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arXiv:2305.13827 · Physical review. B./Physical review. B

Layer-by-layer disentangling two-dimensional topological quantum codes

Mohammad Hossein Zarei, Mohsen Rahmani Haghighi

Published Jul 10, 2023• 1 citation
arXivHTMLPDF

Abstract

While local unitary transformations are used for identifying quantum states that are in the same topological class, nonlocal unitary transformations are also important for studying the transition between different topological classes. In particular, it is an important task to find suitable nonlocal transformations that systematically sweep different topological classes. Here, regarding the role of dimension in the topological classes, we introduce partially local unitary transformations, namely Greenberger-Horne-Zeilinger (GHZ) disentanglers, which reduce the dimension of the initial topological model by a layer-by-layer disentangling mechanism. We justify the importance of such a mechanism for characterizing topological phases by considering two important examples. To this end, we apply GHZ disentanglers to two-dimensional (2D) topological quantum codes and show that they are converted to many copies of Kitaev's ladders. It implies that the above disentanglers cause a transition from an intrinsic topological phase to a symmetry-protected topological phase. Then, we show that while Kitaev's ladders are building blocks of both color code and toric code, there are different patterns of entangling ladders in 2D color code and toric code. It shows that different topological features of these topological codes are reflected in different patterns of entangling ladders. In this regard, we propose that the layer-by-layer disentangling mechanism can be used as a systematic method for classification of topological orders based on finding different patterns of long-range entanglement in topological lattice models.

DOI
10.1103/physrevb.108.035116

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While local unitary transformations are used for identifying quantum states that are in the same topological class, nonlocal unitary transformations are also important for studying the transition between different topological classes. In particular, it is an important task to find suitable nonlocal transformations that systematically sweep different topological classes. Here, regarding the role of dimension in the topological classes, we introduce partially local unitary transformations, namely Greenberger-Horne-Zeilinger (GHZ) disentanglers, which reduce the dimension of the initial topological model by a layer-by-layer disentangling mechanism. We justify the importance of such a mechanism for characterizing topological phases by considering two important examples. To this end, we apply GHZ disentanglers to two-dimensional (2D) topological quantum codes and show that they are converted to many copies of Kitaev's ladders. It implies that the above disentanglers cause a transition from an intrinsic topological phase to a symmetry-protected topological phase. Then, we show that while Kitaev's ladders are building blocks of both color code and toric code, there are different patterns of entangling ladders in 2D color code and toric code. It shows that different topological features of these topological codes are reflected in different patterns of entangling ladders. In this regard, we propose that the layer-by-layer disentangling mechanism can be used as a systematic method for classification of topological orders based on finding different patterns of long-range entanglement in topological lattice models.
Studying equivalence classes under local unitary transformations is one of the most important approaches for the classification of topological quantum states. It has specially attracted much attention for topological quantum codes due to their application in quantum computing. In particular, It has been shown that each D dimensional color code is local unitary equivalence to many copies of D dimensional toric codes. In this paper, we consider such transformations for two- dimensional (2D) topological codes by introducing GHZ disentanglers. We apply the above disentanglers on qubits corresponding to one particular color in the color code defined on a three-colorable honeycomb lattice. Then, we show that it leads to disentangling other colors because the initial color code is converted to two copies of the triangular toric codes. Furtheremore we extend the above transformations for color codes on different three-colorable lattices. We show that by applying GHZ disentanglers corresponding to one particular color, the color code is converted to two toric codes defined on dual lattices corresponding to other colors. This result is also useful for comparing color codes on different lattices regarding the difference between their dual lattices.
Abstract Toric Code, a cornerstone of the topological order, admits numerous realizations on different planar graphs. We prove that all these realizations share a single, universal entanglement structure in a sense that any Toric Code state can be generated by applying a fixed set of non-local unitaries between similar quasi-one-dimensional quantum ladders. In particular, we consider Toric Code states corresponding to various planar graphs and apply non-local dientanglers to qubits corresponding to non-contractible cycles that satisfy a topological constraint. We demonstrate that, independent of the geometry of the underlying graph, disentanglers convert Toric Code states into a tensor product of Kitaev's Ladder states. This exact and graph-independent construction reveals that the long-range entanglement characterizing the Toric Code's topological phase has a precise decomposable structure in a sense that it arises entirely from a specific pattern of entanglement between lower-dimensional subsystems. This structural insight opens new avenues for classification and simulation of topological orders.
Although the topological order is known as a quantum order in quantum many-body systems, it seems that there is not a one-to-one correspondence between topological phases and quantum phases. As a well-known example, it has been shown that all one-dimensional (1D) quantum phases are topologically trivial [X. Chen et al., Phys. Rev. B 83, 035107 (2011)]. By such facts, it seems a challenging task to understand when a quantum phase transition between different topological models necessarily reveals different topological classes of them. In this paper, we make an attempt to consider this problem by studying a phase transition between two different quantum phases which have a universal topological phase. We define a Hamiltonian as interpolation of the toric code model with Z2{Z}_{2}Z2​ topological order and the color code model with ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ topological order on a hexagonal lattice. We show such a model is exactly mapped to many copies of 1D quantum Ising model in transverse field by rewriting the Hamiltonian in a new complete basis. Consequently, we show that the universal topological phase of the color code model and the toric code model reflects in the 1D nature of the phase transition. We also consider the expectation value of Wilson loops by a perturbative calculation and show that behavior of the Wilson loop captures the nontopological nature of the quantum phase transition. The result on the point of phase transition also shows that the color code model is strongly robust against the toric code model.
We demonstrate that two toric code layers on the square lattice coupled by an Ising interaction display two distinct phases with intrinsic topological order. The second-order quantum phase transition between the weakly coupled ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ and the strongly coupled Z2{\mathbb{Z}}_{2}Z2​ topological order can be described by the condensation of bosonic quasiparticles from both sides and belongs to the three-dimensional Ising∗{\mathrm{Ising}}^{*}Ising∗ universality class. This can be shown by an exact duality transformation to the transverse-field Ising model on the square lattice, which builds on the existence of an extensive number of local Z2{\mathbb{Z}}_{2}Z2​ conserved parities. These conserved quantities correspond to the product of two adjacent star operators on different layers. Notably, we show that the low-energy effective model derived about the limit of large Ising coupling is given by an effective single-layer toric code in terms of the conserved quantities of the Ising toric code bilayer. The two topological phases are further characterized by the topological entanglement entropy which serves as a nonlocal order parameter.
In recent years, there have been many studies on local stabilizer codes. Under the assumption of translation and scale invariance Yoshida classified such codes. His result implies that translation invariant 2D color codes are equivalent to copies of toric codes. Independently, Bombin, Duclos-Cianci, and Poulin showed that a local translation invariant 2D topological stabilizer code is locally equivalent to a finite number of copies of Kitaev's toric code. In this paper we focus on 2D topological color codes and relax the assumption of translation invariance. Using a linear algebraic framework we show that any 2D color code (without boundaries) is locally equivalent to two copies of a related surface code. This surface code is induced by color code. We study the application of this equivalence to the decoding of 2D color codes over the bit flip channel as well as the quantum erasure channel. We report the performance of the color code on the square octagonal lattice over the quantum erasure channel. Further, we provide explicit circuits that perform the transformation between 2D color codes and surface codes. Our circuits do not require any additional ancilla qubits.