Quantum Circuit Simplifier: ZX-Calculus Spider Fusion

Circuit

Wires 4

ZX Rewrite Rules

Diagram Size

Spiders

Edges

T-count

Selected Node

Click a node in the diagram to inspect it.

Z-spider (green, computational basis) X-spider (red, Hadamard basis) Boundary (circuit input/output)

How it works

Every quantum circuit built from Hadamard, CNOT and phase gates can be redrawn as a ZX-diagram: green "Z-spiders" act in the computational basis, red "X-spiders" act in the Hadamard basis, and each spider carries a phase 0–2π. Wires connect spiders; a spider's phase is the phase gate it represents.

Spider fusion:  •—θ—• (same color)  ⟶  •(θ+φ)
  connected same-color spiders merge into one,
  phases add — this deletes gates for free.

Identity removal:  a—•(0)—b  ⟶  a—b
  a degree-2 spider with phase 0 does nothing
  and can be deleted, its wire reconnected.

These two rewrite rules are literally provable circuit identities — applying them never changes what the circuit computes, only how many gates it takes. Real quantum compilers (PyZX, Quantinuum's tket) run exactly this kind of graph rewriting to shrink circuits before they run on noisy hardware, because every extra spider is extra error. The T-count readout tracks non-Clifford phases (odd multiples of π/4) specifically, because those are the expensive gates in fault-tolerant quantum computing — minimizing them is often the entire point of the optimization pass.

Drag to orbit · click a node to inspect

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.