Efficient Closest Matrix Product State Learning in Logarithmic Depth

• URL: http://arxiv.org/abs/2510.07798v1
• Date: Thu, 09 Oct 2025 05:25:00 GMT
• Title: Efficient Closest Matrix Product State Learning in Logarithmic Depth
• Authors: Chia-Ying Lin, Nai-Hui Chia, Shih-Han Hung,
• Abstract summary: We show a new efficient MPS learning algorithm that runs in $O(log n)$ depth and has sample complexity $O(n3)$.<n>Our algorithms also improve both sample complexity and circuit depth of previous known algorithms.
• Score: 1.887251485029662
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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 $\Omega(n)$. 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.

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