Interactive quantum advantage with noisy, shallow Clifford circuits
Abstract
Recent work by Bravyi et al. constructs a relation problem that a noisy constantdepth quantum circuit (QNC$^0$) can solve with near certainty (probability $1  o(1)$), but that any bounded fanin constantdepth classical circuit (NC$^0$) fails with some constant probability. We show that this robustness to noise can be achieved in the other lowdepth 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 $p \geq 2$, and a conditional separation between noisy QNC$^0$ circuits and logspace classical machines under a plausible complexitytheoretic conjecture. A key component of this reduction is showing averagecase 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 $\oplus$Lhard to simulate. To do this, we borrow techniques from randomized encodings used in cryptography.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 arXiv:
 arXiv:2102.06833
 Bibcode:
 2021arXiv210206833G
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity
 EPrint:
 32 pages