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Compiling quantum circuits lends itself to an elegant formulation in thelanguage of rewriting systems on non commutative polynomial algebras $\mathbbQ\langle X\rangle$. The alphabet $X$ is the set of the allowed hardware 2-qubitgates. The set of gates that we wish to implement from $X$ are elements of afree monoid $X^*$ (obtained by concatenating the letters of $X$). In thissetting, compiling an idealized gate is equivalent to computing its uniquenormal form with respect to the rewriting system $\mathcal R\subset \mathbbQ\langle X\rangle$ that encodes the hardware constraints and capabilities. Thissystem $\mathcal R$ is generated using two different mechanisms: 1) using theKnuth-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 $X$.

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