Quasi-quantum states and the quasi-quantum PCP theorem

TL;DR

This paper introduces a relaxed class of quantum states called quasi-quantum states, maps their properties to classical distributions, and proves a PCP theorem for local Hamiltonian problems, offering insights into the quantum PCP conjecture.

Contribution

It defines quasi-quantum states, establishes their classical equivalence, and proves a PCP theorem for local Hamiltonian problems in this framework, advancing understanding of quantum complexity.

Findings

Quasi-quantum states relax positivity constraints of quantum states.

Optimization over quasi-quantum states is NP-complete.

A PCP theorem for local Hamiltonian problems over quasi-quantum states is established.

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.