Interactive shallow Clifford circuits: quantum advantage against NC$^1$ and beyond
Abstract
Recent work of Bravyi et al. and followup work by Bene Watts et al. demonstrates a quantum advantage for shallow circuits: constantdepth quantum circuits can perform a task which constantdepth classical (i.e., AC$^0$) circuits cannot. Their results have the advantage that the quantum circuit is fairly practical, and their proofs are free of hardness assumptions (e.g., factoring is classically hard, etc.). Unfortunately, constantdepth classical circuits are too weak to yield a convincing realworld demonstration of quantum advantage. We attempt to hold on to the advantages of the above results, while increasing the power of the classical model. Our main result is a tworound interactive task which is solved by a constantdepth quantum circuit (using only Clifford gates, between neighboring qubits of a 2D grid, with Pauli measurements), but such that any classical solution would necessarily solve $\oplus$Lhard problems. This implies a more powerful class of constantdepth classical circuits (e.g., AC$^0[p]$ for any prime $p$) unconditionally cannot perform the task. Furthermore, under standard complexitytheoretic conjectures, logdepth circuits and logspace Turing machines cannot perform the task either. Using the same techniques, we prove hardness results for weaker complexity classes under more restrictive circuit topologies. Specifically, we give QNC$^0$ interactive tasks on $2 \times n$ and $1 \times n$ grids which require classical simulations of power NC$^1$ and AC$^{0}[6]$, respectively. Moreover, these hardness results are robust to a small constant fraction of error in the classical simulation. We use ideas and techniques from the theory of branching programs, quantum contextuality, measurementbased quantum computation, and Kilian randomization.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.02555
 Bibcode:
 2019arXiv191102555G
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity
 EPrint:
 54 pages