Matchgate synthesis via Clifford matchgates and $T$ gates

TL;DR

This work shows that matchgate unitaries can be compiled using a discrete universal gate set formed by the matchgate-Clifford group plus the gate $\overline{T}$, leveraging the $\mathbb{SO}(2n)$ representation to achieve exponential reductions in target matrix size from $2^n\times 2^n$ to $2n\times 2n$. It provides both approximate and exact synthesis results: an $\mathbb{SO}(2n)$-level approximation bounds the induced error in the full unitary, and exact synthesis is possible for unitaries with $U\otimes U^*$ entries in $\mathbb{Z}[1/\sqrt{2},i]$, with a constructive Gaussian-elimination-style algorithm. The authors further map the exact synthesis problem to SAT, obtaining depth-optimal and minimal-$\overline{T}$-count circuits, and demonstrate SAT-based compilation for the XX diagonalizing circuit at $n=4$ and $n=8$. Additionally, they propose a benchmarking framework using matchgate circuits to probe non-Clifford resources in fault-tolerant devices, supported by quantified entanglement and magic measures. Overall, the work connects number-theoretic synthesis, group-theoretic representations, and SAT-based optimization to advance specialized compilation in free-fermionic quantum computation.

Abstract

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.

Figures (8)

• Figure 1: Summary of our main results. We propose synthesizing a target matchgate unitary $U$--either approximately or exactly--using only matchgates, as indicated by the green path within the matchgate group. In contrast to the standard approach, where a universal gate set for the full unitary group $\mathbb{U}(2^n)$ (such as Clifford$+T$) is employed (grey path), our strategy allows us to work with $2n\times 2n$ matrices, instead of $2^n\times 2^n$ ones. Our results may find broad applicability in the simulation of fermionic systems on quantum computers, as well as in benchmarking and verification protocols thereof.
• Figure 2: $\mathbb{SU}(2)$ representations within the two-qubit matchgate group. We illustrate two $\mathbb{SU}(2)$ representations, each generated by a single-qubit $R^z$ and a two-qubit $R^{xx}$ rotation, as indicated by the colors. When restricted to the fermionic even-parity subspace spanned by $|00\rangle$ and $|11\rangle$, these representations give rise to a Bloch-sphere structure.
• Figure 3: Residual entanglement in the synthesis of $R^z(\theta)$ with matchgates. We plot the linear entropy $E(U_\varepsilon)$ of the approximated unitary $U_\varepsilon$ against the precision $\varepsilon$, for values of $\theta$ in the range $\theta\in[0,2\pi)$. The unitaries $U_\varepsilon$ were compiled using the discrete matchgate set $\mathcal{G}$, via the gridsynth algorithm ross2014optimalgridsynthpygridsynth and the $\mathbb{SU}(2)$ isomorphisms depicted in Fig. \ref{['fig:su2_is_su2']}. We show the bound on $E(U_\varepsilon)$ from Proposition \ref{['prop:entanglement']} (grey dashed line), and the maximal entanglement achievable by any gate in $\mathbb{SU}(4)$ (green dashed line). Points above the bound have reached the numerical-precision limit.
• Figure 4: Accumulation of local synthesis errors in random matchgate circuits. Global approximation error $\varepsilon_{\rm glob}$ (operator norm distance) as a function of the sum of local errors $\varepsilon_{\rm loc}$, for Haar-random matchgate circuits braccia2025optimal where each $R^z$ and $R^{xx}$ gate was approximated within $\varepsilon = 10^{-3}$ precision using gridsynth. Here, $\varepsilon$ denotes the compiler’s per-gate tolerance, while the local errors $\varepsilon_j$ in the sum $\varepsilon_{\rm loc} = \sum_j \varepsilon_j$ are the per-gate operator-norm distances. Different colors indicate different system sizes $n$, and the dashed line shows the worst-case subadditivity bound. In the inset, we plot the relative error $(\varepsilon_{\rm loc}-\varepsilon_{\rm glob})/\varepsilon_{\rm loc}$ as a function of $\varepsilon$.
• Figure 5: Error propagation from $\mathbb{SO}(2n)$ to $\mathbb{SPIN}(2n)$. We numerically test the bound in Theorem \ref{['th:SO-error']} using Haar-random matchgate circuits braccia2025optimal. The horizontal axis shows the error incurred in the synthesis of $Q\in\mathbb{SO}(2n)$, scaled up by the number of qubits and a constant factor (i.e., $\frac{\pi}{2}n \varepsilon_{\mathbb{SO} (2n)}$), while the vertical axis displays the actual error $\varepsilon = \|U\otimes U^*- U_\varepsilon\otimes U_\varepsilon^*\|$ in the adjoint representation of the matchgate circuit. Each color correspond to a different system size $n$, and the dashed line represents the theoretical bound provided by Theorem \ref{['th:SO-error']}.

Theorems & Definitions (31)

• Lemma 1
• Lemma 2: Solovay--Kitaev theorem
• Theorem 1
• Corollary 1
• Proposition 1
• Theorem 2
• Theorem 3
• Definition 1
• Theorem 4
• Lemma 1