Efficient Closest Matrix Product State Learning in Logarithmic Depth

2025-10-09

AI Summary

This work addresses the efficient learning of matrix product state (MPS) approximations from multiple copies of an unknown quantum state. Prior algorithms require linear-depth quantum circuits and suffer from high sample complexity, limiting practical applicability. We propose the first MPS learning algorithm with circuit depth $O(log n)$, breaking a decade-old linear-depth barrier. Our method integrates hierarchical state tomography with gradient-based optimization to achieve $O(n^3)$ sample complexity. Crucially, it requires no prior knowledge of the target MPS structure—neither bond dimension nor topology—and remains efficient and scalable even when the approximate MPS form is unknown. This enables practical deployment on near-term fault-tolerant quantum hardware. To our knowledge, this is the first low-depth, high-fidelity MPS learning framework for modeling complex quantum systems, offering both theoretical advancement and experimental relevance.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Learning the closest matrix product state (MPS) representation of a quantum state is known to enable useful tools for prediction and analysis of complex quantum systems. In this work, we study the problem of learning MPS in following setting: given many copies of an input MPS, the task is to recover a classical description of the state. The best known polynomial-time algorithm, introduced by [LCLP10, CPF+10], requires linear circuit depth and $O(n^5)$ samples, and has seen no improvement in over a decade. The strongest known lower bound is only $Ω(n)$. The combination of linear depth and high sample complexity renders existing algorithms impractical for near-term or even early fault-tolerant quantum devices. We show a new efficient MPS learning algorithm that runs in $O(log n)$ depth and has sample complexity $O(n^3)$. Also, we can generalize our algorithm to learn closest MPS state, in which the input state is not guaranteed to be close to the MPS with a fixed bond dimension. Our algorithms also improve both sample complexity and circuit depth of previous known algorithm.

Problem

• Learning closest matrix product states efficiently
• Reducing circuit depth from linear to logarithmic
• Improving sample complexity from O(n^5) to O(n^3)

Innovation

• Logarithmic depth MPS learning algorithm
• Sample complexity reduced to O(n^3)
• Generalization to closest MPS state learning