Bethe Ansatz, Quantum Circuits, and the F-basis

Abstract

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 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 XXZ model.

v1: 38 pages, 18 figures, 3 tables. v2: 39 pages, 17 figures, 3 tables. Includes corrections and improvements. v3: 39 pages, 17 figures, 3 tables. Matches the published version

References in corpus (44)

• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Real time confinement following a quantum quench to a non-integrable model
• Solving strongly correlated electron models on a quantum computer
• Remarks on the notion of quantum integrability
• Integrable Trotterization: Local Conservation Laws and Boundary Driving
• Correlation functions of the open XXZ chain I
• New Construction of Eigenstates and Separation of Variables for SU(N) Quantum Spin Chains
• Algebraic Bethe Ansatz for a discrete-state BCS pairing model
• Exact solutions of exactly integrable quantum chains by a matrix product ansatz
• Introduction to the thermodynamic Bethe ansatz
• The Bethe ansatz as a matrix product ansatz
• Numerical Contraction of the Tensor Network generated by the Algebraic Bethe Ansatz
• The Algebraic Bethe Ansatz and Tensor Networks
• Integrable models for confined fermions: applications to metallic grains
• Algebraic Bethe Circuits
• The BCS model and the off shell Bethe ansatz for vertex models
• The Staggered Six-Vertex Model: Conformal Invariance and Corrections to Scaling
• Generalization of the matrix product ansatz for integrable chains
• Derivation of Matrix Product Ansatz for the Heisenberg Chain from Algebraic Bethe Ansatz
• On Bethe vectors in ${\mathfrak{gl}}_3$-invariant integrable models
• Preparing exact eigenstates of the open XXZ chain on a quantum computer
• Resolution of the Nested Hierarchy for Rational sl(n) Models
• Bethe states on a quantum computer: success probability and correlation functions
• Drinfeld twists and algebraic Bethe ansatz of the supersymmetric model associated with $U_q(gl(m|n))$
• Drinfeld Twists and Algebraic Bethe Ansatz of the Supersymmetric t-J Model
• Continuous Matrix Product Ansatz for the One-Dimensional Bose Gas with Point Interaction
• Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries
• The Bethe Ansatz as a Quantum Circuit
• Deterministic Bethe state preparation
• Drinfeld Twists and Symmetric Bethe Vectors of Supersymmetric Fermion Models
• Coordinate space wave function from the Algebraic Bethe Ansatz for the inhomogeneous six-vertex model
• Estimating Bethe roots with VQE
• The propagator of the finite XXZ spin-$\frac{1}{2}$ chain
• Quantum encoder for fixed Hamming-weight subspaces
• Factorizing $F$-matrices and the XXZ spin-1/2 chain: A diagrammatic perspective
• Factorizing twists and the universal R-matrix of the Yangian Y(sl_2)
• Construction of Monodromy Matrix in the F- basis and Scalar products in Spin Chains
• The monodromy matrix in the F-basis for arbitrary six-vertex models
• A convenient basis for the Izergin-Korepin model
• Factorizing twists and R-matrices for representations of the quantum affine algebra U_q(\hat sl_2)
• On factorizing $F$-matrices in $Y(s{l}_n)$ and $U_q(\widehat{s{l}_n})$ spin chains
• Computation of partition functions of free fermionic solvable lattice models via permutation graphs
• Quantum Spin Chains and Symmetric Functions
• Efficient Eigenstate Preparation in an Integrable Model with Hilbert Space Fragmentation

Cited by in corpus (5)

• Symmetric quantum states: a review of recent progress
• Fractal decompositions and tensor network representations of Bethe wavefunctions
• Effective Bethe Ansatz for Spin-1 Non-integrable Models
• Spin-$s$ $U(1)$-eigenstate preparation
• Preparation of the single-spinon wave function on a quantum computer