Relative-error t-designs with linear scaling in t

Construct relative-error approximate unitary t-designs on n qubits whose circuit depth scales linearly in t, achieving negligible error for all polynomial t, thereby strengthening linear-depth diamond-error designs to the relative-error regime.

Background

The authors achieve linear-in-t circuit depth for diamond-error t-designs and provide a quadratic-in-t construction for relative-error t-designs via amplification. However, the optimal goal—relative-error designs with linear depth in t—remains unproven.

They note that their current analysis cannot establish relative-error behavior for the PFC ensemble on the non-distinct subspace, hinting at technical obstacles to achieving the desired result.