Nadish de Silva

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The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The groups of Pauli and Clifford gates constitute the first and second 'levels', respectively. Non-Clifford gates from the third level or higher, such as the T gate, are necessary for achieving fault-tolerant universal quantum computation.
Since it was defined twenty-five years ago by Gottesman-Chuang, two questions have been studied by numerous researchers. First, precisely which gates constitute the Clifford hierarchy? Second, which subset of the hierarchy gates admit efficient gate teleportation protocols?
We completely solve both questions in the case of the Clifford hierarchy for gates of one qubit or one qudit of prime dimension. We express every such hierarchy gate uniquely as a product of three simple gates, yielding also a formula for the size of every level. These results are a consequence of our finding that all such hierarchy gates can be expressed in a certain form that guarantees efficient gate teleportation. Our decomposition of Clifford gates as a unique product of three simple gates is of broader applicability.
We then consider the more complex case of two-qudit gates and focus on third-level gates: those that can be most easily implemented fault-tolerantly using magic states. We prove that every two-qudit third-level gate admits efficient gate teleportation. Our proof leverages tools of algebraic geometry which can be applied more widely within quantum information.

Joint work with: Imin Chen and Oscar Lautsch
Based on: https://arxiv.org/abs/2309.15184 and https://arxiv.org/abs/2501.07939

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Simon Fraser University, Vancouver
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Contact: Markus Heinrich