Proper Agnostic Learning of Matrix Product States and Tree Tensor Networks

Abstract:We establish proper agnostic learning of matrix product states and tree tensor networks. Given copies of an arbitrary quantum state $\rho$, our algorithms return a state $|\psi\rangle$ of chosen bond dimension such that $\langle\psi| \rho |\psi\rangle$ is within $\varepsilon$ of the optimum over the model class, without assuming that $\rho$ itself belongs to or is well approximated by that class. The main idea is to use improper learning to compress the mixed-state objective, reducing proper agnostic learning to optimization against a finite collection of explicitly specified pure states. We then introduce a comparator-dual compression procedure that reduces the bond dimension of these targets while uniformly preserving their overlaps with all bounded-bond comparators, with an error independent of system size. For matrix product states, this gives polynomial copy complexity in the system size, local dimension, bond dimension, and $1/\varepsilon$, together with polynomial runtime in the system size for fixed local dimension, bond dimension, and accuracy. The same framework yields proper agnostic learning of tree tensor networks on bounded-degree trees, with both copy complexity and runtime polynomial in the system size when the remaining parameters are fixed.

arXiv:2609.30148 [quant-ph]

(or arXiv:2609.30148v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.30148 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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