Fractal decompositions and tensor network representations of Bethe wavefunctions
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Bibliographic record
Abstract
We investigate the entanglement structure of a generic M <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>M</mml:mi> </mml:math> -particle Bethe wavefunction (not necessarily an eigenstate of an integrable model) on a 1d lattice by dividing the lattice into L <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>L</mml:mi> </mml:math> parts and decomposing the wavefunction into a sum of products of L <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>L</mml:mi> </mml:math> local wavefunctions. Using the fact that a Bethe wavefunction accepts a fractal multipartite decomposition – it can always be written as a linear combination of L^M <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>M</mml:mi> </mml:msup> </mml:math> products of L <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>L</mml:mi> </mml:math> local wavefunctions, where each local wavefunction is in turn also a Bethe wavefunction – we then build exact, analytical tensor network representations with finite bond dimension \chi=2^M <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>χ</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>M</mml:mi> </mml:msup> </mml:mrow> </mml:math> , 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) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mo>log</mml:mo> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mi>/</mml:mi> <mml:mi>M</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> and can be transformed into an adaptive quantum circuit of the same depth, composed of unitary gates acting on 2^M <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>M</mml:mi> </mml:msup> </mml:math> -dimensional qudits and mid-circuit measurements, that deterministically prepares the Bethe wavefunction. Finally, we put forward a much larger class of generalized Bethe wavefunctions, for which the above decompositions, tensor network and quantum circuit representations are also possible.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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