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.

翻译：我们展示一种基于"魔法方"游戏并行重复的关系，该关系可由一维（均匀）深度为2、几何局部、噪声（低于阈值）、扇入为4的量子电路以接近1的概率（最坏情况输入）求解。同时表明，同一关系无法由一维（非均匀）几何局部、亚线性深度的扇入为2的NAND门经典电路以指数小成功概率（均匀随机输入平均）求解。量子与经典电路均允许使用与输入无关的（几何非局部）资源态，即分别对应纠缠与随机性。据我们所知，此前量子与经典电路能力差别的（类似）最佳深度分离为常数对亚对数级——尽管针对的是一般（几何非局部）电路。我们对经典电路的困难性证明基于Jain与Kundu [JK22]关于经典通信协议的直积定理。作为应用，我们提出一个可在NISQ时代展现可验证量子优势的协议。此外，我们给出了结果在高维电路及更广泛贝尔游戏类中的推广。