Block encodings of discrete subgroups on a quantum computer
Henry Lamm, Yingying Li, Jing Shu, Yi-Lin Wang, Bin Xu
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