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 with an underlying measure over which the average is close to a Haar random ensemble up to the first k moments. A particularly strong notion of approximation bounds the distance from Haar randomness in relative error. Such relative-error approximate designs are secure against queries by an adaptive adversary trying to distinguish it from a Haar ensemble. We construct relative-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 averages, giving a bound on the convergence speed to the full averaging 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. We use these constructions as the building blocks of a two-step protocol that achieves a relative-error design in O((logm+log(1/ϵ)+klogk)kpolylog(k)) depth, where m is the number of qudits in the complete system and ϵ the approximation error. This sublinear depth construction answers a variant of [21, Harrow and Mehraban 2023, Section 1.5, Open Question 1] and [21, Harrow and Mehraban 2023, Section 1.5, Open Question 7]. Moreover, entanglement generated by the sublinear depth scheme follows area laws on spatial lattices up to corrections logarithmic in the full system size.