Improved Quantum Algorithms for Reinforcement Learning Under a Generative Model

Reinforcement learning is a subfield of machine learning that studies how an agent interacts with an environment in order to extract as large a reward as possible. A standard approach to study such interaction is through Markov Decision Processes (MDPs) and the task of choosing an optimal policy --- a function that tells the agent which action to take. In this work, we study two types of MDPs --- finite-horizon and infinite-horizon discounted --- and propose new quantum algorithms for computing approximate optimal policies. Our quantum algorithms are based on a new combination of standard value iteration and quantum subroutines like quantum mean estimation and quantum maximum finding, overall enhanced with techniques from sample-optimal classical algorithms. Our resulting query complexities improve upon previous works, thus approaching already established quantum lower bounds.

Quantum Speedups for Stochastic Optimization with Heavy-Tailed Noise

We study stochastic optimization with heavy-tailed gradient noise. We first propose a novel quantum mean estimator for multivariate heavy-tailed random variables that achieves lower query complexity than optimal classical estimators in the low-dimensional regime. We further develop an unbiased quantum mean estimator by applying a generalized multi-level Monte Carlo technique. We prove quantum lower bounds showing that, when the dimension $d$ of the random vector is small and can be viewed as a constant, our quantum estimators are optimal up to logarithmic factors. We further derive stronger dimension-dependent lower bounds for tail index $p>4/3$, showing that a nontrivial dependence on the dimension is unavoidable in the low-dimensional regime. Based on these estimators, we propose a quantum normalized stochastic gradient descent method ($\texttt{QNSGD}$), which finds an $ε$-stationary point using $\tilde{\mathcal{O}}\big(\sqrt d\,ε^{-\frac{5p-4}{2p-2}}\big)$ queries to the quantum stochastic gradient oracle. For a convex objective function, we propose a quantum projected stochastic gradient descent method ($\texttt{QPSGD}$), which computes a solution with $ε$-optimal solution using $\tilde{\mathcal{O}}\big(\sqrt d\,ε^{-\frac{3p-2}{2p-2}}+ε^{-2}\big)$ queries in expectation. These sharper bounds improve upon the classical lower bounds $Ω\big(ε^{-\frac{3p-2}{p-1}}\big)$ for nonconvex problems and $Ω\big(ε^{-\frac{p}{p-1}}\big)$ for convex problems in the low-dimensional regimes $d\lesssimε^{-\frac{p}{p-1}}$ and $d\lesssimε^{-\frac{2-p}{p-1}}$, respectively.

From Meta Idea to Advanced Mathematical Discovery -- Human-AI Co-Discovery of Sign-Embedding Quantum Algorithms

AI-assisted mathematics is often evaluated on solving predefined problems. In practice, however, many important advances begin earlier, when a vague research intuition is transformed into a concrete problem, a promising route, and a theorem family worth proving. This report studies that stage through a case study that led to sign-embedding quantum algorithms for matrix equations and matrix functions, foundational primitives in quantum linear algebra and operator-output quantum algorithms. The project began with a human-originated intuition that rational approximation is especially effective for jump-type functions such as the sign function, and might therefore serve as a design principle for quantum algorithms. Rather than merely assisting after the problem was fixed, AI-assisted exploration, including workflows later integrated into the agentic AI-mathematician system AIM, played a key role in expanding this intuition into a route map, comparing candidate formulations, and converging toward sign embedding as the central framework. AIM then helped connect a known matrix-sign identity to wider classes of matrix equations and matrix functions, and drafted proof and complexity calculations. The decisive scientific judgments remained human: selecting which human-AI-expanded routes were worth pursuing, rejecting a Cayley-trapezoidal approximation when its validity required a hidden condition, and refining the Sylvester implementation from a coarse quadratic-gap query route to the final factorized and scaled analysis. The report argues that human-AI co-discovery workflows, with systems such as AIM as important components, are most valuable not as standalone theorem provers, but as research partners for problem formation, connection discovery, derivation, and skeptical review inside a human-gated research loop.