Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets
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Bibliographic record
Abstract
We generalize an efficient exact synthesis algorithm for single-qubit unitaries over the Clifford+T gate set which was presented by Kliuchnikov, Maslov, and Mosca [Quantum Inf. Comput. 13(7,8), 607–630 (2013)]. Their algorithm takes as input an exactly synthesizable single-qubit unitary—one which can be expressed without error as a product of Clifford and T gates—and outputs a sequence of gates which implements it. The algorithm is optimal in the sense that the length of the sequence, measured by the number of T gates, is smallest possible. In this paper, for each positive even integer n, we consider the “Clifford-cyclotomic” gate set consisting of the Clifford group plus a z-rotation by πn. We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of πn z-rotations. For the Clifford+T case n = 4, the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring Z[eiπn,1/2]. We prove that this characterization holds for a handful of other small values of n but the fraction of positive even integers for which it fails to hold is 100%.
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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 | 0.001 | 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 | 0.001 | 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 it