Barbell code

Barbell code[1]

Description

Member of a family of high-rate, two-dimensionally local CSS QLDPC codes designed so that all two-qubit interactions of the QEC cycle are native to a fixed-connectivity superconducting chip layout whose hardware complexity stays constant as the code distance grows. Barbell codes are tile codes [2,3] equipped with a pairing of \(X\)- and \(Z\)-check qubits such that every data qubit in the support of either check is adjacent, in a given hardware connectivity graph, to one of the two paired check qubits; the family studied is realized on a six-qubit star lattice plus near-local coupler (6QSL+NLC) architecture, also called the Barbell architecture.

Each \(X\)-check qubit is joined to a partner \(Z\)-check qubit by a single near-local coupler, and the two checks’ data-qubit neighborhoods together form a barbell. The corresponding pair of \(X\)- and \(Z\)-type stabilizers is supported on the six hexagonal cells of this barbell. Syndrome information is read out using superdense syndrome extraction [4]: a Bell pair is prepared on the two check qubits via the near-local coupler, the data qubits are entangled, and a Bell measurement is performed. For weight-\(w\) stabilizers this yields a QEC cycle of depth \(w+4\) (depth 12 for weight-8 stabilizers). Because all near-local couplers are parallel and of equal, distance-independent length, they route within a single additional hardware tier without air bridges.

The weight-8 family studied numerically contains the \([[242,16,8]]\), \([[288,16,9]]\), \([[338,16,10]]\), \([[392,16,11]]\), and \([[450,16,14]]\) codes, with \(n=2LM\) for an \(L\times M\) data-qubit lattice and bulk-stabilizer boxes of size \((D+1)\times(D+1)\); two weight-6 families with \(k=7\) and \(k=8\), and a weight-10 family with \(k=16\) (reaching \([[512,16,16]]\)), are also given [1]. With fewer than 30 data qubits per logical qubit, the weight-8 barbell codes reduce qubit overhead by up to a factor of 7.0 relative to the rotated surface code of the same distance, while the weight-10 family reaches the maximal factor of eight [1].

Protection

Protects against Pauli noise. Under uniform depolarizing circuit-level noise, a distance-14 weight-8 barbell code reaches the teraquop regime (logical error rate below \(10^{-12}\) per round, enabling several trillion QEC cycles) at physical error rates above \(10^{-4}\). At a physical error rate of \(10^{-3}\), one patch of the distance-11 barbell code (\([[392,16,11]]\)) and 16 patches of the distance-5 rotated surface code use essentially the same number of data qubits (392 versus 400) to encode 16 logical qubits, but the barbell code achieves a per-round logical error rate of \(8.8\times 10^{-7}\) versus \(9.6\times 10^{-4}\) for the surface code [1].

Rate

High-rate QLDPC family. The weight-8 examples encode \(k=16\) logical qubits using fewer than 30 data qubits per logical qubit, an up-to-sevenfold reduction in physical-qubit overhead compared with the rotated surface code of the same distance.

Gates

Logical operators and derived automorphisms of the underlying tile codes are studied in Ref. [5].

Decoding

Superdense syndrome-extraction circuit [4] of depth \(w+4\) (depth 12 for weight-8 stabilizers), using one near-local coupler per barbell to prepare and later measure 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.Relay-BP decoder [6] applied to detector error models constructed with Stim [7]; \(X\)- and \(Z\)-type errors are decoded separately.

Fault Tolerance

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 only slightly higher than that of the corresponding memory experiment [1].

Notes

The 6QSL+NLC (Barbell) architecture is composed only of experimentally demonstrated components: a six-qubit star lattice of qubits coupled to central elements by tunable couplers [9], supplemented by near-local couplers connecting check-qubit pairs. Its hardware-complexity metric is \(C_{\text{hw}}\approx 1.65\) in the framework of Ref. [10], compared with \(C_{\text{hw}}=1\) for the surface code and \(C_{\text{hw}}>3\) for the \([[144,12,12]]\) bivariate bicycle code.

Cousins

Rotated surface code— Barbell codes match the rotated surface code in logical performance per QEC round at comparable distance for physical error rates at most \(10^{-3}\), while reducing the physical-qubit overhead by up to a factor of 7.0 for the simulated weight-8 codes (up to eight for the weight-10 family) [1].
Bivariate bicycle (BB) code— Bivariate bicycle (BB) codes are the closest QLDPC comparison to barbell codes. Unlike BB codes, whose longest near-local coupler scales with the code distance and whose routing requires multiple tiers (hardware complexity \(C_{\text{hw}}>3\) for the \([[144,12,12]]\) code), the near-local couplers of a barbell code are parallel, equal-length, and distance-independent, fitting into a single additional tier.
2D color code— Barbell syndrome extraction adapts the superdense syndrome-extraction circuits originally developed to implement color codes on a square grid [4].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum

Tile codeLattice stabilizer CSS Stabilizer Hamiltonian-based Qubit QECC Quantum

Parents

Tile code

Barbell codes are a special case of tile codes whose \(X\)- and \(Z\)-type stabilizers are tailored for measurement on the six-qubit star lattice plus near-local coupler (Barbell) architecture, with near-local couplers used exclusively to pair \(X\)- and \(Z\)-check qubits. They are obtained from tile codes by adding ancilla check qubits, translating the qubit positions, and embedding the qubits in the connectivity graph of that architecture.