Improved bounds on spectral gaps of random brickwork circuits

The spectral gap of a random quantum circuit ensemble characterizes the rate at which its moments converge to those of the Haar measure. It can be used to obtain bounds on the compressibility and approximate $t$-design depth of the circuit. We give new bounds which improve on previous results for the 1D brickwork architecture. The main technical innovation is a reduction from the spectral gap of the $N$-site brickwork to that of a $3$-site operator via a block-triangular decomposition. We also exploit a relationship between moment operators for the same circuit at different values of $t$. We show that the resulting bound on the spectral gap is nearly optimal. In addition, we prove that the corresponding bound on the $\epsilon$-approximate $t$-design depth is asymptotically nearly optimal as $\epsilon \rightarrow 0$.