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Paper 1
High-threshold decoding of non-Pauli codes for 2D universality
Julio C. Magdalena de la Fuente, Noa Feldman, Jens Eisert, Andreas Bauer
- Year
- 2026
- Journal
- arXiv preprint
- DOI
- arXiv:2604.02033
- arXiv
- 2604.02033
Topological codes have many desirable properties that allow fault-tolerant quantum computation with relatively low overhead. A core challenge for these codes, however, is to achieve a low-overhead universal gate set with limited connectivity. In this work, we explore a non-Pauli stabilizer code that can be used to complete a universal gate set on topological toric and surface codes in strictly two dimensions. Fault-tolerant syndrome extraction for the non-Pauli code requires mid-circuit $X$ corrections, a key difference to conventional Pauli codes. We construct and benchmark a just-in-time (JIT) matching decoder to reliably decide these corrections. Under a phenomenological error model with equally likely physical and measurement errors, we find a high threshold of $\approx 2.5\,\%$, close to the $\approx 2.9\,\%$ of a decoder with access to the full syndrome history. We also perform a finite-size scaling analysis to estimate how the logical error rate scales below threshold and verify an exponential suppression in both physical error rate and in the system size. A second global decoding step for $Z$ errors is required and the non-Clifford gates in the circuit reduce the threshold from $\approx 2.9\,\%$ to $\approx 1.8\,\%$ with a naive decoder. We show how $Z$ decoding can be improved using knowledge of the $X$ corrections, pushing the threshold to $\approx 2.2\,\%$. Our results suggest non-Clifford logic in 2D codes could perform comparably to 2D quantum memory. Our formalism for efficient benchmarking and decoding directly generalizes to a broader family of CSS codes whose $X$ stabilizers are twisted by diagonal Clifford operators, and spacetime versions thereof, defined by CSS-like circuits enriched by $CCZ$, $CS$, and $T$ gates.
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Majorana-XYZ subsystem code
Tobias Busse, Lauri Toikka
- Year
- 2026
- Journal
- arXiv preprint
- DOI
- arXiv:2603.26311
- arXiv
- 2603.26311
We present a new type of a quantum error correction code, termed Majorana-XYZ code, where the logical quantum information scales macroscopically yet is protected by topologically non-trivial degrees of freedom. It is a $[n,k,g,d]$ subsystem code with $n=L^2$ physical qubits, $k= \lfloor L/2 \rfloor$ logical qubits, $g \sim L^2$ gauge qubits, and distance $d = L$. The physical check operations, i.e. the measurements needed to obtain the error syndrome, are $3$-local and nearest-neighbour. The code detects every 1- and 2-qubit error, and every error of weight 3 and higher (constrained by the distance) that is not a product of the 3-qubit check operations, however, these products act only on the gauge qubits leaving the code space invariant. The undetected weight-3 and higher operators are confined to the gauge group and do not affect logical information. While the code does not have local stabiliser generators, the logical qubits cannot be modified locally by an undetectable error, and in this sense the Majorana-XYZ code combines notions of both topological and local gauge codes while providing a macroscopic number of topological logical qubits. Taken as a non-gauge stabiliser code we can encode $k \sim L^2 - 3L$ logical qubits into $L^2$ physical qubits; however, the check operators then become weight $2L$. The code is derived from an experimentally promising system of Majorana fermions on the honeycomb lattice with only nearest-neighbour interactions.
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