DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Controlled Unitary Access

Definition

Let $C = C^{2}$ be a control qubit with ordered orthonormal basis $∣ 0 ⟩, ∣ 1 ⟩$ , let $H$ be a finite-dimensional complex inner-product space, and let $U : H \to H$ be unitary. On the control-first register $C \otimes H$ , define

$C (U) : = (\begin{array}{cc} I_{H} & 0 \\ 0 & U \end{array}) .$

Thus $C (U) (∣ 0 ⟩ \otimes v) = ∣ 0 ⟩ \otimes v$ and $C (U) (∣ 1 ⟩ \otimes v) = ∣ 1 ⟩ \otimes U v$ . Controlled access means that the chosen operation model explicitly grants a call to this block operator. An ordinary call to a black-box $U$ does not silently grant a controlled call, and no query or gate cost is assigned unless the model states it.

Facts & Assumptions

Given: The finite target space $H$ , the ordered control-first tensor product, and a unitary $U$ on $H$ .

[L1]

Ordered tensor registers, their inner product, and wire permutations defines the ordered factors and product inner product after an ordered orthonormal basis is specified for each factor. Its proof establishes that the pairing is independent of those basis choices.

[L2]

Exact one-qubit gates and rotations fixes the ordered qubit computational basis and exact one-qubit matrices. Its current draft definition body was read; its item is now present in the QC07 manifest and page; the page remains draft and no acceptance is claimed.

[F1]

Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces: a unitary operator on a finite-dimensional complex inner-product space is an invertible linear isometry, so it preserves every vector norm and has an inverse.

[F2]

Every finite-dimensional real or complex inner product space has an orthonormal basis: every finite-dimensional inner-product space has an ordered orthonormal basis.

Verification

1.1givenL1L2F2choose

Choose an ordered orthonormal basis of $H$ using [F2]. Together with the control basis in [L2], it supplies the factor bases required by [L1]; the pairing is independent of this choice by [L1]. Write an arbitrary vector of $C \otimes H$ as $∣ 0 ⟩ \otimes v_{0} + ∣ 1 ⟩ \otimes v_{1}$ . The orthonormal control basis and the product inner product from [L1] give squared norm $∥ v_{0} ∥^{2} + ∥ v_{1} ∥^{2}$ .

2.1step 1.1F1algebra

The block operator sends that vector to $∣ 0 ⟩ \otimes v_{0} + ∣ 1 ⟩ \otimes U v_{1}$ . By [F1], $∥ U v_{1} ∥ = ∥ v_{1} ∥$ , so the squared norm is unchanged.

3.1givenF1algebra∎

The block matrix $diag (I_{H}, U^{- 1})$ is a two-sided inverse for $C (U)$ , so $C (U)$ is invertible as well as norm-preserving. By the unitary definition in [F1], $C (U)$ is unitary.

Depends on

Used by

Amplitude amplification with independently chosen phase rotations Algorithm
Amplitude-estimation expectation estimator Algorithm
Bounded-error evaluation from a dual adversary certificate Algorithm
Coherent SDP matrix-expectation oracle Algorithm
Controlled T using a clean ancilla Algorithm
Costed coherent evaluation of cyclic group powers Algorithm
Exact success amplification by attenuating the good branch Algorithm
Finite amplification from a supplied success lower bound Algorithm
Finite capped quantum quantile minimum Algorithm
Finite CKS sparse QLSP with variable-time inverse estimation Algorithm
Gray-Path Synthesis of a Two-Level Gate Algorithm
HHL finite sine-window phase-estimation algorithm Algorithm
MNRS finite reversible-walk finding Algorithm
One-step oblivious amplitude amplification Algorithm
QC24 State Preparation Algorithm
QFT-based amplitude estimation Algorithm
QSVT regularized pseudoinverse transformation Algorithm
Quantum membership-to-separation procedure Algorithm
Quantum phase estimation Algorithm
Quantum singular-value transformation circuit Algorithm
Sparse-oracle Gibbs-state purification Algorithm
Szegedy-walk marked-set detection Algorithm
Unknown-amplitude randomized search Algorithm
Variance-aware quantum Monte Carlo Algorithm
A bare unitary oracle does not supply controlled powers for free Counterexample
Amplitude estimation and counting input/output contract Definition
Block-encoded QLSP input and singular-gap promise Definition
Block-encoding oracle and implementation cost model Definition
Childs–Kothari–Somma sparse QLSP oracle model Definition
Clifford+T gate library and exact matrices Definition
CNOT and Control-Target Order Definition
Coherent finite sampler and observable value Definition
Coherent variable-time algorithm model Definition
Coherent XOR function oracle Definition
Continuous-time Hamiltonian access cost model Definition
Controlled Phase Rotation Definition
Eigenphase estimation access model Definition
Eigenvalue-estimation access and aliasing convention Definition
Finite circuit resource accounting Definition
Finite cyclic discrete-logarithm problem and costed encoding Definition

…and 50 more results.

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 6: Quantum Algorithms (standard reference, not scraped)