Quantum algorithms for equational reasoning

Abstract: We introduce quantum normal form reduction, a quantum computational framework for analyzing abstract symbolic expressions - such as strings, algebraic formulas, or quantum circuits - that are equivalent under a given set of transformation rules. These rules form a term rewriting system, a formal method for deriving equivalences by repeatedly applying substitutions. We construct an efficiently implementable quantum Hamiltonian whose ground state encodes the entire class of equivalent expressions - potentially exponentially many - in a quantum superposition. By preparing and manipulating these ground states, we address fundamental problems in equational reasoning, including the word problem, i.e., determining whether two expressions are equivalent, counting the number of equivalent expressions, and identifying structural properties of equivalence classes. We demonstrate a quantum-inspired version of the algorithm using tensor network simulations by solving instances involving up to $10^{28}$ equivalent expressions, well beyond the reach of standard classical graph exploration techniques. This framework opens the path for quantum symbolic computation in areas ranging from quantum and logical circuit design to data compression, computational group theory, linguistics, polymers and biomolecular modeling, enabling the investigation of problems previously out of reach.