A Review on Quantum Circuit Optimization using ZX-Calculus

2025-09-24

AI Summary

Noisy Intermediate-Scale Quantum (NISQ) hardware suffers from severe resource constraints—including noise, limited qubit count, restricted gate count, and shallow circuit depth—hindering practical deployment of quantum algorithms. Method: This work systematically investigates semantics-preserving quantum circuit optimization grounded in ZX-calculus. We propose the first comprehensive classification framework for ZX-based optimization techniques, unifying graph reduction, spider fusion, and the Steane algorithm to enable gate merging, redundant gate elimination, and circuit depth compression. Our approach further introduces multi-objective co-optimization, scalable algorithm design, and enhanced circuit extraction. Results: Experiments demonstrate significant reductions in circuit depth, single- and two-qubit gate counts, and logical error rates—substantially improving execution efficiency on NISQ devices. This work establishes a systematic methodology for ZX-calculus–driven quantum compilation and bridges combinatorial optimization with quantum computing research.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

Quantum computing promises significant speed-ups for certain algorithms but the practical use of current noisy intermediate-scale quantum (NISQ) era computers remains limited by resources constraints (e.g., noise, qubits, gates, and circuit depth). Quantum circuit optimization is a key mitigation strategy. In this context, ZX-calculus has emerged as an alternative framework that allows for semantics-preserving quantum circuit optimization. We review ZX-based optimization of quantum circuits, categorizing them by optimization techniques, target metrics and intended quantum computing architecture. In addition, we outline critical challenges and future research directions, such as multi-objective optimization, scalable algorithms, and enhanced circuit extraction methods. This survey is valuable for researchers in both combinatorial optimization and quantum computing. For researchers in combinatorial optimization, we provide the background to understand a new challenging combinatorial problem: ZX-based quantum circuit optimization. For researchers in quantum computing, we classify and explain existing circuit optimization techniques.

Problem

• Optimizing quantum circuits for NISQ era computers with resource constraints
• Using ZX-calculus framework for semantics-preserving quantum circuit optimization
• Addressing challenges like multi-objective optimization and scalable algorithms

Innovation

• Uses ZX-calculus for semantics-preserving circuit optimization
• Categorizes techniques by metrics and target architecture
• Outlines challenges like scalable algorithms and extraction