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arXiv:1510.08887 · International Conference on Sampling Theory and Applications

Phase retrieval using unitary 2-designs

Shelby Kimmel, Yi-Kai Liu

Published Jul 1, 2017• 18 citations
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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)|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.

DOI
10.1109/sampta.2017.8024414

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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.
We develop a method for quantum process tomography that combines the efficiency of compressed sensing with the robustness of randomized benchmarking. Our method is robust to state preparation and measurement errors, and it achieves a quadratic speedup over conventional tomography when the unknown process is a generic unitary evolution. Our method is based on PhaseLift, a convex programming technique for phase retrieval. We show that this method achieves approximate recovery of almost all signals, using measurements sampled from spherical or unitary 2-designs. This is the first positive result on PhaseLift using 2-designs. We also show that exact recovery of all signals is possible using unitary 4-designs. Previous positive results for PhaseLift required spherical 4-designs, while PhaseLift was known to fail in certain cases when using spherical 2-designs.
We study the recovery of low-rank Hermitian matrices from rank-one measurements obtained by uniform sampling from complex projective 3-designs, using nuclear-norm minimization. This framework includes phase retrieval as a special case via the PhaseLift method. In general, complex projective ttt-designs provide a practical means of partially derandomizing Gaussian measurement models. While near-optimal recovery guarantees are known for 444-designs, and it is known that 222-designs do not permit recovery with a subquadratic number of measurements, the case of 333-designs has remained open. In this work, we close this gap by establishing recovery guarantees for (exact and approximate) 333-designs that parallel the best-known results for 444-designs. In particular, we derive bounds on the number of measurements sufficient for stable and robust low-rank recovery via nuclear-norm minimization. Our results are especially relevant in practice, as explicit constructions of 444-designs are significantly more challenging than those of 333-designs.
Linear-optical circuits are elementary building blocks for classical and quantum information processing with light. In particular, due to its monolithic structure, integrated photonics offers great phase-stability and can rely on the large scale manufacturability provided by the semiconductor industry. New devices, based on such optical circuits, hold the promise of faster and energy-efficient computations in machine learning applications and even implementing quantum algorithms intractable for classical computers. However, this technological revolution requires accurate and scalable certification protocols for devices that can be comprised of thousands of optical modes. Here, we present a novel technique to reconstruct the transfer matrix of linear optical networks that is based on the recent advances in low-rank matrix recovery and convex optimisation problems known as PhaseLift algorithms. Conveniently, our characterisation protocol can be performed with a coherent classical light source and photodiodes. We prove that this method is robust to noise and scales efficiently with the number of modes. We experimentally tested the proposed characterisation protocol on a programmable integrated interferometer designed for quantum information processing. We compared the transfer matrix reconstruction obtained with our method against the one provided by a more demanding reconstruction scheme based on two-photon quantum interference. For 5-dimensional random unitaries, the average circuit fidelity between the matrices obtained from the two reconstructions is 0.993.
We show that for any positive integer nnn, the maps x∈Cn↦{∣⟨x,zi⟩∣2}i=14n∈R4nx \in \mathbb{C}^n \mapsto \{\left|\langle x, z_i \rangle \right|^2\}_{i=1}^{4n} \in \mathbb{R}^{4n}x∈Cn↦{∣⟨x,zi​⟩∣2}i=14n​∈R4n, where ziz_izi​ are the columns of four n×nn\times nn×n unitary matrices, are generically injective modulo multiplication by a global phase factor, yielding a family of embeddings of CPn−1\mathbb{C}P^{n-1}CPn−1 into R4n−4\mathbb{R}^{4n-4}R4n−4. In particular, this implies that distribution measurements about a pure state with four generic full-rank observables are informationally complete, which is sharp for n≥6n \geq 6n≥6. To complement this information-theoretic study, we establish in a companion paper that the PhaseLift algorithm yields efficient phase retrieval from quadratic measurements with O(1)O(1)O(1) unitary matrices, with high probability, where the unitaries are iid according to Haar measure.
In this article we propose a dynamic approach to complex vector reconstruction in the context of quantum tomography. There are two underlying assumptions behind our reasoning. The first one claims that the evolution of a d-level pure quantum system is given by the Schrödinger equation with a time-independent Hamiltonian and the other states that the knowledge about the quantum state is provided from projective measurements, also called intensity measurements. The problem of quantum state reconstruction is connected with the notion known as phase retrieval -- recovering a complex vector from modulus of inner product with frame vectors. Phase retrieval is widely studied in many areas of science but still there is a number of problems that remain to be answered. We believe that the dynamic approach can significantly improve the effectiveness of the vector reconstruction as it aims to decrease the number of distinct projectors by taking advantage of the knowledge about the evolution. General conditions and observations are applied to a specific unitary evolution model.
Multi-photon system has been studied by many groups. The challenge facing by most groups is the limited samples are far from enough for detecting quantum entanglement. The problem is even more serious for the quantum state tomography. Adaptive quantum state tomography is one possible way to solve the problem. We develop a distribution scheme of samples and extend it into three step to improve the performance of adaptive quantum state tomography. Mean Square Error of reconstructed density matrix is improved for several states being tested numerically. In addition, Phaselift is applied to save the storage space of measurement operator.