Synthesis and Arithmetic of Single Qutrit Circuits
Bibliographic record
Abstract
In this paper we study single qutrit circuits consisting of words over the Clifford<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>+</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi></mml:mrow></mml:math> cyclotomic gate set, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mtext>diag</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mo>&#x00B1;</mml:mo><mml:msup><mml:mi>&#x03BE;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>a</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>&#x00B1;</mml:mo><mml:msup><mml:mi>&#x03BE;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>b</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>&#x00B1;</mml:mo><mml:msup><mml:mi>&#x03BE;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03BE;</mml:mi></mml:math> is a primitive <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>9</mml:mn></mml:math>-th root of unity and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi></mml:math> are integers. We characterize classes of qutrit unit vectors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi></mml:math> with entries in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>&#x03BE;</mml:mi><mml:mo>,</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>&#x03C7;</mml:mi></mml:mfrac><mml:mo stretchy="false">]</mml:mo></mml:math> based on the possibility of reducing their smallest denominator exponent (sde) with respect to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03C7;</mml:mi><mml:mo>:=</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2013;</mml:mo><mml:mi>&#x03BE;</mml:mi><mml:mo>,</mml:mo></mml:math> by acting an appropriate gate in Clifford<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>+</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi></mml:mrow></mml:math>. We do this by studying the notion of `derivatives mod <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn></mml:math>' of an arbitrary element of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>&#x03BE;</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:math> and using it to study the smallest denominator exponent of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi></mml:mrow><mml:mi>z</mml:mi></mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math> is the qutrit Hadamard gate and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi></mml:mrow></mml:math>. In addition, we reduce the problem of finding all unit vectors of a given sde to that of finding integral solutions of a positive definite quadratic form along with some additional constraints. As a consequence we prove that the Clifford<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>+</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi></mml:mrow></mml:math> gates naturally arise as gates with sde <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn></mml:math> in the group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>&#x03BE;</mml:mi><mml:mo>,</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>&#x03C7;</mml:mi></mml:mfrac><mml:mo stretchy="false">]</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn><mml:mo>&#x00D7;</mml:mo><mml:mn>3</mml:mn></mml:math> unitaries with entries in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>&#x03BE;</mml:mi><mml:mo>,</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>&#x03C7;</mml:mi></mml:mfrac><mml:mo stretchy="false">]</mml:mo></mml:math>. We illustrate the general applicability of these methods to obtain an exact synthesis algorithm for Clifford<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>+</mml:mo><mml:mi>R</mml:mi></mml:math> and recover the previously known exact synthesis algorithm of Kliuchnikov, Maslov, Mosca (2012). The framework developed to formulate qutrit gate synthesis for Clifford<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>+</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi></mml:mrow></mml:math> extends to qudits of arbitrary prime power.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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 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".