bibkey: goodenough2026fusion authors: Kenneth Goodenough, Andrew Landahl, Joon Lee, Antonio Russo and Kevin Thompson year: 2026 title: “Optimal Fusion Strategies for Quantum Computation” doi: null url: https://arxiv.org/abs/2609.02559 claim: “For an [[n,1,d]] stabilizer code, fusion failures on a set W of qubits cause a logical failure iff the substrings of the failure-axis Pauli string on W contain a non-trivial logical operator (Proposition 3.1); the fusion distance equals the size of the largest minimal logical operator, and for graph codes it is the maximum LC-degree of the encoding vertex of the progenitor graph; the Discussion conjectures that every [[n,1,d]] code with a connected progenitor graph has a perfect adaptive strategy.” strata_touched:

• D5/S3/Quantum/Measurements/ConnectedProgenitorPerfectAdaptiveFusion license: citation-only triage: anchor

Optimal fusion strategies for quantum computation

arXiv:2609.02559v2 (primary quant-ph; versions v1 and v2). A logical fusion teleports a state encoded in an $[[n,1,d]]$ code through an encoded Bell pair by physical fusions on pairs of qubits; a failed physical fusion measures both qubits in its failure axis. Proposition 3.1 reads:

Let $\mathcal{C}$ be an $[[n,k,d]]$ stabilizer code, and let $Q$ be a fusion strategy. Then fusion failures on a subset $W$ of qubits lead to a logical failure if and only if $\hat{Q}[W]$ contains a non-trivial logical operator of $\mathcal{C}$.

Section 4 describes an $[[n,1,d]]$ code by a progenitor stabilizer state on $n+1$ qubits with an encoding qubit $e$ incident to at least one edge; the logical operators are the physical parts $s_{\mathrm{phys}}$ of its stabilizers $s=s_{\log }\otimes s_{\mathrm{phys}}$, acting as $s_{\log }$, and up to local Cliffords the progenitor state is a graph state. A strategy is perfect when a single successful fusion guarantees a successful logical fusion (Introduction). Section 6 (Discussion) reads:

Early on in this project we formulated the conjecture that for $k=1$ a connected progenitor graph $G^{\prime }$ indicated the existence of a perfect strategy. This was false and the example of the code given in \Cref{sec:examples} is such an example. However, this code does have a perfect adaptive strategy. So, a natural conjecture and direction of future work is that all $[[n,1,d]]$ codes with connected progenitor graphs have perfect adaptive strategies.

Verified locator

• URL: https://arxiv.org/abs/2609.02559 (v2; source file arxiv_v2.tex, md5 aeae388c3964f292d0911e80481b730c).
• Related: Reiß and van Loock, arXiv:2601.08820. Their Theorem 1 bounds the success probability of a logical Bell measurement by $1-(1-P_B{)}^{\min (n_1,n_2)}$. Their Theorem 2 gives general sufficient conditions for reaching it: a failure-axis sequence used before the first success, a sequence of stabilizer generators, and for each position a pair of logical operators that anticommute only there and commute with the earlier failure axes. They construct such sequences for the quantum parity, five-qubit, standard and rotated planar surface, tree and Steane codes.