Accurate contraction of tensor networks beyond one dimension is essential in various fields including quantum many-body physics. However, existing approaches typically rely on approximate contraction schemes and do not provide certified error bars. In this talk, we introduce an alternative perspective on tensor-network contraction problems via the numerical bootstrap framework. This technique casts the problem 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 our scheme can provide tight bounds on the contraction result. Our work suggests numerical bootstrap could be a possible way forward for the rigorous contraction of higher-dimensional tensor networks.

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