Single-qubit unitary gates by graph scattering

TL;DR

This paper explores how scattering plane-wave states on finite graphs can implement single-qubit gates in quantum computing, identifying various graphs that produce diverse unitary operations.

Contribution

It systematically enumerates graphs with up to 9 vertices that realize single-qubit gates via scattering, revealing exponential growth and diverse rotations including irrational multiples of pi.

Findings

Number of realizable gates grows exponentially with graph size

Most gates for larger graphs are rotations around axes uniformly distributed on the Bloch sphere

Both rational and irrational multiples of pi are achieved as rotation angles

Abstract

We consider the effects of plane-wave states scattering off finite graphs, as an approach to implementing single-qubit unitary operations within the continuous-time quantum walk framework of universal quantum computation. Four semi-infinite tails are attached at arbitrary points of a given graph, representing the input and output registers of a single qubit. For a range of momentum eigenstates, we enumerate all of the graphs with up to $n=9$ vertices for which the scattering implements a single-qubit gate. As $n$ increases, the number of new unitary operations increases exponentially, and for $n>6$ the majority correspond to rotations about axes distributed roughly uniformly across the Bloch sphere. Rotations by both rational and irrational multiples of $\pi$ are found.