Fractal decompositions and tensor network representations of Bethe wavefunctions

Abstract

We investigate the entanglement structure of a generic $M$-particle Bethe wavefunction (not necessarily an eigenstate of an integrable model) on a 1d lattice by dividing the lattice into $L$ parts and decomposing the wavefunction into a sum of products of $L$ local wavefunctions. Using the fact that a Bethe wavefunction accepts a \textit{fractal} multipartite decomposition -- it can always be written as a linear combination of $L^M$ products of $L$ local wavefunctions, where each local wavefunction is in turn also a Bethe wavefunction -- we then build \textit{exact, analytical} tensor network representations with finite bond dimension $\chi =2^M$, for a generic planar tree tensor network (TTN), which includes a matrix product states (MPS) and a regular binary TTN as prominent particular cases. For a regular binary tree, the network has depth ${\log }_2(N/M)$ and can be transformed into an adaptive quantum circuit of the same depth, composed of unitary gates acting on $2^M$-dimensional qudits and mid-circuit measurements, that deterministically prepares the Bethe wavefunction. Finally, we put forward a much larger class of \textit{generalized} Bethe wavefunctions, for which the above decompositions, tensor network and quantum circuit representations are also possible.

v2: expanded introduction, more references and clarifications; v3: Added references and further clarifications; v4: minor clarification added

References in corpus (35)

• Entanglement in quantum critical phenomena
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Entanglement renormalization
• An Area Law for One Dimensional Quantum Systems
• A class of quantum many-body states that can be efficiently simulated
• Matrix product states represent ground states faithfully
• Classical simulation of quantum many-body systems with a tree tensor network
• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
• How Algebraic Bethe Ansatz works for integrable model
• Sequential generation of entangled multi-qubit states
• Remarks on the notion of quantum integrability
• The Schmidt Measure as a Tool for Quantifying Multi-Particle Entanglement
• An introduction to integrable techniques for one-dimensional quantum systems
• Measurement as a shortcut to long-range entangled quantum matter
• Preparation of matrix product states with log-depth quantum circuits
• Holographic quantum algorithms for simulating correlated spin systems
• Computation of static Heisenberg-chain correlators: Control over length and temperature dependence
• Sequential generation of projected entangled-pair states
• The Algebraic Bethe Ansatz and Tensor Networks
• Numerical Contraction of the Tensor Network generated by the Algebraic Bethe Ansatz
• Excited state correlations of the finite Heisenberg chain
• Asymptotic analysis and quantum integrable models
• Algebraic Bethe Circuits
• Preparing Bethe Ansatz Eigenstates on a Quantum Computer
• Generalization of the matrix product ansatz for integrable chains
• Derivation of Matrix Product Ansatz for the Heisenberg Chain from Algebraic Bethe Ansatz
• Preparing exact eigenstates of the open XXZ chain on a quantum computer
• Holographic quantum simulation of entanglement renormalization circuits
• The Bethe Ansatz as a Quantum Circuit
• Introduction to the nested algebraic Bethe ansatz
• Deterministic Bethe state preparation
• Probing critical states of matter on a digital quantum computer
• Bethe Ansatz, Quantum Circuits, and the F-basis
• Preparing Renormalization Group Fixed Points on NISQ Hardware

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