Reconstruction of quantum channel via convex optimization

Abstract

Quantum process tomography is often used to completely characterize an unknown quantum process. However, it may lead to an unphysical process matrix, which will cause the loss of information with respect to the tomography result. Convex optimization, widely used in machine learning, is able to generate a global optimum that best fits the raw data while keeping the process tomography in a legitimate region. Only by correctly revealing the original action of the process can we seek deeper into its properties like its phase transition and its Hamiltonian. Here, we reconstruct the seawater channel using convex optimization and further test it on the seven fundamental gates. We compare our method to the standard-inversion and norm-optimization approaches using the cost function value and our proposed state deviation. The advantages convince that our method enables a more precise and robust estimation of the elements of the process matrix with less demands on preliminary resources. In addition, we examine on a set of non-unitary channels and the reconstructions reach up to $99.5\%$ accuracy. Our method offers a more universal tool for further analyses on the components of the quantum channels and we believe that the crossover between quantum process tomography and convex optimization may help us move forward to machine learning of quantum channels.

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 $99.5\%$, showing that our method can be applied to a general quantum process including unitary and non-unitary ones.