With a Few Square Roots, Quantum Computing Is as Easy as Pi
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Bibliographic record
Abstract
Rig groupoids provide a semantic model of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo>Π</mml:mo> </mml:math> , a universal classical reversible programming language over finite types. We prove that extending rig groupoids with just two maps and three equations about them results in a model of quantum computing that is computationally universal and equationally sound and complete for a variety of gate sets. The first map corresponds to an 8th root of the identity morphism on the unit 1. The second map corresponds to a square root of the symmetry on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . As square roots are generally not unique and can sometimes even be trivial, the maps are constrained to satisfy a nondegeneracy axiom, which we relate to the Euler decomposition of the Hadamard gate. The semantic construction is turned into an extension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo>Π</mml:mo> </mml:math> , called <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msqrt> <mml:mo>Π</mml:mo> </mml:msqrt> </mml:math> , that is a computationally universal quantum programming language equipped with an equational theory that is sound and complete with respect to the Clifford gate set, the standard gate set of Clifford+T restricted to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>≤</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> qubits, and the computationally universal Gaussian Clifford+T gate set.
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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 it