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

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 pseudorandomness. In a seminal paper by Brandão, Harrow, and Horodecki [Commun. Math. Phys. 346, 397 (2016)] it was proven that the $t\mathrm{th}$ moment operator of local random quantum circuits on $n$ qudits with local dimension $q$ has a spectral gap of at least $\mathrm{\Omega }\left(n^{-1}{t}^{-5-3.1/ln\left(q\right)}\right)$, 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 one-dimensional random quantum circuits have a spectral gap scaling as $\mathrm{\Omega }\left(n^{-1}\right)$, provided that $t$ is small compared to the local dimension: $t^2\le O\left(q\right)$. 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 $\mathrm{\Omega }\left[n^{-1}{ln}^{-1}\left(n\right)t^{-\alpha \left(q\right)}\right]$ for random quantum circuits with all-to-all interactions. This improves both the $n$ and $t$ scaling in design depth for the nonlocal model. We show this by proving a recursion relation for the spectral gaps involving an auxiliary random walk. Lastly, we solve the smallest nontrivial case exactly and combine with numerics and Knabe bounds to improve the constants involved in the spectral gap for small values of $t$.

Figure 1

An instance of the auxiliary walk described by ${\nu }_n^{\mathrm{aux}}$ for $n=6$.

Figure 2

Numerically computed Hamiltonian gaps for the second moment $t=2$ of OBC/PBC 1D and nonlocal RQCs, which demonstrate a $1/n^2$ decay for the 1D spectral gaps and an $n$-dependent growth of the gaps for the nonlocal Hamiltonian.