Extending the Graph Formalism to Higher-Order Gates

TL;DR

This paper introduces an algorithm that extends the graph formalism for simulating quantum circuits, enabling efficient handling of higher-order gates like Toffoli and 8 gates, with applications in circuit optimization and magic state representation.

Contribution

It develops an algorithm for simulating quantum circuits with higher-order gates within the graph formalism, including methods for splitting and merging stabilizer states.

Findings

Efficient simulation of 8 and Toffoli gates on stabilizer states.

Conditions for merging stabilizer states in the graph formalism.

Applications to circuit identities and low-rank magic state representations.

Abstract

We present an algorithm for efficiently simulating a quantum circuit in the graph formalism. In the graph formalism, we represent states as a linear combination of graphs with Clifford operations on their vertices. We show how a $\mathcal{C}_3$ gate such as the Toffoli gate or $\frac\pi8$ gate acting on a stabilizer state splits it into two stabilizer states. We also describe conditions for merging two stabilizer states into one. We discuss applications of our algorithm to circuit identities and finding low stabilizer rank representations of magic states.