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Efficient unitary designs and pseudorandom unitaries from permutations
April 26, 2024, 4:11 a.m. | Chi-Fang Chen, Adam Bouland, Fernando G. S. L. Brand\~ao, Jordan Docter, Patrick Hayden, Michelle Xu
cs.CR updates on arXiv.org arxiv.org
Abstract: In this work we give an efficient construction of unitary $k$-designs using $\tilde{O}(k\cdot poly(n))$ quantum gates, as well as an efficient construction of a parallel-secure pseudorandom unitary (PRU). Both results are obtained by giving an efficient quantum algorithm that lifts random permutations over $S(N)$ to random unitaries over $U(N)$ for $N=2^n$. In particular, we show that products of exponentiated sums of $S(N)$ permutations with random phases approximately match the first $2^{\Omega(n)}$ moments of the Haar …
algorithm arxiv construction cs.cr math.mp math-ph parallel poly quant-ph quantum random results work
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