Matrix Concentration Inequalities And Efficiency Of Random Universal Sets Of Quantum Gates

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Piotr Dulian^1,2 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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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 $mathbb{P}left(delta geq x right)leq 2D_t , frac{e^{-|mathcal{S}| x , mathrm{arctanh}(x)}}{(1-x^2)^{|mathcal{S}|/2}} = Oleft( 2D_t left( frac{e^{-x^2}}{sqrt{1-x^2}} right)^{|mathcal{S}|} right)$, 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 $O( delta^{-2}(tlog(d)-log(1-P)))$ 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.

► BibTeX data

@article{Dulian2023matrixconcentration, doi = {10.22331/q-2023-04-20-983}, url = {https://doi.org/10.22331/q-2023-04-20-983}, title = {Matrix concentration inequalities and efficiency of random universal sets of quantum gates}, author = {Dulian, Piotr and Sawicki, Adam}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {983}, month = apr, year = {2023}<br /> }

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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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