Unitary synthesis with optimal brick wall circuits

TL;DR

This paper introduces optimal brick wall quantum circuits that efficiently parametrize $SU(2^n)$ for up to five qubits, demonstrating their universality and extending to specific subgroups, with both numerical and partial theoretical validation.

Contribution

It presents a new optimal brick wall circuit architecture for $SU(2^n)$, providing evidence of universality for up to five qubits and extending to certain subgroups, with a combination of numerical and partial theoretical proofs.

Findings

Circuits are universal for $n \,\leq\, 5$

Achieved state-of-the-art synthesis of $SU(2^n)$ matrices

Extended method to $SO(2^n)$ and $Sp^*(2^n)$ subgroups

Abstract

We present quantum circuits with a brick wall structure using the optimal number of parameters and two-qubit gates to parametrize $SU(2^n)$, and provide evidence that these circuits are universal for $n\leq 5$. For this, we successfully compile random matrices to the presented circuits and show that their Jacobian has full rank almost everywhere in the domain. Our method provides a new state of the art for synthesizing typical unitary matrices from $SU(2^n)$ for $n=3, 4, 5$, and we extend it to the subgroups $SO(2^n)$ and $Sp^\ast(2^n)$. We complement this numerical method by a partial proof, which hinges on an open conjecture that relates universality of an ansatz to it having full Jacobian rank almost everywhere.