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.

中文翻译：

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计算基础上的量子QR分解

在本文中，我们提出了一个量子算法用于近似任何的QR分解\（N \乘N \）矩阵与正在运行的时间\（O（\压裂{1} {\小量^ 2} \）\（N ^ {2.5} \ text {polylog}（N））\），其中\（\ epsilon \）是所需的精度。这种量子算法提供了优于最佳经典算法的多项式加速，后者的运行时间为\（O（N ^ 3）\）。我们的量子算法利用计算基础（QCCB）中的量子计算和可更新的量子内存设置。我们进一步提出了一种应用QCCB来模拟任何量子算法的系统方法。通过这种方法，模拟时间不超过\（O（N ^ 2 \ text {polylog}（N））\）乘以最初使用幅度编码方法设计的量子算法的运行时间的乘积，其中N是问题的大小。