Fractal decompositions and tensor network representations of Bethe wavefunctions
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arXiv
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| Format: | Preprint |
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2024
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| author | Sahu, Subhayan Vidal, Guifre |
| author_facet | Sahu, Subhayan Vidal, Guifre |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00923 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractal decompositions and tensor network representations of Bethe wavefunctions Sahu, Subhayan Vidal, Guifre Quantum Physics Statistical Mechanics Strongly Correlated Electrons Exactly Solvable and Integrable Systems 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. |
| title | Fractal decompositions and tensor network representations of Bethe wavefunctions |
| topic | Quantum Physics Statistical Mechanics Strongly Correlated Electrons Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2412.00923 |