Barbell quantum code[1]
Description
Member of a family of tile codes [2,3] whose \(X\)- and \(Z\)-type check qubits are paired, with each pair joined by one near-local coupler and read out by superdense syndrome extraction [4]. A Bell pair prepared across the coupler lets both check qubits collect syndrome information for both stabilizers of the pair, so a stabilizer need only lie within the union of the two check qubits’ neighborhoods in the connectivity graph [1]. Every two-qubit gate of the QEC cycle is thereby native to a fixed-connectivity layout whose complexity does not grow with the code distance.
The codes are built for the six-qubit star lattice plus near-local coupler (6QSL+NLC) architecture, also called the Barbell architecture, a superconducting chip layout with two connectivity layers [1]. The first layer is a honeycomb of hexagonal cells whose six qubits share one multi-qubit coupler, so that any two qubits in a cell can interact. The second layer holds the near-local couplers, each joining an \(X\)-check qubit to its partner \(Z\)-check qubit. The six cells containing the two check qubits form a barbell, and the corresponding pair of \(X\)- and \(Z\)-type stabilizers is supported on its data qubits. A barbell code is obtained from a tile code in three steps [1]. Dummy stabilizers with empty support are added so that \(X\)- and \(Z\)-type stabilizers correspond one to one, and a check qubit is placed at each stabilizer. Each qubit type is then translated by a fixed vector so that every data qubit in the support of either stabilizer of a pair is adjacent to one of its two check qubits. Because the tile code is translation invariant, all near-local couplers are parallel and of equal, distance-independent length, and can be routed in a single additional connectivity layer without crossings.
The weight-8 family with \(k=16\) has \(n=2LM\) for an \(L\times M\) data-qubit lattice with bulk-stabilizer boxes of size \((D+1)\times(D+1)\), e.g., the \([[450,16,14]]\) code [1]. See Ref. [1; Table 3] for the weight-6, weight-8, and weight-10 families.
Rate
High-rate QLDPC family. The weight-8 examples encode \(k=16\) logical qubits using fewer than 30 data qubits per logical qubit. This is a reduction in physical-qubit overhead by up to a factor of 7.0 compared with the rotated surface code of the same distance [1]. The weight-10 family reaches the maximal factor of eight with the \([[512,16,16]]\) code [1].Gates
For stabilizer tiles not confined to a strip and giving rise to total topological order, derived automorphisms of the underlying tile codes implement products of logical CNOT gates [5]. The automorphisms extend the lattice on one side and shrink it on the other [5].Decoding
Relay-BP decoder [6] applied to detector error models constructed with Stim [7]. The \(X\)- and \(Z\)-type errors are decoded separately.Fault Tolerance
Superdense syndrome-extraction circuit [4] of depth \(w+4\), which is depth 12 for weight-8 stabilizer generators. One near-local coupler per barbell prepares and later measures a Bell pair on the paired \(X\)- and \(Z\)-check qubits. A Pauli-frame correction tracked in software relates the measurement outcomes to the stabilizer syndromes.Under uniform depolarizing circuit-level noise, a distance-14 weight-8 barbell code reaches the teraquop regime, meaning a logical error rate below \(10^{-12}\) per round, at physical error rates above \(10^{-4}\) [1]. At a physical error rate of \(10^{-3}\), one patch of the distance-11 \([[392,16,11]]\) barbell code encodes 16 logical qubits at a per-round logical error rate of \(8.8\times 10^{-7}\) [1]. Sixteen patches of the distance-5 rotated surface code use nearly the same number of data qubits, 400 versus 392, to encode the same 16 logical qubits. Their per-round logical error rate is \(9.6\times 10^{-4}\) [1].The logical multi-qubit Pauli measurement protocol for tile codes [8] applies directly. The per-round logical error rate of a distance-8 logical \(ZZ\) measurement is slightly higher than that of the same code used as a memory [1].Cousin
- 2D color code— Barbell syndrome extraction adapts the superdense syndrome-extraction circuits originally developed to implement color codes on a square grid [4].
Primary Hierarchy
References
- [1]
- S. H. Choe, V. Steffan, F. Vigneau, P. Parrado-Rodríguez, H.-S. Ku, M. Leib, F. R. F. Pereira, and F. Šimkovic, “Barbell Codes: qLDPC Codes for Superconducting Quantum Hardware”, (2026) arXiv:2606.06062
- [2]
- Z. Liang, J. N. Eberhardt, and Y.-A. Chen, “Planar quantum low-density parity-check codes with open boundaries”, (2025) arXiv:2504.08887
- [3]
- V. Steffan, S. H. Choe, N. P. Breuckmann, F. R. F. Pereira, and J. N. Eberhardt, “Tile Codes: High-Efficiency Quantum Codes on a Lattice with Boundary”, (2025) arXiv:2504.09171
- [4]
- C. Gidney and C. Jones, “New circuits and an open source decoder for the color code”, (2023) arXiv:2312.08813
- [5]
- N. P. Breuckmann, S. H. Choe, J. N. Eberhardt, F. R. F. Pereira, and V. Steffan, “Logical Operators and Derived Automorphisms of Tile Codes”, (2025) arXiv:2511.14589
- [6]
- T. Müller, T. Alexander, M. E. Beverland, M. Bühler, B. R. Johnson, T. Maurer, and D. Vandeth, “Improved belief propagation is sufficient for real-time decoding of quantum memory”, (2025) arXiv:2506.01779
- [7]
- C. Gidney, “Stim: a fast stabilizer circuit simulator”, Quantum 5, 497 (2021) arXiv:2103.02202 DOI
- [8]
- Y. Yang, G. Zhang, and Y. Li, “Planar fault-tolerant logical measurements with low qubit overhead”, npj Quantum Information (2026) arXiv:2506.18061 DOI
Page edit log
- Victor V. Albert (2026-09-26) — most recent
- Victor V. Albert (2026-08-26)
- Victor V. Albert (2026-06-11)
- Tomasz Andrzejewski (2026-06-11)
Cite as:
“Barbell quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/barbell, arXiv:2606.11484