Matrix concentration inequalities and efficiency of random universal sets of quantum gates

Abstract

For a random set S⊂U(d) of quantum gates we provide bounds on the probability that S forms a δ-approximate t-design. In particular we have found that for S drawn from an exact t-design the probability that it forms a δ-approximate t-design satisfies the inequality P(δ≥x)≤2Dte−|S|xarctanh(x)(1−x2)|S|/2=O(2Dt(e−x21−x2)|S|), where Dt is a sum over dimensions of unique irreducible representations appearing in the decomposition of U↦U⊗t⊗U¯⊗t. We use our results to show that to obtain a δ-approximate t-design with probability P one needs O(δ−2(tlog⁡(d)−log⁡(1−P))) many random gates. We also analyze how δ concentrates around its expected value Eδ for random S. Our results are valid for both symmetric and non-symmetric sets of gates.