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Real 3-qubit gate decompositions via triality

Brendan Pawlowski

TL;DR

This work delivers a tight real-3-qubit circuit decomposition by leveraging the exotic triality symmetry of PSO(8). It reframes multi-qubit gate synthesis through Cartan decompositions, magic bases, and carefully chosen commuting involutions to reduce CNOT counts, achieving at most 14 CNOTs plus 35 single-qubit rotations for any unimodular real 3-qubit gate. The approach hinges on a constructive triality map that interchanges tensor-product subgroups with block matrices, enabling explicit circuit realizations and aligning Lie-algebraic canonical parameters with circuit parameters. The results provide both a concrete circuit template and a broader Lie-theoretic framework that could influence future optimal-gate decompositions for higher-qubit real gates. Overall, the paper contributes a novel algebraic pathway to more efficient quantum circuit synthesis in the real 3-qubit regime with potential extensions via triality-inspired mappings.

Abstract

We show that any unimodular real 3-qubit gate can be expressed as the product of at most 14 CNOT gates plus single-qubit gates, improving on the bound of 16 CNOTs due to Wei and Di. Our method uses the exotic triality symmetry of $\operatorname{PSO}(8)$, and we explore some of the useful properties of this map in relation to the study of real 3-qubit gates.

Real 3-qubit gate decompositions via triality

TL;DR

This work delivers a tight real-3-qubit circuit decomposition by leveraging the exotic triality symmetry of PSO(8). It reframes multi-qubit gate synthesis through Cartan decompositions, magic bases, and carefully chosen commuting involutions to reduce CNOT counts, achieving at most 14 CNOTs plus 35 single-qubit rotations for any unimodular real 3-qubit gate. The approach hinges on a constructive triality map that interchanges tensor-product subgroups with block matrices, enabling explicit circuit realizations and aligning Lie-algebraic canonical parameters with circuit parameters. The results provide both a concrete circuit template and a broader Lie-theoretic framework that could influence future optimal-gate decompositions for higher-qubit real gates. Overall, the paper contributes a novel algebraic pathway to more efficient quantum circuit synthesis in the real 3-qubit regime with potential extensions via triality-inspired mappings.

Abstract

We show that any unimodular real 3-qubit gate can be expressed as the product of at most 14 CNOT gates plus single-qubit gates, improving on the bound of 16 CNOTs due to Wei and Di. Our method uses the exotic triality symmetry of , and we explore some of the useful properties of this map in relation to the study of real 3-qubit gates.

Paper Structure

This paper contains 16 sections, 23 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matrix notation
  4. Quantum gate conventions
  5. Lie group notation and Cartan decomposition
  6. Commuting involutions
  7. Two-qubit magic
  8. Block matrix subgroups of $\operatorname{PSO}(n)$
  9. Automorphisms and roots
  10. Triality
  11. Triality for $\mathfrak{so}(8)$
  12. Triality for $\operatorname{PSO}(8)$
  13. Some correspondences under triality
  14. Pauli operators and triality
  15. Real circuit elements
  16. ...and 1 more sections

Key Result

Theorem 2.2

helgason With notation as above, we have $\mathcal{G} = \mathcal{K}\mathcal{A}\mathcal{K}$. That is, any $g \in \mathcal{G}$ can be written as $k_1 a k_2$ with $k_1,k_2 \in \mathcal{K}$ and $a \in \mathcal{A}$. Also, $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ and where $\mathop{\mathrm{Ad}}\nolimits_k : \mathfrak{g} \to \mathfrak{g}$ is the derivative of the conjugation map $c_k$ at the id

Figures (1)

  • Figure 1: Triality correspondences

Theorems & Definitions (62)

  • Example 2.1
  • Theorem 2.2: Cartan decomposition
  • Example 2.3
  • Corollary 2.4
  • proof
  • Example 2.5
  • Definition 2.6
  • Example 2.7
  • Example 2.8
  • Lemma 2.9
  • ...and 52 more