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Buildings for Synthesis with Clifford+R

Mark Deaconu, Nihar Gargava, Amolak Ratan Kalra, Michele Mosca, Jon Yard

TL;DR

We address exact synthesis for the qutrit Clifford$+\mathsf{R}$ gate set by connecting circuit synthesis to the geometry of a Bruhat-Tits building for $U_3$ over $\mathbb{Z}[\chi^{-1}]$. The authors construct the building explicitly via lattice chains and Cartan decomposition, and use it to prove arithmeticity of the Clifford$+\mathsf{R}$ gate set in a new way. The main technical result is that the group generated by $H,S,R$ at $p=3$ equals $U_3(\mathbb{Z}[\chi^{-1}])$, and synthesis can be visualized as path traversal on a tree acting on lattice chains. This approach bridges number theory and quantum circuit design, providing both a self-contained Bruhat-Tits framework and potential for efficient exact and approximate synthesis algorithms. The work thus advances understanding of arithmetic gate sets and offers concrete geometric tools for qutrit circuit synthesis.

Abstract

We study the problem of exact synthesis for the Clifford+R gate set and give the explicit structure of the underlying Bruhat-Tits building for this group. In this process, we also give an alternative proof of the arithmetic nature of the Clifford+R gate set.

Buildings for Synthesis with Clifford+R

TL;DR

We address exact synthesis for the qutrit Clifford gate set by connecting circuit synthesis to the geometry of a Bruhat-Tits building for over . The authors construct the building explicitly via lattice chains and Cartan decomposition, and use it to prove arithmeticity of the Clifford gate set in a new way. The main technical result is that the group generated by at equals , and synthesis can be visualized as path traversal on a tree acting on lattice chains. This approach bridges number theory and quantum circuit design, providing both a self-contained Bruhat-Tits framework and potential for efficient exact and approximate synthesis algorithms. The work thus advances understanding of arithmetic gate sets and offers concrete geometric tools for qutrit circuit synthesis.

Abstract

We study the problem of exact synthesis for the Clifford+R gate set and give the explicit structure of the underlying Bruhat-Tits building for this group. In this process, we also give an alternative proof of the arithmetic nature of the Clifford+R gate set.

Paper Structure

This paper contains 28 sections, 28 theorems, 72 equations, 5 figures.

Table of Contents

  1. Introduction
  2. General synthesis problem
  3. Scope of our work
  4. Bruhat-Tits Theory and Gate Synthesis
  5. Thin groups vs arithmeticity
  6. Mathematical preliminaries
  7. Important Gate Sets
  8. Local fields
  9. Galois group
  10. Rational prime and local completion
  11. Hermitian form and unitary group
  12. Lattices over local valuation rings
  13. Dual lattice
  14. Cartan decomposition over local fields
  15. Buildings over unitary groups
  16. ...and 13 more sections

Key Result

Theorem 1

The group generated by the Clifford$+\mathsf{R}$ gate set for $p=3$ is where the group on the right is the set of unitaries with entries in $\mathbb{Z}[{\raisebox{\depth}{$\chi$}}^{-1}]$ and ${\raisebox{\depth}{$\chi$}}$ is a solution to $X^{2}-3X +3$ in $\mathbb{C}$.

Figures (5)

  • Figure 1: An alternating vertex connected to two pure vertices
  • Figure 2: What a loop would look like in $\mathcal{B}$
  • Figure 3: A pure vertex connected to four alternating vertices via the four isotropic lines $V \subset V^\perp$ in $\mathbb{F}_3^3$.
  • Figure 4: Vertices up to distance 6 from the center of $\mathcal{B}$. Note that the pure vertices are denoted by red and the alternating ones by blue.
  • Figure 5: The section $S_v$ of the tree, as well as the dotted path from $e_0$ to $v$.

Theorems & Definitions (80)

  • Theorem
  • Remark 3.1
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Theorem 1: Theorem 5.5 in Kalra2025synthesisarithmetic
  • Remark 3.5
  • Remark 3.6
  • Lemma 3.7
  • ...and 70 more