Knuth-Bendix Completion Algorithm and Shuffle Algebras For Compiling NISQ Circuits

Abstract

Compiling quantum circuits lends itself to an elegant formulation in the language of rewriting systems on non commutative polynomial algebras $\mathbb Q\langle X\rangle$. The alphabet $X$ is the set of the allowed hardware 2-qubit gates. The set of gates that we wish to implement from $X$ are elements of a free monoid $X^*$ (obtained by concatenating the letters of $X$). In this setting, compiling an idealized gate is equivalent to computing its unique normal form with respect to the rewriting system $\mathcal R\subset \mathbb Q\langle X\rangle$ that encodes the hardware constraints and capabilities. This system $\mathcal R$ is generated using two different mechanisms: 1) using the Knuth-Bendix completion algorithm on the algebra $\mathbb Q\langle X\rangle$, and 2) using the Buchberger algorithm on the shuffle algebra $\mathbb Q[L]$ where $L$ is the set of Lyndon words on $X$.