Unitary synthesis with optimal brick wall circuits

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 $n\le 5$. 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 $S{p}^{\ast }(2^n)$. 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.

15 pages, 12 figures, 4 tables. Associated code: https://github.com/XanaduAI/unicirc/ . Comments welcome!