Quantum Langevin Dynamics for Optimization

We initiate the study of utilizing Quantum Langevin Dynamics (QLD) to solve optimization problems, particularly those non-convex objective functions that present substantial obstacles for traditional gradient descent algorithms. Specifically, we examine the dynamics of a system coupled with an infinite heat bath. This interaction induces both random quantum noise and a deterministic damping effect to the system, which nudge the system towards a steady state that hovers near the global minimum of objective functions. We theoretically prove the convergence of QLD in convex landscapes, demonstrating that the average energy of the system can approach zero in the low temperature limit with an exponential decay rate correlated with the evolution time. Numerically, we first show the energy dissipation capability of QLD by retracing its origins to spontaneous emission. Furthermore, we conduct detailed discussion of the impact of each parameter. Finally, based on the observations when comparing QLD with classical Fokker-Plank-Smoluchowski equation, we propose a time-dependent QLD by making temperature and $\hbar$ time-dependent parameters, which can be theoretically proven to converge better than the time-independent case and also outperforms a series of state-of-the-art quantum and classical optimization algorithms in many non-convex landscapes.

Near-Optimal Quantum Coreset Construction Algorithms for Clustering

$k$-Clustering in $\mathbb{R}^d$ (e.g., $k$-median and $k$-means) is a fundamental machine learning problem. While near-linear time approximation algorithms were known in the classical setting for a dataset with cardinality $n$, it remains open to find sublinear-time quantum algorithms. We give quantum algorithms that find coresets for $k$-clustering in $\mathbb{R}^d$ with $\tilde{O}(\sqrt{nk}d^{3/2})$ query complexity. Our coreset reduces the input size from $n$ to $\mathrm{poly}(k\epsilon^{-1}d)$, so that existing $\alpha$-approximation algorithms for clustering can run on top of it and yield $(1 + \epsilon)\alpha$-approximation. This eventually yields a quadratic speedup for various $k$-clustering approximation algorithms. We complement our algorithm with a nearly matching lower bound, that any quantum algorithm must make $\Omega(\sqrt{nk})$ queries in order to achieve even $O(1)$-approximation for $k$-clustering.

Logarithmic-Regret Quantum Learning Algorithms for Zero-Sum Games

We propose the first online quantum algorithm for zero-sum games with $\tilde O(1)$ regret under the game setting. Moreover, our quantum algorithm computes an $\varepsilon$-approximate Nash equilibrium of an $m \times n$ matrix zero-sum game in quantum time $\tilde O(\sqrt{m+n}/\varepsilon^{2.5})$, yielding a quadratic improvement over classical algorithms in terms of $m, n$. Our algorithm uses standard quantum inputs and generates classical outputs with succinct descriptions, facilitating end-to-end applications. As an application, we obtain a fast quantum linear programming solver. Technically, our online quantum algorithm "quantizes" classical algorithms based on the optimistic multiplicative weight update method. At the heart of our algorithm is a fast quantum multi-sampling procedure for the Gibbs sampling problem, which may be of independent interest.

Provably Efficient Exploration in Quantum Reinforcement Learning with Logarithmic Worst-Case Regret

While quantum reinforcement learning (RL) has attracted a surge of attention recently, its theoretical understanding is limited. In particular, it remains elusive how to design provably efficient quantum RL algorithms that can address the exploration-exploitation trade-off. To this end, we propose a novel UCRL-style algorithm that takes advantage of quantum computing for tabular Markov decision processes (MDPs) with $S$ states, $A$ actions, and horizon $H$, and establish an $\mathcal{O}(\mathrm{poly}(S, A, H, \log T))$ worst-case regret for it, where $T$ is the number of episodes. Furthermore, we extend our results to quantum RL with linear function approximation, which is capable of handling problems with large state spaces. Specifically, we develop a quantum algorithm based on value target regression (VTR) for linear mixture MDPs with $d$-dimensional linear representation and prove that it enjoys $\mathcal{O}(\mathrm{poly}(d, H, \log T))$ regret. Our algorithms are variants of UCRL/UCRL-VTR algorithms in classical RL, which also leverage a novel combination of lazy updating mechanisms and quantum estimation subroutines. This is the key to breaking the $\Omega(\sqrt{T})$-regret barrier in classical RL. To the best of our knowledge, this is the first work studying the online exploration in quantum RL with provable logarithmic worst-case regret.

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

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 $\kappa$ 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 $\epsilon$. Second, we develop quantum Metropolis-adjusted Langevin algorithms with query complexity $\widetilde{O}(\kappa^{1/2}d)$ and $\widetilde{O}(\kappa^{1/2}d^{3/2}/\epsilon)$ for log-concave sampling and normalizing constant estimation, respectively, achieving polynomial speedups in $\kappa,d,\epsilon$ over the best known classical algorithms by exploiting quantum analogs of the Monte Carlo method and quantum walks. We also prove a $1/\epsilon^{1-o(1)}$ quantum lower bound for estimating normalizing constants, implying near-optimality of our quantum algorithms in $\epsilon$.

On Quantum Speedups for Nonconvex Optimization via Quantum Tunneling Walks

Classical algorithms are often not effective for solving nonconvex optimization problems where local minima are separated by high barriers. In this paper, we explore possible quantum speedups for nonconvex optimization by leveraging the global effect of quantum tunneling. Specifically, we introduce a quantum algorithm termed the quantum tunneling walk (QTW) and apply it to nonconvex problems where local minima are approximately global minima. We show that QTW achieves quantum speedup over classical stochastic gradient descents (SGD) when the barriers between different local minima are high but thin and the minima are flat. Based on this observation, we construct a specific double-well landscape, where classical algorithms cannot efficiently hit one target well knowing the other well but QTW can when given proper initial states near the known well. Finally, we corroborate our findings with numerical experiments.

Quantum Speedups of Optimizing Approximately Convex Functions with Applications to Logarithmic Regret Stochastic Convex Bandits

We initiate the study of quantum algorithms for optimizing approximately convex functions. Given a convex set ${\cal K}\subseteq\mathbb{R}^{n}$ and a function $F\colon\mathbb{R}^{n}\to\mathbb{R}$ such that there exists a convex function $f\colon\mathcal{K}\to\mathbb{R}$ satisfying $\sup_{x\in{\cal K}}|F(x)-f(x)|\leq \epsilon/n$, our quantum algorithm finds an $x^{*}\in{\cal K}$ such that $F(x^{*})-\min_{x\in{\cal K}} F(x)\leq\epsilon$ using $\tilde{O}(n^{3})$ quantum evaluation queries to $F$. 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 $\tilde{O}(n^{5}\log^{2} T)$ regret, an exponential speedup in $T$ compared to the classical $\Omega(\sqrt{T})$ lower bound. Technically, we achieve quantum speedup in $n$ by exploiting a quantum framework of simulated annealing and adopting a quantum version of the hit-and-run walk. Our speedup in $T$ for zeroth-order stochastic convex bandits is due to a quadratic quantum speedup in multiplicative error of mean estimation.

Adaptive Online Learning of Quantum States

In the fundamental problem of shadow tomography, the goal is to efficiently learn an unknown $d$-dimensional quantum state using projective measurements. However, it is rarely the case that the underlying state remains stationary: changes may occur due to measurements, environmental noise, or an underlying Hamiltonian state evolution. In this paper we adopt tools from adaptive online learning to learn a changing state, giving adaptive and dynamic regret bounds for online shadow tomography that are polynomial in the number of qubits and sublinear in the number of measurements. Our analysis utilizes tools from complex matrix analysis to cope with complex numbers, which may be of independent interest in online learning. In addition, we provide numerical experiments that corroborate our theoretical results.

Quantum Multi-Armed Bandits and Stochastic Linear Bandits Enjoy Logarithmic Regrets

Multi-arm bandit (MAB) and stochastic linear bandit (SLB) are important models in reinforcement learning, and it is well-known that classical algorithms for bandits with time horizon $T$ suffer $\Omega(\sqrt{T})$ regret. In this paper, we study MAB and SLB with quantum reward oracles and propose quantum algorithms for both models with $O(\mbox{poly}(\log T))$ regrets, exponentially improving the dependence in terms of $T$. To the best of our knowledge, this is the first provable quantum speedup for regrets of bandit problems and in general exploitation in reinforcement learning. Compared to previous literature on quantum exploration algorithms for MAB and reinforcement learning, our quantum input model is simpler and only assumes quantum oracles for each individual arm.

Escape saddle points by a simple gradient-descent based algorithm

Escaping saddle points is a central research topic in nonconvex optimization. In this paper, we propose a simple gradient-based algorithm such that for a smooth function $f\colon\mathbb{R}^n\to\mathbb{R}$, it outputs an $\epsilon$-approximate second-order stationary point in $\tilde{O}(\log n/\epsilon^{1.75})$ iterations. Compared to the previous state-of-the-art algorithms by Jin et al. with $\tilde{O}((\log n)^{4}/\epsilon^{2})$ or $\tilde{O}((\log n)^{6}/\epsilon^{1.75})$ iterations, our algorithm is polynomially better in terms of $\log n$ and matches their complexities in terms of $1/\epsilon$. For the stochastic setting, our algorithm outputs an $\epsilon$-approximate second-order stationary point in $\tilde{O}((\log n)^{2}/\epsilon^{4})$ iterations. Technically, our main contribution is an idea of implementing a robust Hessian power method using only gradients, which can find negative curvature near saddle points and achieve the polynomial speedup in $\log n$ compared to the perturbed gradient descent methods. Finally, we also perform numerical experiments that support our results.