Interactive quantum advantage with noisy, shallow Clifford circuits
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Recent work by Bravyi et al. constructs a relation problem that a noisy constant-depth quantum circuit (QNC^0) can solve with near certainty (probability 1 - o(1)), but that any bounded fan-in constant-depth classical circuit (NC^0) fails with some constant probability. We show that this robustness to noise can be achieved in the other low-depth quantum/classical circuit separations in this area. In particular, we show a general strategy for adding noise tolerance to the interactive protocols of Grier and Schaeffer. As a consequence, we obtain an unconditional separation between noisy QNC^0 circuits and AC^0[p] circuits for all primes p ≥ 2, and a conditional separation between noisy QNC^0 circuits and log-space classical machines under a plausible complexity-theoretic conjecture. A key component of this reduction is showing average-case hardness for the classical simulation tasks – that is, showing that a classical simulation of the quantum interactive task is still powerful even if it is allowed to err with constant probability over a uniformly random input. We show that is true even for quantum tasks which are ⊕L-hard to simulate. To do this, we borrow techniques from randomized encodings used in cryptography.
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