Realization of a quantum autoencoder for lossless compression of quantum data

Chang-Jiang Huang, Hailan Ma, Qi Yin, Jun-Feng Tang, Daoyi Dong, Chunlin Chen, Guo-Yong Xiang, Chuan-Feng Li, and Guang-Can Guo

Phys. Rev. A 102, 032412 – Published 17 September 2020

Abstract

As a ubiquitous aspect of modern information technology, data compression has a wide range of applications. Therefore, a quantum autoencoder which can compress quantum information into a low-dimensional space is fundamentally important to achieve automatic data compression in the field of quantum information. Such a quantum autoencoder can be implemented through training the parameters of a quantum device using classical optimization algorithms. In this paper, we demonstrate the condition of achieving a perfect quantum autoencoder and theoretically prove that a quantum autoencoder can losslessly compress high-dimensional quantum information into a low-dimensional space (also called latent space) if the number of maximum linearly independent vectors from input states is no more than the dimension of the latent space. Also, we experimentally realize a universal two-qubit unitary gate and design a quantum autoencoder device by applying a machine learning method. Experimental results demonstrate that our quantum autoencoder is able to compress two two-qubit states into two one-qubit states. Besides compressing quantum information, the quantum autoencoder is used to experimentally discriminate two groups of nonorthogonal states.

Received 1 October 2019
Revised 27 December 2019
Accepted 24 July 2020

DOI:https://doi.org/10.1103/PhysRevA.102.032412

Physics Subject Headings (PhySH)

Machine learning Quantum algorithms Quantum gates Quantum optics

Quantum Information, Science & Technology

Authors & Affiliations

Chang-Jiang Huang^1,2, Hailan Ma^3,4, Qi Yin^1,2, Jun-Feng Tang^1,2, Daoyi Dong^3,*, Chunlin Chen⁴, Guo-Yong Xiang^1,2,†, Chuan-Feng Li^1,2, and Guang-Can Guo^1,2

¹Key Laboratory of Quantum Information, University of Science and Technology of China, Chinese Academy of Sciences, Hefei 230026, China
²Chinese Academy of Sciences Center for Excellence in Quantum Information and Quantum Physics, Hefei 230026, China
³School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia
⁴Department of Control and Systems Engineering, School of Management and Engineering, Nanjing University, Nanjing 210093, China

^*daoyidong@gmail.com
^†gyxiang@ustc.edu.cn

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Vol. 102, Iss. 3 — September 2020

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Images

Figure 1
(a) A graphical representation of encoding and decoding process. The map $E$ encodes the input data (yellow dots) into a lower-dimensional space (red dots). The decoder $D$ can reconstruct the input data at the output (green dots). (b) The hybrid scheme for training a quantum autoencoder [17]. The input state $| φ_{i} 〉$ is compressed by a parametrized unitary operator $U^{j} (p_{1}, p_{2}, ..., p_{n})$ at iteration $j$ . When the overlaps between the trash state and the reference state for all states in the input set are collected, a classical learning algorithm computes and sets a new group of parameters to generate new unitary operator $U^{j + 1} (p_{1}, p_{2}, ..., p_{n})$ . (c) Universal two-qubit unitary gate composed of two beam splitters, two mirrors, and four same single-qubit parts (V1, V2, VR, and VL). (d) Each single-qubit part is composed of two QWPs, an HWP, and a phase shifter (PS).
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Figure 2
Experimental setup for realizing a quantum autoencoder. The setup consists of three modules. (a), (b) State preparation: Photon pairs are created by type-I SPDC through a BBO. One photon is set as a trigger and the other photon is prepared in the state $| H 〉$ through a PBS. Then an HWP along with a PBS can control the path bit of the photon. In each path, an HWP and a QWP are used to control the polarization of the photon. (c)–(e) Parametrized unitary $U$ and measurements: The second Sagnac interferometer contains four unitary polarization operators $V_{1}, V_{2}, V_{R},$ and $V_{L}$ . Due to the structure of the Sagnac interferometer, produced two-qubit states go through the NBS-coated surface twice. Thus, a parametrized universal two-qubit unitary gate is achieved. Then, a QWP, an HWP, and a PBS can form any local measurements on polarization. (f) Classical optimization algorithm: The algorithm is carried out mainly by a computer and electronic-controlled devices.
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Figure 3
Characterization of experimentally realized gates. Here we plot the real elements in Fig. 3 [Fig. 3] and the imaginary elements in Fig. 3 [Fig. 3] of the cnot [swap] gate. We use red to represent positive and blue to represent negative. The fidelities of the cnot and swap gates are $0.9574 \pm 0.0006$ and $0.9482 \pm 0.0007$ , respectively.
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Figure 4
The results of encoding two two-qubit states into two qubit states. Here we show the results of encoding different initial states into different qubits (path and polarization). (a) Encode { $| R H 〉, | L V 〉$ } into the path qubit. (b) Encode { $| R H 〉, | L V 〉$ } into the polarization qubit. (c) Encode { $\frac{\sqrt{2}}{4} | R H 〉 + \frac{\sqrt{2}}{4} | R V 〉 + \frac{\sqrt{3}}{2} | L V 〉, | L V 〉$ } into the path qubit. (d) Encode { $\frac{\sqrt{2}}{4} | R H 〉 + \frac{\sqrt{2}}{4} | R V 〉 + \frac{\sqrt{3}}{2} | L V 〉, | L V 〉$ } into the polarization qubit. Here infidelity is the cost function in our algorithm and iterations indicate the training process.
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Figure 5
The results of discriminating two different groups of nonorthogonal states. The bound for (a) and (b) is plotted in blue dashed line. (a) Encode { $cos θ_{1 / 2} | R H 〉 + sin θ_{1 / 2} | R V 〉, θ_{1 / 2} = \pm 4^{\circ}$ } and { $cos θ_{3 / 4} | R H 〉 + sin θ_{3 / 4} | R V 〉, θ_{3 / 4} = 60^{\circ} \pm 4^{\circ}$ } into different polarization qubits. (b) Encode { $cos θ_{1 / 2} | R H 〉 + sin θ_{1 / 2} | R V 〉, θ_{1 / 2} = \pm 2^{\circ}$ } and { $cos θ_{3 / 4} | R H 〉 + sin θ_{3 / 4} | R V 〉, θ_{3 / 4} = 30^{\circ} \pm 2^{\circ}$ } into different polarization qubits. Here infidelity is the cost function in our algorithm.
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