Tensor Train Quantum State Tomography using Compressed Sensing

TL;DR

This work targets the exponential scaling of quantum state tomography by introducing a Block-TT (tensor-train) density-matrix representation that inherently preserves positive semidefiniteness. It recasts QST as a non-convex optimization over TT cores, minimizing a least-squares fit to observed measurements with TT-SVD projections to maintain a compact, left-orthogonal form. Empirical results show the method matches or outperforms existing low-rank approaches while offering substantially improved memory and computational efficiency, scaling favorably with system size up to 12 qubits. The approach leverages tensor-network contractions to efficiently compute observables, addressing the curse of dimensionality in large-scale QST and enabling practical tomography for complex quantum devices.

Abstract

Quantum state tomography (QST) is a fundamental technique for estimating the state of a quantum system from measured data and plays a crucial role in evaluating the performance of quantum devices. However, standard estimation methods become impractical due to the exponential growth of parameters in the state representation. In this work, we address this challenge by parameterizing the state using a low-rank block tensor train decomposition and demonstrate that our approach is both memory- and computationally efficient. This framework applies to a broad class of quantum states that can be well approximated by low-rank decompositions, including pure states, nearly pure states, and ground states of Hamiltonians.

Figures (6)

• Figure 1: Basic tensor diagrams.
• Figure 2: Tensor diagram of a TTM decomposition.
• Figure 3: Decomposition of $\bm{\rho}_{\mathrm{TT}}$ into Block-TT format, i.e., $\bm{\rho}_{\mathrm{TT}} = \mathbf{\mathcal{A}}\bullet_{(N-1)} \mathbf{\mathcal{A}}^{\mathrm{H}},$ where $\mathbf{\mathcal{A}} \in \mathbb{C}^{2 \times 2 \times \cdots \times K \times 2}.$
• Figure 4: Tensor network contraction for computing the expectation value $\bm{\rho}_{\mathrm{TT}} \bar{\bullet}\mathbf{\mathcal{E}}_m,$ with $\bm{\rho}_{\mathrm{TT}}$ and $\mathbf{\mathcal{E}}_m$ represented in TT format.
• Figure 5: The proposed method outperforms MLE and has an overall similar performance to LR. The CVX method performs best at low sampling ratios ($M/D^2 < 0.3$), but this approach solves the original problem \ref{['eqn:sdpQST']} and becomes very expensive as the problem size increases.