The Classification of Clifford Gates over Qubits

TL;DR

This paper classifies all possible Clifford gate sets over qubits into 57 distinct classes, providing a comprehensive framework for understanding their structure and the bases they preserve, advancing quantum gate set theory.

Contribution

It introduces the first complete classification of Clifford unitaries generated by specific gate sets, extending classical reversible gate classification to the quantum domain.

Findings

Identified exactly 57 classes of Clifford unitaries.

Characterized classes by basis preservation and tableau invariants.

Provided a decomposition method for Clifford gates using tableau representations.

Abstract

We examine the following problem: given a collection of Clifford gates, describe the set of unitaries generated by circuits composed of those gates. Specifically, we allow the standard circuit operations of composition and tensor product, as well as ancillary workspace qubits as long as they start and end in states uncorrelated with the input, which rule out common "magic state injection" techniques that make Clifford circuits universal. We show that there are exactly 57 classes of Clifford unitaries and present a full classification characterizing the gate sets which generate them. This is the first attempt at a quantum extension of the classification of reversible classical gates introduced by Aaronson et al., another part of an ambitious program to classify all quantum gate sets. The classification uses, at its center, a reinterpretation of the tableau representation of Clifford gates to give circuit decompositions, from which elementary generators can easily be extracted. The 57 different classes are generated in this way, 30 of which arise from the single-qubit subgroups of the Clifford group. At a high level, the remaining classes are arranged according to the bases they preserve. For instance, the CNOT gate preserves the X and Z bases because it maps X-basis elements to X-basis elements and Z-basis elements to Z-basis elements. The remaining classes are characterized by more subtle tableau invariants; for instance, the T_4 and phase gate generate a proper subclass of Z-preserving gates.