Eigenphase estimation access model

Let $H$ be a finite-dimensional complex inner-product space, $U:H\to H$ a unitary, and $\psi \in H$ a unit vector with a promised phase $\theta \in [0,1)$ such that $U\psi =\exp (2\pi i\theta )\psi \mathrm{.}$ For an integer $m\ge 1$, let $Q=2^m$. The phase register is $({\mathbb{C}}^2{)}^{\otimes m}$ with ordered computational basis indexed big-endian by $x={\sum }_{k=0}^{m-1}{b}_k{2}^{m-1-k}$, where $b_k\in \{0,1\}$ (Finite registers and their ordered computational bases, Big-endian encoding of ordered bit strings). Thus the wire of weight $2^j$ is $q_{m-1-j}$.

There are distinct forward-access options:

1. Controlled-power access. The model explicitly supplies one controlled $U^{2^j}$ operation for each $j=0,\dots ,m-1$. Each use is counted as one controlled-power query; an implementation's ordinary gate cost is recorded separately.
2. Single controlled-unitary access. The model supplies calls to $C(U)=\mathrm{diag}(I_H,U)$ in the control-first order of Controlled Unitary Access. A controlled $U^{2^j}$ may be implemented by $2^j$ sequential calls to $C(U)$; those calls are charged individually.

Optional controlled inverse-power access is a separate capability that explicitly supplies $C(U^{-2^j})$ for each required $j$. It is not implied by either forward option. It is needed only by a consuming procedure that reverses the controlled-power portion of a coherent QPE prefix. Reversing the inverse-QFT gate list does not use $U^{-1}$.

Finite ordered-register syntax is specified by Finite circuit syntax and ordered wire labels, gate action by Denotational semantics of a finite layered circuit, and projective measurement outcomes by Finite projective measurements. The definition is an access-model promise: it does not assert that the supplied controlled powers have an efficient construction or that $\theta$ has an exact $m$-bit expansion.