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.
Smallest square root of two denominator exponent
Definition
For nonzero , define ; exponents may be negative. Set . For a matrix , the denominator statistic used in the synthesis statements below is , the exponent of the squared modulus of its upper-left entry. The base regime in this convention is exponent at most .
Remarks
The set whose minimum is taken is nonempty and bounded below for each nonzero by Denominator exponent is a well-defined integer. This is the square-root of-two convention of Kliuchnikov–Maslov–Mosca, not an exponent of .
Depends on
Used by
- Cyclotomic scaled grid and norm-equation solver Algorithm
- Exact single-qubit Clifford+T synthesis Algorithm
- Clifford+T denominator descent and finite base table Lemma
- Denominator exponent and optimal T count for determinant-one unitaries Lemma
- Exponent growth of the Rz scaled grid Lemma
- Finite coefficient bounds for the scaled cyclotomic grid Lemma
- Reduced Hadamard words and their exact denominators Lemma
- Size and residue of the Rz candidate norm integer Lemma
- Squared entry denominator bound Lemma
- Exact single-qubit Clifford+T ring characterization Theorem
- The exact single-qubit compiler returns the input matrix Theorem
Dependency tree · two levels
5 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
- Kliuchnikov, Maslov, and Mosca, Practical approximation of single-qubit unitaries by single-qubit quantum circuits (standard reference, not scraped)