T-count and T-depth of any multi-qubit unitary
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Bibliographic record
Abstract
Abstract We design an algorithm to determine the (minimum) T-count of any n -qubit ( n ≥ 1) unitary W of size 2 n × 2 n , over the Clifford+T gate set. The space and time complexity of our algorithm are $$O\left({2}^{2n}\right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> and $$O\left({2}^{2n{{{{\mathcal{T}}}}}_{\epsilon }(W)+4n}\right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:msub> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>W</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mn>4</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> , respectively. $${{{{\mathcal{T}}}}}_{\epsilon }(W)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>W</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> ( ϵ -T-count) is the (minimum) T-count of an exactly implementable unitary U ( $${{{\mathcal{T}}}}(U)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>U</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> ), such that d ( U , W ) ≤ ϵ and $${{{\mathcal{T}}}}(U)\le {{{\mathcal{T}}}}({U}^{{\prime} })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>U</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:mi>T</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>U</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>′</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> where $${U}^{{\prime} }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>U</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>′</mml:mo> </mml:mrow> </mml:msup> </mml:math> is any exactly implementable unitary with $$d({U}^{{\prime} },W)\le \epsilon$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>U</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>′</mml:mo> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>W</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:math> . d (. , .) is the global phase invariant distance. Our algorithm can also be used to determine the (minimum) T-depth as well as the minimum non-Clifford-gate count or depth required to implement any multi-qubit unitary with a finite universal gate set like Clifford+CS, Clifford+V, etc. For small enough ϵ , we can synthesize the optimal circuits.
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 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