DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.

Inverse Unitary Access

Definition

For a finite-dimensional unitary $U$ , inverse access is the explicitly granted operation $U^{†} = U^{- 1}$ . Its availability and query or gate charge are recorded separately from forward access, including any controlled access in Controlled Unitary Access. For a chronological sequence of length $k \geq 0$ that applies $U_{1}$ first and $U_{k}$ last, the total operator is $W = U_{k} \dots U_{1}$ when $k > 0$ and $W = I_{H}$ when $k = 0$ . For $k > 0$ , the inverse sequence applies $U_{k}^{†}$ first, then $U_{k - 1}^{†}$ , down to $U_{1}^{†}$ ; for $k = 0$ it is the empty sequence.

Facts & Assumptions

Given: Finite-dimensional unitaries $U_{1}, \dots, U_{k}$ and an explicit access model for the operations named in the Definition.

[L1]

Controlled Unitary Access fixes the target unitary whose forward and controlled capabilities are explicitly declared. Its carrier is present in this batch.

[F1]

For an endomorphism in finite dimension, preserving lengths, preserving inner products, carrying orthonormal bases to orthonormal bases, and $T^{*} T = I$ are equivalent: for a finite-dimensional endomorphism, preserving norms implies $U^{*} U = U U^{*} = I$ and $U^{- 1} = U^{*}$ .

[F2]

Adjoints satisfy $(S + T)^{*} = S^{*} + T^{*}$ , $(λ T)^{*} = \overline{λ} T^{*}$ , $(S T)^{*} = T^{*} S^{*}$ , and $T^{* *} = T$ : for composable finite-dimensional maps, $(S T)^{*} = T^{*} S^{*}$ .

[F3]

The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ : the adjoint is characterized by $⟨ T v, w ⟩ = ⟨ v, T^{*} w ⟩$ .

Verification

1.1givenL1F1F3

Each $U_{j}$ is an invertible isometry by the unitary convention in [L1] and [F1]. Thus its adjoint exists in finite dimension and is its inverse, so the inverse call specified in the Definition has the stated mathematical meaning.

1.2givenF2algebra

For $k > 0$ , repeated application of [F2] gives $(U_{k} \dots U_{1})^{*} = U_{1}^{*} \dots U_{k}^{*}$ . Acting on a vector from right to left, this applies $U_{k}^{*}$ first and $U_{1}^{*}$ last, exactly reversing the original chronology.

1.3givenalgebra

For $k = 0$ , the specified total map is $I_{H}$ and its inverse is also $I_{H}$ ; there are no calls or operations in either sequence. This covers the empty-sequence endpoint without interpreting an undefined product.

2.1L1step 1.2∎

The matrix identity determines the inverse map but says nothing about whether the access model grants or charges its calls. The Definition therefore records inverse availability and cost explicitly instead of inferring either from forward access.

Depends on

Used by

Amplitude amplification with a fixed integer iteration count Algorithm
Amplitude amplification with independently chosen phase rotations Algorithm
Amplitude-estimation expectation estimator Algorithm
Coherent SDP matrix-expectation oracle 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
HHL finite sine-window phase-estimation algorithm Algorithm
MNRS finite reversible-walk finding Algorithm
One-step oblivious amplitude amplification Algorithm
QFT-based amplitude estimation Algorithm
QSVT regularized pseudoinverse transformation Algorithm
Quantum membership-to-separation procedure 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
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
Coherent finite sampler and observable value Definition
Coherent variable-time algorithm model Definition
Continuous-time Hamiltonian access cost model Definition
Eigenphase estimation access model Definition
Eigenvalue-estimation access and aliasing convention Definition
Finite circuit resource accounting Definition
Finite row-preparation access model for a Szegedy walk Definition
Hamiltonian simulation error and resources Definition
HHL state-preparation and simulation access model Definition
LCU decomposition and access model Definition
Orthogonal-Projector Reflection and Access Contract Definition
Prepared unitary and success subspace for amplitude amplification Definition
Promise query instance and oracle family Definition
Quantum convex membership and weak-separation model Definition
Quantum query algorithm Definition
Qubitization access model Definition
Sparse SDP matrix access and computational model Definition

…and 20 more results.

Dependency tree · two levels

17 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)