Phase Retrieval Using Unitary 2-Designs

TL;DR

This paper extends phase retrieval to the matrix setting of unitary matrices, demonstrating that PhaseLift can effectively reconstruct unitaries using measurements from unitary 4- and 2-designs, with implications for quantum process tomography.

Contribution

It adapts PhaseLift to the unitary matrix setting and proves its effectiveness with measurements from unitary 4- and 2-designs, including the first positive results with 2-designs.

Findings

PhaseLift can reconstruct all unitaries with 4-design measurements.

PhaseLift recovers almost all signals with 2-design measurements, up to a small error.

Measurements from 2-designs are practical for quantum process tomography.

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.