Alphabeta MathQuantum Computing
This proposed category develops quantum computing as finite-dimensional mathematics. It begins with complex inner-product spaces, ordered registers, tensor products and finite outcome laws, then defines pure and mixed states, measurements, instruments and completely positive trace-preserving maps. It builds qubit and gate conventions before giving circuits a precise syntax and denotational semantics, with their wire order, classical control, inverse access and ancilla costs made explicit. Reversible computation, exact and approximate gate synthesis, uniform circuit families, promise problems and bounded-error models then establish the interfaces used by quantum algorithms. Separate branches develop Pauli and stabilizer methods, teleportation, classical and quantum error correction, fault-tolerant computation, measurement-based computation, circuit resource limits and state-preparation costs. The scope retains every one of the twenty-four commissioned QC topics while keeping unrelated branches separate in the dependency graph. Existing root linear algebra, finite probability, algebra, Boolean-circuit and computation items are reused by exact ID. Physics postulates, experiments and observations are never logical suppliers: physical motivations may appear only in explicit nonlogical remarks, and no physical-realization claim is needed. This file is a draft overview for a prose-planning proposal; it registers no page or item carrier and makes no proof-acceptance claim.
An exhaustive draft-contract audit supplies explicit finite compiler and simulator instructions, exact resource models and local prerequisite proof routes, including isometry completion and preparation precision. The 358 planned contracts remain unauthored mathematical work; structural consistency and source receipts supply no proof acceptance. Substantial finite-precision synthesis, universal fault-tolerant gadget and inherited root-prerequisite obligations stay visible in the audit evidence before any later build is authorized.
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
3 pagesThis page fixes the finite-dimensional complex inner-product conventions used by the quantum-computing track.
2 definitions, 1 proposition, 7 theoremsExamples & counterexamples →This page defines finite-dimensional complex registers, fixes the ordered computational basis, and proves the big-endian encoding convention.
5 definitions, 2 lemmas, 6 theoremsExamples & counterexamples →This page builds multiplicity-aware fusion spaces in a unitary fusion category, defines F and pullback R coordinates, and proves their unitarity and the pentagon and hexagon…
9 definitions, 1 lemma, 7 theorems, 2 examples, 1 counterexampleExamples & counterexamples →
Level 1
2 pagesThis page develops classical linear codes over an explicitly named finite field and finite-register quantum code subspaces.
8 definitions, 6 lemmas, 7 theorems, 1 exampleExamples & counterexamples →A quantum state on a nonzero finite-dimensional complex inner-product space is represented here by a positive semidefinite trace-one operator.
4 definitions, 1 lemma, 10 theoremsExamples & counterexamples →
Level 2
2 pages- Measurements, POVMs, and Instruments16 results
This page defines finite projective measurements and POVMs as algebraic families of operators, then proves positivity and normalization of their outcome laws.
5 definitions, 1 lemma, 10 theoremsExamples & counterexamples → A qubit is the ordered two-dimensional complex register with basis |0⟩,|1⟩ and a state is a positive semidefinite trace-one operator.
4 definitions, 6 theoremsExamples & counterexamples →
Level 3
2 pagesThis page develops finite-dimensional complex-linear maps between full operator spaces.
5 definitions, 4 lemmas, 2 propositions, 8 theoremsExamples & counterexamples →This page fixes control-first block conventions and separate access contracts for controlled and inverse unitary calls.
9 definitions, 6 theoremsExamples & counterexamples →
Level 4
4 pagesThis page defines finite layered circuit syntax on ordered qubit wires, the local embedding of gates, and the chronological denotation of a circuit.
5 definitions, 5 lemmas, 5 theoremsExamples & counterexamples →This page proves the finite-dimensional Knill–Laflamme recovery criterion, constructs a trace-preserving syndrome recovery, and extends the criterion to a protected logical factor with a gauge subsystem.
3 definitions, 7 lemmas, 3 theorems, 1 corollary, 1 exampleExamples & counterexamples →This page fixes the phase convention P(k,x,z)=iᵏXˣZᶻ, with tensor factors ordered and the binary label ordered (x,z).
5 definitions, 5 lemmas, 6 theoremsExamples & counterexamples →For finite-dimensional density operators, this page fixes the root-fidelity convention F(ρ,σ)=‖√ρ√σ‖ 1 and the state trace distance D(ρ,σ)=1/2‖ρ-σ‖ 1.
5 definitions, 10 lemmas, 20 theoremsExamples & counterexamples →
Level 5
3 pagesThis page defines the four ordered Bell vectors, proves their preparation by Hadamard followed by control-first CNOT and their decoding by the reverse map, then gives complete teleportation and dense-coding procedures and proofs.
2 definitions, 2 algorithms, 7 theoremsExamples & counterexamples →This page develops reversible Boolean computation with explicit finite circuits and ordered input, work, and target registers.
3 definitions, 1 algorithm, 4 lemmas, 6 theoremsExamples & counterexamples →Develop the finite phase-aware stabilizer formalism for ordered qubit registers.
5 definitions, 1 algorithm, 13 lemmas, 3 theoremsExamples & counterexamples →
Level 6
2 pagesThis page develops exact matrix synthesis from continuous one-qubit rotations and CNOT gates.
2 definitions, 2 algorithms, 8 lemmas, 10 theoremsExamples & counterexamples →This page develops stabilizer code spaces from commuting Hermitian phase-aware Pauli checks on n≥1 physical qubits, defines their syndrome and logical-Pauli quotient, and proves the projector dimension and distance criteria.
5 definitions, 1 algorithm, 1 lemma, 8 theorems, 1 exampleExamples & counterexamples →
Level 7
3 pagesThis page develops discrete approximate universality and the single-qubit Solovay–Kitaev recursion in operator norm.
2 definitions, 1 algorithm, 9 lemmas, 4 theoremsExamples & counterexamples →This page defines finite graph states over ordered qubit registers and gives their computational-basis and stabilizer descriptions.
6 definitions, 11 lemmas, 6 theoremsExamples & counterexamples →This page constructs the periodic square-lattice toric stabilizer code over F 2, proves its check relations, two encoded qubits, e/m string syndromes, logical algebra, distance and recovery threshold.
6 definitions, 7 lemmas, 8 theorems, 1 counterexampleExamples & counterexamples →
Level 8
3 pagesThis page treats exact one-qubit Clifford+T synthesis, controlled-T with and without a clean ancilla, and ancilla-free on-the-nose Rz approximation.
5 definitions, 5 algorithms, 25 lemmas, 1 proposition, 10 theorems, 1 corollaryExamples & counterexamples →Define finite ordered channel schedules and independent or local-stochastic fault models.
8 definitions, 13 lemmas, 5 theoremsExamples & counterexamples →This page treats state and isometry preparation as finite mathematical input and circuit problems.
4 definitions, 2 algorithms, 5 lemmas, 9 theoremsExamples & counterexamples →
Level 9
1 pageThis page fixes full-list polynomial-time generation: one deterministic machine prints each complete circuit description from unary input length.
6 definitions, 1 algorithm, 2 lemmas, 3 theoremsExamples & counterexamples →
Level 10
2 pagesThis page separates exact optimization tasks from locally terminating rewrites and construction upper bounds.
3 definitions, 1 algorithm, 17 lemmas, 10 theoremsExamples & counterexamples →This page develops unitary braid actions on fixed-total-charge fusion spaces, then separates projective phase, logical block error, and leakage.
8 definitions, 9 lemmas, 9 theorems, 2 examples, 1 counterexampleExamples & counterexamples →