy-tailed ones, under relaxed assumptions beyond strong convexity, such as dissipativity. We develop an unbiased extension of quantum-accelerated multilevel Monte Carlo (QA-MLMC) to eliminate all biases from discretization and time truncation, together with introducing a change-of-measure approach and the Girsanov theorem via Radon-Nikodym derivatives. As a result, our approach achieves quantum complexity O~(ε−1)\widetilde{\mathcal{O}}(ε^{-1})O(ε−1) within error εεε, whereas the classical MLMC requires O~(ε−2)\widetilde{\mathcal{O}}(ε^{-2})O(ε−2) and existing quantum algorithms yield biased estimators under stronger assumptions. Furthermore, our unified framework enables unbiased quantum sampling and estimation for certain heavy-tailed distributions after transformation. We provide several concrete applications of our approach in statistics, machine learning, and finance, towards more practical scenarios of the quantum acceleration of stochastic processes.
We initiate the study of quantum algorithms for optimizing approximately convex functions. Given a convex set K⊆Rn{\cal K}\subseteq\mathbb{R}^{n}K⊆Rn and a function F ⁣:Rn→RF\colon\mathbb{R}^{n}\to\mathbb{R}F:Rn→R such that there exists a convex function f ⁣:K→Rf\colon\mathcal{K}\to\mathbb{R}f:K→R satisfying sup⁡x∈K∣F(x)−f(x)∣≤ε/n\sup_{x\in{\cal K}}|F(x)-f(x)|\leq ε/nsupx∈K​∣F(x)−f(x)∣≤ε/n, our quantum algorithm finds an x∗∈Kx^{*}\in{\cal K}x∗∈K such that F(x∗)−min⁡x∈KF(x)≤εF(x^{*})-\min_{x\in{\cal K}} F(x)\leqεF(x∗)−minx∈K​F(x)≤ε using O~(n3)\tilde{O}(n^{3})O~(n3) quantum evaluation queries to FFF. This achieves a polynomial quantum speedup compared to the best-known classical algorithms. As an application, we give a quantum algorithm for zeroth-order stochastic convex bandits with O~(n5log⁡2T)\tilde{O}(n^{5}\log^{2} T)O~(n5log2T) regret, an exponential speedup in TTT compared to the classical Ω(T)Ω(\sqrt{T})Ω(T​) lower bound. Technically, we achieve quantum speedup in nnn by exploiting a quantum framework of simulated annealing and adopting a quantum version of the hit-and-run walk. Our speedup in TTT for zeroth-order stochastic convex bandits is due to a quadratic quantum speedup in multiplicative error of mean estimation.
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