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

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 to. 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 hypergraphH. 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.

Phase transition in a noisy Kitaev toric code model

“…It is well-known that the partition function of a classical spin model can be mapped to an inner product of a product state and an entangled state [12]. We have recently provided such a mapping using a duality transformation for CSS states which are mapped to classical spin models [29]. In this section, we review such mapping between the TC state and 2D Ising model.…”

“…Most such studies were based on a specific mappings between classical-quantum models. However, we have recently introduced a canonical relation as a duality mapping where any given CSS quantum state can be mapped, via hypergraph representations, to an arbitrary classical spin model [29].…”

“…Now, since there is a correspondence between such classical spin models and entangled quantum states, one would have to wonder what the consequences of such phase transitions are on the quantum states. It is our intention to take a step in this direction by considering the well-known ferromagnetic phase transition in a 2D Ising model and its consequences on the Kitaev toric code (TC) [32], which we have previously shown to be related via a duality mapping [29]. The TC state is of particular interest since it has a topological order [33,34] with a robust nature [35][36][37][38] as well as an important application in quantum error correction [39][40][41].…”

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

“…Therefore, it seems that such a result can be obtained in other topological CSS codes which are related to classical spin models by the dual correspondence [12]. It means that we have a useful tool for finding error thresholds in topological CSS codes; by using the hypergraph duality which was introduced in [12], one is able to find the classical spin model that is associated to an arbitrary topological CSS state. Then, one should numerically calculate the magnetization of the corresponding random bond spin model to find the transition point on the Nishimori line.…”

Ising order parameter and topological phase transitions: Toric code in a uniform magnetic field

“…However, in comparison with ordinary quantum phase transitions one can expect there is also an order parameter with a non-local nature for a topological phase transition [36][37][38][39][40]. Interestingly, there are mappings between topological phase transitions and ordinary quantum phase transitions which have been introduced for lattice gauge theories in [41,42] and also for Calderbank-Stean-Shor (CSS) code models in [31], see also [43] for a mapping to classical phase transitions. By such mappings, one can expect that the wellestablished knowledge of quantum phase transitions are used for understanding their corresponding topological phase transitions.…”