# Tensor Train Quantum State Tomography using Compressed Sensing

arXiv: https://arxiv.org/abs/2506.23560 · HTML: https://mlsanity.com/paper/2506.23560/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 37 references that are in the map.
- First posted on arXiv 30 Jun 2025.

## Builds on (most cited first, 5 of 5)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Dropping Convexity for Faster Semi-definite Optimization](https://mlsanity.com/paper/1509.03917/) (2015)
- [Quantum State Tomography for Matrix Product Density Operators](https://mlsanity.com/paper/2306.09432/) (2023)
- [Fast quantum state reconstruction via accelerated non-convex programming](https://mlsanity.com/paper/2104.07006/) (2021)
- [A Riemannian rank-adaptive method for higher-order tensor completion in the tensor-train format](https://mlsanity.com/paper/2402.12182/) (2024)

## Built on by (most cited first, 5 of 5)

- [Tensor Train Completion from Fiberwise Observations Along a Single Mode](https://mlsanity.com/paper/2509.18149/) (2025)
- [Tomography by Design: An Algebraic Approach to Low-Rank Quantum States](https://mlsanity.com/paper/2602.15202/) (2026)
- [StoTAM: Stochastic Alternating Minimization for Tucker-Structured Tensor Sensing](https://mlsanity.com/paper/2601.13522/) (2026)
- [Fixed-Protocol Amortized MPS Tomography with Conformalized Predictive Uncertainty](https://mlsanity.com/paper/2607.11273/) (2026)
- [A Block Tensor Train Burer-Monteiro Framework for Low-Rank Quantum State Tomography](https://mlsanity.com/paper/2609.09457/) (2026)

## Abstract (arXiv)

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.