2015/10/31 by Shelby Kimmel, Yi-Kai Liu · 13 citations
Computer Science · Engineering · Materials Science · Mathematics · Physics and Astronomy · #Advanced X-ray Imaging Techniques #Advancements in Photolithography Techniques #Circular ensemble #Electron and X-Ray Spectroscopy Techniques #Matrix (chemical analysis) #Orthogonal matrix #Quantum Fourier transform #Quantum phase estimation algorithm #Unitary group #Unitary matrix #Unitary state #Unitary transformation #cs.IT #math.IT #math.ST #quant-ph #stat.TH
paper · pdf · doi:10.1109/sampta.2017.8024414
published as International Conference on Sampling Theory and Applications (SampTA), July 3-7, 2017, Tallinn, Estonia, pp.345-349 · 21 pages; v3: minor revisions, to appear at SampTA 2017; v2: rewritten to focus on phase retrieval, with new title, improved error bounds, and numerics; v1: original version, titled "Quantum Compressed Sensing Using 2-Designs"
openalex created_date 2017/03/16 · arxiv created 2017/05/12 · openalex publication_date 2017/07/01 · arxiv updated 2018/03/07 · openalex updated_date 2026/08/05
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)|2with 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.