QC24 Clean Ancilla Output Projection

Let $n,a\ge 0$ be integers and $D=2^n$. Let the data and workspace registers have ordered computational spaces ${\mathbb{C}}^D$ and ${\mathbb{C}}^{2^a}$, with the ordered tensor factors and product basis of Ordered tensor registers, their inner product, and wire permutations. For a circuit unitary $U$ on their tensor product, its all-zero-workspace projection on a fixed input $\mathrm{\mid }{0}^n⟩\mathrm{\mid }{0}^a⟩$ is

$v=(I_D\otimes ⟨0^a\mathrm{\mid })U(\mathrm{\mid }{0}^n⟩\otimes \mathrm{\mid }{0}^a⟩)\in {\mathbb{C}}^D\mathrm{.}$

The projected joint vector is $v\otimes \mathrm{\mid }{0}^a⟩$. An exact clean-workspace output on this input means that the full, unconditional output equals $\lambda \mathrm{\mid }\psi ⟩\otimes \mathrm{\mid }{0}^a⟩$ for a unit data vector $\psi$ and a unit scalar $\lambda$; then $v=\lambda \psi$. A merely nonzero $v$ defines only the zero-workspace projection branch and does not assert that the workspace is deterministically clean.