# Speeding up quantum circuits simulation using ZX-Calculus

**Authors:** Tristan Cam, Simon Martiel  
**arXiv:** [2305.02669](https://arxiv.org/abs/2305.02669)  
**Date:** 2023-05-04  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2305.02669v1/

## Key points

- Authors propose a simple and efficient method to reduce contraction cost of tensor network for simulating quantum circuits
- Circuit represented as ZX-diagram and sparsified using simplification and local complementation rules
- Optimizing graph-like ZX-diagrams significantly improves existing state-of-the-art contraction cost by several orders of magnitude
- Motivation is to understand limits of classical simulation of quantum circuits as quantum computing advances
- Small circuits can be simulated using direct linear algebra simulation, larger circuits require more sophisticated approaches
- Matrix Product State (MPS) formalism allows lossy simulation of large quantum systems in shallow circuit regimes
- Restricting set of gates enables polynomial-time simulation
- Method achieves significant improvements in contraction cost compared to existing techniques
- For Sycamore circuits with depth 20, average contraction cost is 1180 times better than previous methods, up to 4200 times better at peak performance
- Contributes to advancing field of quantum circuit simulation by reducing contraction costs through graph-like ZX-diagrams
- Implications for improving efficiency and scalability of simulating complex quantum computations

## Summary

In this paper, the authors propose a simple and efficient method to reduce the contraction cost of a tensor network for simulating quantum circuits. They start by representing the circuit as a ZX-diagram and then apply simplification and local complementation rules to sparsify it. The authors find that optimizing graph-like ZX-diagrams significantly improves the existing state-of-the-art contraction cost by several orders of magnitude. The motivation behind this research is to better understand the limits of classical simulation of quantum circuits as the field of quantum computing continues to advance. While small circuits can be simulated using direct linear algebra simulation, larger circuits require more sophisticated approaches. The Matrix Product State (MPS) formalism allows for lossy simulation of large quantum systems in shallow circuit regimes, while restricting the set of gates can enable polynomial-time simulation. The authors demonstrate that their method achieves significant improvements in contraction cost compared to existing techniques. For Sycamore circuits with a depth of 20, they achieve an average contraction cost that is 1180 times better than previous methods and up to 4200 times better at peak performance. Overall, this research contributes to advancing the field of quantum circuit simulation by providing a novel approach to reducing contraction costs through graph-like ZX-diagrams. This has implications for improving the efficiency and scalability of simulating complex quantum computations.