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:1410.8863

Parameter-dependent Gaussian $(z,N)$-generalized Yang-Baxter operators

Eric C. Rowell

Published Oct 31, 2014• 3 citations
arXivHTMLPDF

Abstract

We find unitary solutions $\tilde{R}(a)$ to the (multipicative parameter-dependent) $(z,N)$-generalized Yang-Baxter equation that carry the standard measurement basis to $m$-level $N$-partite states that generalize the Bell states corresponding to $\tilde{R}(0)$ in the case $m=N=2$. This is achieved by a careful study of solutions to the Yang-Baxter equation discovered by Fateev and Zamolodchikov in 1982.

DOI
10.48550/arxiv.1410.8863

Related papers

Gaussian (N, z)-generalized Yang-Baxter operators

Eric C. Rowell

We find unitary matrix solutions R˜(a) to the (multiplicative parameter-dependent) (N, z)-generalized Yang-Baxter equation that carry the standard measurement basis to m-level N-partite entangled states that generalize the 2-level bipartite entangled Bell states. This is achieved by a careful study of solutions to the Yang-Baxter equation discovered by Fateev and Zamolodchikov in 1982.

Quantum Information and Computation•Jan 1, 2016•2 citations

New spectral-parameter dependent solutions of the Yang-Baxter equation

Alexander. S. Garkun, Suvendu Barik, Aleksey K. Fedorov +1 more

The Yang-Baxter Equation (YBE) plays a crucial role for studying integrable many-body quantum systems. Many known YBE solutions provide various examples ranging from quantum spin chains to superconducting systems. Models of solvable statistical mechanics and their avatars are also based on YBE. Therefore, new solutions of the YBE could be used to construct new interesting 1D quantum or 2D classical systems with many other far-reaching applications. In this work, we attempt to find (almost) exhaustive set of solutions for the YBE in the lowest dimensions corresponding to a two-qubit case. We develop an algorithm, which can potentially be used for generating new higher-dimensional solutions of the YBE.

arXiv:2401.12710•Jan 23, 2024•3 citations

GHZ States, Almost-Complex Structure and Yang--Baxter Equation (I)

Yong Zhang, Mo‐Lin Ge

Recent study suggests that there are natural connections between quantum information theory and the Yang--Baxter equation. In this paper, in terms of the generalized almost-complex structure and with the help of its algebra, we define the generalized Bell matrix to yield all the GHZ states from the product base, prove it to form a unitary braid representation and present a new type of solution of the quantum Yang--Baxter equation. We also study Yang-Baxterization, Hamiltonian, projectors, diagonalization, noncommutative geometry, quantum algebra and FRT dual algebra associated with this generalized Bell matrix.

arXiv:quant-ph/0701244•Jan 31, 2007•4 citations

Quantum entanglement, supersymmetry, and the generalized Yang-Baxter equation

Pramod Padmanabhan, Fumihiko Sugino, Diego Trancanelli

Entangled states, such as the Bell and GHZ states, are generated from separable states using matrices known to satisfy the Yang-Baxter equation and its generalization. This remarkable fact hints at the possibility of using braiding operators as quantum entanglers, and is part of a larger speculated connection between topological and quantum entanglement. We push the analysis of this connection forward, by showing that supersymmetry algebras can be used to construct large families of solutions of the spectral parameter-dependent generalized Yang-Baxter equation. We present a number of explicit examples and outline a general algorithm for arbitrary numbers of qubits. The operators we obtain produce, in turn, all the entangled states in a multi-qubit system classified by the Stochastic Local Operations and Classical Communication protocol introduced in quantum information theory.

Quantum Information and Computation•Feb 1, 2020•9 citations

Unitary and entangling solutions to the parametric Yang–Baxter equation in all dimensions

Arash Pourkia

We present a new class of solutions to the parameter-dependent Yang–Baxter equation across all dimensions, which includes a significant subclass of unitary and entangling solutions. In any dimension d ≥ 2 , we construct a well-structured matrix R ( u ) that exhibits intriguing and useful symmetries. A key feature of R ( u ) is its composition from four monomial-based layers: R ( u ) = R a ( u ) + R b ( u ) + R x ( u ) + R y ( u ) , where a ( u ) , b ( u ) , x ( u ) and y ( u ) are four sets of monomial complex functions of the form, a u n a , b u n b , x u n x and y u n y , respectively. This well-designed structure plays a crucial role in demonstrating that R ( u ) is indeed a solution to the Yang–Baxter equation and in establishing the conditions for its unitarity and entangling properties. Additionally, it allows us to identify interesting non-trivial subfamilies with these properties. As is widely recognized, unitary and entangling solutions to the Yang–Baxter equation serve as universal quantum logic gates for qudit quantum computing . Moreover, the search for new solutions to the Yang–Baxter equation in higher dimensions is a common endeavor in both mathematics and physics .

Physics Open•Apr 7, 2025•1 citation

Symmetries of the Generalized Yang--Baxter Equations

Pramod Padmanabhan, Somnath Maity, Akash Sinha +1 more

The generalized Yang-Baxter equations are multi-site versions of the standard Yang-Baxter equation. When spectral parameters are included, such equations are expected to lead to integrable Hamiltonians with local interactions involving multiple degrees of freedom. In this work we characterize both the continuous and discrete symmetries of these equations required to establish an equivalence class of solutions. We find that the set of such symmetries depends on the number of sites on which the equation is supported. In several cases there are more symmetries than the standard Yang-Baxter equation, thus placing heavy constraints on the number of inequivalent solutions and the associated integrable models. As an application we show how to restrict the generic $16\times16$ ansatz to the 30-dimensional subspace invariant under a chosen discrete symmetry subgroup. An explicit $16\times 16$ solution is also written down. 21 pages + References ; v2 31 pages, Includes several new symmetries, with an exhaustive account of permutation symmetries, and a new section on application of these methods, Published in "A Special Issue in Honour of Jarmo Hietarinta''

arXiv:2606.26510•Open Communications in Nonlinear Mathematical Physics•Sep 10, 2026•0 citations
Parameter-dependent Gaussian (z,N)(z,N)(z,N)-generalized Yang-Baxter operators
We find unitary solutions R~(a)\tilde{R}(a)R~(a) to the (multipicative parameter-dependent) (z,N)(z,N)(z,N)-generalized Yang-Baxter equation that carry the standard measurement basis to mmm-level NNN-partite states that generalize the Bell states corresponding to R~(0)\tilde{R}(0)R~(0) in the case m=N=2m=N=2m=N=2. This is achieved by a careful study of solutions to the Yang-Baxter equation discovered by Fateev and Zamolodchikov in 1982.
The Yang-Baxter Equation (YBE) plays a crucial role for studying integrable many-body quantum systems. Many known YBE solutions provide various examples ranging from quantum spin chains to superconducting systems. Models of solvable statistical mechanics and their avatars are also based on YBE. Therefore, new solutions of the YBE could be used to construct new interesting 1D quantum or 2D classical systems with many other far-reaching applications. In this work, we attempt to find (almost) exhaustive set of solutions for the YBE in the lowest dimensions corresponding to a two-qubit case. We develop an algorithm, which can potentially be used for generating new higher-dimensional solutions of the YBE.
Recent study suggests that there are natural connections between quantum information theory and the Yang--Baxter equation. In this paper, in terms of the generalized almost-complex structure and with the help of its algebra, we define the generalized Bell matrix to yield all the GHZ states from the product base, prove it to form a unitary braid representation and present a new type of solution of the quantum Yang--Baxter equation. We also study Yang-Baxterization, Hamiltonian, projectors, diagonalization, noncommutative geometry, quantum algebra and FRT dual algebra associated with this generalized Bell matrix.
Entangled states, such as the Bell and GHZ states, are generated from separable states using matrices known to satisfy the Yang-Baxter equation and its generalization. This remarkable fact hints at the possibility of using braiding operators as quantum entanglers, and is part of a larger speculated connection between topological and quantum entanglement. We push the analysis of this connection forward, by showing that supersymmetry algebras can be used to construct large families of solutions of the spectral parameter-dependent generalized Yang-Baxter equation. We present a number of explicit examples and outline a general algorithm for arbitrary numbers of qubits. The operators we obtain produce, in turn, all the entangled states in a multi-qubit system classified by the Stochastic Local Operations and Classical Communication protocol introduced in quantum information theory.
We present a new class of solutions to the parameter-dependent Yang–Baxter equation across all dimensions, which includes a significant subclass of unitary and entangling solutions. In any dimension d ≥ 2 , we construct a well-structured matrix R ( u ) that exhibits intriguing and useful symmetries. A key feature of R ( u ) is its composition from four monomial-based layers: R ( u ) = R a ( u ) + R b ( u ) + R x ( u ) + R y ( u ) , where a ( u ) , b ( u ) , x ( u ) and y ( u ) are four sets of monomial complex functions of the form, a u n a , b u n b , x u n x and y u n y , respectively. This well-designed structure plays a crucial role in demonstrating that R ( u ) is indeed a solution to the Yang–Baxter equation and in establishing the conditions for its unitarity and entangling properties. Additionally, it allows us to identify interesting non-trivial subfamilies with these properties. As is widely recognized, unitary and entangling solutions to the Yang–Baxter equation serve as universal quantum logic gates for qudit quantum computing . Moreover, the search for new solutions to the Yang–Baxter equation in higher dimensions is a common endeavor in both mathematics and physics .
The generalized Yang-Baxter equations are multi-site versions of the standard Yang-Baxter equation. When spectral parameters are included, such equations are expected to lead to integrable Hamiltonians with local interactions involving multiple degrees of freedom. In this work we characterize both the continuous and discrete symmetries of these equations required to establish an equivalence class of solutions. We find that the set of such symmetries depends on the number of sites on which the equation is supported. In several cases there are more symmetries than the standard Yang-Baxter equation, thus placing heavy constraints on the number of inequivalent solutions and the associated integrable models. As an application we show how to restrict the generic 16×1616\times1616×16 ansatz to the 30-dimensional subspace invariant under a chosen discrete symmetry subgroup. An explicit 16×1616\times 1616×16 solution is also written down. 21 pages + References ; v2 31 pages, Includes several new symmetries, with an exhaustive account of permutation symmetries, and a new section on application of these methods, Published in "A Special Issue in Honour of Jarmo Hietarinta''