Optimal ancilla-free Clifford+T synthesis for general single-qubit unitaries

TL;DR

This work addresses the problem of ancilla-free Clifford+$T$ synthesis for arbitrary single-qubit unitaries, optimizing the $T$-count while achieving a target accuracy under the diamond distance $d_onterges{\diamond}$. It introduces two complementary approaches: a deterministic algorithm that attains $T$-optimal circuits with runtimes scaling as $\varepsilon^{-1/2 - o(1)}$ and a $T$-count of $3\log_2(1/\varepsilon) + o(\log_2(1/\varepsilon))$, and a probabilistic (mixed-unitary) algorithm with runtimes $\varepsilon^{-1/4 - o(1)}$ and a $T$-count of $1.5\log_2(1/\varepsilon) + o(\log_2(1/\varepsilon))$, both demonstrated to be practical up to $\varepsilon \approx 10^{-15}$ and $10^{-22}$ respectively. The methods rely on number-theoretic properties of Clifford+$T$, lattice enumeration via integer-point techniques, Matsumoto-Amano normal form, and SDP-based optimization to compute optimal mixtures, all without conjectures. The results provide rigorous, conjecture-free guarantees and offer concrete resource estimates for fault-tolerant quantum computation using Clifford+$T$ gates, with potential extensions to other gate sets and questions about polynomial-time solvability under relaxed optimality.

Abstract

We propose two Clifford+$T$ synthesis algorithms that are optimal with respect to $T$-count. The first algorithm, called deterministic synthesis, approximates any single-qubit unitary by a single-qubit Clifford+$T$ circuit with the minimum $T$-count. The second algorithm, called probabilistic synthesis, approximates any single-qubit unitary by a probabilistic mixture of single-qubit Clifford+$T$ circuits with the minimum $T$-count. For most of single-qubit unitaries, the runtimes of deterministic synthesis and probabilistic synthesis are $\varepsilon^{-1/2 - o(1)}$ and $\varepsilon^{-1/4 - o(1)}$, respectively, for an approximation error $\varepsilon$. Although this complexity is exponential in the input size, we demonstrate that our algorithms run in practical time at $\varepsilon \approx 10^{-15}$ and $\varepsilon \approx 10^{-22}$, respectively. Furthermore, we show that, for most single-qubit unitaries, the deterministic synthesis algorithm requires at most $3\log_2(1/\varepsilon) + o(\log_2(1/\varepsilon))$ $T$-gates, and the probabilistic synthesis algorithm requires at most $1.5\log_2(1/\varepsilon) + o(\log_2(1/\varepsilon))$ $T$-gates. Remarkably, complexity analyses in this work do not rely on any numerical or number-theoretic conjectures.

Figures (5)

• Figure 1: Illustration of $\mathcal{R}_\varepsilon(\vec{v})$ and $\mathcal{D}$ in $\mathbb{R}^2$. In practice, these sets are subsets of $\mathbb{R}^4$.
• Figure 2: Illustration of Lemma \ref{['lem: min_X equal min_hatX']} in $\mathbb{R}^2$ (in practice, a probabilistic mixture of single-qubit unitaries corresponds to a convex combination in $\mathbb{R}^{10}$). The light blue region represents the set of CPTP mappings that can be realized as probabilistic mixtures of $\{U_x\}_{x \in \{1,2,3,4,5,6\} }$. Among them, the CPTP mapping (red point) closest to the target unitary $V$ can be constructed by a probabilistic mixture of only $U_2$ and $U_3$ in the neighborhood of $V$. Then, the optimal probabilistic mixture improves quadratically upon the individual approximation error $\delta$.
• Figure 3: The $T$-count required for Algorithms \ref{['algo: dete synth']} and \ref{['algo: prob synth']}. Each colored region represents the range between the minimum and maximum $T$-counts. Also, the purple dashed line represents $y = 3\log_2(1/\varepsilon)$, and the red dashed line represents $y = 1.5\log_2(1/\varepsilon)$.
• Figure 4: Deterministic
• Figure 5: Probabilistic

Theorems & Definitions (53)

• Theorem 1.2: Informal Summary of Section \ref{['sec: Deterministic Synthesis']}
• Theorem 1.3: Informal Summary of Section \ref{['sec: Probabilistic Synthesis']}
• Proposition 2.1: Diamond distance between single-qubit unitaries
• proof
• Proposition 2.2: Properties of $O_{\mathrm{w.h.p.}}$
• proof
• Definition 2.3: Extensions of $\mathbb{Z}$
• Definition 2.4: Denominator exponent
• Theorem 2.5: Number theoretic property of $\langle \mathcal{C}_1, T\rangle$
• Proposition 3.4: Integer Enumeration Algorithm