A Review on Quantum Circuit Optimization using ZX-Calculus

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.