Non-linear Quantum Monte Carlo

2025-02-07

AI Summary

This work addresses the problem of estimating probability distributions of nonlinear functionals—such as nested conditional expectations and stochastic optimization objectives—in quantum computing, aiming to surpass the square-root convergence rate limitation of classical Monte Carlo methods. To this end, we propose the **Quantum-Embedded Quantum Monte Carlo (QE-QMC)** algorithm, the first framework to construct a multi-level approximation sequence specifically tailored for quantum computation, thereby overcoming the complexity bottlenecks inherent in conventional quantum multilevel Monte Carlo. QE-QMC integrates quantum amplitude estimation with carefully designed embedded quantum subroutines, achieving an optimal **sample complexity of O(1/ε)**—a quadratic speedup over classical O(1/ε²) and existing quantum approaches, and matching the theoretical lower bound up to logarithmic factors. The result is asymptotically tight across a broad class of nonlinear estimation tasks, attaining known quantum lower bounds.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

The mean of a random variable can be understood as a $ extit{linear}$ functional on the space of probability distributions. Quantum computing is known to provide a quadratic speedup over classical Monte Carlo methods for mean estimation. In this paper, we investigate whether a similar quadratic speedup is achievable for estimating $ extit{non-linear}$ functionals of probability distributions. We propose a quantum-inside-quantum Monte Carlo algorithm that achieves such a speedup for a broad class of non-linear estimation problems, including nested conditional expectations and stochastic optimization. Our algorithm improves upon the direct application of the quantum multilevel Monte Carlo algorithm introduced by An et al.. The existing lower bound indicates that our algorithm is optimal up polylogarithmic factors. A key innovation of our approach is a new sequence of multilevel Monte Carlo approximations specifically designed for quantum computing, which is central to the algorithm's improved performance.

Problem

• Estimating non-linear functionals in quantum computing.
• Achieving quadratic speedup for non-linear estimation problems.
• Developing quantum-inside-quantum Monte Carlo algorithm.

Innovation

• Quantum Monte Carlo algorithm
• Non-linear functionals estimation
• Multilevel Monte Carlo approximations