2026/02/05 by Berta Casas, Paolo Braccia, Élie Gouzien +2 · 1 voice
#quant-ph
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 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 \mathbbSO(2n), which reduces the compilation from 2n× 2n unitaries to 2n×2n ones, thus reducing exponentially the size of the target matrix. Moreover, we rigorously show that this scheme is efficient, as an approximation error ε_\mathbbSO(2n) incurred in this smaller-dimensional representation translates at most into an O(n ε_\mathbbSO(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⊗ U^* has entries in the ring ℤ[1/√ 2,i] can be exactly synthesized by a finite sequence of gates from the matchgate-Clifford+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.