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

• URL: http://arxiv.org/abs/2012.05259v1
• Date: Wed, 9 Dec 2020 19:00:50 GMT
• Title: Improved spectral gaps for random quantum circuits: large local dimensions and all-to-all interactions
• Authors: Jonas Haferkamp, Nicholas Hunter-Jones
• Abstract summary: We 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: $t2leq O(q)$. Our second result is an unconditional spectral gap bounded below by $Omega(n-1log-1(n) t-alpha(q))$ for random quantum circuits with all-to-all interactions.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $t$.

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