Abstract

I will sketch two recent results to do with i) stabiliser state geometry and ii) efficient gate teleporatation of qudit gates.

i) Arbitrary quantum states admit many stabiliser decompositions: ways of being expressed as asuperposition of stabiliser states. Understanding of the structure of stabiliser decompositions has applications in simulating and verifying near-term quantum computers. We introduce and study the vector space of linear dependencies of n-qubit stabiliser states. These spaces have canonical bases containing vectors whose size grows exponentially in n. We construct elegant bases of dependencies of constant size three.

ii) Third-level gates are those that can be most easily implemented fault-tolerantly using magic states. Those that are semi-Clifford can be implemented with far more efficient use of resources. We prove that every third-level gate of up to two qudits is semi-Clifford. Our proof introduces techniques from algebraic geometry which can be applied more widely within quantum information.