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arXiv:2510.08312

Clifford+V synthesis for multi-qubit unitary gates

Soichiro Yamazaki, Seiseki Akibue

Published Oct 9, 2025• 0 citations
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Abstract

We developed a general framework for synthesizing target gates by using a finite set of basic gates, which is a crucial step in quantum compilation. When approximating a gate in SU($n$), a naive brute-force search requires a computational complexity of $O(1/\varepsilon^{(n^2 - 1)})$ to achieve an approximation with error $\varepsilon$. In contrast, by using our method, the complexity can be reduced to $O(-n^2 \log\varepsilon/\varepsilon^{((n^2 - 1)/2)})$. This method requires almost no assumptions and can be applied to a variety of gate sets, including Clifford+$T$ and Clifford+$V$. Further, we introduce a suboptimal but short run-time algorithm for synthesizing multi-qubit controlled gates. This approach highlights the role of subgroup structures in reducing synthesis complexity and opens a new direction of study on the compilation of multi-qubit gates. The framework is broadly applicable to different universal gate sets, and our analysis suggests that it can serve as a foundation for resource-efficient quantum compilation in near-term architectures.

DOI
10.48550/arxiv.2510.08312

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We developed a general framework for synthesizing target gates by using a finite set of basic gates, which is a crucial step in quantum compilation. When approximating a gate in SU(nnn), a naive brute-force search requires a computational complexity of O(1/ε(n2−1))O(1/\varepsilon^{(n^2 - 1)})O(1/ε(n2−1)) to achieve an approximation with error ε\varepsilonε. In contrast, by using our method, the complexity can be reduced to O(−n2log⁡ε/ε((n2−1)/2))O(-n^2 \log\varepsilon/\varepsilon^{((n^2 - 1)/2)})O(−n2logε/ε((n2−1)/2)). This method requires almost no assumptions and can be applied to a variety of gate sets, including Clifford+TTT and Clifford+VVV. Further, we introduce a suboptimal but short run-time algorithm for synthesizing multi-qubit controlled gates. This approach highlights the role of subgroup structures in reducing synthesis complexity and opens a new direction of study on the compilation of multi-qubit gates. The framework is broadly applicable to different universal gate sets, and our analysis suggests that it can serve as a foundation for resource-efficient quantum compilation in near-term architectures.
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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+T+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‾\overline{T}T gate (a TTT unitary up to a phase) is universal for the matchgate group. Our approach leverages the connection between nnn-qubit matchgate circuits and the standard representation of SO(2n)\mathbb{SO}(2n)SO(2n), which reduces the compilation from 2n×2n2^n\times 2^n2n×2n unitaries to 2n×2n2n\times2n2n×2n ones, thus reducing exponentially the size of the target matrix. Moreover, we rigorously show that this scheme is efficient, as an approximation error εSO(2n)\varepsilon_{\mathbb{SO}(2n)}εSO(2n)​ incurred in this smaller-dimensional representation translates at most into an O(n εSO(2n))O(n \,\varepsilon_{\mathbb{SO}(2n)})O(nεSO(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 UUU such that U⊗U∗U\otimes U^*U⊗U∗ has entries in the ring Z[1/2,i]\mathbb{Z}\big[1/\sqrt 2,i\big]Z[1/2​,i] can be exactly synthesized by a finite sequence of gates from the matchgate-Clifford+T‾+\overline{T}+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 XXXXXX Hamiltonian on n=4, 8n=4,\,8n=4,8 qubits.
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We present two deterministic compilation algorithms for single-qutrit unitaries with $O(log1/\ensuremath{\varepsilon})$ gate depth. Each algorithm selects a nearby approximation to the target unitary and then exactly synthesizes the approximation over the Clifford +++ R\mathbf{R}R basis. The first algorithm exhaustively searches over the group; while the second algorithm searches only for Householder reflections. The exhaustive search algorithm yields an average R\mathbf{R}R count of $2.193(11)+8.621(7){log}_{10}(1/\ensuremath{\varepsilon})$, albeit with a time complexity of $O({\ensuremath{\varepsilon}}^{\ensuremath{-}4.4})$. The Householder search algorithm results in a larger average R\mathbf{R}R count of $3.20(13)+10.77(3){log}_{10}(1/\ensuremath{\varepsilon})$ at a reduced time complexity of $O({\ensuremath{\varepsilon}}^{\ensuremath{-}0.42})$, greatly extending the reach in $\ensuremath{\varepsilon}$. These costs correspond asymptotically to 35% and 69% more non-Clifford gates compared with synthesizing the same unitary with two qubits. Such initial results are encouraging for using the R\mathbf{R}R gate as the nontransversal gate for qutrit-based computation.
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Approximate synthesis of general single-qubit unitaries over the Clifford+T\sqrt{T}T​ gate set
For the standard Clifford+TTT gate set, deterministic, ancilla-free synthesis now attains the minimal TTT-count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The T\sqrt{T}T​ gate rotates by half the angle of TTT, generating a finer lattice of implementable operations. Access to this magic state is expected to lower the cost of deterministic and ancilla-free synthesis of general single-qubit unitaries (Kliuchnikov et al., arXiv:2203.10064), but no direct Clifford+T\sqrt{T}T​ algorithm existed for this case. We provide one by extending the integer lattice-point enumeration method of Morisaki et al. We adopt a resource state cost model based on the magic-state catalysis approach of Gidney and Fowler (arXiv:1812.01238). On Haar-random targets synthesized to precisions ranging from ε=10−3\varepsilon=10^{-3}ε=10−3 to 10−810^{-8}10−8, the cost of Clifford+T\sqrt{T}T​ circuits scales as 2.4log⁡2(1/ε)2.4\log_2(1/\varepsilon)2.4log2​(1/ε) compared to 3.0log⁡2(1/ε)3.0\log_2(1/\varepsilon)3.0log2​(1/ε) for TTT-count-optimal Clifford+TTT circuits. Once a one-time catalyst state is amortized, the Clifford+T\sqrt{T}T​ circuits are never costlier than their Clifford+TTT counterparts.