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arXiv:2412.21055 · PRX Quantum

The Surface Code beyond Pauli Channels: Logical Noise Coherence, Information-Theoretic Measures, and Errorfield-Double Phenomenology

Jan Behrends, Benjámin Béri

Published Oct 21, 2025• 3 citations
arXivHTMLPDF

Abstract

We consider the surface code under errors featuring both coherent and incoherent components and study the coherence of the corresponding logical noise channel and how this impacts information-theoretic measures of code performance, namely coherent information and quantum relative entropy. Using numerical simulations and developing a phenomenological field theory, focusing on the most general single-qubit X -error channel, we show that, for any nonzero incoherent noise component, the coherence of the logical noise is exponentially suppressed with the code distance. We also find that the information-theoretic measures require this suppression to detect optimal thresholds for Pauli recovery; for this they thus require increasingly large distances for increasing error coherence and ultimately break down for fully coherent errors. To obtain our results, we develop a statistical mechanics mapping and a corresponding matrix-product-state algorithm for approximate syndrome sampling. These methods enable the large-scale simulation of these non-Pauli errors, including their maximum-likelihood thresholds, away from the limits captured by previous approaches.

DOI
10.1103/psf5-b6j2

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arXiv:1710.02270•npj Quantum Information•Oct 25, 2018•157 citations
We consider the surface code under errors featuring both coherent and incoherent components and study the coherence of the corresponding logical noise channel and how this impacts information-theoretic measures of code performance, namely coherent information and quantum relative entropy. Using numerical simulations and developing a phenomenological field theory, focusing on the most general single-qubit X -error channel, we show that, for any nonzero incoherent noise component, the coherence of the logical noise is exponentially suppressed with the code distance. We also find that the information-theoretic measures require this suppression to detect optimal thresholds for Pauli recovery; for this they thus require increasingly large distances for increasing error coherence and ultimately break down for fully coherent errors. To obtain our results, we develop a statistical mechanics mapping and a corresponding matrix-product-state algorithm for approximate syndrome sampling. These methods enable the large-scale simulation of these non-Pauli errors, including their maximum-likelihood thresholds, away from the limits captured by previous approaches.
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.
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 ZZZ 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 ZZZ-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.
The surface code, one of the leading candidates for quantum error correction, is known to protect encoded quantum information against stochastic, i.e., incoherent errors. The protection against coherent errors, such as from unwanted gate rotations, is however understood only for special cases, such as rotations about the X or Z axes. Here we consider generic single-qubit coherent errors in the surface code, i.e., rotations by angle α about an axis that can be chosen arbitrarily. We develop a statistical mechanical mapping for such errors and perform entanglement analysis in transfer matrix space to numerically establish the existence of an error-correcting phase, which we chart in a subspace of rotation axes to estimate the corresponding maximum-likelihood thresholds α th . The classical statistical mechanics model we derive is a random-bond Ising model with complex couplings and four-spin interactions (i.e., a complex-coupled Ashkin-Teller model). The error-correcting phase, α < α th , where the logical error rate decreases exponentially with code distance, is shown to correspond in transfer matrix space to a gapped one-dimensional quantum Hamiltonian exhibiting spontaneous breaking of a Z 2 symmetry. Our numerical results rest on two key ingredients: (i) we show that the state evolution under the transfer matrix, a nonunitary ( 1 + 1 ) -dimensional quantum circuit, can be efficiently numerically simulated using matrix product states; and (ii) based on this approach, we also develop an algorithm to (approximately) sample syndromes based on their Born probability. The α th values we find show that the maximum-likelihood thresholds for coherent errors are larger than those for the corresponding incoherent errors (from the Pauli twirl), and significantly exceed the values found using minimum weight perfect matching.
Scalable realisation of quantum computing is reliant on the development of fault tolerant devices. Analysis of quantum error correction protocols typically considers incoherent noise models or noise-free syndrome measurements. While this is simple to simulate classically and straightforward to compute analytically, these simplifications are unable to capture the full dynamics of a noisy quantum system. In this work we use advanced hybrid stabilizer-tensor network simulation techniques to simulate coherent quantum crosstalk noise during syndrome extraction on a surface code. We show that the inclusion of coherence increases logical error rates and lowers the code threshold. In addition, we show that the specific distribution of the noise can quantitatively change logical error rates. The methods in this work allow simulation of quantum error correction with noise models previously inaccessible to classical simulation, providing new insights on the effect of crosstalk noise on quantum error correction codes.
A key appeal of quantum low-density parity check (qLDPC) codes is their ability to suppress stochastic Pauli noise below nonzero thresholds. Coherent errors are fundamentally different: they produce superpositions of error patterns whose amplitudes can interfere even after syndrome measurement. Rigorous understanding of coherent errors remains limited. Here we show that general qLDPC codes admit a nonzero code capacity threshold against local coherent noise and more generally local channel noise. For any family of qLDPC codes with distance d=Ω(log⁡n)d=Ω(\log n)d=Ω(logn), we show that there is a constant noise strength below which the logical recovery error in diamond distance decays exponentially with the code distance. The result is established for optimal recovery as well as the minimum-weight decoder. The key technical ingredient is what we call a \emph{cluster resummation}: rather than bounding superposed error configurations one by one, we isolate a large connected error cluster in the channel expansion and exactly resum all errors disconnected from it before taking norms. Standard cluster counting then yields exponential suppression. This work resolves a longstanding challenge in fault tolerance theory, providing general robustness guarantees for qLDPC codes against coherent noise and laying a rigorous foundation for future studies of fault-tolerant quantum technologies.
Abstract Surface codes are building blocks of quantum computing platforms based on 2D arrays of qubits responsible for detecting and correcting errors. The error suppression achieved by the surface code is usually estimated by simulating toy noise models describing random Pauli errors. However, Pauli noise models fail to capture coherent processes such as systematic unitary errors caused by imperfect control pulses. Here we report the first large-scale simulation of quantum error correction protocols based on the surface code in the presence of coherent noise. We observe that the standard Pauli approximation provides an accurate estimate of the error threshold but underestimates the logical error rate in the sub-threshold regime. We find that for large code size the logical-level noise is well approximated by random Pauli errors even though the physical-level noise is coherent. Our work demonstrates that coherent effects do not significantly change the error correcting threshold of surface codes. This gives more confidence in the viability of the fault-tolerance architecture pursued by several experimental groups.