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Quantum QR decomposition in the computational basis

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Abstract

In this paper, we propose a quantum algorithm for approximating the QR decomposition of any \(N\times N\) matrix with a running time \(O(\frac{1}{\epsilon ^2}\) \(N^{2.5}\text {polylog}(N))\), where \(\epsilon \) is the desired precision. This quantum algorithm provides a polynomial speedup over the best classical algorithm, which has a running time \(O(N^3)\). Our quantum algorithm utilizes the quantum computation in the computational basis (QCCB) and a setting of updatable quantum memory. We further present a systematic approach to applying the QCCB to simulate any quantum algorithm. By this approach, the simulation time does not exceed \(O(N^2\text {polylog}(N))\) times the running time of the quantum algorithm originally designed with the amplitude encoding method, where N is the size of the problem.

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Notes

  1. Generally, the time required to compute the QR decomposition of an \(N\times N\) matrix is O\((N^3)\); unless, the matrix has some special structures, such as Vandermonde-like form.

  2. The approach to efficiently implementing the conditional rotations can be found in [25, Lemma 2]

  3. Every time before computing inner product in Algorithm 3, we can apply maximum-finding algorithm [1] and count algorithm [4] to investigate the distribution of the sum terms to obtain the best \(\kappa \).

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Acknowledgements

We thank the reviewers for their constructive and valuable suggestions.

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Authors and Affiliations

  1. Key Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Guangsheng Ma & Jiman Zhao

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Hongbo Li

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  1. Guangsheng Ma
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  2. Hongbo Li
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Correspondence to Jiman Zhao.

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H. Li supported by NSFC (Grant No. 11671388), National Key Research Program 2018YFA0704705. J. Zhao supported by National Natural Science Foundation of China (Grant Nos. 11471040 and 11761131002).

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Ma, G., Li, H. & Zhao, J. Quantum QR decomposition in the computational basis. Quantum Inf Process 19, 271 (2020). https://doi.org/10.1007/s11128-020-02777-4

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