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arXiv:2410.22436 · PRX Quantum

Statistical Mechanical Mapping and Maximum-Likelihood Thresholds for the Surface Code under Generic Single-Qubit Coherent Errors

Jan Behrends, Benjámin Béri

Published Jul 28, 2025• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/gskb-t5ql

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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.
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.
Statistical mechanics mappings provide key insights on quantum error correction. However, existing mappings assume incoherent noise, thus ignoring coherent errors due to, e.g., spurious gate rotations. We map the surface code with coherent errors, taken as XXX or ZZZ rotations (replacing bit or phase flips), to a two-dimensional (2D) Ising model with complex couplings, and further to a 2D Majorana scattering network. Our mappings reveal both commonalities and qualitative differences in correcting coherent and incoherent errors. For both, the error-correcting phase maps, as we explicitly show by linking 2D networks to 1D fermions, to a Z2{\mathbb{Z}}_{2}Z2​-nontrivial 2D insulator. However, beyond a rotation angle ${\ensuremath{\phi}}_{\mathrm{th}}$, instead of a Z2{\mathbb{Z}}_{2}Z2​-trivial insulator as for incoherent errors, coherent errors map to a Majorana metal. This ${\ensuremath{\phi}}_{\mathrm{th}}$ is the theoretically achievable storage threshold. We numerically find ${\ensuremath{\phi}}_{\mathrm{th}}\ensuremath{\approx}0.14\ensuremath{\pi}$. The corresponding bit-flip rate ${\mathrm{sin}}^{2}({\ensuremath{\phi}}_{\mathrm{th}})\ensuremath{\approx}0.18$ exceeds the known incoherent threshold ${p}_{\mathrm{th}}\ensuremath{\approx}0.11$.
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.
The surface code is a promising platform for a quantum memory, but its threshold under coherent errors remains incompletely understood. We study maximum-likelihood decoding of the square-lattice surface code in the presence of single-qubit unitary rotations that create electric anyon excitations. We microscopically derive a non-linear sigma model with target space SO(2n)/U(n)\mathrm{SO}(2n)/\mathrm{U}(n)SO(2n)/U(n) as the effective long-distance theory of this decoding problem, with distinct replica limits: n→1n\to1n→1 for optimal decoding, which assumes knowledge of the coherent rotation angle, and n→0n\to0n→0 for suboptimal decoding with imperfect angle information. This exposes a sharp distinction between the two decoders. The suboptimal decoder supports a "thermal-metal" phase, a non-decodable regime that is qualitatively distinct from the conventional non-decodable phase of the surface code under incoherent Pauli errors. By contrast, the metal phase cannot arise in optimal decoding, since the metallic fixed-point becomes unstable in the n→1n\to 1n→1 replica limit. We argue that optimal decoding may be possible up to the maximally-coherent rotation angle. Within the sigma model description, we show that the decoding fidelity is related to twist defects of the order-parameter field, yielding quantitative predictions for its system-size dependence near the metallic fixed point for both decoders. We examine our analytic predictions for the decoding fidelity as well as other physical observables with extensive numerical simulations. We discuss how the symmetries and the target space for the sigma model rely on the lattice of the surface code, and how a stable thermal metal phase can arise in optimal decoding when the syndromes reside on a non-bipartite lattice.
The surface code represents a promising candidate for fault-tolerant quantum computation due to its high error threshold and experimental accessibility with nearest-neighbor interactions. However, current exact surface code threshold analyses are based on the assumption of independent and identically distributed (i.i.d.) errors. Though there are numerical studies for the threshold with correlated error, they are only the lower bound rather than the exact value, which offers potential for higher error thresholds. Here, we establish an error-edge map, which allows for the mapping of quantum error correction to a square-octagonal random bond Ising model. We then present the maximum-likelihood threshold under a realistic noise model that combines independent single-qubit errors with correlated errors between nearest-neighbor data qubits. Our method is applicable for any ratio of nearest-neighbor correlated errors to i.i.d. errors. We present analytical constraints for the error correction threshold of surface codes, and by applying these constraints, we numerically obtain the error threshold higher than the existing values. Our error threshold is both an upper bound and achievable.
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.