Classical circuit depth log2(s) for exact solutions to the 2D Hidden Linear Function problem

Determine whether, for exactly solving the two-dimensional Hidden Linear Function (HLF) problem, the minimum classical circuit depth required to implement a Boolean function that depends on all s inputs using two-input, one-output gates equals log2(s), without imposing any constraints on circuit connectivity. This establishes a near-term classical lower bound to compare against shallow quantum circuits that solve the 2D HLF problem.

Background

Bravyi et al. proved that certain 2D Hidden Linear Function (HLF) problems can be solved exactly by constant-depth quantum circuits, while any classical solution requires logarithmic depth, establishing a quantum-classical separation in the asymptotic regime. However, those bounds become relevant only for very large instances (n ≳ 10^6 qubits), making them unsuitable for direct comparison with near-term experimental devices.

To enable practical benchmarking against classical resources for the sizes accessible today, the authors propose a conjectured classical depth formula based on two-input, one-output gates that counts the required two-bit gate layers to exactly solve the HLF problem. Validating or refuting this conjecture would provide a concrete, connectivity-agnostic lower bound for classical circuits in the near-term regime and facilitate more direct comparisons with the shallow quantum circuits implemented on current hardware.