Coherent errors and readout errors in the surface code

Áron Márton; János K. Asbóth

doi:10.22331/q-2023-09-21-1116

Coherent errors and readout errors in the surface code

Áron Márton¹ and János K. Asbóth^1,2

¹Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
²Wigner Research Centre for Physics, H-1525 Budapest, P.O. Box 49., Hungary

Published:	2023-09-21, volume 7, page 1116
Eprint:	arXiv:2303.04672v3
Doi:	https://doi.org/10.22331/q-2023-09-21-1116
Citation:	Quantum 7, 1116 (2023).

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Abstract

We consider the combined effect of readout errors and coherent errors, i.e., deterministic phase rotations, on the surface code. We use a recently developed numerical approach, via a mapping of the physical qubits to Majorana fermions. We show how to use this approach in the presence of readout errors, treated on the phenomenological level: perfect projective measurements with potentially incorrectly recorded outcomes, and multiple repeated measurement rounds. We find a threshold for this combination of errors, with an error rate close to the threshold of the corresponding incoherent error channel (random Pauli-Z and readout errors). The value of the threshold error rate, using the worst case fidelity as the measure of logical errors, is 2.6%. Below the threshold, scaling up the code leads to the rapid loss of coherence in the logical-level errors, but error rates that are greater than those of the corresponding incoherent error channel. We also vary the coherent and readout error rates independently, and find that the surface code is more sensitive to coherent errors than to readout errors. Our work extends the recent results on coherent errors with perfect readout to the experimentally more realistic situation where readout errors also occur.

Featured image: Our numerical results show that Quantum Error Correction (QEC) using the Surface Code works well (green area: better protection if more qubits used) even if the readout error rate is quite high, assuming the rate of coherent errors is not too high.

Popular summary

To perform long computations, the quantum information that quantum computers work on has to be protected against environmental noise. This requires quantum error correction (QEC), whereby each logical qubit is encoded into collective quantum states of many physical qubits. We studied, using numerical simulation, how well the most promising quantum error correcting code, the so-called Surface Code can protect quantum information against a combination of so-called coherent errors (a type of calibration errors) and readout errors. We found that the Surface Code provides better protection as the code is scaled up, as long as the error levels are below a threshold. This threshold is close to the well-known threshold of another combination of errors: incoherent errors (a type of error arising from entanglement with a quantum environment) and readout errors. We also found (as shown in the accompanying image) that the Surface Code is more robust against readout errors than coherent errors. Note that we used the so-called phenomenological error model: we modeled the noise channels very precisely, but did not do a modeling of the code on the quantum circuit level.

► BibTeX data

@article{Marton2023coherenterrors,
  doi = {10.22331/q-2023-09-21-1116},
  url = {https://doi.org/10.22331/q-2023-09-21-1116},
  title = {Coherent errors and readout errors in the surface code},
  author = {M{\'{a}}rton, {\'{A}}ron and Asb{\'{o}}th, J{\'{a}}nos K.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1116},
  month = sep,
  year = {2023}
}

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[2] Áron Márton and János K. Asbóth, "Optimal number of stabilizer measurement rounds in an idling surface code patch", Quantum 9, 1767 (2025).

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[7] Andras Di Giovanni, Adrian Skasberg Aasen, Jürgen Lisenfeld, Martin Gärttner, Hannes Rotzinger, and Alexey V. Ustinov, "Benchmarking the quality of multiplexed qubit readout beyond assignment fidelity", Physical Review Applied 24 4, 044043 (2025).

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