Random quantum circuits are approximate unitary $t$-designs in depth $O\left(nt^{5+o(1)}\right)$

TL;DR

This paper improves the theoretical understanding of how deep random quantum circuits need to be to approximate unitary t-designs, reducing the depth from previous bounds and employing advanced mathematical techniques.

Contribution

It provides a tighter lower bound on the spectral gap of moment operators, leading to a more efficient depth estimate for random quantum circuits to form approximate t-designs.

Findings

Lower bound on spectral gap improved to 54(n^{-1}t^{-4-o(1)})

Random quantum circuits generate approximate t-designs in depth O(nt^{5+o(1)})

Proved fast convergence to Haar measure for certain random Clifford unitaries.

Abstract

The applications of random quantum circuits range from quantum computing and quantum many-body systems to the physics of black holes. Many of these applications are related to the generation of quantum pseudorandomness: Random quantum circuits are known to approximate unitary $t$-designs. Unitary $t$-designs are probability distributions that mimic Haar randomness up to $t$th moments. In a seminal paper, Brand\~{a}o, Harrow and Horodecki prove that random quantum circuits on qubits in a brickwork architecture of depth $O(n t^{10.5})$ are approximate unitary $t$-designs. In this work, we revisit this argument, which lower bounds the spectral gap of moment operators for local random quantum circuits by $\Omega(n^{-1}t^{-9.5})$. We improve this lower bound to $\Omega(n^{-1}t^{-4-o(1)})$, where the $o(1)$ term goes to $0$ as $t\to\infty$. A direct consequence of this scaling is that random quantum circuits generate approximate unitary $t$-designs in depth $O(nt^{5+o(1)})$. Our techniques involve Gao's quantum union bound and the unreasonable effectiveness of the Clifford group. As an auxiliary result, we prove fast convergence to the Haar measure for random Clifford unitaries interleaved with Haar random single qubit unitaries.