Surface codes—leading candidates for quantum error correction (QEC)—and entanglement phases—a key notion for many-body quantum dynamics—have heretofore been unrelated. Here we establish a link between the two. We map two-dimensional (2D) surface codes under a class of incoherent or coherent errors (bit flips or uniaxial rotations) to (1+1)D free-fermion quantum circuits via Ising models. We show that the error-correcting phase implies a topologically nontrivial area law for the circuit's 1D long-time state |Ψ∞〉 . Above the error threshold, we find a topologically trivial area law for incoherent errors and logarithmic entanglement in the coherent case. In establishing our results, we formulate 1D parent Hamiltonians for |Ψ∞〉 via linking Ising models and 2D scattering networks, the latter displaying respective insulating and metallic phases and setting the 1D fermion gap and topology via their localization length and topological invariant. We expect our results to generalize to a duality between the error-correcting phase of ( d+1 )D topological codes and d -dimensional area laws; this can facilitate assessing code performance under various errors. The approach of combining Ising models, scattering networks, and parent Hamiltonians can be generalized to other fermionic circuits and may be of independent interest. Published by the American Physical Society 2024
Surface codes—leading candidates for quantum error correction (QEC)—and entanglement phases—a key notion for many-body quantum dynamics—have heretofore been unrelated. Here we establish a link between the two. We map two-dimensional (2D) surface codes under a class of incoherent or coherent errors (bit flips or uniaxial rotations) to (1+1)D free-fermion quantum circuits via Ising models. We show that the error-correcting phase implies a topologically nontrivial area law for the circuit's 1D long-time state |Ψ∞〉 . Above the error threshold, we find a topologically trivial area law for incoherent errors and logarithmic entanglement in the coherent case. In establishing our results, we formulate 1D parent Hamiltonians for |Ψ∞〉 via linking Ising models and 2D scattering networks, the latter displaying respective insulating and metallic phases and setting the 1D fermion gap and topology via their localization length and topological invariant. We expect our results to generalize to a duality between the error-correcting phase of ( d+1 )D topological codes and d -dimensional area laws; this can facilitate assessing code performance under various errors. The approach of combining Ising models, scattering networks, and parent Hamiltonians can be generalized to other fermionic circuits and may be of independent interest. Published by the American Physical Society 2024
Surface codes$\unicode{x2014}$leading candidates for quantum error correction (QEC)$\unicode{x2014}$and entanglement phases$\unicode{x2014}$a key notion for many-body quantum dynamics$\unicode{x2014}$have heretofore been unrelated. Here, we establish a link between the two. We map two-dimensional (2D) surface codes under a class of incoherent or coherent errors (bit flips or uniaxial rotations) to (1+1)D free-fermion quantum circuits via Ising models. We show that the error-correcting phase implies a topologically nontrivial area law for the circuit's 1D long-time state ∣Ψ∞⟩. Above the error threshold, we find a topologically trivial area law for incoherent errors and logarithmic entanglement in the coherent case. In establishing our results, we formulate 1D parent Hamiltonians for ∣Ψ∞⟩ via linking Ising models and 2D scattering networks, the latter displaying respective insulating and metallic phases and setting the 1D fermion gap and topology via their localization length and topological invariant. We expect our results to generalize to a duality between the error-correcting phase of (d+1)D topological codes and d-dimensional area laws; this can facilitate assessing code performance under various errors. The approach of combining Ising models, scattering networks, and parent Hamiltonians can be generalized to other fermionic circuits and may be of independent interest.
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 X or Z 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-nontrivial 2D insulator. However, beyond a rotation angle ${\ensuremath{\phi}}_{\mathrm{th}}$, instead of a 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 ${p}_{\mathrm{th}}\ensuremath{\approx}0.11$.
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.
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.
Quantum error correcting (QEC) codes protect quantum information from decoherence as long as error rates fall below critical error thresholds. In general, obtaining thresholds implies simulating the QEC procedure using, in general, suboptimal decoding strategies. In a few cases and for sufficiently simple noise models, optimal decoding of QEC codes can be framed as a phase transition in disordered classical spin models. In both situations, accurate estimation of thresholds demands intensive computational resources. Here we use the coherent information of the mixed state of noisy QEC codes to accurately estimate the associated optimal QEC thresholds already from small-distance codes at moderate computational cost. We show the effectiveness and versatility of our method by applying it first to the topological surface and color code under bit-flip and depolarizing noise. We then extend the coherent information based methodology to phenomenological and quantum circuit level noise settings. For all examples considered, we obtain highly accurate estimates of optimal error thresholds from small, low-distance instances of the codes, in close agreement with threshold values reported in the literature. Our findings establish the coherent information as a reliable competitive tool for the calculation of optimal thresholds of state-of-the-art QEC codes under realistic noise models. Published by the American Physical Society 2024