Alphabeta MathHow 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 consistency reflection uses two bit queries
Statement
For every x and every control value, the controlled input-consistency reflection circuit makes exactly two standard XOR bit-query calls. Equality tests, the phase operation, and scratch-register operations are nonoracle gates and are not included in this query count.
Facts & Assumptions
Given: The controlled consistency-reflection algorithm from Controlled input-consistency reflection from a bit query.
Controlled input-consistency reflection from a bit query: its circuit makes one XOR query to compute x_i into a clean scratch bit and a second to uncompute it.
Quantum query complexity: one invocation of the designated standard oracle is one query; nonoracle gates are outside this count.
Proof
The algorithm contains exactly two designated calls to O_x: one before the equality-controlled phase and one after it. Both calls occur for every value of the control bit; the inactive branch is protected by its scratch/response state but does not remove a designated invocation.
The second call is valid uncomputation because O_x is self-inverse, as recorded in [F1]. The equality test, phase gate, control logic and scratch preparation are nonoracle operations, so [F2] excludes them from query cost. Therefore the cost is exactly two queries.
Depends on
Used by
Dependency tree · two levels
14 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
- Andrew M. Childs, Lecture Notes on Quantum Algorithms (17 April 2025) (standard reference, not scraped)
- Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (standard reference, not scraped)