Phase Retrieval Using Unitary 2-Designs

Published

September 4, 2017

Author(s)

Yi-Kai Liu, Shelby Kimmel

Abstract

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 near-optimal 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.

Proceedings Title

International Conference on Sampling Theory and Applications (SampTA)

Conference Dates

July 3-7, 2017

Conference Location

Tallinn

Pub Type

Conferences

Download Paper

https://doi.org/10.1109/SAMPTA.2017.8024414

Keywords

Phase retrieval, quantum process tomography, unitary designs

Citation

Liu, Y. and Kimmel, S. (2017), Phase Retrieval Using Unitary 2-Designs, International Conference on Sampling Theory and Applications (SampTA), Tallinn, -1, [online], https://doi.org/10.1109/SAMPTA.2017.8024414 (Accessed October 10, 2025)