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arXiv:1911.10494 · Physical Review A

Noisy Toric code and random-bond Ising model: The error threshold in a dual picture

Mohammad Hossein Zarei, Abolfazl Ramezanpour

Published Dec 10, 2019• 4 citations
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Abstract

It is known that noisy topological quantum codes are related to random-bond Ising models where the order-disorder phase transition in the classical model is mapped to the error threshold of the corresponding topological code. On the other hand, there is a dual mapping between classical spin models and quantum Calderbank-Shor-Steane (CSS) states where the partition function of a classical model defined on a hypergraph $H$ is written as an inner product of a product state and a CSS state on dual hypergraph $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$. It is then interesting to see what the interpretation is of the classical phase transition in the random-bond Ising model within the framework of the above duality, and whether such an interpretation has any connection to the error threshold of the corresponding topological CSS code. In this paper, we consider the above duality relation specifically for a two-dimensional random-bond Ising model. We show that the order parameter of this classical system is mapped to a coherence order parameter in a noisy Toric code model. In particular, a quantum phase transition from a coherent phase to a noncoherent phase occurs when the initial coherent state is affected by two sequences of bit-flip quantum channels where a quenched disorder is induced by measurement of the errors after the first channel. On the other hand, the above transition is directly related to the error threshold of the Toric code model. Accordingly, and since the noisy process can be applied to other topological CSS states, we conclude that the dual correspondence can also provide a useful tool for the study of error thresholds in different topological CSS codes.

DOI
10.1103/physreva.100.062313

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Many aspects of the well-known mapping between the partition function of a classical spin model and the quantum entangled state have been studied in recent years. However, the consequences of the existence of a classical (critical) phase transition on the corresponding quantum state have been mostly ignored. In this paper, we consider this problem for an important example of the Kitaev toric code model which has been shown to correspond to the two-dimensional (2D) Ising model though a duality transformation. We show that the temperature on the classical side is mapped to bit-flip noise on the quantum side. It is then shown that a transition from a coherent superposition of a given quantum state to a noncoherent mixture corresponds exactly to paramagnetic-ferromagnetic phase transition in the Ising model. To identify such a transition further, we define an order parameter to characterize the decoherence of such a mixture and show that it behaves similar to the order parameter (magnetization) of the 2D Ising model, a behavior that is interpreted as a robust coherence in the toric code model. Furthermore, we consider other properties of the noisy toric code model exactly at the critical point. We show that there is a relative stability to noise for the toric code state at the critical noise which is revealed by a relative reduction in susceptibility to noise.
The correspondence between classical spin models and quantum states has attracted much attention in recent years. However, it remains an open problem as to which specific spin model a given (well-known) quantum state maps. Here, we provide such an explicit correspondence for an important class of quantum states where a duality relation is proved between classical spin models and quantum Calderbank-Shor-Steane (CSS) states. In particular, we employ graph-theoretic methods to prove that the partition function of a classical spin model on a hypergraph HHH is equal to the inner product of a product state with a quantum CSS state on a dual hypergraph $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$. We next use this dual correspondence to prove that the critical behavior of the classical system corresponds to a relative stability of the corresponding CSS state to bit-flip (or phase-flip) noise, thus called critical stability. We finally conjecture that such critical stability is related to the topological order in quantum CSS states, thus providing a possible practical characterization of such states.
We investigate how the lattice geometry influences the error thresholds of two-dimensional toric codes. In the presence of bit- or phase-flip errors, the toric code maps onto the two-dimensional random-bond Ising model (RBIM). We determine the critical behaviour of the RBIM using replica-exchange Monte Carlo simulations. While previous studies have explored thresholds for general lattice geometries under suboptimal decoders such as minimum-weight perfect matching, the rapid development of near-optimal decoders makes resolving the ultimate, maximum-likelihood code capacity relevant. We compute these optimal error thresholds on the Nishimori line for the square, honeycomb, triangular, dice, and kagome lattices. Owing to lattice duality, the phase-flip threshold on a given lattice is equivalent to the bit-flip threshold on its dual. We find that the optimal thresholds of dual-lattice pairs display a characteristic duality-driven splitting around the self-dual square-lattice, mirroring the qualitative behaviour observed for suboptimal decoders. While the average coordination number has the strongest impact on the error threshold, our results demonstrate that the detailed arrangement of vertices and plaquettes also plays a significant role in determining its precise value.
Quantum error correction (QEC) codes protect quantum information from errors due to decoherence. Many of them also serve as prototypical models for exotic topological quantum matters. Investigating the behavior of the QEC codes under decoherence sheds light on not only the codes' robustness against errors but also new out-of-equilibrium quantum phases driven by decoherence. The phase transitions, including the error threshold, of the decohered QEC codes can be probed by the systems' R\'enyi entropies SR{S}_{R}SR​ with different R\'enyi indices RRR. In this paper, we study the general construction of the statistical models that characterize the R\'enyi entropies of QEC codes decohered by Pauli noise. We show that these statistical models can be organized into a ``tapestry'' woven by rich duality relations among them. For Calderbank-Shor-Steane (CSS) codes with bit-flip and phase-flip errors, we show that each R\'enyi entropy is captured by a pair of dual statistical models with randomness. For $R=2,3,\ensuremath{\infty}$, there are additional dualities that map between the two error types, relating the critical bit-flip and phase-flip error rates of the decoherence-induced phase transitions in the CSS codes. For CSS codes with an ``ememem symmetry'' between the XXX-type and the ZZZ-type stabilizers, the dualities with $R=2,3,\ensuremath{\infty}$ become self-dualities with super-universal self-dual error rates. These self-dualities strongly constrain the phase transitions of the code signaled by ${S}_{R=2,3,\ensuremath{\infty}}$. For general stabilizer codes decohered by generic Pauli noise, we also construct the statistical models that characterize the systems' entropies and obtain general duality relations between Pauli noise with different error rates.
Stabilizer codes lie at the heart of modern quantum-error-correcting codes (QECCs). Of particular importance is a class called Calderbank-Shor-Steane (CSS) codes, which includes many important examples such as toric codes, color codes, and fractons. Recent studies have revealed that the decoding transition for these QECCs could be intrinsically captured by calculating information-theoretic quantities from the mixed state. Here we perform a simple analytic calculation of the coherent information for general CSS codes under local incoherent Pauli errors via diagonalization of the density matrices and mapping to classical statistical mechanical (SM) models. Our result establishes a rigorous connection between the decoding transition of the quantum code and the phase transition in the random classical SM model. It is also directly confirmed for CSS codes that exact error correction is possible if and only if the maximum-likelihood decoder always succeeds in the thermodynamic limit. Thus, the fundamental threshold is saturated by the optimal decoder.
We study the error thresholds of a new topological quantum error-correcting code under the independent bit-flip and phase-flip noisy channel. By mapping to classical spin-glass models, we find that if Pauli X and Z errors are equally probable, the threshold of vertex syndromes is improved by 30% compared to that of the toric code, while the threshold of plaquette syndromes is lowered. Both thresholds increase when the Z component is less likely than the X component, reaching 0.5 if the Z component is absent. An analytical proof justifies this special case of perfect correctability, and a disordered version of Peierls's argument establishes a lower bound for the vertex-syndrome threshold.