Clifford+V synthesis for multi-qubit unitary gates

TL;DR

The paper tackles the problem of efficiently synthesizing multi-qubit unitaries using Clifford+$V$ (and Clifford+$T$) gate sets, a central challenge for resource-aware quantum compilation. It introduces two complementary strategies: a meet-in-the-middle exhaustive search that leverages nearest-neighbor data structures to achieve near-optimal decompositions with average $V$-count $3\log_5(1/\varepsilon)$ and average runtime $O(\log(1/\varepsilon)/\varepsilon^{1.5})$, and a subgroup-guided search that exploits a structured set of 12 bases to further accelerate synthesis of conditional gates with provable reductions in the search space. A specialized, suboptimal algorithm for multi-qubit-controlled-unitary gates lowers the $V$-count significantly in practical cases (e.g., about 30% for two qubits) and improves log-error by up to 60% relative to direct MITM approaches, with generalization to $n$-qubit generalized controlled gates. Collectively, these results establish a subgroup-first design principle for multi-qubit compilation that reduces search dimensionality, enables parallelization, and provides a practical path toward resource-efficient quantum compilation in near-term architectures.

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.

Figures (2)

• Figure 1: V-count for ten Haar random target SU(2)s for each accuracy $\epsilon$. Blue lines show the results of synthesizing the targets and orange lines show those of synthesizing the CC(target)s. Solid lines are the averages of the individual results, dashed lines are linear regressions of each result, and shaded regions indicate the range of V-count from min to max. The slopes of the blue and orange dashed lines are 3.00 and 2.15.
• Figure 2: V-count scaling of different synthesis strategies as a function of the target accuracy, plotted against $\log_{10}(1/\varepsilon)$. Dots show individual data points and dashed lines show linear regressions to those averages. For 24 Haar-random targets, we fixed the V-count and computed the minimum error $\varepsilon$ for each, repeating this procedure for several even values of the V-count. Black - Result of Meet-in-the-Middle Exhaustive Search for targets in SU(4). Cyan - Result of Meet-in-the-Middle Exhaustive Search for targets in SU(4) controlled gate. Blue - Result of Subgroup-Guided Search for targets in SU(4) generalized controlled gate. Red - Result of Meet-in-the-Middle Exhaustive Search for targets in SU(4) generalized controlled gate. Orange - Result of Subgroup-Guided Search and SU(2) synthesis search for targets in SU(4) generalized controlled gate. Each point with an odd V-count indicates that a sequence of the corresponding odd number of V gates can approximate the target more accurately than a sequence with one additional V gate. Each slope and intercept of the linear fits are summarized in Table \ref{['tab:comparison']}.