Computational complexity of injective projected entangled pair states

Projected entangled pair states (PEPS) constitute a variational family of quantum states with area-law entanglement. PEPS are particularly relevant and successful for studying ground states of spatially local Hamiltonians. However, computing local expectation values in these states is known to be \postBQP-hard. Injective PEPS, where all constituent tensors fulfil an injectivity constraint, are generally believed to be better behaved, because they are unique ground states of spatially local Hamiltonians. In this work, we therefore examine how the computational hardness of contraction depends on the injectivity. We establish that below a constant positive injectivity threshold, evaluating local observables remains \postBQP-complete, while above a different constant nontrivial threshold there exists an efficient classical algorithm for the task, resolving an open question from (Anshu et al., STOC `24). We do this by proving that noisy postselected quantum computation can be made fault-tolerant.