“Approximate Unitary k-Designs from Shallow, Low-Communication Circuits”
Abstract: Random unitaries are useful in quantum information and related fields but hard to generate with limited resources. An approximate unitary k-design is an ensemble of unitaries and measure over which the average is close to a Haar (uniformly) random ensemble up to the first k moments. A particularly strong notion of approximation bounds the distance from Haar randomness in relative error: the approximate design's induced weighted twirl can be written as a convex combination involving a uniform twirl and vice versa. We construct multiplicative-error approximate unitary k-design ensembles for which communication between subsystems is O(1) in the system size. These constructions use the alternating projection method to analyze overlapping Haar twirls, giving a bound on the convergence speed to the full twirl with respect to the 2-norm. Using von Neumann subalgebra indices to replace system dimension, the 2-norm distance converts to relative error without introducing any additional dimension dependence. Via recursion on these constructions, we construct a scheme yielding relative error design unitaries in O( (log n + log 1/ϵ + k log k ) k polylog k ) depth, where n is the number of qudits in the complete system and ϵ the approximation error. This sublinear depth construction answers one variant of [Harrow and Mehraban 2023, Open Problem 1] and of [Harrow and Mehraban 2023, Open Problem 7]. Moreover, entanglement generated by the sublinear depth scheme follows area laws on spatial lattices up to corrections logarithmic in the full system size. This is joint work with Felix Leditzky at UIUC.
Bio: Nicholas LaRacuente received a PhD in physics from UIUC in 2020, following a concurrent mathematics Master's in 2018. He worked with Marius Junge on entropy inequalities and quantum Markov semigroups. Following an IBM postdoc at the University of Chicago and Chicago Quantum Exchange supported by IBM, LaRacuente is now an assistant professor of computer science at Indiana University Bloomington. Laracuente's research merges ideas from quantum information theory, mathematical physics, and computational complexity.
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