Low-Degree Testing for Quantum States, and a Quantum Entangled Games PCP for QMA

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 ≥ 2, we give such a game in which questions from the verifier are O(log n) bits long, and answers are poly(log log n) bits long. We show that for any constant ε ≥ 0, any strategy that succeeds with probability at least 1 − ε in the test must use a state that is within distance δ(ε) = O(ε 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 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 techniques from the classical PCP literature. More specifically, we use constructions of PCPs of proximity introduced by Ben-Sasson et al. (CCC'05), and crucially rely 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 constraint satisfaction quantum PCP conjecture.give two alternate formulations of Theorem 1.2 that would also establish the same QMA-hardness result, under deterministic reductions, provided that either:(i) it is QMA-hard to approximate the minimum energy of a local Hamiltonian in Y-free form (Definition 6.8) to within constant accuracy (this is a variant of the quantum PCP conjecture for local Hamiltonians), or (ii) it is QMA hard to approximate the ground energy of (not necessarily local) frustration-free Hamiltonian whose every term is a tensor product of generalized Pauli τ X or τ Z observables.Note that point (i) amounts to a deterministic reduction from Conjecture 1.2 to the constraint satisfaction quantum PCP conjecture, and establishes the first proven relation between the two conjectures (see [GKP16] for an incomparable result that relates stronger variants of both conjectures). Point (ii) is arguably a weaker assumption, as the gap is not required to be a constant and the terms of the Hamiltonian are not required to be local. However, due to the restriction that the Hamiltonian is frustration-free, it is currently not known whether the problem is QMA...

Self-testing of quantum systems: a review

“…A combination of self-testing based on nonlocal games with the quantum version of the linearity test from [BLR93], named Pauli braiding test [NV17] led to the first self-test of n EPR pairs in which robustness does not get worse if the num-ber of EPR pairs tested increases. Another parallel self-test keeping this desirable property is presented in [NV18]. The test can be seen as a quantum version of the classical plane-vs-point test for multivariate low-degree polynomials [RS97].…”

“…Inputs size (in bits) Outputs size (in bits) [BŠCA18b] poly(n, ) O(n) n [CRSV18] poly(n, ) O(log n) 1 [Col17] poly(n, ) O(n) n [CS17b] poly(n, ) O(n) n [CGJV17] poly( ) O(n log n) 2 [CN16] poly(n, ) O(n) n [McK16b] poly(n, ) O(n) n [McK17] poly(n, ) O(n) n [NV17] poly( ) O(n) 2 [NV18] poly( ) O(poly(log n)) poly(log log n) [OV16] poly(n, ) O(log n) 1 The other relevant property is its complexity, in terms of the size of the inputs. The size of the outputs is also a relevant factor, especially in possible applications for randomness expansion.…”

“…The size of the outputs is also a relevant factor, especially in possible applications for randomness expansion. For now, the self-testing protocol presented in [NV18] has the best properties in terms of the total number of inputs (polynomial) and robustness bounds (independent on n). The papers use different distance measures, but all the bounds given here are in terms of the Euclidean distance using definition 4.…”

Verifier-on-a-Leash: New Schemes for Verifiable Delegated Quantum Computation, with Quasilinear Resources

“…In this section we propose a new two-prover game for QMA, which is based on the Dog-Walker protocol. Such type of games are important in the context of the Quantum PCP conjecture [AAV13], more specifically to its game version that was recently proved [NV18].…”

Test for a large amount of entanglement, using few measurements

“…Natarajan and Vidick [NV17] leverage a quantum version of the linearity test by Blum et al [BLR93], and obtain robustness that is independent of n. Their test is rather complex and requires the verifier to choose among exponentially many possible questions. This was recently improved [NV18] to a test that achieves simultaneously poly(n) questions and constant robustness (O( √ δ)). Subsequently to our work, Ostrev and Vidick [OV16] apply our Theorem 2.1 to analyze a simpler XOR game than ours, resulting in similar number of possible questions, but slightly weaker robustness guarantees.…”