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Fractal decompositions and tensor network representations of Bethe wavefunctions

Subhayan Sahu, Guifre Vidal

TL;DR

The paper reveals that Bethe wavefunctions on a 1D lattice possess fractal bipartite and multipartite decompositions that restrict their entanglement, leading to exact tensor-network representations with bond dimension $\chi=2^M$ for planar TTNs (including MPS and regular binary TTN). From these representations, an adaptive quantum circuit of depth $O(\log N)$ can deterministically prepare the Bethe state, providing a substantial depth advantage over prior MPS-based schemes. The authors extend these constructions to a broad class of generalized Bethe wavefunctions (GBW), showing that the same TTN and quantum-circuit frameworks apply, thereby offering compact, scalable tools for both classical simulation and quantum-state preparation of Bethe-type states. These results illuminate the entanglement-structure of Bethe states beyond integrable models and suggest practical pathways for efficient computation of norms, overlaps, and high-order correlations, as well as potential experimental realizations on quantum devices. The work also connects to recent literature on MPS representations and deterministic quantum circuits for Bethe states, highlighting a coherent framework that unifies exact tensor-network decompositions with short-depth quantum circuit implementations.

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 $χ=2^M$, 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.

Fractal decompositions and tensor network representations of Bethe wavefunctions

TL;DR

The paper reveals that Bethe wavefunctions on a 1D lattice possess fractal bipartite and multipartite decompositions that restrict their entanglement, leading to exact tensor-network representations with bond dimension for planar TTNs (including MPS and regular binary TTN). From these representations, an adaptive quantum circuit of depth can deterministically prepare the Bethe state, providing a substantial depth advantage over prior MPS-based schemes. The authors extend these constructions to a broad class of generalized Bethe wavefunctions (GBW), showing that the same TTN and quantum-circuit frameworks apply, thereby offering compact, scalable tools for both classical simulation and quantum-state preparation of Bethe-type states. These results illuminate the entanglement-structure of Bethe states beyond integrable models and suggest practical pathways for efficient computation of norms, overlaps, and high-order correlations, as well as potential experimental realizations on quantum devices. The work also connects to recent literature on MPS representations and deterministic quantum circuits for Bethe states, highlighting a coherent framework that unifies exact tensor-network decompositions with short-depth quantum circuit implementations.

Abstract

We investigate the entanglement structure of a generic -particle Bethe wavefunction (not necessarily an eigenstate of an integrable model) on a 1d lattice by dividing the lattice into parts and decomposing the wavefunction into a sum of products of 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 products of 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 , 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 and can be transformed into an adaptive quantum circuit of the same depth, composed of unitary gates acting on -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.

Paper Structure

This paper contains 30 sections, 9 theorems, 145 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Bethe wavefunctions
  3. Generic M-particle wavefunctions
  4. M-particle Bethe wavefunction
  5. Bethe data and choice vectors
  6. Fractal bipartite decomposition
  7. Bipartition with particle number conservation
  8. Factorization of Bethe wavefunction amplitudes
  9. Fractal bipartite decomposition
  10. Fractal multipartite decomposition
  11. Revamped notation for bipartite decomposition
  12. Fractal tripartite decomposition
  13. Fractal multipartite decomposition
  14. Contiguous multipartite decomposition
  15. Tensor Network representation
  16. ...and 15 more sections

Key Result

Theorem 3.1

Given a bipartition of the 1d lattice $\mathcal{L}$ into left and right parts $A$ and $B$, the $M$-particle Bethe wavefunction $\ket{\Phi_{M}}$ in Eq. eq:BW_def, with data $(\vec{k}, \boldsymbol{\theta})$, can be decomposed into a sum of products of two local Bethe wavefunctions $\ket{\Phi_{m}^{\vec

Figures (17)

  • Figure 1: The kinematic constraints in a Bethe wavefunction lead to an exact tree tensor network representation with bond dimension $\chi=2^M$ and a quantum circuit with depth $\log_2(N/M)$ (made of unitary gates on $2^M$-dimensional qudits and mid-circuit measurements) for deterministically preparing such states using a quantum computer.
  • Figure 2: (a) Hierarchy of Bethe wavefunctions: constrained (fulfilling integrability constraints), unconstrained (or simply Bethe wavefunctions; see definition \ref{['def:bw']}; not fulfilling integrability constraints) and generalized (see definition \ref{['def:GBW']}). (b) Our explicit tensor network representations for Bethe wavefunctions cover any planar tree tensor network (planar TTN), which includes the matrix product states (MPS) and the regular binary tree tensor network (regular binaty TTN) as particular cases. (c) In this work we propose fractal decompositions, as well as planar TTN representations (including MPS and regular binary TTN representations) for Bethe wavefunctions, then also for generalized Bethe wavefunctions, thus covering the 12 entries of this table. Several important particular cases had been previously covered in the literature, most notably the MPS representation of constrained Bethe wavefunctions in Refs. Alcaraz_2006Katsura_2010Murg_2012 and more recently of unconstrained Bethe wavefunctions in Refs. Ruiz_2024ruiz2024betheansatzquantumcircuits and a version of the generalized Bethe wavefunction in Ref. ruiz2024betheansatzquantumcircuits. As explained in a note added at the end of the paper, the fractal bipartite decomposition for constrained Bethe wavefunctions (in the algebraic Bethe wavefunction formalism) had been previously proposed in Ref. Korepin_Bogoliubov_Izergin_1993.
  • Figure 3: Example of permutation diagrams. All the permutations for $M =3$ symbols are described using strands and their crossings, and their corresponding scattering amplitudes are noted below.
  • Figure 4: A Bethe wavefunction for $M$ particles on lattice $\mathcal{L}$ (here for $M=3$) is characterized by (a) M quasi-momenta $\vec{k}= (k_1, k_2, \cdots, k_M)$ and (b) $M(M-1)/2$ scattering angles $\theta_{j_2j_1}$. From each quasi-momentum $k_j$ we generate a single-particle plain wave with amplitude $e^{ik_jx}$, and to each pair of plain waves we associate a scattering angle $\theta_{j_2j_1}$. (c) Set of choice vectors for an $M=3$ particle bethe wavefunction, as described in Sec. \ref{['sec:choice']}.
  • Figure 5: Types of multi-partitions we consider in this paper.
  • ...and 12 more figures

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1: (Fractal bipartite decomposition of a Bethe wavefunction
  • proof
  • Corollary 3.1
  • Theorem 4.1: Fractal multipartite decomposition of a Bethe wavefunction
  • proof
  • Corollary 4.1
  • Theorem 4.2: Contiguous multipartite decomposition of a Bethe wavefunction
  • Corollary 4.2
  • ...and 4 more