Stitched surface codes

Circuits and data for M. A. Shalby, A. Bourassa, A. H. Karamlou, L. Martin, A. Morvan, N. Noll, C. Gidney, and L. P. Pryadko, Stitched surface codes: planar concatenation through GHZ corridors, arXiv:2610.21357 (2026).

The stitched code concatenates four rotated surface-code patches of distance t with the [[4,2,2]] code. It encodes two logical qubits at distance d = 2t in N = 2d² + 4d − 3 qubits on a square lattice with nearest-neighbor couplings. The two weight-4t generators of the outer code are measured with verified GHZ states prepared on corridors of ancillary qubits between the patches.

Circuit generator

stitched_surface_code.py depends only on Stim.

from stitched_surface_code import build_circuit

circuit = build_circuit(t=3, rv=2, ro=1, memory="ZZ")

t ≥ 2 is the patch distance, rv ≥ 1 the number of GHZ verification rounds per corridor, ro ≥ 1 the number of outer-code rounds (each measures both generators of the outer code), and memory the basis, "ZZ" or "XX", in which both logical qubits are prepared and measured. The circuit is noiseless and declares all detectors and both logical observables. Its circuit distance is min(2t, r_v + 3) for r_v ≥ 2 (Eq. 8 and Table III of the paper).

Simulation

memory_experiment.py runs the simulation pipeline of the paper for one stitched code and for two independent surface codes of distance 2t − 1: compilation to CZ gates with StimFlow, SI1000 noise, and decoding with PyMatching through sinter.

pip install stim==1.16.0 sinter==1.16.0 pymatching==2.3.1 stimflow==1.17.dev0
python memory_experiment.py --t 3 --rv 2 --p 2e-3

The pinned versions are those used for the paper (Python 3.12).

The reported quantity is the logical error rate per surface-code cycle, μ_tot, of the four-state Markov chain of the two logical qubits (Appendix B of the paper), obtained from the failure probabilities of L0, of L1 and of either logical qubit after τ cycles with an ideal initial state. For two independent surface codes with failure probability P₁ each, μ_tot = −ln(1 − 2P₁)/τ. The stitched circuits have τ = 3r_v + 4 cycles and the surface-code circuits τ = 3d; the latter are Stim's rotated_memory_z circuits without the initial reset layer, so that both codes start from a noiseless |0⟩.

Simulation data

File	Rows	Columns
data/stitched_si1000.csv, data/stitched_uniform.csv	every simulated (t, rv, p) under the SI1000 and the uniform depolarizing model	t, d, r_v, p, cycles, shots, errors, errors_L0, errors_L1, selected, mu_tot_per_cycle
data/surface_si1000.csv, data/surface_uniform.csv	every simulated (d, p)	d, p, cycles, shots, errors, mu_tot_per_cycle_two_codes
data/verification_depth_t7_si1000.csv	t = 7, rv = 2–6 (Fig. 4 of the paper)	t, noise, mode, r_v, p, cycles, shots, errors, errors_L0, errors_L1, mu_tot_per_cycle, ci95_lo, ci95_hi, mu_01, mu_10, mu_11

At each (t, p), every r_v from 1 up to one above the best value was sampled. The value used in the figures of the paper (selected = 1) has the lowest μ_tot; among values tied with it within one combined standard error, the one with the closest L0 and L1 error counts. Rows with few errors are screening runs. For the surface code, shots and errors refer to one copy. The uniform depolarizing model assigns the probability p to every gate, idle step, measurement and reset (stimflow.NoiseModel.uniform_depolarizing(p)). In the verification-depth data, mode = corridor_only denotes SI1000 noise restricted to the GHZ-corridor operations.

Hardware data

The code was run on a 105-qubit Willow processor at d = 4 (t = 2, r_v = 1) and d = 6 (t = 3, r_v = 2), in both memory bases, for r_o = 1–8 outer-code rounds, with 200,000 shots per circuit.

experiment/logical_outcomes.csv gives, for each of the 32 circuits and for both decoders (minimum-weight perfect matching and correlated matching), the number of shots with no logical error, with an error on L0 only, on L1 only, and on both. cycles is the number of surface-code cycles, τ = 1 + (r_v + 1)(2r_o + 1).

The 32 executed circuits are in experiment/stim/t<t>/<basis>/ (Stim format, one file per circuit, named circuit_t<t>rv<r_v>_ro<r_o>_<basis>.stim as in the archive below). Converting the archive's measurement outcomes and sweep bits with these circuits (stim m2d) reproduces its detection events and logical observables.

The complete data set, hardware_data.zip (1.3 GB), is attached to the v1 release. Circuits are named t<t>rv<r_v>_ro<r_o>_<basis>. For every circuit the archive contains

File	Content
raw_shots/shots_*.npz	meas_shots (all measurement outcomes), sweep_bits (random initial bits of the data qubits), det_shots (detection events), obs_shots (measured logical observables); one row per shot
decoded/dem/dem_*.dem	detector error model used for decoding, SI1000 noise at p = 10⁻³
decoded/predictions/{std,corr}_pred_*.npz	corrected_obs: decoder prediction XOR measured observable, per shot and logical qubit, for minimum-weight perfect matching (std) and correlated matching (corr), PyMatching 2.3.1

A shot has a logical error on L0 (L1) when column 0 (1) of corrected_obs is true; experiment/logical_outcomes.csv tabulates these arrays.

Citation

If you use these circuits or data, please cite the paper:

@misc{Shalby2026stitched,
  title         = {Stitched surface codes: planar concatenation through {GHZ} corridors},
  author        = {Shalby, Mohamed A. and Bourassa, Alexandre and Karamlou, Amir H. and Martin, Leigh and
                   Morvan, Alexis and Noll, Nicholas and Gidney, Craig and Pryadko, Leonid P.},
  year          = {2026},
  eprint        = {2610.21357},
  archivePrefix = {arXiv},
  primaryClass  = {quant-ph}
}

Licensing

• Code & Scripts: Licensed under the Apache License, Version 2.0.
• Data and figures: Licensed under Creative Commons Attribution 4.0 International (CC-BY-4.0).