On the power of geometrically-local classical and quantum circuits

We show a relation, based on parallel repetition of the Magic Square game, that can be solved, with probability exponentially close to $1$ (worst-case input), by $1D$ (uniform) depth $2$, geometrically-local, noisy (noise below a threshold), fan-in $4$, quantum circuits. We show that the same relation cannot be solved, with an exponentially small success probability (averaged over inputs drawn uniformly), by $1D$ (non-uniform) geometrically-local, sub-linear depth, classical circuits consisting of fan-in $2$ NAND gates. Quantum and classical circuits are allowed to use input-independent (geometrically-non-local) resource states, that is entanglement and randomness respectively. To the best of our knowledge, previous best (analogous) depth separation for a task between quantum and classical circuits was constant v/s sub-logarithmic, although for general (geometrically non-local) circuits. Our hardness result for classical circuits is based on a direct product theorem about classical communication protocols from Jain and Kundu [JK22]. As an application, we propose a protocol that can potentially demonstrate verifiable quantum advantage in the NISQ era. We also provide generalizations of our result for higher dimensional circuits as well as a wider class of Bell games.