A Random Matrix Model for Random Approximate t-Designs

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.