Group designs need more qubits as depth falls, study shows

Maxwell West of Los Alamos National Laboratory and the University of Melbourne, alongside colleagues, has demonstrated fundamental limits to how efficiently quantum circuits can mimic random processes. The research establishes that while approximate designs scaling logarithmically in qubit number are possible for unitary groups, no such sublinear-depth designs exist for the orthogonal group. The work derives a lower bound on the accuracy of any circuit ensemble attempting to approximate the Haar measure over various groups, revealing that designs on a D-dimensional lattice require depths scaling at least as n1/D. These findings also show a constant separation between the complexity of approximating and exactly implementing random matchgate circuits. The matchgate group, a specific configuration of quantum gates, demands circuit depths scaling linearly…