Low-degree testing for quantum states, and a quantum entangled games PCP for QMA

TL;DR

This paper proves that distinguishing entangled strategies in certain multiplayer games is QMA-hard, advancing the quantum PCP conjecture by constructing tests for maximally entangled states with efficient questions and answers.

Contribution

It introduces a novel family of two-player tests for maximally entangled states, linking quantum low-degree tests with PCP constructions to establish QMA-hardness.

Findings

QMA-hardness of distinguishing entangled strategies in multiplayer games

Construction of efficient quantum tests for maximally entangled states

Reduction from the games quantum PCP conjecture to the Hamiltonian quantum PCP conjecture

Abstract

We show that given an explicit description of a multiplayer game, with a classical verifier and a constant number of players, it is QMA-hard, under randomized reductions, to distinguish between the cases when the players have a strategy using entanglement that succeeds with probability 1 in the game, or when no such strategy succeeds with probability larger than 1/2. This proves the "games quantum PCP conjecture" of Fitzsimons and the second author (ITCS'15), albeit under randomized reductions. The core component in our reduction is a construction of a family of two-player games for testing $n$-qubit maximally entangled states. For any integer $n\geq2$, we give a test in which questions from the verifier are $O(\log n)$ bits long, and answers are $\mathrm{poly}(\log\log n)$ bits long. We show that for any constant $\varepsilon\geq0$, any strategy that succeeds with probability at least $1-\varepsilon$ in the test must use a state that is within distance $O(\varepsilon^c)$ from a state that is locally equivalent to a maximally entangled state on $n$ qubits, for some universal constant $c>0$. The construction is based on the classical plane-vs-point test for multivariate low-degree polynomials of Raz and Safra (STOC'97). We extend the classical test to the quantum regime by executing independent copies of the test in the generalized Pauli $X$ and $Z$ bases over $\mathbb{F}_q$, where $q$ is a sufficiently large prime power, and combine the two through a test for the Pauli twisted commutation relations. Our main complexity-theoretic result is obtained by combining this family of games with constructions of PCPs of proximity introduced by Ben-Sasson et al. (CCC'05), and crucially relies on a linear property of such PCPs. Another consequence of our results is a deterministic reduction from the games quantum PCP conjecture to a suitable formulation of the Hamiltonian quantum PCP conjecture.