Constant-depth quantum advantage for efficiently verifiable problems

Determine whether there exists a computational problem whose solutions can be verified by a constant-depth quantum circuit and that exhibits a quantum advantage over classical circuits of small depth; specifically, construct a problem that is solvable by a constant-depth quantum circuit but not by any small-depth classical circuit while remaining efficiently verifiable by constant-depth quantum circuitry.

Background

The paper discusses prior work showing quantum advantage for shallow circuits (Bravyi–Gosset–König) and its lifting to the distributed setting. However, the underlying problem used to demonstrate this separation is not efficiently verifiable by constant-depth quantum circuits. This motivates a central unresolved question in quantum circuit complexity: whether an analogous separation can be obtained for problems whose solutions can themselves be efficiently verified by shallow quantum circuits.

Because this question remains unresolved, the authors note that a simple lifting strategy from circuit results to distributed results cannot be applied to their setting, underscoring the importance of settling the circuit-level question.