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Journal of Physics Conference Series

Bethe ansatz, quantum computers, and unitary geometry

Tadeusz Lulek

Published Mar 1, 2008• 3 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/104/1/012043

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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 main purpose of this report is a thorough analysis of completeness of solutions of the one-dimensional Heisenberg Hamiltonian through the hypothesis of strings. A somehow astonishing conclusion emerges from studying of the structure of the classical configuration space of this system. Namely, all allowed information concerning quantum states, which are exact solutions of the Bethe equations, encoded in quantum numbers, are predictable via a bijection between the set of the magnetic configurations and the string configurations. This startling and beautiful observation constitutes the proof of the completeness of the eigenstates of the Heisenberg Hamiltonian, deduced in a purely combinatorial way. We interpret the set of all magnetic configurations with a fixed number r of spin deviations as the classical configuration space of a hypothetic system of r Bethe pseudoparticles, which move, in a stroboscopic manner, on the magnetic ring. The geometry of this configuration space, induced by the action of Heisenberg Hamiltonian and the translation symmetry group of the ring, implies the structure of a locally r -dimensional hypercubic lattice with well defined F -dimensional boundaries, 1 ⩽ F ⩽ r . We demonstrate that rigged string configurations originate from these boundaries, depending upon the island structure of spin deviations. We show that a relatively simple combinatoric definition of rigged strings reproduces completely exact results of Bethe Ansatz. It is expressed in terms of a combined bijection: Robinson-Schensted with Kerov- Kirillov-Reshetikhin (RSKKR) which produces a geography of exact Bethe Ansatz solutions on the classical configuration space.
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.
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)
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.
We demonstrate that the seminal one-dimensional model of the Heisenberg magnet, consisting of N spins 1/2 with the nearest-neighbour isotropic interaction, solved exactly by Bethe ansatz, admits an interpretation of a system of r = N/2 -M pseudoparticles (spin deviations) which are indistinguishable, have hard cores and move on the chain by local hoppings.Such an approach allows us to construct a manifold with some boundaries, which is genericly r-dimensional, and whose F -dimensional regions, 0 < F < r, point out all l-strings.The latter classify, in terms of rigged string configurations of Kerov, Kirillov and Reshetikhin, all exact Bethe eigenfunctions.In this way, we interpret these eigenfunctions in terms of the classical configuration space, in particular on the structure of islands of adjacent spin deviations, in a way independent of the size N .
We demonstrate the way in which composition of two famous combinatorial bijections, of Robinson-Schensted and Kerov-Kirillov-Reshetikhin, applied to the Heisenberg model of magnetic ring with spin 1/21/21/2, defines the geography of rigged strings (which lab