Efficient Closest Matrix Product State Learning in Logarithmic Depth

TL;DR

The paper tackles the problem of efficiently learning the closest Matrix Product State (MPS) representation of an unknown $n$-qudit state from multiple copies, addressing the impractical deep circuits of prior approaches.It introduces a binary-tree disentangling framework that reduces circuit depth to $O(\log n)$ by using block-wise unitaries with rank constraints $\le D^{2}$, and leverages rank-bounded tomography to control error propagation, achieving a total sample complexity of $O\left(\frac{D^{6} d^{2} n^{3} \log(n/\delta)}{(\log_d D)^{3} \epsilon^{4}}\right)$ for exact MPS learning.The framework extends to learning the closest MPS (for inputs not exactly in $\text{MPS}(D)$) with a higher, but still polynomial, sample complexity bound involving $D^{12} n^{7}$ terms and a detailed dependence on $\epsilon$ and $\delta$, and yields a succinct circuit description with $O(n d^{4} D^{8})$ parameters.By reusing earlier ideas and proving correctness for the new constructions (Algorithms A2 and A4), the work demonstrates both depth reduction and improved sample complexity relative to prior linear-depth approaches, with practical implications for near-term quantum devices.

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.

Figures (2)

• Figure 1: The quantum circuits used in MPS tomography in (A) the work of Landon-Cardinal et al. landoncardinal2010efficient the work of Bakshi et al. bakshi2025learning, and in (B) this paper.
• Figure 2: An example of the circuit generated from Algorithm \ref{['alg:A2']} with $n=29, p=2$. Each rectangle represents a unitary $U_i^j$, where the superscript $j$ denotes the layer index and the subscript $i$ labels the unitary within that layer (with indexing restarted at each layer). The vertical lines correspond to qudits, ordered from left to right. The gates are arranged vertically to indicate temporal progression, with lower gates applied after those above. The support sets ${B}_i^1$ of qubits for the unitaries in the first layer and the sets of qubits $\widetilde{\mathcal{B}}_i^j$ carried forward to the next layer are also indicated in this figure.

Theorems & Definitions (46)

• Definition 1: Matrix Product State with Bond Dimension $D$ (MPS($D$))
• Theorem 1
• Theorem 2
• Definition 2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 3: Quantum State Learning Problem
• Theorem 3: Theorem 1 in anshu2024survey