Matrix permanent and determinant from a spin system
Bibliographic record
Abstract
In contrast to the determinant, no algorithm is known for the exact determination of the permanent of a square matrix that runs in time polynomial in its dimension. Consequently, noninteracting fermions are classically efficiently simulatable while noninteracting bosons are not, underpinning quantum supremacy arguments for sampling the output distribution of photon interferometer arrays. This work introduces a graph-theoretic framework that bridges both the determinant and permanent. The only nonzero eigenvalues of a sparse non-Hermitian operator $\stackrel{\ifmmode \u\else \textasciibreve \fi{}}{M}$ for $n$ spin-$1/2$ particles are the $n\mathrm{th}$ roots of the permanent or determinant of an $n\ifmmode\times\else\texttimes\fi{}n$ matrix $M$, interpreting basis states as bosonic or fermionic occupation states, respectively. This operator can be used to design a simple and straightforward method for the classical determination of the permanent that matches the efficiency of the best-known algorithm. Gauss-Jordan elimination for the determinant of $M$ is then equivalent to the successive removal of the generalized zero eigenspace of the fermionic $\stackrel{\ifmmode \u\else \textasciibreve \fi{}}{M}$, equivalent to the deletion of some nodes and reweighting of the remaining edges in the graph such that only $n$ nodes survive after the last step. In the bosonic case, the successive removal of generalized zero eigenspaces for $\stackrel{\ifmmode \u\else \textasciibreve \fi{}}{M}$ is also equivalent to node deletion, but new edges are added during this process, which gives rise to the higher complexity of computing the permanent. Our analysis may point the way to new strategies for classical and quantum evaluation of the permanent.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.014 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".