ArticleReconstruction of quantum channel via convex optimization
Graphical abstract
Introduction
Quantum technologies have obtained great advances in recent years, including quantum computation [1], [2], [3] and quantum simulation [4], as well as quantum communication [5], [6], [7]. Either in quantum computation or in quantum communication, a primary task is to give a mathematical characterization of the physical system. Generally, there are two difficulties for this mission, that is, decoherence of quantum states and loss of information in states manipulation. In order to delve deeper into the evolution mechanism of the system, a reveal of the real quantum process is of paramount importance.
In the realm of quantum information science [1], such kind of quantum system characterization is commonly known as quantum tomography, like quantum detector tomography (QDT) [8], [9], [10], quantum state tomography (QST) [11], [12] and quantum process tomography (QPT) [13]. For the process estimation, QST and QPT are two mainly used techniques. QST is an indispensable method in QPT and the information that lies in the process can be transformed into a mathematical mapping, that is the process matrix, between the input and output sides in QPT. These techniques are widely employed in state estimation [14], [15], [16], [17], [18], [19], process reconstruction [20], [21], [22], [23], [24], [25], [26], [27], [28] and Hamiltonian estimation [29], [30] of various systems.
The procedure of QPT requires well preparation of input states, control over the interaction of qubits and following the projection measurement of output states. A systematical resource analysis [13] is investigated using different kinds of tomography ways, like standard quantum process tomography, ancilla-assisted quantum process tomography, etc. Through these methods, we can rebuild the quantum process of systems for different quantum tasks, for example, quantum communication channels, giving it a complete characterization.
Quantum communication, which harnesses the non-cloning theorem, is regarded as an approach for unconditional communication security. Experimental efforts have been made in free-space [5], fiber [6] and underwater [7] channels. The key rate and the security rely on the error in channels, which can be caused by decoherence and/or man-made eavesdropping, thus we need to have a full knowledge of the channel. However, standard reconstruction of the channel will lead to an unphysical matrix (Fig. 1a) [1], [11], like non-completely-positive and trace-increasing, which means the inversion is not the optimal fitting of the raw tomography result. If the reconstruction does not lie in a correct and physical region, it will result in a misjudgement on the channels. What’s worse, in terms of multipartite communication scheme which requires a participation of many channels, such kind of errors will accumulate exponentially. Even for a standard point-to-point quantum communication channel (Fig. 2b), we find that the unphysical issue is nonnegligible, which forces us to derive an approach to reconstruct the actual channel in a physical fashion.
We resort to some modern-information-processing technologies, like convex optimization [31], one of the kernels in machine learning whose task is to find the parameters in the model that best fits the prior information. The general optimization in machine learning will lead to a local optimum while the convex optimization generates a global optimum. By applying the convex optimization, we will be able to extract the full and correct information from our measurement result. In addition, the combination of convex optimization and quantum information may arouse some interesting applications such as neural-network quantum state tomography [32] and quantum state classifier [33].
In this article, we first reconstruct our seawater channel via standard inversion and find it unphysical. Then we propose the convex-optimization-based QPT by formulating the reconstruction as a least square optimization problem [31]. Solving the optimization problem of a convex function leads to the global yet physical optima, which means we can accurately determine the true action of the seawater channel. Further, we test the advantages of our method on reconstructing the seven fundamental gates. We compare our method to the standard reconstruction [1] and the norm optimization [25], [34] via cost function values and state deviations, convincing that our method is able to estimate the elements of the process matrices more precisely with less demand on preliminary resources. At last, we discuss the constraints on the process and measure them on a set of non-unitary channels, the accuracy of which reaches up to , showing that our method can be applied to a general quantum process including unitary and non-unitary ones.
Section snippets
Convex-optimization quantum process tomography
The purpose of QPT [1], [13] is to determine the unknown process , which can be expressed in the operator-sum representation, such that, , where are the operator elements of the process operation . During the procedure of QPT, it needs to prepare the quantum system in a set of quantum states and subject them respectively to the channel we need to characterize. After that, QST is performed to measure the output states and then the operation is fully characterized by a
Experiments and results
As is shown in Fig. 1b, we reconstruct a seawater channel of 3.3 meters by applying standard inversion QPT [1] which was also used in Ref. [7]. By using a blue-violet coherent light at 405 nm, the needed four input states in single-photon level are prepared with the polarization beam splitter, the half-wave plate and the quarter-wave plate. Before mapping into the channel, we do a state tomography to determine the input state, which will be used in the convex optimization procedure rather than
Conclusion
In summary, we reconstruct the true seawater channel whose action comprises different operations by using convex-optimization QPT. Further, we test our method on these different operations, i.e., the seven fundamental gates. We employ the cost function value and state deviation, to compare our method with the standard inversion and norm optimization. As denoted by these metrics, our method outperforms previous methods in terms of precision and robustness. Previous methods also require more
Conflict of interest
The authors declare that they have no conflict of interest.
Acknowledgments
This work was supported by the National Key R&D Program of China (2019YFA0308700, 2017YFA0303700), the National Natural Science Foundation of China (61734005, 11761141014, and 11690033), the Science and Technology Commission of Shanghai Municipality (15QA1402200, 16JC1400405, and 17JC1400403), and the Shanghai Municipal Education Commission (16SG09 and 2017-01-07-00-02-E00049). X.-M.J. acknowledges additional support from a Shanghai Talent Program. The authors thank Jian-Wei Pan for helpful
Author contributions
Xian-Min Jin conceived and supervised the project. Xuan-Lun Huang and Xian-Min Jin designed the experiment. Xuan-Lun Huang, Zhi-Qiang Jiao, Zeng-Quan Yan, Zhe-Yong Zhang, Dan-Yang Chen, Xi Zhang and Ling Ji performed the experiment. Xuan-Lun Huang and Xian-Min Jin analysed the data and wrote the paper.
Xuan-Lun Huang is a master student at the School of Physics and Astronomy, Shanghai Jiao Tong University. He received his bachelor degree in South China University of Technology. His research interests include quantum tomography, Anderson localization and artificial intelligence.
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Xuan-Lun Huang is a master student at the School of Physics and Astronomy, Shanghai Jiao Tong University. He received his bachelor degree in South China University of Technology. His research interests include quantum tomography, Anderson localization and artificial intelligence.
Xian-Min Jin is currently a professor at the School of Physics and Astronomy, Shanghai Jiao Tong University (SJTU). He received his Ph.D. degree at University of Science and Technology of China (USTC) in 2008. After two-year postdoctoral research in Hefei National Laboratory for Physical Sciences at the Microscale, he joined the Clarendon Laboratory and University of Oxford as a research associate, and became Marie Curie Fellow in 2012. He started establishing the Laboratory of Integrated Photonics and Quantum Information in SJTU in 2012 and was promoted as a full professor in 2019. His interests cover a broad spectrum ranging from quantum computing, quantum communication and quantum metrology with special focus on the subject of building large-scale quantum systems, via integrated photonics and quantum memory.