Proposition: Postselection and amplification ledgers

In an ancilla-free reflection implementation, the postselected construction and every fixed-$r$ amplification construction have

There is no external buffer in this batch convention.

The postselected construction uses one $U^{\ast }$ call and no $U^T$ call. The amplified construction uses

No call acts on the actual remnant. Original-evolution calls satisfy $n_U=n_{U^{†}}=0$ when the stated conjugate operations are the native oracle services.

At the end, destructive readout consumes $D{D}^{\prime }{C}^{\prime }$, a total of $2d+c$ qubits, and returns the $k$-qubit $\hat{M}$. Retaining all raw readout bits and a success flag costs at most $2d+c+1$ classical bits.

Proof. The displayed live register list accounts for every supplied and allocated factor. Reflections preserve those factors. Each $P$-reflection uses one $U^T$ and one $U^{\ast }$; the initial preparation uses one $U^{\ast }$. The final readouts and output consume or return every factor.