QLunch: Marco Fanizza

Speaker: Marco Fanizza

Title: Learning finitely correlated states: stability of the spectral reconstruction

Abstract: Matrix product operators allow efficient descriptions (equivalently, realizations) of states on a 1D lattice. We consider the task of estimating a realization from copies of an unknown state, measuring the error of the reconstructed matrix product operator in trace norm. For finitely correlated, translation-invariant states on an infinite spin-chain, a realization of minimal dimension can be exactly recovered via linear algebra operations from the knowledge of a finite size marginal, whose size is bounded by a function of the realization dimension. We thus consider an algorithm that, given approximate knowledge of a marginal of large enough size, estimates a candidate realization and reconstructs an approximation of the marginal of size t of the unknown state.  We bound the error as a function of t and other relevant parameters, proving that the sample and computational complexity of the task are polynomial in t. The theory of operator systems plays a central role in the analysis. We give refined bounds for states generated by quantum memories, and we extend the bounds to the finite-size, non-translation invariant case under additional assumptions.

Based on:

Fanizza, M., Galke, N., Lumbreras, J., Rouzé, C., & Winter, A. Learning finitely correlated states: stability of the spectral reconstruction. arXiv:2312.07516.