Learning finitely correlated states: stability of the spectral reconstruction

TL;DR

The paper tackles learning a minimal-dimension matrix product density-operator (MPDO) realization of a translation-invariant finitely correlated state from finite marginals. It introduces LearnFCS, a spectral-reconstruction tomography algorithm that recovers realization parameters from marginals and produces an MPDO estimate with controlled trace-norm error on chains of length $t$, with a provable $O(t^2)$ sample complexity and explicit dependence on site and memory dimensions and a stability parameter. The authors prove error-propagation bounds within an operator-system generalization of quantum channels, giving polynomial-scaling guarantees for both translation-invariant finite chains and non-homogeneous cases, and they show refinements for $C^*$-finitely correlated states. The results establish robustness to states near finitely correlated ones and extend to certain matrix-product-density-operator classes reconstructible from local marginals, with potential applicability to 1D Gibbs states and broader tensor-network families. Overall, the work provides a rigorous, scalable framework for learning structured quantum states in 1D via spectral techniques, linking quantum information, operator-system theory, and classical spectral learning methods.

Abstract

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 $O(t^2)$ 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 $\tilde{O}(t^3)$. 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.

Figures (6)

• Figure 1: Sketch for learning a matrix product operator representation of a finitely correlated state. In the measurement phase we aquire data from $n$ copies of a finite size marginal of the state through local or global measurements. Then the spectral reconstruction phase is a classical post-processing technique that allows us to obtain approximations of the parameters of a linear model that generates the state. Using this approximations we then can generate an accurate estimation of an MPO representation. This allows us to obtain estimations of expected values of local observables or correlations functions on larger marginals of the state.
• Figure 2: Schematic of a finitely correlated state. The marginal state corresponding the dotted rectangle completely determines the state. An approximate estimate of the marginal can be used to reconstruct an estimate of the state. We bound the error in trace distance of the reconstruction up to a desired size.
• Figure 3: LearnFCS
• Figure 4: Scaling of trace norm error $TD(\omega_k,\hat{\omega}_k ) = \frac{1}{2}\| \rho_k - \hat{\rho}_k \|_1$ between the true reduced density matrix $\rho_k$ of the ground state of the AKLT model and the reconstructed one $\hat{\rho}_k$ from the maps $\hat{\Omega},\hat{\Omega}_{(\cdot )}$. We study the scaling for different errors $\epsilon = \|\hat{\Omega} - \Omega \|_2 = \| \hat{\Omega}_{(\cdot)} - \Omega_{(\cdot)} \|_2$Left plot: each line corresponds to a fixed $\epsilon$ and the reconstruction of $\omega_k$ is done using the same $\hat{\Omega},\hat{\Omega}_{(\cdot)}$. This plot illustrates the exponential behavior of the error propagation Theorem \ref{['thmerrorpropagation']} coming from the factor $(1+\Delta)^t$. Right plot: each dot corresponds to an independent experiment and the error is normalized by the number of sites. This plot illustrates the linear behavior if we renormalize the errors since $(1+\Delta/k)^k \leq 1+2k$ (for $\Delta\leq 1/2$).
• Figure 5: Representation of a $C^*$-finitely correlated state.

Theorems & Definitions (77)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Remark 2.9
• Definition A.1