Quantum Topology Express Method

We introduce a numerical method for identifying topological order in two-dimensional models based on one-dimensional bulk operators. The idea is to identify approximate symmetries supported on thin strips through the bulk that behave as string operators associated to an anyon model. We can express these ribbon operators in a matrix product form and define a cost function that allows us to efficiently optimize over this ansatz class. We test this method on spin models with Abelian topological order by finding ribbon operators for ${\mathbb{Z}}_{d}$ quantum double models with local fields and Ising-like terms. In addition, we identify ribbons in the Abelian phase of Kitaev's honeycomb model which serve as the logical operators of the encoded qubit for the quantum error-correcting code. We further identify the topologically encoded qubit in the quantum compass model, and show that despite this qubit, the model does not support topological order. Finally, we discuss how the method supports generalizations for detecting non-Abelian topological order.We introduce a numerical method for identifying topological order in two-dimensional models based on one-dimensional bulk operators. The idea is to identify approximate symmetries supported on thin strips through the bulk that behave as string operators associated to an anyon model. We can express these ribbon operators in a matrix product form and define a cost function that allows us to efficiently optimize over this ansatz class. We test this method on spin models with Abelian topological order by finding ribbon operators for ${\mathbb{Z}}_{d}$ quantum double models with local fields and Ising-like terms. In addition, we identify ribbons in the Abelian phase of Kitaev's honeycomb model which serve as the logical operators of the encoded qubit for the quantum error-correcting code. We further identify the topologically encoded qubit in the quantum compass model, and show that despite this qubit, the model does not support topological order. Finally, we discuss how the method supports generalizations for detecting non-Abelian topological order.

arXiv:1603.02275Physical review. B./Physical review. BNov 14, 201619 citations

Topology and Quantum Computing

Louis Hirsch Kauffman, Samuel J. Lomonaco

Lecture notes in physicsNov 7, 20084 citations

Quantum Computing and Quantum Topology

Louis Hirsch Kauffman, Samuel J. Lomonaco

Chapman & Hall/CRC applied mathematics and nonlinear science seriesSep 19, 20070 citations

Dynamic Topological Geometry: An Adaptive Topological Geometry Approach

Jincheng Zhang

Dynamic Topological Geometry (DTG) presents a novel framework for geometric computation and modeling, leveraging the principles of quantum entanglement to achieve real-time adjustments to topological structures. This paper introduces a novel method for constructing and manipulating complex geometric forms through the application of quantum entanglement. The core mechanism involves establishing a quantum entanglement system that dynamically modifies the topological landscape, enabling rapid and adaptable geometric operations. This approach promises to significantly enhance the scalability and expressiveness of geometric modeling, particularly in areas requiring complex transformations and simulations. The study explores the potential of DTG to address limitations of traditional topological geometry, offering a paradigm shift in computational geometry.Dynamic Topological Geometry (DTG) presents a novel framework for geometric computation and modeling, leveraging the principles of quantum entanglement to achieve real-time adjustments to topological structures. This paper introduces a novel method for constructing and manipulating complex geometric forms through the application of quantum entanglement. The core mechanism involves establishing a quantum entanglement system that dynamically modifies the topological landscape, enabling rapid and adaptable geometric operations. This approach promises to significantly enhance the scalability and expressiveness of geometric modeling, particularly in areas requiring complex transformations and simulations. The study explores the potential of DTG to address limitations of traditional topological geometry, offering a paradigm shift in computational geometry.

Aug 31, 20260 citations

REDUCED QUANTUM TOPOLOGICAL TRAJECTORY IN COHERENCE AND DECOHERENCE

Xuguang Shi, Yi-Shi Duan

We have studied the topological expression of the reduced quantum trajectory using ϕ-mapping topological current theory and reduced density matrix theory. Moreover, we have proposed a new topological expression called the reduced quantum topological trajectory. Ignoring the coefficients of the winding number, we have found that the expression for the reduced quantum topological trajectory is the same for coherence as it is for decoherence. Under certain conditions, the vorticity of the reduced quantum topological trajectory is a topological invariant.We have studied the topological expression of the reduced quantum trajectory using ϕ-mapping topological current theory and reduced density matrix theory. Moreover, we have proposed a new topological expression called the reduced quantum topological trajectory. Ignoring the coefficients of the winding number, we have found that the expression for the reduced quantum topological trajectory is the same for coherence as it is for decoherence. Under certain conditions, the vorticity of the reduced quantum topological trajectory is a topological invariant.

Modern Physics Letters AMay 30, 20113 citations

{\mathbb{Z}}_{d}

quantum double models with local fields and Ising-like terms. In addition, we identify ribbons in the Abelian phase of Kitaev's honeycomb model which serve as the logical operators of the encoded qubit for the quantum error-correcting code. We further identify the topologically encoded qubit in the quantum compass model, and show that despite this qubit, the model does not support topological order. Finally, we discuss how the method supports generalizations for detecting non-Abelian topological order.