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Quantum algorithm for Laplacian eigenmap via Rayleigh quotient iteration

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Abstract

Laplacian eigenmap (LE) is a geometrically motivated algorithm for dimensionality reduction. However, as the data dimensionality increasing, it is intractable for classical computers to solve the generalized eigen-problem which is a crucial subroutine of the LE. In this work, we propose a quantum algorithm for Laplacian eigenmap (QLE). Compared with classical LE, our QLE can provide an exponential speedup in eigenmapping step over classical counterparts under some mild condition. In our scheme, we first present a quantum subroutine for adjacency graph construction based on the controlled SWAP test and maximum searching algorithm. Second, a variational quantum generalized eigensolver (VQGE) is proposed to solve the generalized eigen-problem \( A\varvec{x} = \lambda B \varvec{x}\). The proposed VQGE avoids the determination of the partial derivatives of the Rayleigh quotient and the difficulty of selecting hyperparameter. Finally, as a proof-of-principle, we conduct a numerical simulation to solve the variational quantum Laplacian eigenmap of randomly generated data points.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 61871111), the Social Development Projects of Jiangsu Science and Technology Department (No. BE2018704) and the Fundamental Research Funds for the Central Universities (No. 2242021k30053).

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Authors and Affiliations

  1. State Key Laboratory of Millimeter Waves, Southeast University, Nanjing, 210096, China

    Ze-Tong Li & Xu-Tao Yu

  2. Quantum Information Center, Southeast University, Nanjing, 210096, China

    Ze-Tong Li, Fan-Xu Meng, Xu-Tao Yu & Zai-Chen Zhang

  3. National Mobile Communications Research Laboratory, Southeast University, Nanjing, 210096, China

    Fan-Xu Meng

  4. Purple Mountain Lab, Nanjing, 211111, China

    Xu-Tao Yu & Zai-Chen Zhang

  5. National Mobile Communications Research Laboratory, Southeast University, Nanjing, 211189, China

    Zai-Chen Zhang

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  1. Ze-Tong Li
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Correspondence to Xu-Tao Yu.

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Ze-Tong Li and Fan-Xu Meng have contributed equally to this work.

Hardware efficient ansatz

Hardware efficient ansatz

The hardware efficient ansatz is a kind of structured circuit consist of hardware implementable parameterized elementary gates. Based on the IBM QX’s elementary gate set, we use the ansatz as shown in Figs. 5 and 6.

Fig. 5
figure 5

The unit of hardware efficient ansatz

Full size image
Fig. 6
figure 6

The hardware efficient ansatz with l layers

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In Fig. 5, the gates \(R_y(\theta )\), \(R_x(\theta )\) are Pauli rotation gates whose mathematical representation are

$$\begin{aligned} R_y(\theta ) = e^{-i\frac{\theta }{2}Y},\,\, R_z(\theta ) = e^{-i\frac{\theta }{2}Z}. \end{aligned}$$
(40)

It is clear that the unit can be written by

$$\begin{aligned} (U^i_3\otimes U^i_4)CNOT(U^i_1\otimes U^i_2), \end{aligned}$$
(41)

where \(R_z(\theta ^i_{3j - 2})R_y(\theta ^i_{3j - 1})R_z(\theta ^i_{3j})\) is the Euler decomposition of \(U^i_j\).

There are 12 parameters per unit, and \(12\left\lfloor \frac{n}{2}\right\rfloor \) parameters per layer, where n is the number of qubits.

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Li, ZT., Meng, FX., Yu, XT. et al. Quantum algorithm for Laplacian eigenmap via Rayleigh quotient iteration. Quantum Inf Process 21, 11 (2022). https://doi.org/10.1007/s11128-021-03347-y

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