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

TL;DR

This paper presents a method to construct approximate unitary $k$-designs with relative-error bounds using shallow, low-communication quantum circuits, enabling efficient and secure quantum randomness generation.

Contribution

It introduces a novel construction of relative-error approximate unitary $k$-designs with $O(1)$ communication, using the alternating projection method and von Neumann algebra techniques, achieving sublinear depth.

Findings

Constructed relative-error approximate $k$-designs with $O(1)$ communication.

Achieved a sublinear depth protocol for $k$-designs with explicit bounds.

Generated entanglement obeying area laws on spatial lattices.

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 $O \big ( (\log m + \log(1/\epsilon) + k \log k ) k\, \text{polylog}(k) \big )$ depth, where $m$ is the number of qudits in the complete system and $\epsilon$ the approximation error. This sublinear depth construction answers a variant of [Harrow and Mehraban 2023, Section 1.5, Open Questions 1 and 7]. Moreover, entanglement generated by the sublinear depth scheme follows area laws on spatial lattices up to corrections logarithmic in the full system size.