Non-Clifford Gates between Stabilizer Codes via Non-Abelian Topological Order

We propose protocols to implement non-Clifford logical gates between stabilizer codes by entangling into a non-Abelian topological order as an intermediate step. Generalizing previous approaches, we provide a framework that generates a large class of non-Clifford and nondiagonal logical gates between qudit surface codes by gauging the topological symmetry of symmetry-enriched topological orders. As our main example, we concretely detail a protocol that utilizes the quantum double of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>S</a:mi> <a:mn>3</a:mn> </a:msub> </a:math> to generate a controlled-charge-conjugation ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>C</c:mi> <c:mrow> <c:mi mathvariant="script">C</c:mi> </c:mrow> </c:math> ) gate between a qubit and qutrit surface code. Both the preparation of non-Abelian states and logical state injection between the Abelian and non-Abelian codes are executed via finite-depth quantum circuits with measurement and feedforward. We discuss aspects of the fault tolerance of our protocol, presenting insights into how to construct a heralded decoder for the quantum double of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:msub> <f:mi>S</f:mi> <f:mn>3</f:mn> </f:msub> </f:math> . We also outline how analogous protocols can be used to obtain logical gates between qudit surface codes by entangling into <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mrow> <h:mi mathvariant="script">D</h:mi> </h:mrow> <h:mo stretchy="false">(</h:mo> <h:mi>G</h:mi> <h:mo stretchy="false">)</h:mo> <h:mo>,</h:mo> </h:math> where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>G</m:mi> </m:math> is a semidirect product of Abelian groups. This work serves as a step toward classifying the computational power of non-Abelian quantum phases beyond the paradigm of anyon braiding on near-term quantum devices.