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The Clifford hierarchy for one qubit or qudit

Nadish de Silva, Oscar Lautsch

TL;DR

This work completely resolves the Clifford hierarchy for a single qubit or one-qudit of prime dimension, proving that every hierarchy gate is semi-Clifford and admitting a unique MDC decomposition $G = M D C$ with $M$ from a finite set, $D$ diagonal, and $C$ Clifford. By combining a new normal form for Clifford gates with a geometric Pauli-support criterion, the authors show that all hierarchy gates are simplifiable and provide a precise size formula $|[ ext{C}_k]| = d^3(d^2-1)(d^{k-1}+d^{k-2}-d)$ for the $k^{th}$ level. These results yield a complete, implementable description of fault-tolerant gate teleportation schemes in the one-qudit setting and enable efficient gate synthesis via the MDC normal form. The findings have practical implications for constructing universal fault-tolerant quantum computers with qudits and offer a foundation for extending the classification to multi-qudit and higher-qudit systems.

Abstract

The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The groups of Pauli and Clifford gates constitute the first and second 'levels', respectively. Non-Clifford gates from the third level or higher, such as the $T$ gate, are necessary for achieving fault-tolerant universal quantum computation. Since it was defined twenty-five years ago by Gottesman-Chuang, two questions have been studied by numerous researchers. First, precisely which gates constitute the Clifford hierarchy? Second, which subset of the hierarchy gates admit efficient gate teleportation protocols? We completely solve both questions in the practically-relevant case of the Clifford hierarchy for gates of one qubit or one qudit of prime dimension. We express every such hierarchy gate uniquely as a product of three simple gates, yielding also a formula for the size of every level. These results are a consequence of our finding that all such hierarchy gates can be expressed in a certain form that guarantees efficient gate teleportation. Our decomposition of Clifford gates as a unique product of three elementary Clifford gates is of broad applicability.

The Clifford hierarchy for one qubit or qudit

TL;DR

This work completely resolves the Clifford hierarchy for a single qubit or one-qudit of prime dimension, proving that every hierarchy gate is semi-Clifford and admitting a unique MDC decomposition with from a finite set, diagonal, and Clifford. By combining a new normal form for Clifford gates with a geometric Pauli-support criterion, the authors show that all hierarchy gates are simplifiable and provide a precise size formula for the level. These results yield a complete, implementable description of fault-tolerant gate teleportation schemes in the one-qudit setting and enable efficient gate synthesis via the MDC normal form. The findings have practical implications for constructing universal fault-tolerant quantum computers with qudits and offer a foundation for extending the classification to multi-qudit and higher-qudit systems.

Abstract

The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The groups of Pauli and Clifford gates constitute the first and second 'levels', respectively. Non-Clifford gates from the third level or higher, such as the gate, are necessary for achieving fault-tolerant universal quantum computation. Since it was defined twenty-five years ago by Gottesman-Chuang, two questions have been studied by numerous researchers. First, precisely which gates constitute the Clifford hierarchy? Second, which subset of the hierarchy gates admit efficient gate teleportation protocols? We completely solve both questions in the practically-relevant case of the Clifford hierarchy for gates of one qubit or one qudit of prime dimension. We express every such hierarchy gate uniquely as a product of three simple gates, yielding also a formula for the size of every level. These results are a consequence of our finding that all such hierarchy gates can be expressed in a certain form that guarantees efficient gate teleportation. Our decomposition of Clifford gates as a unique product of three elementary Clifford gates is of broad applicability.
Paper Structure (24 sections, 33 theorems, 54 equations, 4 figures)

This paper contains 24 sections, 33 theorems, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Prior work
  3. Summary of main results
  4. Background
  5. Notational conventions
  6. Pauli gates
  7. The Pauli basis
  8. Conjugate pairs
  9. Clifford gates
  10. The Clifford hierarchy
  11. Diagonal hierarchy gates
  12. Semi-Clifford gates
  13. Technical preliminaries
  14. A normal form for Clifford gates
  15. Almost diagonal, simplified, and simplifiable gates
  16. ...and 9 more sections

Key Result

Lemma 1

For any $U,V\in M_d(\mathbb{C})$ and $p, q\in \mathbb{Z}_d$, we have

Figures (4)

  • Figure 1: The Pauli support of the linear combination $X^2+3Z^2X-Z^3X^3$. The powers of $Z$ range from $0$ to $d-1$ along the horizontal axis, while the powers of $X$ range along the vertical axis.
  • Figure 2: The compact gate teleportation protocol for semi-Clifford gates.
  • Figure 3: The Pauli support of a diagonal gate is contained in the linear subspace $\{(p, 0) \ | \ p\in \mathbb{Z}_d\}$, and in general, the Pauli support of $DX^{q_0}$ is contained in $\{(p,q_0) \ | \ p\in \mathbb{Z}_d\}$ if $D$ is diagonal.
  • Figure 4: Proof of Theorem \ref{['cor: Pauli support of conj. pairs']}. The horizontal lines in the left diagram contain the Pauli supports of $U'$ and $V'$, which are mapped to the parallel lines on the right side of the diagram by $C$.

Theorems & Definitions (72)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Definition 4
  • Lemma 2
  • Definition 5
  • Lemma 3: Discrete Stone-von Neumann theorem, single-qudit version
  • Definition 6
  • Definition 7
  • ...and 62 more