Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1103.4606 · New Journal of Physics

Universal topological phase of two-dimensional stabilizer codes

Héctor Bombín, Guillaume Duclos-Cianci, David Poulin

Published Jul 25, 2012• 121 citations
arXivHTMLPDF

Abstract

Two topological phases are equivalent if they are connected by a local unitary transformation. In this sense, classifying topological phases amounts to classifying long-range entanglement patterns. We show that all 2D topological stabilizer codes are equivalent to several copies of one universal phase: Kitaev's topological code. Error correction benefits from the corresponding local mappings.

DOI
10.1088/1367-2630/14/7/073048

Related papers

Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order

Xie Chen, Zheng‐Cheng Gu, Xiao-Gang Wen

Two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states. On the other hand, gapped ground states remain within the same phase under local unitary transformations. Therefore, local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems. Since local unitary transformations can remove local entanglement, the above equivalence/universality classes correspond to pattern of long-range entanglement, which is the essence of topological order. The local unitary transformation also allows us to define a wave function renormalization scheme, under which a wave function can flow to a simpler one within the same equivalence/universality class. Using such a setup, we find conditions on the possible fixed-point wave functions where the local unitary transformations have finite dimensions. The solutions of the conditions allow us to classify this type of topological orders, which generalize the string-net classification of topological orders. We also describe an algorithm of wave function renormalization induced by local unitary transformations. The algorithm allows us to calculate the flow of tensor-product wave functions which are not at the fixed points. This will allow us to calculate topological orders as well as symmetry-breaking orders in a generic tensor-product state.

arXiv:1004.3835•Physical Review B•Oct 26, 2010•890 citations

On the Local Equivalence of 2D Color Codes and Surface Codes with Applications

Arun B. Aloshious, Arjun Nitin Bhagoji, Pradeep Kiran Sarvepalli

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.

arXiv:1804.00866•Apr 3, 2018•3 citations

Equivalence of topological color codes (without translational symmetry) to surface codes.

Arjun Nitin Bhagoji, Pradeep Kiran Sarvepalli

In a recent work, Bombin, Duclos-Cianci, and Poulin showed that every local translationally 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 color codes and relax the constraint on translation invariance. We show that any color code can be mapped to exactly two copies of a related surface code. The surface code in our map is induced by the color code and easily derived from the color code. Furthermore, our map does not require any ancilla qubits for the surface codes. We also indicate the various degrees of freedom in constructing the map and the consequent variations.

•Mar 10, 2015•0 citations

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

Mohammad Hossein Zarei, Mohsen Rahmani Haghighi

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.

arXiv:2305.13827•Physical review. B./Physical review. B•Jul 10, 2023•1 citation

Equivalence of 2D color codes (without translational symmetry) to surface codes

Arjun Nitin Bhagoji, Pradeep Kiran Sarvepalli

In a recent work, Bombin, Duclos-Cianci, and Poulin showed that every local translationally invariant 2D topological stabilizer code is locally equivalent to a finite number of copies of Kitaev's toric code. For 2D color codes, Delfosse relaxed the constraint on translation invariance and mapped a 2D color code onto three surface codes. In this paper, we propose an alternate map based on linear algebra. We show that any 2D color code can be mapped onto exactly two copies of a related surface code. The surface code in our map is induced by the color code and easily derived from the color code. Furthermore, our map does not require any ancilla qubits for the surface codes.

arXiv:1503.03009•IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2015•10 citations

Universal representation of the long-range entanglement in the family of Toric Code states

Mohammad Hossein Zarei, Mohsen Rahmani Haghighi

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.

arXiv:2509.03422•Europhysics Letters (EPL)•Jun 29, 2026•0 citations
Two topological phases are equivalent if they are connected by a local unitary transformation. In this sense, classifying topological phases amounts to classifying long-range entanglement patterns. We show that all 2D topological stabilizer codes are equivalent to several copies of one universal phase: Kitaev's topological code. Error correction benefits from the corresponding local mappings.
Two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states. On the other hand, gapped ground states remain within the same phase under local unitary transformations. Therefore, local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems. Since local unitary transformations can remove local entanglement, the above equivalence/universality classes correspond to pattern of long-range entanglement, which is the essence of topological order. The local unitary transformation also allows us to define a wave function renormalization scheme, under which a wave function can flow to a simpler one within the same equivalence/universality class. Using such a setup, we find conditions on the possible fixed-point wave functions where the local unitary transformations have finite dimensions. The solutions of the conditions allow us to classify this type of topological orders, which generalize the string-net classification of topological orders. We also describe an algorithm of wave function renormalization induced by local unitary transformations. The algorithm allows us to calculate the flow of tensor-product wave functions which are not at the fixed points. This will allow us to calculate topological orders as well as symmetry-breaking orders in a generic tensor-product state.
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.
In a recent work, Bombin, Duclos-Cianci, and Poulin showed that every local translationally 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 color codes and relax the constraint on translation invariance. We show that any color code can be mapped to exactly two copies of a related surface code. The surface code in our map is induced by the color code and easily derived from the color code. Furthermore, our map does not require any ancilla qubits for the surface codes. We also indicate the various degrees of freedom in constructing the map and the consequent variations.
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.
In a recent work, Bombin, Duclos-Cianci, and Poulin showed that every local translationally invariant 2D topological stabilizer code is locally equivalent to a finite number of copies of Kitaev's toric code. For 2D color codes, Delfosse relaxed the constraint on translation invariance and mapped a 2D color code onto three surface codes. In this paper, we propose an alternate map based on linear algebra. We show that any 2D color code can be mapped onto exactly two copies of a related surface code. The surface code in our map is induced by the color code and easily derived from the color code. Furthermore, our map does not require any ancilla qubits for the surface codes.
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.