Classical and Quantum Algorithms for Characters of the Symmetric Group
Bibliographic record
Abstract
Characters of irreducible representations are ubiquitous in group theory. However, computing characters of some groups such as the symmetric group <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>S</a:mi> <a:mi>n</a:mi> </a:msub> </a:math> is a challenging problem known to be #P-hard in the worst case. Here we describe a matrix product state (MPS) algorithm for characters of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>S</c:mi> <c:mi>n</c:mi> </c:msub> </c:math> . The algorithm computes an MPS encoding all irreducible characters of a given permutation. It relies on a mapping from characters of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>S</e:mi> <e:mi>n</e:mi> </e:msub> </e:math> to quantum spin chains proposed by Crichigno and Prakash. We also provide a simpler derivation of this mapping. We complement this result by presenting a <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>poly</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>n</g:mi> <g:mo stretchy="false">)</g:mo> </g:math> size quantum circuit that prepares the corresponding MPS obtaining an efficient quantum algorithm for certain sampling problems based on characters of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mi>S</k:mi> <k:mi>n</k:mi> </k:msub> </k:math> . To assess classical hardness of these problems, we present a general reduction from strong simulation (computing a given probability) to weak simulation (sampling with a small error). This reduction applies to any sampling problem with a certain granularity structure and may be of independent interest.
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How this classification was reachedexpand
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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".