Parity vs. AC0 with Simple Quantum Preprocessing

A recent line of work [5, 27, 12, 6, 28] has shown the unconditional advantage of constant-depth quantum computation, or QNC(0), over NC0, AC(0), and related models of classical computation. Problems exhibiting this advantage include search and sampling tasks related to the parity function, and it is natural to ask whether QNC(0) can be used to help compute parity itself. Namely, we study AC(0) circle QNC(0) - a hybrid circuit model where AC(0) operates on measurement outcomes of a QNC(0) circuit - and we ask whether Par is an element of AC(0) circle QNC(0). We believe the answer is negative. In fact, we conjecture AC(0) circle QNC(0) cannot even achieve Omega(1) correlation with parity. As evidence for this conjecture, we prove: When the QNC(0) circuit is ancilla-free, this model can achieve only negligible correlation with parity, even when AC(0) is replaced with any function having LMN-like decay in its Fourier spectrum. For the general (non-ancilla-free) case, we show via a connection to nonlocal games that the conjecture holds for any class of postprocessing functions that has approximate degree o(n) and is closed under restrictions. Moreover, this is true even when the QNC(0) circuit is given arbitrary quantum advice. By known results [8], this confirms the conjecture for linear-size AC(0) circuits. Another approach to proving the conjecture is to show a switching lemma for AC(0) circle QNC(0). Towards this goal, we study the effect of quantum preprocessing on the decision tree complexity of Boolean functions. We find that from the point of view of decision tree complexity, nonlocal channels are no better than randomness: a Boolean function f precomposed with an n-party nonlocal channel is together equal to a randomized decision tree with worst-case depth at most DTdepth[f]. Taken together, our results suggest that while QNC(0) is surprisingly powerful for search and sampling tasks, that power is "locked away" in the global correlations of its output, inaccessible to simple classical computation for solving decision problems.