Branch-independent residuals for phase-estimation uncomputation

Determine whether a phase-estimation circuit can be constructed whose residual operator after factoring out the bitwise garbage-label operation is independent of the eigenvalue branch, thereby permitting reuse of the inverse phase-estimation oracle for deterministic measurement-based phase correction.

Background

The paper shows that applying the conventional inverse quantum phase-estimation circuit after measuring the eigenvalue register cancels the measurement-induced phases but generally leaves branch-dependent residual garbage states. For textbook phase estimation, these residuals depend on the difference between modular integer subtraction and bitwise XOR, so the inverse QPE cannot generally be reused as a deterministic phase-correction oracle.

The analysis derives failure for the textbook implementation but does not exclude every possible phase-estimation circuit. A circuit with a branch-independent residual would satisfy the structural condition needed for deterministic measurement-based uncomputation, so determining whether such an implementation exists remains unresolved.