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Synthesis of Single Qutrit Circuits from Clifford+R

Erik J. Gustafson, Henry Lamm, Diyi Liu, Edison M. Murairi, Shuchen Zhu

TL;DR

Two deterministic algorithms to approximate single-qutrit gates using the Clifford + $\mathbf{R}$ group to find the best approximation of diagonal rotations are presented.

Abstract

We present two deterministic algorithms to approximate single-qutrit gates. These algorithms utilize the Clifford + $\mathbf{R}$ group to find the best approximation of diagonal rotations. The first algorithm exhaustively searches over the group; while the second algorithm searches only for Householder reflections. The exhaustive search algorithm yields an average $\mathbf{R}$ count of $2.193(11) + 8.621(7) \log_{10}(1 / \varepsilon)$, albeit with a time complexity of $\mathcal{O}(\varepsilon^{-4.4})$. The Householder search algorithm results in a larger average $\mathbf{R}$ count of $3.20(13) + 10.77(3) \log_{10}(1 / \varepsilon)$ at a reduced time complexity of $\mathcal{O}(\varepsilon^{-0.42})$, greatly extending the reach in $\varepsilon$. These costs correspond asymptotically to 35% and 69% more non-Clifford gates compared to synthesizing the same unitary with two qubits. Such initial results are encouraging for using the $\mathbf{R}$ gate as the non-transversal gate for qutrit-based computation.

Synthesis of Single Qutrit Circuits from Clifford+R

TL;DR

Two deterministic algorithms to approximate single-qutrit gates using the Clifford + group to find the best approximation of diagonal rotations are presented.

Abstract

We present two deterministic algorithms to approximate single-qutrit gates. These algorithms utilize the Clifford + group to find the best approximation of diagonal rotations. The first algorithm exhaustively searches over the group; while the second algorithm searches only for Householder reflections. The exhaustive search algorithm yields an average count of , albeit with a time complexity of . The Householder search algorithm results in a larger average count of at a reduced time complexity of , greatly extending the reach in . These costs correspond asymptotically to 35% and 69% more non-Clifford gates compared to synthesizing the same unitary with two qubits. Such initial results are encouraging for using the gate as the non-transversal gate for qutrit-based computation.

Paper Structure

This paper contains 10 sections, 1 theorem, 62 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical Background
  3. Exhaustive search algorithm
  4. Householder reflection search
  5. Exact Synthesis
  6. Numerical Results
  7. Conclusion
  8. Enumeration algorithm for
  9. Solving the norm equation
  10. Bound on

Key Result

Lemma 1

For $\mathbf{y} \in \mathcal{D}(\mathbf{u}, f, \varepsilon)$, $|y_4| \leq \sqrt{r^2_2 - r^2_1}$.

Figures (2)

  • Figure 1: Search regions for: (left) Exhaustive Search Algorithm. The search regions of Eq. (\ref{['eq:approx-constraints-full-search']}) are: the blue corresponds to $x^\prime_1$; the green region corresponds to $y^\prime_2$ and the orange region corresponds to $z^\prime_3$; (right) Householder Search Algorithm. $\mathbf{u} = \frac{1}{\sqrt{2}}(\cos\theta/2, \sin\theta/2, -1, 0)^T$. The yellow region corresponds to the search region in Eq. (\ref{['eq:approx-lattice-version']}).
  • Figure 2: Scaling of the number of non-Clifford gates, $N_{\mathcal{G}}$ ($\mathbf{T}$ or $\mathbf{R}$ depending on qudit dimension) against the infidelity $\varepsilon$. Angles are uniformly sampled in the region $\theta\in(-\pi/2 , \pi/2)$.

Theorems & Definitions (2)

  • Lemma 1
  • proof