On the moments of random quantum circuits and robust quantum complexity

TL;DR

This paper establishes new lower bounds on the growth of robust quantum circuit complexity for random circuits, showing linear and square-root growth depending on the approximation error, using moment bounds of an auxiliary random walk.

Contribution

It provides novel lower bounds on quantum circuit complexity growth for random circuits without relying on unitary t-designs, using a new approach based on auxiliary random walks.

Findings

Proves linear complexity growth for small approximation error $ heta(2^{-n})$.

Establishes square-root complexity growth for constant approximation error.

Introduces a simple conjecture linking Fourier support of Boolean functions to complexity growth.

Abstract

We prove new lower bounds on the growth of robust quantum circuit complexity -- the minimal number of gates $C_{\delta}(U)$ to approximate a unitary $U$ up to an error of $\delta$ in operator norm distance. More precisely we show two bounds for random quantum circuits with local gates drawn from a subgroup of $SU(4)$. First, for $\delta=\Theta(2^{-n})$, we prove a linear growth rate: $C_{\delta}\geq d/\mathrm{poly}(n)$ for random quantum circuits on $n$ qubits with $d\leq 2^{n/2}$ gates. Second, for $ \delta=\Omega(1)$, we prove a square-root growth of complexity: $C_{\delta}\geq \sqrt{d}/\mathrm{poly}(n)$ for all $d\leq 2^{n/2}$. Finally, we provide a simple conjecture regarding the Fourier support of randomly drawn Boolean functions that would imply linear growth for constant $\delta$. While these results follow from bounds on the moments of random quantum circuits, we do not make use of existing results on the generation of unitary $t$-designs. Instead, we bound the moments of an auxiliary random walk on the diagonal unitaries acting on phase states. In particular, our proof is comparably short and self-contained.