Toward bootstrapping tensor-network contractions

TL;DR

This paper introduces a numerical bootstrap framework that transforms tensor-network contraction problems into convex optimization tasks, providing certified bounds and improving accuracy in quantum many-body physics computations.

Contribution

It develops a convex optimization approach for tensor-network contractions, enabling certified bounds and demonstrating its effectiveness for matrix product states.

Findings

Second-order-cone relaxation yields tight bounds under canonical form.

Semidefinite programming can provide similar bounds without canonical form.

The approach is polynomial in computational cost.

Abstract

Accurate contraction of tensor networks beyond one dimension is essential in various fields including quantum many-body physics. Existing approaches typically rely on approximate contraction schemes and do not provide certified error bars. We introduce a numerical bootstrap framework which casts the problem of tensor-network contractions into a convex optimization problem, thereby yielding certified lower and upper bounds on expectation values of physical observables. As a proof-of-principle, we construct such constraints explicitly for translationally invariant matrix product states and demonstrate that, assuming a canonical form, second-order-cone relaxation can provide tight bounds on the contraction result. We further demonstrate that when the requirement on canonical form is lifted, a more general semidefinite-programming approach could yield similar tight bounds at higher but still polynomial computational cost. Our work suggests numerical bootstrap could be a possible way forward for the rigorous contractions of tensor networks.