Coherent finite sampler and observable value

Let $X$ be a finite nonempty set with probability law $p_x\ge 0$, ${\sum }_{x\in X}{p}_x=1$, and let the value register contain an exact finite encoding of a real function $g:X\to [a,b]$, where $a\le b$ are finite rationals. A coherent finite sampler for $(X,p,g)$ is an explicitly supplied finite-dimensional unitary $V$ on ordered sample, value, and work registers such that $V\mathrm{\mid }0⟩={\sum }_{x\in X}\sqrt{p_x}\mathrm{\mid }x⟩\mathrm{\mid }g(x)⟩\mathrm{\mid }{w}_x⟩,$ where the sample labels $\mathrm{\mid }x⟩$ are distinct computational-basis vectors and each work state $\mathrm{\mid }{w}_x⟩$ has norm one. Zero-probability labels may occur and need no special branch. The registers and their order use Finite registers and their ordered computational bases and Ordered tensor registers, their inner product, and wire permutations; each output is a pure state in the sense of Pure state and ray. The separate item Measurement law of a coherent sampler verifies the induced measurement law.

The access contract lists the actual circuits or oracle interfaces for $V$, $V^{†}$, controlled $V$ and controlled $V^{†}$, value decoding, reversible arithmetic, any coherent predicate and controlled rotation, and the reflections used by the chosen estimator. Their availability is explicit under Controlled Unitary Access and Inverse Unitary Access. A forward unitary alone does not grant an inverse or a controlled call.

The resource record follows Circuit-family and computation cost model: charge state preparation, inverse and controlled access, data loading, value comparison and arithmetic, controlled-rotation synthesis and precision, good-state and initial-state reflections, gates, width, ancillas, and classical postprocessing separately. The state-preparation circuit and its costs are inputs; a state description is not a free QRAM or preparation oracle.