A random matrix model for random approximate $t$-designs

TL;DR

This paper introduces a random matrix model to describe the distribution of approximate t-designs in quantum computing, providing theoretical bounds and numerical evidence for its accuracy.

Contribution

The authors propose a novel random matrix model for approximate t-designs and prove its convergence properties and spectral gap conjecture satisfaction.

Findings

Operator norm of the matrix converges in distribution as the set size grows.

Explicit bounds on tail probabilities for the approximation measure.

Numerical simulations support the model's accuracy across different set sizes.

Abstract

For a Haar random set $\mathcal{S}\subset U(d)$ of quantum gates we consider the uniform measure $\nu_\mathcal{S}$ whose support is given by $\mathcal{S}$. The measure $\nu_\mathcal{S}$ can be regarded as a $\delta(\nu_\mathcal{S},t)$-approximate $t$-design, $t\in\mathbb{Z}_+$. We propose a random matrix model that aims to describe the probability distribution of $\delta(\nu_\mathcal{S},t)$ for any $t$. Our model is given by a block diagonal matrix whose blocks are independent, given by Gaussian or Ginibre ensembles, and their number, size and type is determined by $t$. We prove that, the operator norm of this matrix, $\delta({t})$, is the random variable to which $\sqrt{|\mathcal{S}|}\delta(\nu_\mathcal{S},t)$ converges in distribution when the number of elements in $\mathcal{S}$ grows to infinity. Moreover, we characterize our model giving explicit bounds on the tail probabilities $\mathbb{P}(\delta(t)>2+\epsilon)$, for any $\epsilon>0$. We also show that our model satisfies the so-called spectral gap conjecture, i.e. we prove that with the probability $1$ there is $t\in\mathbb{Z}_+$ such that $\sup_{k\in\mathbb{Z}_{+}}\delta(k)=\delta(t)$. Numerical simulations give convincing evidence that the proposed model is actually almost exact for any cardinality of $\mathcal{S}$. The heuristic explanation of this phenomenon, that we provide, leads us to conjecture that the tail probabilities $\mathbb{P}(\sqrt{\mathcal{S}}\delta(\nu_\mathcal{S},t)>2+\epsilon)$ are bounded from above by the tail probabilities $\mathbb{P}(\delta(t)>2+\epsilon)$ of our random matrix model. In particular our conjecture implies that a Haar random set $\mathcal{S}\subset U(d)$ satisfies the spectral gap conjecture with the probability $1$.