  The ubiquity of stabilizer circuits in the design and operation of quantum
computers makes techniques to verify their correctness essential. The
simulation of stabilizer circuits, which aims to replicate their behavior using
a classical computer, is known to be efficient and provides a means of testing
correctness. However, simulation is limited in its ability to examine the
exponentially large space of possible measurement outcomes. We propose a
comprehensive set of efficient classical algorithms to fully characterize and
exhaustively verify stabilizer circuits with Pauli unitaries conditioned on
parities of measurements. We introduce, as a practical characterization, a
general form for such circuits and provide an algorithm to find a general form
of any stabilizer circuit. We then provide an algorithm for checking the
equivalence of stabilizer circuits. When circuits are not equivalent our
algorithm suggests modifications for reconciliation. Next, we provide an
algorithm that characterizes the logical action of a (physical) stabilizer
circuit on an encoded input. All of our algorithms provide relations of
measurement outcomes among corresponding circuit representations. Finally, we
provide an analytic description of the logical action induced by measuring a
stabilizer group, with application in correctness proofs of code-deformation
protocols including lattice surgery and code switching.
