TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Clean Ancilla Multicontrolled Tradeoff

Statement

For $m \geq 1$ controls, reuse the exact no-workspace $O (m^{2})$ and one-clean-bit $O (m)$ constructions from QC11, with total widths $m + 1$ and $m + 2$ and exact return of the clean bit. These compare construction upper bounds rather than optimal complexities.

Facts & Assumptions

Given: a controlled operation with $m$ controls and one target, exact elementary gates, and the stated clean-workspace convention.

[A1]

Multicontrolled Unitaries with and without a Clean Work Bit proves the no-workspace $O (m^{2})$ and one-clean-bit $O (m)$ construction bounds; its current proof body and exact borrowed-wire return argument were inspected.

[F1]

Clean and dirty ancillas defines a clean ancilla as initialized to $∣ 0 ⟩$ and returned exactly to $∣ 0 ⟩$ by the complete block for every data input.

[F2]

Finite circuit resource accounting defines width as the number of wires and size as the number of gate occurrences in the declared circuit.

Proof

1.1A1F2givenalgebra

The no-workspace construction in [A1] uses the $m$ controls and one target, so its width is $m + 1$ by [F2]. Its $O (m^{2})$ one-qubit and CNOT gates are $O (m^{2})$ gate occurrences under the declared circuit-size coordinate [F2].

1.2A1F1F2givenalgebra

The one-clean-bit construction in [A1] adds one work bit, so its width is $m + 2$ by [F2]. Its $O (m)$ one-qubit and CNOT gates are $O (m)$ gate occurrences [F2]. The compute/apply/uncompute sequence returns the bit to zero on every computational-basis input; because the overall operation is linear, the same identity holds on every superposition. By [F1] this is exactly the clean-ancilla return condition.

2.1A1step 1.1step 1.2given∎

The two upper bounds use different widths and have no matching lower bounds in this item. They therefore establish construction bounds, not an optimality tradeoff.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Barenco et al., Elementary gates for quantum computation (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)