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

Journal of Physics Conference Series

Magnonic qudit and algebraic Bethe Ansatz

Barbara Lulek, Tadeusz Lulek

Published Mar 1, 2010• 2 citations
Publisher

Abstract

A magnonic qudit is proposed as the memory unit of a register of a quantum computer. It is the N-dimensional space, extracted from the 2 N -dimensional space of all quantum states of the magnetic Heisenberg ring of N spins 1/2, as the space of all states of a single magnon. Three bases: positional, momentum, and that of Weyl duality are described, together with appropriate Fourier and Kostka transforms. It is demonstrated how exact Bethe Ansatz (BA) eigenfunctions, classified in terms of rigged string configurations, can be coded using a collection of magnonic qudits. To this aim, the algebraic BA is invoked, such that a single magnonic qudit is prepared in a state corresponding to a magnon in one of the states provided by spectral parameters emerging from the corresponding BA equations.

DOI
10.1088/1742-6596/213/1/012009

Related papers

Schwinger Geometry, Bethe Ansatz, and a Magnonic Qudit

Paweł Jakubczyk, S. Topolewicz, Andrzej Wal +1 more

Schwinger approach of unitary geometry for a finite-dimensional Hilbert space is interpreted in terms of a magnonic qudit — a hypothetic elementary unit of memory of a quantum computer. The space is interpreted within the Heisenberg model for a magnetic ring, its calculational basis as the classical configuration space for a single spin deviation, treated as a Bethe pseudoparticle, and the dual basis corresponds to quasimomenta, so that the classical phase space spans the quantum algebra of observables. Effects of the Schur-Weyl duality and Bethe ansatz exact eigenstates of the Heisenberg Hamiltonian for the XXX model on properties of the magnonic qudit are presented.

Open Systems & Information Dynamics•Aug 13, 2009•7 citations

Bethe ansatz, quantum computers, and unitary geometry

Tadeusz Lulek

The one-dimensional ring of N magnetic nodes, each node with the spin 1/2, can be seen as a faithful prototype of a quantum computer with N qubits. We point out here some analogies between typical problems and known solutions of the Heisenberg model of a magnet on this ring, and the related questions posed within the midst of quantum information processing. In particular, we relate such notions as the memory, gates, or entanglement of quantum states with exact results of Bethe Ansatz, expressed in combinatoric terms of rigged string configurations. A promising and transparent interpretation of relation between Bethe Ansatz and quantum computation can be provided by the unitary geometry of Schwinger, presented in our previous SSPCM schools by prof. Vourdas.

Journal of Physics Conference Series•Mar 1, 2008•3 citations

The role of Robinson-Schensted and Kerov-Kirillov-Reshetikhin bijections in Bethe ansatz

Dorota Jakubczyk, Tadeusz Lulek, Paweł Jakubczyk +1 more

The Robinson-Schensted and Kerov-Kirillov-Reshetikhin (RSKKR) bijections allow us to confirm the completeness of solutions of the eigenproblem of the one-dimensional Heisenberg Hamiltonian in a purely combinatorial manner, by studying the structure of the classical configuration space of the system. The combined bijection relates two sets, namely the basis of magnetic configurations and the set of combinatorial objects called rigged string configurations. The former serve as the initial basis for quantum computations, whereas the latter classify the exact Bethe Ansatz (BA) eigenstates. We discuss in this report the application of this bijection in the procedure of construction of two-particle states within Clebsch-Gordan scheme. This bijection provides the irreducible bases for the decomposition of the tensor product of transitive representations R ( Σ N :(Σ N-1 × Σ 1 ) within the scheme of a Hopf algebra of symmetric groups. We point out the role of true physical BA eigenstates, as well as fictituous configurations corresponding to doubly occupied sites (the diagonal of the cartesian square of the one-magnon classical configuration space) and to antisymmetric states.

Journal of Physics Conference Series•Mar 1, 2008•0 citations

A combinatoric description of Bethe Ansatz solutions for nanoscopic systems

Tadeusz Lulek

Abstract Bethe Ansatz provides an exact classification of eigenstates of the Heisenberg Hamiltonian for a finite magnetic ring, consisting of N nodes, each with the spin s = ½ (with some extensions to an arbitrary spin s) in terms of rigged string configurations. The latter are some combinatorial objects which serve as classification labels for solutions of Bethe equations. An astonishing feature is existence of Robinson–Schensted (RS) and Kerov–Kirillov–Reshetikhin (KKR) bijections between sets of (i) all magnetic configurations, (ii) all pairs of standard Young and Weyl tableaux of N boxes and n = 2s + 1 rows, (iii) all rigged string configurations, for given N and s. These bijections allow to point out an exact correspondence between physically admissible solutions of highly nonlinear Bethe Ansatz equations and the initial basis of quantum calculations – magnetic configurations which are just possible distributions of spins over the nodes. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

physica status solidi (b)•Jan 17, 2005•4 citations

Algebraic Bethe Ansatz forN= 4

Dorota Jakubczyk, Tadeusz Lulek, Paweł Jakubczyk +1 more

In the present paper we explore some useful devices for discussing the exact solutions of XXX isotropic Heisenberg Hamiltonian for the single node spin equal to 1/2 as the tensor product states. Our aim is to presents the monodromy matrix and the Lax operator in the contex of the Bethe Ansatz, signed from now on BA. We construct the former, for N equal to 4 nodes of the magnet, using the so called auxiliary space which is taken as a copy of 2 . The form of this matrix in the basis of orbits of the translation group C 4 reveals its block structure of all possible nonzero elements. Each block has its meaning in the language of creation and anihilation of the magnon. This fact implies, that one can think about appropriate operators and create the theory very similar to that of the quantum oscillator.

Journal of Physics Conference Series•Mar 1, 2010•0 citations

A complete set of commuting operators for the Bethe ansatz

Tadeusz Lulek, S. Topolewicz

We have derived an explicit formula for a complete set of commuting operators for the XXX model of the Heisenberg magnetic ring, using an algebraic Bethe ansatz approach of Faddeev and Takhtajan. Each operator turns out to be the sum of increasing l -cycles in the symmetric group acting on the set of nodes of the ring. It is demonstrated that the resulting algebra of operators encloses the total spin S , the Hamiltonian and the quasimomentum. We point out that it is a maximal Abelian subalgebra in the algebra of the symmetric group, associated with the basis of exact Bethe ansatz eigenstates, the latter classified by rigged string configurations of Kerrov, Kirillov and Reshetikhin. This algebra is also conjugated to the Jucys–Murphy algebra, responsible for the Young orthogonal basis of standard tableaux along the Schur–Weyl duality.

Journal of Physics A Mathematical and Theoretical•Nov 11, 2010•0 citations
A magnonic qudit is proposed as the memory unit of a register of a quantum computer. It is the N-dimensional space, extracted from the 2 N -dimensional space of all quantum states of the magnetic Heisenberg ring of N spins 1/2, as the space of all states of a single magnon. Three bases: positional, momentum, and that of Weyl duality are described, together with appropriate Fourier and Kostka transforms. It is demonstrated how exact Bethe Ansatz (BA) eigenfunctions, classified in terms of rigged string configurations, can be coded using a collection of magnonic qudits. To this aim, the algebraic BA is invoked, such that a single magnonic qudit is prepared in a state corresponding to a magnon in one of the states provided by spectral parameters emerging from the corresponding BA equations.
Schwinger approach of unitary geometry for a finite-dimensional Hilbert space is interpreted in terms of a magnonic qudit — a hypothetic elementary unit of memory of a quantum computer. The space is interpreted within the Heisenberg model for a magnetic ring, its calculational basis as the classical configuration space for a single spin deviation, treated as a Bethe pseudoparticle, and the dual basis corresponds to quasimomenta, so that the classical phase space spans the quantum algebra of observables. Effects of the Schur-Weyl duality and Bethe ansatz exact eigenstates of the Heisenberg Hamiltonian for the XXX model on properties of the magnonic qudit are presented.
The one-dimensional ring of N magnetic nodes, each node with the spin 1/2, can be seen as a faithful prototype of a quantum computer with N qubits. We point out here some analogies between typical problems and known solutions of the Heisenberg model of a magnet on this ring, and the related questions posed within the midst of quantum information processing. In particular, we relate such notions as the memory, gates, or entanglement of quantum states with exact results of Bethe Ansatz, expressed in combinatoric terms of rigged string configurations. A promising and transparent interpretation of relation between Bethe Ansatz and quantum computation can be provided by the unitary geometry of Schwinger, presented in our previous SSPCM schools by prof. Vourdas.
The Robinson-Schensted and Kerov-Kirillov-Reshetikhin (RSKKR) bijections allow us to confirm the completeness of solutions of the eigenproblem of the one-dimensional Heisenberg Hamiltonian in a purely combinatorial manner, by studying the structure of the classical configuration space of the system. The combined bijection relates two sets, namely the basis of magnetic configurations and the set of combinatorial objects called rigged string configurations. The former serve as the initial basis for quantum computations, whereas the latter classify the exact Bethe Ansatz (BA) eigenstates. We discuss in this report the application of this bijection in the procedure of construction of two-particle states within Clebsch-Gordan scheme. This bijection provides the irreducible bases for the decomposition of the tensor product of transitive representations R ( Σ N :(Σ N-1 × Σ 1 ) within the scheme of a Hopf algebra of symmetric groups. We point out the role of true physical BA eigenstates, as well as fictituous configurations corresponding to doubly occupied sites (the diagonal of the cartesian square of the one-magnon classical configuration space) and to antisymmetric states.
Abstract Bethe Ansatz provides an exact classification of eigenstates of the Heisenberg Hamiltonian for a finite magnetic ring, consisting of N nodes, each with the spin s = ½ (with some extensions to an arbitrary spin s) in terms of rigged string configurations. The latter are some combinatorial objects which serve as classification labels for solutions of Bethe equations. An astonishing feature is existence of Robinson–Schensted (RS) and Kerov–Kirillov–Reshetikhin (KKR) bijections between sets of (i) all magnetic configurations, (ii) all pairs of standard Young and Weyl tableaux of N boxes and n = 2s + 1 rows, (iii) all rigged string configurations, for given N and s. These bijections allow to point out an exact correspondence between physically admissible solutions of highly nonlinear Bethe Ansatz equations and the initial basis of quantum calculations – magnetic configurations which are just possible distributions of spins over the nodes. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
In the present paper we explore some useful devices for discussing the exact solutions of XXX isotropic Heisenberg Hamiltonian for the single node spin equal to 1/2 as the tensor product states. Our aim is to presents the monodromy matrix and the Lax operator in the contex of the Bethe Ansatz, signed from now on BA. We construct the former, for N equal to 4 nodes of the magnet, using the so called auxiliary space which is taken as a copy of 2 . The form of this matrix in the basis of orbits of the translation group C 4 reveals its block structure of all possible nonzero elements. Each block has its meaning in the language of creation and anihilation of the magnon. This fact implies, that one can think about appropriate operators and create the theory very similar to that of the quantum oscillator.
We have derived an explicit formula for a complete set of commuting operators for the XXX model of the Heisenberg magnetic ring, using an algebraic Bethe ansatz approach of Faddeev and Takhtajan. Each operator turns out to be the sum of increasing l -cycles in the symmetric group acting on the set of nodes of the ring. It is demonstrated that the resulting algebra of operators encloses the total spin S , the Hamiltonian and the quasimomentum. We point out that it is a maximal Abelian subalgebra in the algebra of the symmetric group, associated with the basis of exact Bethe ansatz eigenstates, the latter classified by rigged string configurations of Kerrov, Kirillov and Reshetikhin. This algebra is also conjugated to the Jucys–Murphy algebra, responsible for the Young orthogonal basis of standard tableaux along the Schur–Weyl duality.