Clean and dirty ancillas

Let $D$ be a finite ordered quantum register and let $A$ be an ordered register of $k\ge 0$ ancilla qubits, with Hilbert spaces ${\mathcal{H}}_D$ and ${\mathcal{H}}_A$. Write $\mathrm{\mid }{0}_A⟩=\mathrm{\mid }{0}^k⟩$, including $\mathrm{\mid }{0}_A⟩=1$ when $k=0$. Let $U$ be a unitary on ${\mathcal{H}}_D\otimes {\mathcal{H}}_A$ and let $V$ be a unitary on ${\mathcal{H}}_D$.

The register $A$ is a clean ancilla register for $U$ and $V$ when

for every vector $\mathrm{\mid }\psi {⟩}_D$. Thus the data operation is independent of the input vector and all clean wires return exactly to zero.

The register $A$ is a dirty ancilla register for $U$ and $V$ when the operator identity $U=V\otimes I_A$ holds on ${\mathcal{H}}_D\otimes {\mathcal{H}}_A$. Equivalently, every product input $\mathrm{\mid }\psi {⟩}_D\otimes \mathrm{\mid }\eta {⟩}_A$ is mapped to $V\mathrm{\mid }\psi {⟩}_D\otimes \mathrm{\mid }\eta {⟩}_A$, with the same $V$ for every data and ancilla vector. This requires exact restoration of an arbitrary, initially unknown ancilla state and forbids residual data–ancilla correlation.

These are exact contracts for the complete subroutine. A temporary intermediate alteration of an ancilla is permitted if the full unitary satisfies the stated identity.

The reference-register and arbitrary-matrix consequences are proved in Clean and dirty ancilla identities extend to references and mixed states.