Implementation of single-qubit measurement-based t-designs using IBM
  processors
Random unitary matrices sampled from the uniform Haar ensemble have a number
of important applications both in cryptography and in the simulation of a
variety of fundamental physical systems. Since the Haar ensemble is very
expensive to sample, pseudorandom ensembles in the form of t-designs are
frequently used as an efficient substitute, and are sufficient for most
applications. We investigate t-designs generated using a measurement-based
approach on superconducting quantum computers. In particular, we implemented an
exact single-qubit 3-design on IBM quantum processors by performing
measurements on a 6-qubit graph state. By analysing channel tomography results,
we were able to show that the ensemble of unitaries realised was a 1-design,
but not a 2-design or a 3-design under the test conditions set, which we show
to be a result of depolarising noise during the measurement-based process. We
obtained improved results for the 2-design test by implementing an approximate
2-design, in which measurements were performed on a smaller 5-qubit graph
state, but the test still did not pass for all states. This suggests that the
practical realisation of measurement-based t-designs on superconducting quantum
computers will require further work on the reduction of depolarising noise in
these devices.
Conrad Strydom Mark Tame