Low-degree testing for quantum states

For any integer $n\geq 2$ we construct a one-round two-player game $G_n$, with communication that scales poly-logarithmically with $n$, having the following properties. First, there exists an entangled strategy that wins with probability $1$ in $G_n$ and in which the players' outcomes are determined by performing generalized Pauli measurements on their respective share of an $n$-qudit maximally entangled state, with qudits of local dimension $q = \mathrm{poly}\log(n)$. Second, any strategy that succeeds with probability at least $1-\varepsilon$ in $G_n$ must be within distance $O((\log n)^c\varepsilon^{1/d})$, for universal constants $c,d\geq 1$, of the perfect strategy, up to local isometries. This is an exponential improvement on the size of any previously known game certifying $\Omega(n)$ qudits of entanglement with comparable robustness guarantees. The construction of the game $G_n$ is based on the classical test for low-degree polynomials of Raz and Safra, which we extend to the quantum regime. Combining this game with a variant of the sum-check protocol, we obtain the following consequences. First, we show that is QMA-hard, under randomized reductions, to approximate up to a constant factor the maximum acceptance probability of a multiround, multiplayer entangled game with $\mathrm{poly}\log(n)$ bits of classical communication. Second, we give a quasipolynomial reduction from the multiplayer games quantum PCP conjecture to the constraint satisfaction quantum PCP conjecture. Third, we design a multiplayer protocol with polylogarithmic communication and constant completeness-soundness gap for deciding the minimal energy of a class of frustration-free nonlocal Hamiltonians up to inverse polynomial accuracy.