Jordi Weggemans

We define a general formulation of quantum PCPs, which captures adaptivity and multiple unentangled provers, and give a detailed construction of the quantum reduction to a local Hamiltonian with a constant promise gap. The reduction turns out to be a versatile subroutine to prove properties of quantum PCPs, allowing us to show: (i) Non-adaptive quantum PCPs can simulate adaptive quantum PCPs when the number of proof queries is constant. In fact, this can even be shown to hold when the non-adaptive quantum PCP picks the proof indices simply uniformly at random from a subset of all possible index combinations, answering an open question by Aharonov, Arad, Landau and Vazirani (STOC '09). (ii) If the q-local Hamiltonian problem with constant promise gap can be solved in 𝖰𝖢𝖬𝖠, then 𝖰𝖯𝖢𝖯[q] is in 𝖰𝖢𝖬𝖠 for any constant q. (iii) If 𝖰𝖬𝖠(k) has a quantum PCP for any k=poly(n), then 𝖰𝖬𝖠(2) = 𝖰𝖬𝖠, connecting two of the longest-standing open problems in quantum complexity theory. Moreover, we also show that there exists (quantum) oracles relative to which certain quantum PCP statements are false. Hence, any attempt to prove the quantum PCP conjecture requires, just as was the case for the classical PCP theorem, (quantumly) non-relativizing techniques.

The quantum analogue of the equality function, known as the quantum state identity problem, is the task of deciding whether n unknown quantum states are equal or unequal, given the promise that all states are either pairwise orthogonal or identical. Under the one-sided error requirement, it is known that the permutation test is optimal for this task, and for two input states this coincides with the well-known Swap test. Until now, the optimal measurement in the general two-sided error regime was unknown. Under more specific promises, the problem can be solved approximately or even optimally with simpler tests, such as the circle test. This work attempts to capture the underlying structure of (fine-grained formulations of) the quantum state identity problem. Using tools from semi-definite programming and representation theory, we (i) give an optimal test for any input distribution without the one-sided error requirement by writing the problem as an SDP, giving the exact solutions to the primal and dual programs and showing that the two values coincide; (ii) propose a general G-test which uses an arbitrary subgroup G of S_n, giving an analytic expression of the performance of the specific test, and (iii) give an approximation of the permutation test using only a classical permutation and n−1 Swap tests.