A Random Matrix Model for Random Approximate t-Designs

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