Matchgate synthesis via Clifford matchgates and $T$ gates

Matchgate unitaries are ubiquitous in quantum computation due to their relation to non-interacting fermions and because they can be used to benchmark quantum computers. Implementing such unitaries on fault-tolerant devices requires first compiling them into a discrete universal gate set, typically Clifford$+T$. Here, we propose a different approach for their synthesis: compile matchgate unitaries using only matchgate gates. To this end, we first show that the matchgate-Clifford group (the intersection of the matchgate and Clifford groups) plus the $\overline{T}$ gate (a $T$ unitary up to a phase) is universal for the matchgate group. Our approach leverages the connection between $n$-qubit matchgate circuits and the standard representation of $\mathbb{SO}(2n)$, which reduces the compilation from $2^n\times 2^n$ unitaries to $2n\times2n$ ones, thus reducing exponentially the size of the target matrix. Moreover, we rigorously show that this scheme is efficient, as an approximation error $\varepsilon_{\mathbb{SO}(2n)}$ incurred in this smaller-dimensional representation translates at most into an $O(n \,\varepsilon_{\mathbb{SO}(2n)})$ error in the exponentially large unitary. In addition, we study the exact version of the matchgate synthesis problem, and we prove that all matchgate unitaries $U$ such that $U\otimes U^*$ has entries in the ring $\mathbb{Z}\big[1/\sqrt 2,i\big]$ can be exactly synthesized by a finite sequence of gates from the matchgate-Clifford$+\overline{T}$ set, without ancillas. We then use this insight to map optimal exact matchgate synthesis to Boolean satisfiability, and compile the circuits that diagonalize the free-fermionic $XX$ Hamiltonian on $n=4,\,8$ qubits.