Efficient Clifford+T approximation of single-qubit operators
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Bibliographic record
Abstract
We give an efficient randomized algorithm for approximating an arbitrary element of $\SU(2)$ by a product of Clifford+$T$ operators, up to any given error threshold $\epsilon>0$. Under a mild hypothesis on the distribution of primes, the algorithm's expected runtime is polynomial in $\log(1/\epsilon)$. If the operator to be approximated is a $z$-rotation, the resulting gate sequence has $T$-count $K+4\log_2(1/\epsilon)$, where $K$ is approximately equal to $10$. We also prove a worst-case lower bound of $K+4\log_2(1/\epsilon)$, where $K=-9$, so that our algorithm is within an additive constant of optimal for certain $z$-rotations. For an arbitrary member of $\SU(2)$, we achieve approximations with $T$-count $K+12\log_2(1/\epsilon)$. By contrast, the Solovay-Kitaev algorithm achieves $T$-count $O(\log^c(1/\epsilon))$, where $c$ is approximately $3.97$.
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 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