The maximum likelihood (ML) decoder in the two-dimensional surface code with generic unitary errors is governed by a statistical mechanics model with complex weights, which can be simulated via (1+1)D transfer matrix contraction. Information loss with an increasing error rate manifests as a ferromagnetic-to-paramagnetic transition in the contraction dynamics. In this work, we establish entanglement as a separate obstruction to decoding; it can undergo a transition from area- to volume-law scaling in the transfer matrix contraction with increasing unitary error rate. In particular, the volume-law entanglement can coexist with ferromagnetic order, giving rise to a phase in which the encoded information is retained yet is effectively undecodable. We numerically simulate the ML decoding in the surface code subject to both single- and two-qubit Pauli-X rotations and obtain a phase diagram that contains a ferromagnetic area-law, a paramagnetic volume-law, and a potential ferromagnetic volume-law phase. We further show that, starting from the paramagnetic volume-law phase, tilting the single-qubit rotation away from the X-axis couples the stat-mech models for X and Z errors and can lead to a ferromagnetic volume-law phase in which, although Z errors remain in principle correctable, the encoded classical information is hard to recover. To perform numerical simulations, we develop an algorithm for syndrome sampling based on an isometric tensor network representation of the surface code.
The maximum likelihood (ML) decoder in the two-dimensional surface code with generic unitary errors is governed by a statistical mechanics model with complex weights, which can be simulated via (1+1)D transfer matrix contraction. Information loss with an increasing error rate manifests as a ferromagnetic-to-paramagnetic transition in the contraction dynamics. In this work, we establish entanglement as a separate obstruction to decoding; it can undergo a transition from area- to volume-law scaling in the transfer matrix contraction with increasing unitary error rate. In particular, the volume-law entanglement can coexist with ferromagnetic order, giving rise to a phase in which the encoded information is retained yet is effectively undecodable. We numerically simulate the ML decoding in the surface code subject to both single- and two-qubit Pauli-X rotations and obtain a phase diagram that contains a ferromagnetic area-law, a paramagnetic volume-law, and a potential ferromagnetic volume-law phase. We further show that, starting from the paramagnetic volume-law phase, tilting the single-qubit rotation away from the X-axis couples the stat-mech models for X and Z errors and can lead to a ferromagnetic volume-law phase in which, although Z errors remain in principle correctable, the encoded classical information is hard to recover. To perform numerical simulations, we develop an algorithm for syndrome sampling based on an isometric tensor network representation of the surface code.
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.
Topological stabilizer codes, such as the toric and surface codes, are leading candidates for fault-tolerant quantum computation. While their decodability under stochastic noise has been extensively studied, the effects of coherent errors, which involve quantum interference, remain less explored. In this work, we study the decodability of toric codes on honeycomb and square lattices subject to X- and Z-type coherent errors generated by the X- and Z-rotations on each qubit. We establish a duality between these decoding problems and 1+1D monitored dynamics of non-interacting Majorana fermions. This duality shows that the Altland-Zirnbauer symmetry class of the dual Majorana dynamics governs the universal structure of the decodability phase diagram. We show that the honeycomb-lattice toric code (hTC) with X-type error is dual to class-DIII dynamics, while the hTC with Z-type error and the square-lattice toric code (sTC) with both error types are dual to class-D dynamics. The key distinction arises from time-reversal symmetry. In class DIII, the generic transition out of the decodable phase is dual to a measurement-induced transition between dynamical phases with area-law and logarithmic entanglement scaling. In contrast, in class D, the generic decodability transition corresponds to a transition between two topologically distinct area-law phases. To explore these transitions in microscopic models, we consider hTC and sTC with X-type errors as representatives and introduce a minimal two-parameter coherent error model with spatially varying rotation angles. Using analytical and numerical methods, we map out the decodability phase diagrams and characterize the universal behavior of the transitions. We find that the decodability of sTC is more vulnerable to spatially varying coherent errors than uniform ones.
Unitary k -designs are distributions of unitary gates that match the Haar distribution up to its k th statistical moment. They are a crucial resource for randomized quantum protocols. However, their implementation on encoded logical qubits is nontrivial due to the need for magic gates, which can require a large resource overhead. In this work, we propose an efficient approach to generate unitary designs for encoded qubits in surface codes by applying local unitary rotations (“coherent errors”) on the physical qubits followed by syndrome measurement and error correction. We prove that, under some conditions on the coherent errors (notably including all single-qubit unitaries) and on the error-correcting code, this process induces a unitary transformation of the logical subspace. We numerically show that the ensemble of logical unitaries (indexed by the random syndrome outcomes) converges to a unitary design in the thermodynamic limit, provided that the density or strength of coherent errors is above a finite threshold. This “unitary design” phase transition coincides with the code’s coherent error threshold under optimal decoding. Furthermore, we propose a classical algorithm to simulate the protocol based on a “staircase” implementation of the surface code encoder and decoder circuits. This enables a mapping to a ( 1 + 1 ) -dimensional monitored circuit, where we observe an entanglement phase transition (and thus a classical complexity phase transition of the decoding algorithm) coinciding with the aforementioned unitary design phase transition. Our results provide a practical way to realize unitary designs on encoded qubits, with applications including quantum state tomography and benchmarking in error-correcting codes.
Coherent noise can give rise to unconventional regimes of quantum information recovery and mixed-state order. We study these phenomena in two settings related by Wegner duality: the triangular-lattice surface code subject to single-qubit Pauli-Z rotations followed by syndrome measurements, and a Z2 paramagnet subject to strongly symmetric decoherence. Increasing the rotation angle drives a transition from a decodable phase to a \emph{partially decodable} critical phase in the code, and from a strongly symmetric phase to a phase with \emph{algebraic} Z2 strong-to-weak spontaneous symmetry breaking (SWSSB) in the dual paramagnet. We develop a common description of these phenomena in terms of a weakly coupled SO(n) nonlinear sigma model in the replica limit n→1. Taking this replica limit analytically, we derive universal signatures of the critical phase: an optimal Pauli decoding fidelity that approaches a universal function of the system's aspect ratio in the thermodynamic limit, and a dual fidelity correlator that decays as a universal power law. These observables relate to correlations of twist operators in the sigma model description. Large-scale numerical simulations confirm these analytical predictions. We also argue that the algebraic SWSSB phase cannot be realized by any Pauli dephasing channel acting on the Z2 paramagnet. We probe this "non-classical" character through the output state's sensitivity to well-separated Z2 fluxes inserted in the initial state, which reveals persistent coherence between distinct Z2 charge trajectories at long distances.
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) as the effective long-distance theory of this decoding problem, with distinct replica limits: n→1 for optimal decoding, which assumes knowledge of the coherent rotation angle, and n→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→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 maximum likelihood decoder based on tensor networks has proven highly successful for the 2D surface code, achieving the optimal decoding success rate. However, existing tensor network decoders are typically designed for independent single-qubit error models, and their performance under inter-qubit correlated error models remains unexplored. This is due to two major challenges. The first challenge lies in constructing the tensor network for correlated errors, since the same final Pauli error can arise from many different combinations of independent and correlated errors, preventing a direct factorization of the error probability. The second challenge is that even after a tensor network is constructed, it generally contains huge-dimensional tensors and is therefore not efficiently contractible. In this work, to address the first difficulty, we introduce additional binary indices and two transformations to construct a multi-index tensor network for maximum-likelihood decoding with correlated errors. To address the second difficulty, we use reparametrization, elimination, and index classification to decompose the huge-dimensional tensors into lower-dimensional tensors. This yields an efficiently contractible tensor network for error models satisfying the tractability conditions derived in this work. We perform numerical simulations for a representative correlated error model and show that the maximum-likelihood decoder implemented with our multi-index tensor network construction achieves a higher finite-size threshold than the widely used MWPM decoder.