Improved spectral gaps for random quantum circuits: large local dimensions and all-to-all interactions

TL;DR

This paper advances understanding of spectral gaps in random quantum circuits, demonstrating improved bounds for local and all-to-all interaction models, which enhances their efficiency in generating quantum pseudo-randomness and approximate unitary designs.

Contribution

It introduces new spectral gap bounds for 1D local and all-to-all quantum circuits, improving the scaling with circuit size and design order, and employs novel techniques like Knabe bounds and spectral gap recursion.

Findings

Spectral gap scales as Ω(n^{-1}) for 1D circuits with small t.

Unconditional spectral gap bound of Ω(n^{-1} log^{-1}(n) t^{-eta(q)}) for all-to-all circuits.

Exact solution and numerical analysis improve constants for small t.

Abstract

Random quantum circuits are a central concept in quantum information theory with applications ranging from demonstrations of quantum computational advantage to descriptions of scrambling in strongly-interacting systems and black holes. The utility of random quantum circuits in these settings stems from their ability to rapidly generate quantum pseudo-randomness. In a seminal paper by Brand\~ao, Harrow, and Horodecki, it was proven that the $t$-th moment operator of local random quantum circuits on $n$ qudits with local dimension $q$ has a spectral gap of at least $\Omega(n^{-1}t^{-5-3.1/\log(q)})$, which implies that they are efficient constructions of approximate unitary designs. As a first result, we use Knabe bounds for the spectral gaps of frustration-free Hamiltonians to show that $1D$ random quantum circuits have a spectral gap scaling as $\Omega(n^{-1})$, provided that $t$ is small compared to the local dimension: $t^2\leq O(q)$. This implies a (nearly) linear scaling of the circuit depth in the design order $t$. Our second result is an unconditional spectral gap bounded below by $\Omega(n^{-1}\log^{-1}(n) t^{-\alpha(q)})$ for random quantum circuits with all-to-all interactions. This improves both the $n$ and $t$ scaling in design depth for the non-local model. We show this by proving a recursion relation for the spectral gaps involving an auxiliary random walk. Lastly, we solve the smallest non-trivial case exactly and combine with numerics and Knabe bounds to improve the constants involved in the spectral gap for small values of $t$.