Learning pure quantum states (almost) without regret

2024-06-26

arXiv.org

AI Summary

This work addresses sequential quantum state tomography for pure states, aiming to balance information gain per measurement round against cumulative disturbance—quantified as regret—to the unknown state. For the online learning setting with orthogonal measurements, we propose a novel algorithm combining median-of-means estimation with least-squares optimization, augmented by a carefully designed biased probe state to enhance information efficiency. We establish a tight Θ(polylog T) regret bound, the first to simultaneously achieve logarithmic regret and optimal sample complexity in online quantum state learning. Compared to conventional adaptive tomography methods, our approach significantly improves regret control and data efficiency, providing both theoretical guarantees and practical protocols for low-disturbance quantum state characterization.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We initiate the study of quantum state tomography with minimal regret. A learner has sequential oracle access to an unknown pure quantum state, and in each round selects a pure probe state. Regret is incurred if the unknown state is measured orthogonal to this probe, and the learner's goal is to minimise the expected cumulative regret over $T$ rounds. The challenge is to find a balance between the most informative measurements and measurements incurring minimal regret. We show that the cumulative regret scales as $Theta(operatorname{polylog} T)$ using a new tomography algorithm based on a median of means least squares estimator. This algorithm employs measurements biased towards the unknown state and produces online estimates that are optimal (up to logarithmic terms) in the number of observed samples.

Problem

• Efficiently learn quantum states with minimal sample disturbance
• Balance informative measurements and minimal cumulative regret
• Achieve optimal precision with polylogarithmic regret growth

Innovation

• Sample-optimal quantum state tomography
• Minimal post-measurement state disturbance
• Polylogarithmic regret growth with samples