Primitive quantum gates for an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> discrete subgroup: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>36</mml:mn> <mml:mo>×</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math>
Erik Gustafson, Yao Ji, Henry Lamm, Edison M. Murairi, Sebastian Osorio Perez, Shuchen Zhu
Abstract
We construct the primitive gate set for the digital quantum simulation of the 108-element <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="normal">Σ</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>36</a:mn> <a:mo>×</a:mo> <a:mn>3</a:mn> <a:mo stretchy="false">)</a:mo> </a:math> group. This is the first time a non-Abelian crystal-like subgroup of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>S</f:mi> <f:mi>U</f:mi> <f:mo stretchy="false">(</f:mo> <f:mn>3</f:mn> <f:mo stretchy="false">)</f:mo> </f:math> has been constructed for quantum simulation. The gauge link registers and necessary primitives—the inversion gate, the group multiplication gate, the trace gate, and the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mi mathvariant="normal">Σ</j:mi> <j:mo stretchy="false">(</j:mo> <j:mn>36</j:mn> <j:mo>×</j:mo> <j:mn>3</j:mn> <j:mo stretchy="false">)</j:mo> </j:math> Fourier transform—are presented for both an eight-qubit encoding and a heterogeneous three-qutrit plus two-qubit register. For the latter, a specialized compiler was developed for decomposing arbitrary unitaries onto this architecture. Published by the American Physical Society 2024