Complete set of circuit equations for stabilizer quantum mechanics

We find a sufficient set of equations between quantum circuits from which we can derive any other equation between stabilizer quantum circuits. To establish this result, we rely upon existing work on the completeness of the graphical ZX language for quantum processes, a two-coloured logical calculus for multi-qubit systems. The complexity of the circuit equations, as opposed to the very intuitive reading of the much smaller number of ZX equations, advocates the latter for performing computations with quantum circuits.

Figure 1

Generating diagrams for the ZX network.

Figure 2

Diagrammatic rules for the ZX network.

Figure 3

Quantum circuit interpretation of the ZX network elements.

Figure 4

Sound and complete set of circuit equations for stabilizer quantum mechanics.

Figure 5

Alternative ZX axioms in a form resembling quantum circuit equations.