Ancilla-free logarithmic-depth threshold in higher dimensions

Determine whether logarithmic depth is the threshold for generating ancilla-free pseudorandom quantum states on higher-dimensional local brickwork architectures.

Background

The paper notes that higher-dimensional architectures are less understood than the one-dimensional case. Although structured constructions can exploit greater connectivity and ancillary workspace to reduce depth, the best known ancilla-free cryptographic constructions remain at polylogarithmic depth.

Consequently, it remains unresolved whether logarithmic depth suffices—or is necessary as the threshold—for pseudorandomness without ancillary qubits in higher-dimensional brickwork architectures. The paper’s learning result at $\ell d=O(\log n)$ provides evidence relevant to this threshold but does not settle the pseudorandomness question.