Related papers: Matrix concentration inequalities and efficiency of random universal sets of quantum gates

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

URL: http://arxiv.org/abs/2202.05371v3
Date: Wed, 12 Apr 2023 16:06:19 GMT
Title: Matrix concentration inequalities and efficiency of random universal sets of quantum gates
Authors: Piotr Dulian and Adam Sawicki
Abstract summary: For a random set $mathcalS subset U(d)$ of quantum gates we provide bounds on the probability that $mathcalS$ forms a $delta$-approximate $t$-design. We show that for $mathcalS$ drawn from an exact $t$-design the probability that it forms a $delta$-approximate $t$-design satisfies the inequality $mathbbPleft(delta geq x right)leq 2D_t
Score: 0.0
License: http://creativecommons.org/licenses/by/4.0/
Abstract: 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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