A wavelet basis within Bethe ansatz

Tadeusz Lulek, Ryszard Stagraczyński, M. Łabuz

A method of separation of subspaces with definite values of the total spin, its z-projection and quasimomentum, within the state space of the XXX model, i.e. the N-th tensor power of a single qubit, is presented. We construct a complete system of orthogonal projectors onto such subspaces by use of the basis of wavelets and chosen measuring operators based on the total spin. We also demonstrate, on examples of N=6 and 8 magnetic spin rings, that the constructed projectors can be further decomposed into the orthogonal sum of density matrices of eigenstates of the XXX model in the corresponding subspaces. This decomposition exhibits a Galois symmetry, stemming from roots of indecomposable factors of characteristic polynomial of the Heisenberg Hamiltonian. In this way, we propose an alternative for Bethe Ansatz for moderate lengths of spin chains, by a return to the original spectral problem, avoiding the highly nonlinear system of Bethe Ansatz equations. Such results are easy to adapt in quantum information processing.

Yupeng Wang, Wen‐Li Yang, Junpeng Cao +1 more

Fabio Franchini

Paweł Jakubczyk, Dorota Jakubczyk, Tadeusz Lulek

Abstract We propose some operator techniques, applicable in the procedure of an immediate diagonalization for the Heisenberg chain, such as projection operators for subspaces with definite quasimomentum, total or partial spin, quasimomenta of strings etc. These techniques yield, in particular, matrices of appropriate Clebsch–Gordan coefficients, and reduce significantly the size of an effective Hamiltonian matrix. We apply the basis of wavelets to exploit the translational symmetry, along the Schwinger scheme of unitary geometry, applied to a single N‐dimensional magnon space (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Roberto Ruiz, Alejandro Sopena, Esperanza López +2 more

The Bethe Ansatz is a method for constructing exact eigenstates of quantum-integrable spin chains. Recently, deterministic quantum algorithms, referred to as “algebraic Bethe circuits”, have been developed to prepare Bethe states for the spin- 1/2 1 / 2 XXZ model. These circuits represent a unitary formulation of the standard algebraic Bethe Ansatz, expressed using matrix-product states that act on both the spin chain and an auxiliary space. In this work, we systematize these previous results, and show that algebraic Bethe circuits can be derived by a change of basis in the auxiliary space. The new basis, identical to the “F-basis” known from the theory of quantum-integrable models, generates the linear superposition of plane waves that is characteristic of the coordinate Bethe Ansatz. We explain this connection, highlighting that certain properties of the F-basis (namely, the exchange symmetry of the spins) are crucial for the construction of algebraic Bethe circuits. We demonstrate our approach by presenting new quantum circuits for the inhomogeneous spin- 1/2 1 / 2 XXZ model.