CS-count-optimal quantum circuits for arbitrary multi-qubit unitaries
Bibliographic record
Abstract
Abstract In quantum computing there are quite a few universal gate sets, each having their own characteristics. In this paper we study the Clifford+CS universal fault-tolerant gate set. The CS gate is used is many applications and this gate set is an important alternative to Clifford+T. We introduce a generating set in order to represent any unitary implementable by this gate set and with this we derive a bound on the CS-count of arbitrary multi-qubit unitaries. Analysing the channel representation of the generating set elements, we infer $${\mathcal {J}}_n^{CS}\subset {\mathcal {J}}_n^T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>J</mml:mi> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>CS</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>⊂</mml:mo> <mml:msubsup> <mml:mi>J</mml:mi> <mml:mi>n</mml:mi> <mml:mi>T</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> , where $${\mathcal {J}}_n^{CS}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>J</mml:mi> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>CS</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> and $${\mathcal {J}}_n^T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>J</mml:mi> <mml:mi>n</mml:mi> <mml:mi>T</mml:mi> </mml:msubsup> </mml:math> are the set of unitaries exactly implementable by the Clifford+CS and Clifford+T gate sets, respectively. We develop CS-count optimal synthesis algorithms for both approximately and exactly implementable multi-qubit unitaries. With the help of these we derive a CS-count-optimal circuit for Toffoli, implying $${\mathcal {J}}_n^{Tof}={\mathcal {J}}_n^{CS}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>J</mml:mi> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>Tof</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>J</mml:mi> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>CS</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> , where $${\mathcal {J}}_n^{Tof}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>J</mml:mi> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>Tof</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> is the set of unitaries exactly implementable by the Clifford+Toffoli gate set. Such conclusions can have an important impact on resource estimates of quantum algorithms.
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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 | 0.002 | 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 | 0.001 | 0.000 |
| Scholarly communication | 0.003 | 0.001 |
| 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".