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arXiv:1710.01902 · Physical Review A

Dual correspondence between classical spin models and quantum Calderbank-Shor-Steane states

Mohammad Hossein Zarei, Afshin Montakhab

Published Jul 31, 2018• 15 citations
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Abstract

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 $H$ 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.

DOI
10.1103/physreva.98.012337

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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 H\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}H\ifmmode~\else˜\fi​. 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.
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 HHH 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.
We establish an important duality correspondence between topological order in quantum many-body systems and criticality in ferromagnetic classical spin systems. We show how such a correspondence leads to a classical and simple procedure for characterization of topological order in an important set of quantum entangled states, namely the Calderbank-Shor-Steane (CSS) states. To this end, we introduce a particular quantum Hamiltonian which allows us to consider the existence of a topological phase transition from quantum CSS states to a magnetized state. We study the ground state fidelity in order to find nonanalyticity in the wave function as a signature of a topological phase transition. Since hypergraphs can be used to map any arbitrary CSS state to a classical spin model, we show that fidelity of the quantum model defined on a hypergraph HHH is mapped to the heat capacity of the classical spin model defined on dual hypergraph $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$. Consequently, we show that a ferromagnetic-paramagnetic phase transition in a classical model is mapped to a topological phase transition in the corresponding quantum model. We also show that magnetization does not behave as a local order parameter at the transition point while the classical order parameter is mapped to a nonlocal measure on the quantum side, further indicating the nonlocal nature of the transition. Our procedure not only opens the door for identification of topological phases via the existence of a local and classical quantity, i.e., critical point, but also offers the potential to classify various topological phases through the concept of universality in phase transitions.
Graphs and recently hypergraphs have been known as an important tool for considering different properties of quantum many-body systems. In this paper, we study a mapping between an important class of quantum systems, namely quantum Calderbank-Shor-Steane (CSS) codes, and Ising-like systems by using hypergraphs. We show that the Hamiltonian corresponding to a CSS code on a hypergraph HHH which is perturbed by a uniform magnetic field is mapped to Hamiltonian of a Ising-like system on dual hypergraph $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ in a transverse field. Interestingly, we show that a strong regime of couplings in one of the systems is mapped to a weak regime of couplings in another one. We also give some applications for such a mapping where we study robustness of different topological CSS codes against a uniform magnetic field including Kitaev's toric codes defined on graphs and color codes in different dimensions. We show that a perturbed Kitaev's toric code on an arbitrary graph is mapped to an Ising model in a transverse field on the same graph and a perturbed color code on a DDD colex is mapped to a Ising-like model on a DDD-simplicial lattice in a transverse field. In particular, we use these results to explicitly compare the robustness of toric codes to uniform magnetic-field perturbations on different graphs. Interestingly, our results show that the robustness of such topological codes defined on graphs decreases with increasing dimension. Furthermore, we also use the duality mapping for some self-dual models where we exactly derive the point of phase transition.
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.
Calderbank-Shor-Steane (CSS) codes are a versatile quantum error-correcting family built out of commuting XXX- and ZZZ-type checks. We introduce CSS-like codes on GGG-valued qudits for any finite group GGG that reduce to qubit CSS codes for G=Z2G = \mathbb{Z}_2G=Z2​ yet generalize the Kitaev quantum double model for general groups. The XXX-checks of our group-CSS codes correspond to left and/or right multiplication by group elements, while ZZZ-checks project onto solutions to group word equations. We describe quantum-double models on oriented two-dimensional CW complexes (which need not cellulate a manifold) and prove that, when GGG is non-Abelian and simple, every GGG-covariant group-CSS code with suitably upper-bounded ZZZ-check weight and lower-bounded ZZZ-distance reduces to a CW quantum double. We describe the codespace and logical operators of CW quantum doubles via the same intuition used to obtain logical structure of surface codes. We obtain distance bounds for codes on non-Abelian simple groups from the graph underlying the CW complex, and construct intrinsically non-Abelian code families with asymptotically optimal rate and distances. Adding "ghost vertices" to the CW complex generalizes quantum double models with defects and rough boundary conditions whose logical structure can be understood without reference to non-Abelian anyons or defects. Several non-invertible symmetry-protected topological states, both with ordinary and higher-form symmetries, are the unique codewords of simply-connected CW quantum doubles with a single ghost vertex.