Prepared unitary and success subspace for amplitude amplification

Let $H$ be a nonzero finite-dimensional complex inner-product space, let $\mathrm{\mid }0⟩\in H$ be a specified unit input vector, let $A:H\to H$ be a supplied measurement-free unitary, and let ${\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$ be an orthogonal projection on $H$. The access contract supplies and charges calls to $A$, its inverse $A^{†}$, the good phase reflection $S_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}:=I_H-2{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}},$ and the zero-input phase reflection $S_0:=I_H-2\mathrm{\mid }0⟩⟨0\mathrm{\mid }\mathrm{.}$ The signs are part of the contract: $S_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$ is $-I$ on $\mathrm{ran}{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$ and $+I$ on its orthogonal complement. Reflection about the prepared state is the composite $A{S}_0{A}^{†}$, with its component calls charged unless a separate direct-reflection access is supplied.

Put $\mathrm{\mid }\psi ⟩=A\mathrm{\mid }0⟩$ and define the initial success weight $a:=\mathrm{\parallel }{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}\mathrm{\mid }\psi ⟩{\mathrm{\parallel }}^2\in [0,1]\mathrm{.}$ When $a>0$, define $\mathrm{\mid }G⟩:={\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}\mathrm{\mid }\psi ⟩\mathrm{/}\sqrt{a}$. When $a<1$, define $\mathrm{\mid }B⟩:=(I_H-{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}})\mathrm{\mid }\psi ⟩\mathrm{/}\sqrt{1-a}$. These vectors are normalized and lie respectively in the range and kernel of ${\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$. If $a=0$ or $a=1$, the corresponding zero component is omitted and no vector is normalized from it. With this convention, $\mathrm{\mid }\psi ⟩=\sqrt{a}\mathrm{\mid }G⟩+\sqrt{1-a}\mathrm{\mid }B⟩$ means the sum of the nonzero component terms. Under the projective measurement $\{{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}},I_H-{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}\}$, the probability of the good outcome is $a$; the finite-dimensional rank-one density and projective-measurement outcome laws are A unit-vector outer product is a density operator and Projective measurement outcome weights are probabilities.

The output task may return the conditional quantum state after the good outcome; that state is defined only on a positive-probability branch and is supported in $\mathrm{ran}{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$ by Conditional state for a projective measurement. A classical output additionally requires ${\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$ to be diagonal in a specified output basis and a costed coherent or ordinary verifier. An arbitrary projection does not supply a classical predicate.

The unitary must be supplied coherently. A measured randomized procedure is not an admissible $A$ unless a separate coherent embedding is supplied. Controlled access to $A$ is not assumed. A controlled verifier, if used, is a distinct charged access capability.