Quasi-quantum states and the quasi-quantum PCP theorem

Abstract

We introduce $k$-local quasi-quantum states: a superset of the regular quantum states, defined by relaxing the positivity constraint. We show that a $k$-local quasi-quantum state on $n$ qubits can be 1-1 mapped to a distribution of assignments over $n$ variables with an alphabet of size $4$, which is subject to non-linear constraints over its $k$-local marginals. Therefore, solving the $k$-local Hamiltonian over the quasi-quantum states is equivalent to optimizing a distribution of assignment over a classical $k$-local CSP. We show that this optimization problem is essentially classical by proving it is NP-complete. Crucially, just as ordinary quantum states, these distributions lack a simple tensor-product structure and are therefore not determined straightforwardly by their local marginals. Consequently, our classical optimization problem shares some unique aspects of Hamiltonian complexity: it lacks an easy search-to-decision reduction, and it is not clear that its 1D version can be solved with dynamical programming (i.e., it could remain NP-hard). Our main result is a PCP theorem for the $k$-local Hamiltonian over the quasi-quantum states in the form of a hardness-of-approximation result. The proof suggests the existence of a subtle promise-gap amplification procedure in a model that shares many similarities with the quantum local Hamiltonian problem, thereby providing insights on the quantum PCP conjecture.

50 pages, 17 figures, comments are welcome. Updated version with improved exposition (results are the same)