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Asymptotic Gate Count Bounds for Ancilla-Free Single-Qubit Synthesis with Arithmetic Gates

Kaoru Sano, Hayata Morisaki, Seiseki Akibue

TL;DR

This work analyzes ancilla-free synthesis of single-qubit unitaries using Clifford+$G$ gates, connecting quantum circuit synthesis to number theory. It develops deterministic and probabilistic synthesis frameworks and derives three main asymptotic bounds on the $G$-count required to achieve a target accuracy, with Haar-random unitaries attaining a tight $3\log_p(1/\varepsilon)$ (and a probabilistic $\tfrac{3}{2}\log_p(1/\varepsilon)$) rate and structured unitaries reaching higher lower bounds, while some edge cases exhibit non-convergent behavior. A central methodological advancement is the employment of Diophantine approximation and the Subspace Theorem to prove lower bounds (e.g., $4\log_p(1/\varepsilon)$ or $2\log_p(1/\varepsilon)$) for arithmetic gate families, together with a Liouville-type phenomenon showing undefined exact rates in certain constructions. The results partially resolve a generalized Ross--Selinger conjecture and provide a rigorous connection between gate synthesis resource counts and arithmetic properties of the target unitaries, clarifying scaling implications for fault-tolerant quantum computation with arithmetic gate sets.

Abstract

We study ancilla-free approximation of single-qubit unitaries $U\in {\rm SU}(2)$ by gate sequences over Clifford+$G$, where $G\in\{T,V\}$ or their generalization. Let $p$ denote the characteristic factor of the gate set (e.g., $p=2$ for $G=T$ and $p=5$ for $G=V$). We prove three asymptotic bounds on the minimum $G$-count required to achieve approximation error at most $\varepsilon$. First, for Haar-almost every $U$, we show that $3\log_{p}(1/\varepsilon)$ $G$-count is both necessary and sufficient; moreover, probabilistic synthesis improves the leading constant to $3/2$. Second, for unitaries whose ratio of matrix elements lies in a specified number field, $4\log_p(1/\varepsilon)$ $G$-count is necessary. Again, the leading constant can be improved to $2$ by probabilistic synthesis. Third, there exist unitaries for which the $G$-count per $\log_{p}(1/\varepsilon)$ fails to converge as $\varepsilon\to 0^+$. These results partially resolve a generalized form of the Ross--Selinger conjecture.

Asymptotic Gate Count Bounds for Ancilla-Free Single-Qubit Synthesis with Arithmetic Gates

TL;DR

This work analyzes ancilla-free synthesis of single-qubit unitaries using Clifford+ gates, connecting quantum circuit synthesis to number theory. It develops deterministic and probabilistic synthesis frameworks and derives three main asymptotic bounds on the -count required to achieve a target accuracy, with Haar-random unitaries attaining a tight (and a probabilistic ) rate and structured unitaries reaching higher lower bounds, while some edge cases exhibit non-convergent behavior. A central methodological advancement is the employment of Diophantine approximation and the Subspace Theorem to prove lower bounds (e.g., or ) for arithmetic gate families, together with a Liouville-type phenomenon showing undefined exact rates in certain constructions. The results partially resolve a generalized Ross--Selinger conjecture and provide a rigorous connection between gate synthesis resource counts and arithmetic properties of the target unitaries, clarifying scaling implications for fault-tolerant quantum computation with arithmetic gate sets.

Abstract

We study ancilla-free approximation of single-qubit unitaries by gate sequences over Clifford+, where or their generalization. Let denote the characteristic factor of the gate set (e.g., for and for ). We prove three asymptotic bounds on the minimum -count required to achieve approximation error at most . First, for Haar-almost every , we show that -count is both necessary and sufficient; moreover, probabilistic synthesis improves the leading constant to . Second, for unitaries whose ratio of matrix elements lies in a specified number field, -count is necessary. Again, the leading constant can be improved to by probabilistic synthesis. Third, there exist unitaries for which the -count per fails to converge as . These results partially resolve a generalized form of the Ross--Selinger conjecture.

Paper Structure

This paper contains 20 sections, 26 theorems, 87 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Results
  3. Preliminaries
  4. Deterministic and probabilistic unitary synthesis
  5. Strong approximation theory
  6. Results
  7. Elementary gate sets
  8. Notions for asymptotic $G$-count
  9. Theorems on asymptotic $G$-count
  10. Big hole for arithmetic gates
  11. Diophantine approximations
  12. Preparation of absolute values
  13. Subspace theorem
  14. Proof of Proposition \ref{['prop: approximation']}
  15. Arithmetic quantum matrices and its generalized counts
  16. ...and 5 more sections

Key Result

Lemma 1

akibue2024probabilistic For a target single-qubit unitary channel $\mathcal{U}$ and a finite set $\{\mathcal{V}_x\}_x$ of single-qubit unitary channels, it holds that

Figures (1)

  • Figure 1: Illustration of the difference between (a) the previous research and (b) our research. In both cases, we simultaneously increase the number of exactly synthesizable unitaries (dark gray dots) in compensation for the larger $G$-count and decrease the acceptable error, as shown in the figure from left to right. (a) Previous researchPARZANCHEVSKI2018869: The blue region represents target unitaries that exactly synthesizable unitaries cannot approximate within the acceptable error. If the ratio between the approximation error and the number of synthesizable unitaries is appropriately chosen, the area of this region converges to zero. However, this research cannot capture whether a fixed target unitary is contained in the blue region or not. (b) Our research: The focus is on the number of synthesizable unitaries near a specific target unitary (located at the origin), which changes as $6\rightarrow6\rightarrow2$ in the figure. This is illustrated by zooming into the region around the origin. The red disc represents the region around the target unitary for each level of acceptable error. The light gray dots represent the exactly synthesizable unitaries obtained by increasing $G$-count by one.

Theorems & Definitions (55)

  • Conjecture : ross2014optimal
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 45 more