Block encodings of discrete subgroups on a quantum computer

Abstract

We introduce a block encoding method for mapping discrete subgroups to qubits on a quantum computer. This method is applicable to general discrete groups, including crystal-like subgroups such as <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi mathvariant="double-struck">BI</a:mi></a:mrow></a:math> of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi>S</d:mi><d:mi>U</d:mi><d:mo stretchy="false">(</d:mo><d:mn>2</d:mn><d:mo stretchy="false">)</d:mo></d:math> and <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi mathvariant="double-struck">V</h:mi></h:math> of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>S</k:mi><k:mi>U</k:mi><k:mo stretchy="false">(</k:mo><k:mn>3</k:mn><k:mo stretchy="false">)</k:mo></k:math>. We detail the construction of primitive gates—the inversion gate, the group multiplication gate, the trace gate, and the group Fourier gate—utilizing this encoding method for <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mrow><o:mi mathvariant="double-struck">BT</o:mi></o:mrow></o:math> and for the first time <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mrow><r:mi mathvariant="double-struck">BI</r:mi></r:mrow></r:math> group. We also provide resource estimations to extract the gluon viscosity. The inversion gates for <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:mi mathvariant="double-struck">BT</u:mi></u:mrow></u:math> and <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mrow><x:mi mathvariant="double-struck">BI</x:mi></x:mrow></x:math> are benchmarked on the quantum computer with estimated fidelities of <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msubsup><ab:mn>40</ab:mn><ab:mrow><ab:mo>−</ab:mo><ab:mn>4</ab:mn></ab:mrow><ab:mrow><ab:mo>+</ab:mo><ab:mn>5</ab:mn></ab:mrow></ab:msubsup><ab:mo>%</ab:mo></ab:math> and <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msubsup><cb:mn>4</cb:mn><cb:mrow><cb:mo>−</cb:mo><cb:mn>3</cb:mn></cb:mrow><cb:mrow><cb:mo>+</cb:mo><cb:mn>5</cb:mn></cb:mrow></cb:msubsup><cb:mo>%</cb:mo></cb:math>, respectively. Published by the American Physical Society 2024