Coherent errors and readout errors in the surface code

Áron Márton1 and János K. Asbóth1,2

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

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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.

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.

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Cited by

[1] Dávid Pataki, Áron Márton, János K. Asbóth, and András Pályi, "Coherent errors in stabilizer codes caused by quasistatic phase damping", Physical Review A 110 1, 012417 (2024).

[2] Balint Pato, Judd Will Staples, and Kenneth R. Brown, "Logical coherence in 2D compass codes", arXiv:2405.09287, (2024).

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