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SUMMARY:The one-qudit Clifford hierarchy - Oscar Lautsch\, Simon Fraser Un
 iversity
DTSTART:20241122T123000Z
DTEND:20241122T133000Z
UID:TALK224614@talks.cam.ac.uk
CONTACT:Subhayan Roy Moulik
DESCRIPTION:The Clifford hierarchy defines those quantum gates which can b
 e performed fault-tolerantly via gate teleportation. Semi-Clifford gates a
 re those gates which can be performed using a more efficient protocol. Des
 pite their importance\, the structures of the Clifford hierarchy and of th
 e semi-Clifford subset remain poorly understood\, especially in the qudit 
 case.  While every gate in the one-qubit Clifford hierarchy was shown to b
 e semi-Clifford\, the analogous result for qudits was only known to hold f
 or third-level gates. \n    \nUsing the technique of conjugate pairs and t
 he action of the symplectic group on the Pauli basis\, we show that for an
 y odd prime dimension\, the single-qudit Clifford hierarchy coincides with
  the semi-Clifford subset. This allows us to give a normal form for all Cl
 ifford hierarchy gates\, expressing each one uniquely as product of three 
 simple gates\, as well as a precise count of the number of gates at each l
 evel of the hierarchy.  We thereby completely classify the single-qudit Cl
 ifford hierarchy. This is joint work with Nadish de Silva.
LOCATION:MR15
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