Controlled not connectivity in the Clifford group

TL;DR

This paper analyzes the structure of the two-qubit Clifford group by classifying its elements into orbits under local Clifford equivalence, revealing detailed orbit and intersection properties.

Contribution

It introduces a novel classification of the two-qubit Clifford group into 20 orbits and characterizes their intersections, enhancing understanding of its algebraic structure.

Findings

Clifford group C2 splits into 20 orbits under local Clifford equivalence.

Each orbit contains 4608 elements.

Intersections of orbits have 512 matrices each.

Abstract

The Clifford group is the set of gates generated by the CZ gate, and the two local gates: the Hadamard and the Pi/2 phase shift gate. It is known that, for a two qubit system, the Clifford group C2 is a subgroup of order 92160 of the group of 4 by 4 unitary matrices. It is also known that the local Clifford gates LC2 is a subgroup of order 4608 of the group C2. In order to better understand the set C2, we make two matrices U1 and U2 in C2 equivalent if U_1=VU_2 for some V in LC2. We show that this equivalence relation splits C2 into 20 orbits, O1, ..., O20, each with 4608 elements. Moreover, for each orbit Oi, CZOi intersects 9 different orbits Oi1, ...,Oi9. Moreover, the intersection of Oij and CZOi has 512 matrices for each j=1,2, ..., ,9. The link https://www.youtube.com/watch?v=lcYtB2tnXFw&t=685s leads you to a YouTube video that explains the most important results in this paper.