Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus
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We present a new circuit-to-circuit optimisation routine based on an equational theory called the ZX-calculus. We first interpret quantum circuits as ZX-diagrams, a flexible, lower-level language for describing quantum computations graphically. Then, using the rules of the ZX-calculus, we derive a terminating simplification procedure for ZX-diagrams inspired by the graph-theoretic transformations of local complementation and pivoting. Finally, we use the preservation of a graph invariant called focused gFlow to derive a deterministic strategy for re-extracting a (simpler) quantum circuit from the resultant ZX-diagram. This procedure enables us to temporarily break the rigid quantum circuit structure and "see around" obstructions (namely non-Clifford quantum gates) to produce more simplifications than naive circuit transformation techniques alone.
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