  We introduce a framework for the formal specification and verification of
quantum circuits based on the Feynman path integral. Our formalism, built
around exponential sums of polynomial functions, provides a structured and
natural way of specifying quantum operations, particularly for quantum
implementations of classical functions. Verification of circuits over all
levels of the Clifford hierarchy with respect to either a specification or
reference circuit is enabled by a novel rewrite system for exponential sums
with free variables. Our algorithm is further shown to give a polynomial-time
decision procedure for checking the equivalence of Clifford group circuits. We
evaluate our methods by performing automated verification of optimized
Clifford+T circuits with up to 100 qubits and thousands of T gates, as well as
the functional verification of quantum algorithms using hundreds of qubits. Our
experiments culminate in the automated verification of the Hidden Shift
algorithm for a class of Boolean functions in a fraction of the time it has
taken recent algorithms to simulate.
