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Fibonacci B6 approximates one logical two-qubit gate in both charge blocks

Statement

Assume ${A C}_{ω}$ . Let $U \in U (C^{4})$ and $δ > 0$ . There is one normalized six-strand Fibonacci braid word $b$ such that its actions $B_{1} (b)$ and $B_{τ} (b)$ on the total-charge- $1$ and total-charge- $τ$ blocks are simultaneously within $δ$ of $T_{1} = U \oplus (\det U)^{- 1} \in S U (5), T_{τ} = U \oplus diag ((\det U)^{- 1}, 1, 1, 1) \in S U (8) .$ For the triple-triple computational subspace in either block, the logical block error and leakage amplitude are each at most $δ$ . The corresponding physical categorical braid differs from the normalized word by the common scalar phase specified by The braid phase is a common scalar character.

Facts & Assumptions

Given: The four-dimensional code $C_{a}$ in each six-anyon block, the same ordered path basis in both blocks, $U \in U (4)$ , $δ > 0$ , and ${A C}_{ω}$ .

[F1]

The normalized simultaneous braid closure contains $S U (5) \times S U (8)$ under ${A C}_{ω}$ . Joint SU(5)×SU(8) closure from B6 path blocks

[F2]

The two code blocks have dimensions $4$ , and their leakage complements have dimensions $1$ and $4$ . Six-anyon triple-encoded subspace and leakage complement

[F3]

If a unitary is within $δ$ of a code-preserving block target, its logical block error and leakage are at most $δ$ . Block-target approximation bounds logical error and leakage

[F4]

The original categorical word has one common scalar phase across both charge blocks. The braid phase is a common scalar character

[A1]

${A C}_{ω}$ is countable choice. The Axiom of Countable Choice ( ${A C}_{ω}$ )

Proof

1.1givenF2algebra

Put $λ = (\det U)^{- 1}$ . Since $∣ \det U ∣ = 1$ , $∣ λ ∣ = 1$ . Therefore $\det (T_{1}) = \det (U) λ = 1$ and $\det (T_{τ}) = \det (U) λ = 1$ ; both targets are unitary by their orthogonal direct-sum definitions. Hence $T_{1} \in S U (5)$ and $T_{τ} \in S U (8)$ .

2.1A1F1step 1.1choose

The open product of operator-norm $δ$ -balls around $(T_{1}, T_{τ})$ meets the normalized braid image because its closure contains $S U (5) \times S U (8)$ by [A1] and [F1]. Thus one word $b$ satisfies both block bounds at once. This is simultaneous density; separate choices of words in the two blocks are not used.

3.1F2F3step 2.1algebra

In each block, $T_{a}$ acts as $U$ on the four-dimensional code and as a unitary on its orthogonal complement. Apply [F3] to $B_{1} (b)$ and $T_{1}$ , then to $B_{τ} (b)$ and $T_{τ}$ . The logical block error and leakage amplitude are at most $δ$ in both blocks.

4.1F2F4step 3.1∎

By [F4], the physical categorical braid is the same scalar multiple of both normalized blocks; multiplying by a unit scalar leaves leakage unchanged and is recorded as a projective phase for uncontrolled computation. It must be tracked under coherent control.

Depends on

Used by

B6 exact rank certificates and encoded leakage budget Example

Dependency tree · two levels

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Sources

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