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Quantum tomography
An important task in quantum physics is to fully learn a quantum state or quantum process from measurements. This is called quantum state or quantum process tomography, respectively.
Most quantum state or processes of interest are compressible: They can be described by matrices of low rank. Hence, low-rank matrix reconstruction from compressed sensing can be used to reduce the number of measurements.
Quantum states
Ongoing projects:
Self-certification, see here
Basis measurements
Quantum processes
The optimal scaling from quantum state tomography carries over to quantum process tomography:
Guaranteed recovery of quantum processes from few measurements
Measurements based on Clifford group operations in a randomized benchmarking setup:
Recovering quantum gates from few average gate fidelities
Ongoing projects:
Measurements based on Clifford group operations in for natural measurements, see arXiv:1701.03135
Self-certification, see here
Efficient algorithms