Approximate Unitary k-Designs from Shallow, Low-Communication Circuits

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 can be written as a convex combination involving an exact design 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 designs in O ( (k k + m + (1/) ) k\, polylog (k) ) depth, where m is the number of qudits in the complete system and the approximation error. This sublinear depth construction answers one variant of Harrow and Mehraban 2023, Open Problem 1. Moreover, entanglement generated by the sublinear depth scheme follows area laws on spatial lattices up to corrections logarithmic in the full system size.