Phased outcome-complete simulation

We generalize the polynomial-time outcome-complete simulation algorithm for stabilizer circuits in arXiv:2309.08676 to track global phases exactly, yielding what we call phased outcome-complete simulation. The original algorithm enabled equivalence checking of stabilizer circuits with intermediate measurements and conditional Pauli corrections for all input states and all measurement outcomes simultaneously, but it tracked quantum states only up to a global phase. Our generalization removes this limitation and enables equivalence checking for an important family of non-stabilizer circuits: stabilizer circuits augmented with single-qubit rotations $exp(ialpha Z)$ by symbolic angles. Two such circuits are equivalent if they implement the same quantum channel for all values of the symbolic angles and all measurement outcomes, given a one-to-one correspondence between rotation angles in the two circuits and a mapping between measurement outcomes. This model enables testing of compilation algorithms that transform the Clifford portions of a computation while preserving rotation angles. Examples include Pauli-based computation, edge-disjoint path compilation for surface codes, and custom compilation strategies for reversible circuits such as adders, multipliers, and table lookups. Our efficient classical verification methods extend naturally to circuits with outcome-parity-conditional Pauli gates and intermediate measurements, features that are ubiquitous in fault-tolerant quantum computing but are rarely addressed by existing equivalence-checking approaches.