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Near-linear constructions of exact unitary 2-designs

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arxiv 1501.04592 v3 pith:RW2PPP4V submitted 2015-01-19 quant-ph

Richard Cleve , Debbie Leung , Li Liu , Chunhao Wang This is my paper · ORCID
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keywords unitaryexactdepthdesignsquantumanalogueapproximateapproximations
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A unitary 2-design can be viewed as a quantum analogue of a 2-universal hash function: it is indistinguishable from a truly random unitary by any procedure that queries it twice. We show that exact unitary 2-designs on n qubits can be implemented by quantum circuits consisting of ~O(n) elementary gates in logarithmic depth. This is essentially a quadratic improvement in size (and in width times depth) over all previous implementations that are exact or approximate (for sufficiently strong approximations).

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  1. Dynamical Lie Algebras Cannot Describe Shallow QAOA: Cragged Terrains, Barren Plateaus, and Empirical Hardness Models

    quant-ph 2026-08 conditional novelty 6.0 of 10

    For shallow QAOA on maximum independent set, loss landscape variance increases with system size instead of vanishing, contradicting dynamical Lie algebra predictions.

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