# Learning finitely correlated states: stability of the spectral reconstruction

arXiv: https://arxiv.org/abs/2312.07516 · HTML: https://mlsanity.com/paper/2312.07516/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 64 references that are in the map.
- First posted on arXiv 12 Dec 2023.

## Builds on (most cited first, 9 of 9)

- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Efficient quantum tomography II](https://mlsanity.com/paper/1612.00034/) (2016)
- [Sample-efficient learning of quantum many-body systems](https://mlsanity.com/paper/2004.07266/) (2020)
- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Learning quantum Hamiltonians at any temperature in polynomial time](https://mlsanity.com/paper/2310.02243/) (2023)
- [Learning quantum states and unitaries of bounded gate complexity](https://mlsanity.com/paper/2310.19882/) (2023)
- [The Minimax Learning Rates of Normal and Ising Undirected Graphical Models](https://mlsanity.com/paper/1806.06887/) (2018)
- [Quantum State Tomography for Matrix Product Density Operators](https://mlsanity.com/paper/2306.09432/) (2023)
- [Quantum Tensor Networks, Stochastic Processes, and Weighted Automata](https://mlsanity.com/paper/2010.10653/) (2020)

## Built on by (most cited first, 4 of 4)

- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Efficient Hamiltonian, structure and trace distance learning of Gaussian states](https://mlsanity.com/paper/2411.03163/) (2024)
- [Efficient learning of bosonic Gaussian unitaries](https://mlsanity.com/paper/2510.05531/) (2025)
- [Reinforcement learning for quantum processes with memory](https://mlsanity.com/paper/2603.25138/) (2026)

## Abstract (arXiv)

Matrix product operators allow efficient descriptions (or realizations) of states on a 1D lattice. We consider the task of learning a realization of minimal dimension from copies of an unknown state, such that the resulting operator is close to the density matrix in trace norm. For finitely correlated translation-invariant states on an infinite chain, a realization of minimal dimension can be exactly reconstructed via linear algebra operations from the marginals of a size depending on the representation dimension. We establish a bound on the trace norm error for an algorithm that estimates a candidate realization from estimates of these marginals and outputs a matrix product operator, estimating the state of a chain of arbitrary length t. This bound allows us to establish an O(t2) upper bound on the sample complexity of the learning task, with an explicit dependence on the site dimension, realization dimension and spectral properties of a certain map constructed from the state. A refined error bound can be proven for C^*-finitely correlated states, which have an operational interpretation in terms of sequential quantum channels applied to the memory system. We can also obtain an analogous error bound for a class of matrix product density operators on a finite chain reconstructible by local marginals. In this case, a linear number of marginals must be estimated, obtaining a sample complexity of tildeO(t3). The learning algorithm also works for states that are sufficiently close to a finitely correlated state, with the potential of providing competitive algorithms for other interesting families of states.