Arts & crafts: Strong random unitaries and geometric locality

TL;DR

This paper develops new methods for constructing strong approximate unitary $k$-designs on high-dimensional grids, improving flexibility and optimality over previous results, with applications to pseudorandom unitaries.

Contribution

It introduces two constructions for strong unitary $k$-designs on grids, one flexible with suboptimal depth and one direct with optimal depth, extending prior work to higher dimensions.

Findings

Provided a flexible construction for strong $k$-designs in arbitrary connectivities.

Achieved a direct construction with provably optimal depth for $D$-dimensional grids.

Extended the construction to strong pseudorandom unitaries with optimal depth.

Abstract

We study the problem of constructing strong approximate unitary $k$-designs on $D$-dimensional grids (and more generally on Cartesian products of graphs), building on the work of Schuster et al. arXiv:2509.26310 which establishes strong unitary designs in 1D and in all-to-all connectivity. We provide two constructions. The first construction leverages the existing all-to-all connectivity result with general routing theory to provide flexible (but slightly suboptimal) strong $k$-designs in arbitrary connectivities. The second construction is more direct, requires no auxiliaries and has provably optimal depth (in the number of qubits $n$) for $D$-dimensional grids with constant dimension. Combining these techniques also allows us to construct strong pseudorandom unitaries on $D$-dimensional grids with provably optimal depth.