Dynamical Decoupling Exposed a Spurious Subthreshold Result on IBM Heavy-Hex

Add dynamical decoupling to a surface code running on IBM heavy-hex hardware and an apparent subthreshold scaling trend can vanish. That reversal is the sharpest caution in a Nature Communications paper published on 29 July 2026, and it raises the bar for what counts as a reported distance-scaling result.

Why weight-4 stabilizers do not fit a heavy-hex lattice

The surface code measures weight-4 stabilizers, so it assumes a data qubit can reach four neighbours. IBM's heavy-hex processors, Heron included, do not: each qubit has two or three neighbours (Nature Communications). In 2021 IBM described the lattice as using three distinct frequency assignments and placing the control qubit only on edges connected to target qubits, which it says minimizes frequency collisions and the spectator errors generated by qubits not directly participating in a two-qubit gate (IBM Quantum).

That topology arrived with a code designed for it. Chamberland and colleagues introduced the heavy-hexagon code, a hybrid surface and Bacon-Shor subsystem code mapped onto the lattice, where flag qubits let a modified matching decoder correct to the full code distance under the limited connectivity (Physical Review X 10, 011022). Running a plain surface code on heavy-hex means paying in SWAP gates for connectivity the hardware never promised.

The fold-unfold embedding that holds a stabilizer round to depth 7

Vezvaee, Benito, Morford-Oberst, Bermudez and Lidar co-design the code embedding and the control pulses together (Nature Communications 17, 9201). A folding step applies CNOTs between data qubits to compress a weight-4 stabilizer into a weight-2 operator, an ancilla measures that weight-2 parity, and an unfolding step restores the original stabilizer.

Next-nearest-neighbour CNOTs, which heavy-hex does not couple directly, route through bridge qubits using SWAP gates. Those bridge qubits are never measured. The embedding holds a stabilizer measurement round to a depth-7 circuit, and because only half the stabilizers measure in parallel, a full syndrome extraction cycle takes two rounds.

The codes tested occupied 37 qubits at distance (3,3) and 65 qubits at both (3,5) and (5,3), run mainly on ibm_aachen and ibm_marrakesh; the main text reports ibm_aachen, the best-performing device in the study. Both are 156-qubit Heron devices, which leaves the hardware 6 qubits short of a fully isotropic step from distance 3 to distance 5.

The subthreshold trend that dynamical decoupling reversed

Without DD, the paper reports that logical error rates can decrease as code distance grows, which reads as subthreshold behaviour. Apply DD and that trend may reverse. Once each code was optimised separately for DD inclusion, the subthreshold scaling result on ibm_aachen disappeared. On ibm_marrakesh, the DD-optimised comparison still showed an onset of subthreshold scaling in late cycles, holding at 95% confidence only in the final cycle.

Idle time is the mechanism. Qubits waiting through a syndrome extraction cycle accumulate coherent ZZ crosstalk and non-Markovian dephasing, which the authors name as the main hardware error sources. DD suppresses both in every code but helps the smaller code disproportionately, so a (3,3) reference run without DD looked worse than that code can do, and the larger code's advantage over it was spurious.

Gap-aware means the pulse count is tuned to each idle window rather than fixed once. The authors searched the UR sequence family for pulse counts from 6 to 18 in steps of two, and used XY4 and RGA8a on shorter gaps, with up to roughly 484 single-qubit DD pulses added per cycle. Applying one sequence uniformly across every gap degraded performance, so the authors optimise the sequences experimentally for each data point.

What anisotropic distance scaling bought and what it cost

A (5,5) code does not fit on a 156-qubit Heron device, so the paper grows the X and Z distances one at a time and measures what each step buys. A (3,5) code adds protection against phase flips, which corrupt X-basis states, and a (5,3) code against bit flips, which corrupt Z-basis states; on ibm_aachen both reached an aligned-basis suppression factor of 1.23 to 1.46 against their averaged (3,3) sublattices.

The basis-averaged factor sat well below 1. Protection bought this way is directional: a logical state in the orthogonal basis degrades without a matching benefit, so the code helps only if you already know which error type your algorithm is exposed to. Measured against the best single (3,3) sublattice instead of the average, the aligned-basis factor narrows to between 1.01 and 1.10.

For scale, Google Quantum AI reported a suppression factor of 2.14 ± 0.02 per two units of distance on Willow, whose square grid matches the code, culminating in a 101-qubit distance-7 memory at 0.143% error per cycle and a logical lifetime 2.4 times its best physical qubit (Nature 638, 920).

The authors call their aligned-basis factors on par with the isotropic results reported before, Willow's included (Nature Communications). They price the lattice mismatch in time instead: idle gaps stretch every heavy-hex cycle, so their 10 cycles took about 80 µs, roughly 72 Willow cycles.

What would push a heavy-hex surface code below threshold

The authors put a number on the gap. A calibrated noise reduction of around 30%, together with a slightly larger heavy-hex device, would deliver genuine subthreshold scaling by their projection. Both conditions are incremental, and the paper's main result is already positive: each anisotropic code beat the averaged (3,3) code on the error type it was grown against.

Until that lands, treat any heavy-hex distance-scaling claim as incomplete unless it states how DD was optimised for each code separately. The two error channels the paper blames, coherent ZZ crosstalk and non-Markovian dephasing across idle gaps, are ones calibration benchmarks underestimate: the supplement concludes that the errors keeping these devices from a scaling advantage are precisely those the current benchmarking procedure does not capture.

SuperconducTED works on a neighbouring gap, calibration drift, with fuzzy inference over IBM calibration time series, and the earlier post on error detection with probabilistic error cancellation covers the mitigation side of the same hardware. Read the SuperconducTED calibration-drift approach before trusting a noise model across a recalibration.