# Techniques: Correcting non-independent and non-identically distributed errors with surface codes

## What the paper contributes (one paragraph)

The paper adapts the surface code to realistic device noise that is neither independent nor identical across qubits. Given measured per-qubit Pauli error rates, it applies local single-qubit Clifford conjugations to the standard CSS stabilisers, producing noise-tailored MHHM and MMHH codes whose syndromes stay compatible with fast minimum-weight perfect-matching decoding. For qubits with varying total error rates it instead makes the matching decoder noise-aware, weighting the matching graph by shortest weighted paths (Dijkstra) computed from local rates. For correlated nearest-neighbour two-qubit errors it shows that simply accounting for syndrome and code degeneracy in the matching weights suppresses logical error rates, even with no knowledge of the noise magnitudes. All methods use only single-qubit rotations on data qubits, add no qubit or gate overhead, and yield higher thresholds and exponentially suppressed sub-threshold failure rates versus the standard surface code.

## Techniques offered

- **Noise-aware local Clifford deformation of the surface code (MHHM / MMHH codes)** - Tailors the surface-code stabilisers per qubit to known non-identically distributed Pauli noise via single-qubit Clifford conjugations, so the dominant error syndromes stay matchable, supplying a tailored fault-tolerant code at no qubit or gate overhead.
  - guarantee: statistical bound; exponentially suppressed sub-threshold logical error rate in code distance and a monotonically higher threshold as Pauli noise becomes less uniform, with no extra qubit or gate cost.
  - quote: "Firstly, we design a Clifford-deformed version of the surface code tailored to efficiently correct non-identically distributed Pauli errors when used in conjunction with a scalable perfect-matching decoder."

- **Noise-aware minimum-weight perfect-matching decoder (shortest-weighted-path / Dijkstra weights)** - Sets matching-graph edge weights from each qubit's measured local error rate using shortest weighted paths, improving correction of non-identically distributed total error rates without changing the code.
  - guarantee: statistical bound; higher error-correction thresholds and more favourable sub-threshold scaling as the spread in total qubit error rates grows, relative to standard Manhattan-distance matching.
  - quote: "Secondly, we show that similar improvements in error correcting capabilities of the surface code can be achieved by modifying input parameters of a matching decoder in a situation where qubits experience non-identically distributed total error rates."

- **Degeneracy-aware matching for correlated two-qubit errors** - Adds a syndrome- and code-degeneracy term to the matching weights (with Clifford deformation to keep the syndrome matchable for other correlation types) so a standard MWPM decoder corrects local two-qubit cross-talk errors without knowing the noise magnitudes.
  - guarantee: statistical bound; exponentially suppressed sub-threshold logical error rate under local two-qubit noise, requiring no information about the single- versus two-qubit noise magnitudes.
  - quote: "As we have shown above, a MWPM decoder can efficiently handle correlated nearest-neighbor errors of certain types without the need of adjusting the decoding procedure to a known structure of the noise."

## Where it could apply

- Target primitive(s): none (code-capacity memory experiments; not a circuit primitive such as VQE, QAOA, QPE, sampling, or Trotterised simulation)
- Target application group(s): error_correction; platforms_tooling
- Code family / hardware assumptions: surface code (CSS, XZZX, XXZZ, and the new MHHM / MMHH deformations), rotated and non-rotated geometries; code-capacity noise with perfect syndrome measurement; relevant to superconducting, trapped-ion, silicon spin, and photonic qubits; ideas stated to carry over to 3D topological codes and qLDPC codes.

## Caveats

Code-capacity model only: errors act on data qubits with perfect, noise-free stabiliser measurements, so the results do not yet cover circuit-level noise or measurement errors. Demonstrations use deliberately simple test noise models (random Pauli permutations and truncated-Gaussian rate variations), not calibrated device data, and assume the per-qubit noise channel is known or measurable. Decoding is restricted to standard minimum-weight perfect matching; advanced or alternative decoders are left to future work.