01920nas a2200121 4500008004100000245004400041210004300085260001200128520157400140100001901714700001601733856004901749 2017 eng d00aPhase retrieval using unitary 2-designs0 aPhase retrieval using unitary 2designs c2017/073 a

We consider a variant of the phase retrieval problem, where vectors are replaced by unitary matrices, i.e., the unknown signal is a unitary matrix U, and the measurements consist of squared inner products |tr(C†U)|2 with unitary matrices C that are chosen by the observer. This problem has applications to quantum process tomography, when the unknown process is a unitary operation. We show that PhaseLift, a convex programming algorithm for phase retrieval, can be adapted to this matrix setting, using measurements that are sampled from unitary 4- and 2-designs. In the case of unitary 4-design measurements, we show that PhaseLift can reconstruct all unitary matrices, using a nearoptimal number of measurements. This extends previous work on PhaseLift using spherical 4-designs. In the case of unitary 2-design measurements, we show that PhaseLift still works pretty well on average: it recovers almost all signals, up to a constant additive error, using a near-optimal number of measurements. These 2-design measurements are convenient for quantum process tomography, as they can be implemented via randomized benchmarking techniques. This is the first positive result on PhaseLift using 2-designs.

1 aKimmel, Shelby1 aLiu, Yi-Kai uhttp://ieeexplore.ieee.org/document/8024414/