Quasi-quantum states and the quasi-quantum PCP theorem

• URL: http://arxiv.org/abs/2410.13549v1
• Date: Thu, 17 Oct 2024 13:43:18 GMT
• Title: Quasi-quantum states and the quasi-quantum PCP theorem
• Authors: Itai Arad, Miklos Santha,
• Abstract summary: We show that solving the $k$-local Hamiltonian over the quasi-quantum states is equivalent to optimizing a distribution of assignment over a classical $k$-local CSP. 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.
• Score: 0.21485350418225244
• License:
• 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.

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