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

Piotr Dulian1,2 and Adam Sawicki1

1Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
2Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland

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For a random set $\mathcal{S} \subset U(d)$ of quantum gates we provide bounds on the probability that $\mathcal{S}$ forms a $\delta$-approximate $t$-design. In particular we have found that for $\mathcal{S}$ drawn from an exact $t$-design the probability that it forms a $\delta$-approximate $t$-design satisfies the inequality $\mathbb{P}\left(\delta \geq x \right)\leq 2D_t \, \frac{e^{-|\mathcal{S}| x \, \mathrm{arctanh}(x)}}{(1-x^2)^{|\mathcal{S}|/2}} = O\left( 2D_t \left( \frac{e^{-x^2}}{\sqrt{1-x^2}} \right)^{|\mathcal{S}|} \right)$, where $D_t$ is a sum over dimensions of unique irreducible representations appearing in the decomposition of $U \mapsto U^{\otimes t}\otimes \bar{U}^{\otimes t}$. We use our results to show that to obtain a $\delta$-approximate $t$-design with probability $P$ one needs $O( \delta^{-2}(t\log(d)-\log(1-P)))$ many random gates. We also analyze how $\delta$ concentrates around its expected value $\mathbb{E}\delta$ for random $\mathcal{S}$. Our results are valid for both symmetric and non-symmetric sets of gates.

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Cited by

[1] Dmitry Grinko and Maris Ozols, "Linear programming with unitary-equivariant constraints", arXiv:2207.05713, (2022).

[2] Piotr Dulian and Adam Sawicki, "A random matrix model for random approximate $t$-designs", arXiv:2210.07872, (2022).

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