ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generated

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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.
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Identity target in the two B6 charge blocks

Example

Take the logical two-qubit target $U = I_{4}$ . Its determinant is one, so the special-unitary extensions are $T_{1} = I_{5}$ and $T_{τ} = I_{8}$ . The empty six-strand braid has normalized block matrices $I_{5}, I_{8}$ and exponent sum zero. It therefore certifies exact zero block error, zero logical error and zero leakage; its physical common phase is $R_{1}^{0} = 1$ . This is the identity boundary case of the shared-word B6 interface.

Facts & Assumptions

Given: The empty Artin word, the Fibonacci normalized block convention, and the six-anyon code embeddings.

[F1]

The encoded two-qubit theorem uses determinant-corrected targets $T_{1}, T_{τ}$ and bounds the code error and leakage by the block error. One B6 braid approximates an encoded two-qubit gate

[F2]

The determinant-character phase for exponent sum $k$ is $R_{1}^{k}$ . The braid phase is a common scalar character

Verification

1.1F1givenalgebra

Since $\det I_{4} = 1$ , both determinant-corrected extensions in [F1] are identity matrices; their determinants are one and their code restrictions are $I_{4}$ .

2.1F1F2step 1.1givenalgebra∎

The empty word acts as the identity in each block and has exponent sum zero, so both operator differences vanish. It fixes every code vector, hence its restriction to the code is exactly $I_{4}$ and its projection from the code to the orthogonal leakage complement is zero. Thus the logical error and leakage are zero directly, while [F2] gives physical phase $R_{1}^{0} = 1$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)
Michael H. Freedman, Michael J. Larsen, and Zhenghan Wang, A Modular Functor Which Is Universal for Quantum Computation (standard reference, not scraped)