# Vanishing 2-Qubit Gates with Non-Simplification ZX-Rules

**Authors:** Ryan Krueger  
**arXiv:** [2209.06874](https://arxiv.org/abs/2209.06874)  
**Date:** 2022-09-14  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2209.06874v1/

## Key points

- Circuit optimization is crucial in quantum computing to reduce the number of gates required for a given algorithm.
- Traditional optimization techniques involve optimizing circuits at the gate level, which can be time-consuming and challenging.
- The ZX-calculus allows circuits to be translated into graphical representations known as ZX-diagrams, which can be simplified using specific rules of the calculus.
- Current extraction procedures for obtaining a simplified circuit from a ZX-diagram can lead to an increase in the number of 2-qubit gates required.
- Krueger and Kissinger propose a new strategy for optimizing quantum circuits using non-simplification rewrite rules based on graph-theoretic notions of local complementation and pivoting.
- Their approach generates local variants of a simplified ZX-diagram using these congruences and explores their space with simulated annealing and genetic algorithms to obtain a simplified circuit with fewer 2-qubit gates.
- Their method outperforms state-of-the-art optimization techniques for low-qubit circuits (<10) on randomly generated circuits and consistently reduces overall circuit complexity by an additional ~15–30% and eliminates up to 46% of 2-qubit gates compared to off-the-shelf methods on previously reported benchmark circuits with up to 14 qubits.
- This work highlights how leveraging mathematical concepts from graph theory can provide novel solutions for complex problems in quantum computing.

## Summary

In the field of quantum computing, circuit optimization is a crucial task that aims to reduce the number of gates required to implement a given quantum algorithm. Traditional optimization techniques involve directly optimizing the circuit at the gate level, which can be a time-consuming and challenging process. Alternatively, researchers have developed the ZX-calculus, which allows circuits to be translated into graphical representations known as ZX-diagrams. These diagrams can then be simplified using specific rules of the calculus, resulting in a simplified circuit. However, current extraction procedures for obtaining a simplified circuit from a ZX-diagram can lead to an increase in the number of 2-qubit gates required. In this work by Ryan Krueger and Aleks Kissinger, they propose a new strategy for optimizing quantum circuits using non-simplification rewrite rules based on graph-theoretic notions of local complementation and pivoting. By generating local variants of a simplified ZX-diagram using these congruences and exploring their space with simulated annealing and genetic algorithms, they aim to obtain a simplified circuit with fewer 2-qubit gates. Their approach takes advantage of the fact that local changes in a ZX-diagram can significantly affect the complexity of the extracted circuit. The authors demonstrate that their method outperforms state-of-the-art optimization techniques for low-qubit circuits (<10) on randomly generated circuits. Furthermore, on previously reported benchmark circuits with up to 14 qubits, their method consistently reduces overall circuit complexity by an additional ~15–30% and eliminates up to 46% of 2-qubit gates compared to off-the shelf methods. These preliminary results serve as proof–of–concept for a new optimization strategy in the ZX–calculus that could potentially lead to significant improvements in quantum circuit design. Overall, this work highlights how leveraging mathematical concepts from graph theory can provide novel solutions for complex problems in quantum computing.