Quantum magic and multipartite entanglement in the structure of nuclei

Motivated by the Gottesman-Knill theorem, we present a detailed study of the quantum complexity of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>p</a:mi></a:math>-shell and <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>s</b:mi><b:mi>d</b:mi></b:mrow></b:math>-shell nuclei. Valence-space nuclear shell-model wave functions generated by the code are mapped to qubit registers using the Jordan-Wigner mapping (12 qubits for the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>p</c:mi></c:math> shell and 24 qubits for the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>s</d:mi><d:mi>d</d:mi></d:mrow></d:math> shell), from which measures of the many-body entanglement (<e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>n</e:mi></e:math>-tangles) and magic (nonstabilizerness) are determined. While exact evaluations of these measures are possible for nuclei with a modest number of active nucleons, Monte Carlo simulations are required for the more complex nuclei. The broadly applicable Pauli-string <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mover accent="true"><f:mi>I</f:mi><f:mo>̂</f:mo></f:mover><f:mover accent="true"><f:mi>Z</f:mi><f:mo>̂</f:mo></f:mover></f:mrow></f:math> exact (PSIZe) Markov chain Monte Carlo (MCMC) technique is introduced to accelerate the evaluation of measures of magic in deformed nuclei (with hierarchical wave functions), by factors of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mo>≈</i:mo><i:mn>8</i:mn></i:mrow></i:math> for some nuclei. Significant multinucleon entanglement is found in the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>s</j:mi><j:mi>d</j:mi></j:mrow></j:math> shell, dominated by proton-neutron configurations, along with significant measures of magic. This is evident not only for the deformed states, but also for nuclei on the path to instability via regions of shape coexistence and level inversion. These results indicate that quantum-computing resources will accelerate precision simulations of such nuclei and beyond.