On the controlled-NOT complexity of controlled-NOT–phase circuits
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Bibliographic record
Abstract
Abstract We study the problem of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>CNOT</mml:mi> </mml:math> -optimal quantum circuit synthesis over gate sets consisting of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>CNOT</mml:mi> </mml:math> and Z -basis rotations of arbitrary angles. We show that the circuit-polynomial correspondence relates such circuits to Fourier expansions of pseudo-Boolean functions, and that for certain classes of functions this expansion uniquely determines the minimum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>CNOT</mml:mi> </mml:math> cost of an implementation. As a corollary we prove that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>CNOT</mml:mi> </mml:math> minimization over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>CNOT</mml:mi> </mml:math> and phase gates is at least as hard as synthesizing a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>CNOT</mml:mi> </mml:math> -optimal circuit computing a set of parities of its inputs. We then show that this problem is NP-complete for two restricted cases where all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>CNOT</mml:mi> </mml:math> gates are required to have the same target, and where the circuit inputs are encoded in a larger state space. The latter case has applications to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>CNOT</mml:mi> </mml:math> optimization over more general Clifford+ T circuits. We further present an efficient heuristic algorithm for synthesizing circuits over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>CNOT</mml:mi> </mml:math> and Z -basis rotations with small <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>CNOT</mml:mi> </mml:math> cost. Our experiments show a 23% reduction of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>CNOT</mml:mi> </mml:math> gates on average across a suite of Clifford+ T benchmark circuits, with a maximum reduction of 43%.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.007 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| 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