A Review on Quantum Circuit Optimization using ZX-Calculus
Tobias Fischbach, Pierre Talbot, Pascal Bouvry
TL;DR
This paper surveys quantum circuit optimization via ZX-calculus, framing it as a semantics-preserving, diagrammatic alternative to gate-based optimization. It presents a three-stage ZX workflow—convert to ZX-diagrams, apply rewrite rules, and extract back to circuits—underpinned by the potential to reduce depth, T-gates, and two-qubit gates, while balancing architecture-specific constraints. The authors classify techniques into heuristics, metaheuristics, reinforcement learning, and tree-search approaches, and discuss their targets, including architecture-aware variants and the role of gflow/cflow in enabling polynomial-time extraction. They identify key challenges, notably scalability, multi-objective optimization, and efficient circuit extraction, and outline directions such as intermediate representations, surrogate models, and architecture-aware ZX-based synthesis, to bridge diagrammatic optimization with practical hardware constraints.
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.