Quantum error correction below the surface code threshold

Shiro Tamiya, Masato Koashi

Threshold analyses of quantum error-correcting codes are well established for stochastic error models, in which errors occur randomly with given probabilities. However, errors in actual devices can also be coherent, such as unwanted $Z$ rotations due to imperfect control, which are not captured by stochastic error models. For the surface code, numerical studies have indicated threshold behavior even under coherent errors, but a rigorous proof of threshold existence is lacking. Here we prove that a positive threshold for coherent $Z$-rotation errors exists for quantum low-density parity-check codes with a bounded number of logical qubits, including the surface code and other topological codes. Specifically, we show that the maximum-likelihood Pauli recovery suppresses the entanglement infidelity exponentially in the code distance up to a prefactor linear in the number of physical qubits whenever the rotation angles lie below a constant that is independent of the code size. The proof combines Fourier analysis to retain the interference among the amplitudes of coherent errors with the cluster expansion of abstract polymer models. Our results expand the theoretical foundation of quantum error correction and offer a statistical-mechanical description of quantum error correction beyond stochastic errors.

Christian Kraglund Andersen, Ants Remm, Stefania Lazar +6 more

Physics and Astronomy, REU/FRI Program, Thakur, Pritesh

Fang Zhang, Jianxin Chen

Error correction is an essential part of the theory of quantum computation. However, new quantum computation students may find the theories of error correction and fault tolerance daunting, or they may be stuck with theoretical/outdated schemes (such as the one in the original proof of the threshold theorem by Aharonov and Ben-or) with unrealistically low thresholds and/or high overhead. In this article, we describe an adequately modern approach to fault-tolerant quantum computation based on the surface code and lattice surgery. The reader is assumed to have a basic understanding of quantum computation (state vectors, unitary gates, and measurements, etc.), but no prior knowledge about quantum codes or quantum error correction is needed.

Christian Kraglund Andersen, Ants Remm, Stefania Lazar +6 more

Oscar Higgott, Thomas C. Bohdanowicz, Aleksander Kubica +2 more

Realizing the full potential of quantum computation requires quantum error correction (QEC), with most recent breakthrough demonstrations of QEC using the surface code.QEC codes use multiple noisy physical qubits to encode information in fewer logical qubits, enabling the identification of errors through a decoding process.This process increases the logical fidelity (or accuracy) making the computation more reliable.However, most fast (efficient run-time) decoders neglect important noise characteristics, thereby reducing their accuracy.In this work, we introduce decoders that are both fast and accurate, and can be used with a wide class of QEC codes including the surface code.Our decoders, named belief-matching and belief-find, exploit all noise information and thereby unlock higher accuracy demonstrations of QEC.Using the surface code threshold as a performance metric, we observe a threshold at 0.94% error probability for our decoders, outperforming the 0.82% threshold for a standard minimum-weight perfect matching decoder.We also test our belief-matching decoders in a theoretical case study of codes tailored to a biased noise model.We find that the decoders lead to a much higher threshold and lower qubit overhead in the tailored surface code with respect to the standard, square surface code.Surprisingly, in the wellbelow-threshold regime, the rectangular surface code becomes more resource efficient than the tailored surface code due to a previously unnoticed phenomenon that we call "fragile boundaries."Our decoders outperform all other fast decoders in terms of threshold and accuracy, enabling better results in current quantum-error-correction experiments and opening up new areas for theoretical case studies.