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.
Nearest Neighbor CNOT Overhead
Statement
For the unoptimized recursive QSD circuit in the contiguous interval layout—each multiplexor target is an endpoint, recursive children use the remaining contiguous interval, and terminal two-qubit blocks act on adjacent wires—with both directed orientations of adjacent CNOT allowed, the explicit line routing uses at most nine times the original CNOT count. This is an architecture-specific gate-count bound, not a depth bound or a guarantee for other layouts.
Facts & Assumptions
Given: the unoptimized recursive QSD block decomposition, the contiguous interval layout stated above, and both directed orientations of adjacent CNOT.
Ninefold nearest-neighbor multiplexor routing proves the factor-nine count for each multiplexed rotation and its sum over the recursive construction, using the actual QC09 circuit interfaces.
Proof
Apply the local routing construction in [F1] to every multiplexed rotation in the recursively emitted QSD circuit and leave its terminal adjacent two-qubit blocks unchanged. Since gate count is additive over this explicit sequential list, the total routed count is at most nine times the original count.
The construction specifies only CNOT occurrences on an ordered line. It gives no schedule or parallelization argument, so it implies no depth bound; it also says nothing about layouts without the stated line order.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
7 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
- Shende, Bullock, and Markov, Synthesis of Quantum Logic Circuits, §6 (standard reference, not scraped)
- Ronald de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)