Spectral properties versus magic generation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>-doped random Clifford circuits

We study the emergence of complexity in deep random <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>N</a:mi></a:math>-qubit <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>T</b:mi></b:math>-gate-doped Clifford circuits, as reflected in their spectral properties and in magic generation, characterized by the stabilizer Rényi entropy distribution and the nonstabilizing power of the circuit. For pure (undoped) Clifford circuits, a unique periodic orbit structure in the space of Pauli strings implies peculiar spectral correlations and level statistics with large degeneracies. <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>T</c:mi></c:math>-gate doping induces an exponentially fast transition to chaotic behavior, described by random matrix theory. We compare these complexity indicators with magic generation properties of the Clifford <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mo>+</d:mo><d:mspace width="0.28em"/><d:mi>T</d:mi></d:mrow></d:math> ensemble, and determine the distribution of magic, as well as the average nonstabilizing power of the quantum circuit ensemble. In the dilute limit, <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:msub><f:mi>N</f:mi><f:mi>T</f:mi></f:msub><f:mo>≪</f:mo><f:mi>N</f:mi></f:mrow></f:math>, magic generation is governed by single-qubit behavior. Magic is generated in approximate quanta, increases approximately linearly with the number of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>T</g:mi></g:math> gates, <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:msub><h:mi>N</h:mi><h:mi>T</h:mi></h:msub></h:math>, and displays a discrete distribution for small <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:msub><i:mi>N</i:mi><i:mi>T</i:mi></i:msub></i:math>. At <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:msub><j:mi>N</j:mi><j:mi>T</j:mi></j:msub><j:mo>≈</j:mo><j:mi>N</j:mi></j:mrow></j:math>, the distribution becomes quasicontinuous, and for <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:msub><k:mi>N</k:mi><k:mi>T</k:mi></k:msub><k:mo>≫</k:mo><k:mi>N</k:mi></k:mrow></k:math>, it converges to that of Haar-random unitaries, and averages to a finite magic density, <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:msub><l:mi>m</l:mi><l:mn>2</l:mn></l:msub><l:mo>,</l:mo><l:mo> </l:mo><l:mrow><l:msub><l:mo movablelimits="true" form="prefix">lim</l:mo><l:mrow><l:mi>N</l:mi><l:mo>→</l:mo><l:mi>∞</l:mi></l:mrow></l:msub><l:msub><l:mrow><l:mo>〈</l:mo><l:msub><l:mi>m</l:mi><l:mn>2</l:mn></l:msub><l:mo>〉</l:mo></l:mrow><l:mtext>Haar</l:mtext></l:msub><l:mo>=</l:mo><l:mn>1</l:mn></l:mrow></l:math>. This is in contrast to the spectral transition, where <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mrow><o:mi mathvariant="script">O</o:mi><o:mo>(</o:mo><o:mn>1</o:mn><o:mo>)</o:mo></o:mrow><o:mo> </o:mo><o:mi>T</o:mi></o:math> gates suffice to remove spectral degeneracies and to induce a transition to chaotic behavior in the thermodynamic limit. Magic is therefore a more sensitive indicator of complexity.