Low-rank tensor method cuts quantum state estimation costs
A Block Tensor Train Burer-Monteiro Framework for Low-Rank Quantum State Tomography
Machine Learning
Summary
Estimating the full state of a quantum system becomes extremely difficult as the system grows larger due to the huge size of the data involved. The authors introduce a way to compress this data using a special mathematical format called block tensor train, enabling faster and more memory-efficient estimation of quantum states. Their method ensures that key mathematical properties of quantum states are always preserved and allows for flexible adjustments during the calculation. Experiments show the approach can accurately reconstruct quantum states from fewer measurements, using significantly less computing resources.
What this means in practice
- •For quantum hardware engineers: Estimate and verify the states of quantum devices efficiently using compressed measurement data without requiring exponential computational resources.
- •For compressed sensing developers: Implement scalable algorithms for reconstructing low-rank quantum states that reduce memory and time costs in practical tomography tasks.
Authors
Shakir Showkat Sofi, Charlotte Vermeylen, Fatemeh Mohammadi, Lieven De Lathauwer
Abstract
Quantum state tomography 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 computationally prohibitive as the system size increases due to the exponential growth of the density matrix, describing a quantum state, with the number of qubits. We propose a low-rank tensor-network framework for mixed-state quantum state tomography based on a block tensor train (Block-TT) factorization. Specifically, the density matrix is represented as the contraction of a Block-TT with its Hermitian transpose, yielding a TT analogue of the Burer-Monteiro factorization. This parameterization guarantees Hermiticity and positive semidefiniteness by construction while compressing the number of optimization variables from exponential to linear in the number of qubits. Building on this representation, we develop single-site and two-site density matrix renormalization group (DMRG) algorithms for estimating quantum states from compressed measurements. The resulting methods operate directly on the compressed parameterization, support adaptive rank refinement, and exploit efficient tensor-network contractions for expectation-value evaluation. The framework is applicable to a broad class of low-rank quantum states, including pure states, nearly pure states, and ground states that admit accurate tensor-network approximations. Numerical experiments demonstrate accurate state reconstruction from limited measurements together with substantial reductions in memory requirements and computational cost compared with conventional low-rank tomography methods.