Global symmetries in tensor network states: Symmetric tensors versus minimal bond dimension
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Bibliographic record
Abstract
Tensor networks offer a variational formalism to efficiently represent wave functions of extended quantum many-body systems on a lattice. In a tensor network $\mathcal{N}$, the dimension $\ensuremath{\chi}$ of the bond indices that connect its tensors controls the number of variational parameters and associated computational costs. In the absence of any symmetry, the minimal bond dimension ${\ensuremath{\chi}}^{\mathrm{min}}$ required to represent a given many-body wave function $|\ensuremath{\Psi}\ensuremath{\rangle}$ leads to the most compact, computationally efficient tensor network description of $|\ensuremath{\Psi}\ensuremath{\rangle}$. In the presence of a global, on-site symmetry, one can use a tensor network ${\mathcal{N}}_{\mathrm{sym}}$ made of symmetric tensors. Symmetric tensors allow one to exactly preserve the symmetry and to target specific quantum numbers, while their sparse structure leads to a compact description and lowers computational costs. In this paper we explore the trade-off between using a tensor network $\mathcal{N}$ with minimal bond dimension ${\ensuremath{\chi}}^{\mathrm{min}}$ and a tensor network ${\mathcal{N}}_{\mathrm{sym}}$ made of symmetric tensors, where the minimal bond dimension ${\ensuremath{\chi}}_{\mathrm{sym}}^{\mathrm{min}}$ might be larger than ${\ensuremath{\chi}}^{\mathrm{min}}$. We present two technical results. First, we show that in a tree tensor network, which is the most general tensor network without loops, the minimal bond dimension can always be achieved with symmetric tensors, so that ${\ensuremath{\chi}}_{\mathrm{sym}}^{\mathrm{min}}={\ensuremath{\chi}}^{\mathrm{min}}$. Second, we provide explicit examples of tensor networks with loops where replacing tensors with symmetric ones necessarily increases the bond dimension, so that ${\ensuremath{\chi}}_{\mathrm{sym}}^{\mathrm{min}}>{\ensuremath{\chi}}^{\mathrm{min}}$. We further argue, however, that in some situations there are important conceptual reasons to prefer a tensor network representation with symmetric tensors (and possibly larger bond dimension) over one with minimal bond dimension.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
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 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it