Improved quantum algorithm for A-optimal projection

Abstract

Dimensionality reduction (DR) algorithms, which reduce the dimensionality of a given data set while preserving the information of the original data set as well as possible, play an important role in machine learning and data mining. Duan \emph{et al}. proposed a quantum version of the A-optimal projection algorithm (AOP) for dimensionality reduction [Phys. Rev. A 99, 032311 (2019)] and claimed that the algorithm has exponential speedups on the dimensionality of the original feature space $n$ and the dimensionality of the reduced feature space $k$ over the classical algorithm. In this paper, we correct the time complexity of Duan \emph{et al}.'s algorithm to $O(\frac{κ^{4s}\sqrt{k^s}} {ε^{s}}\mathrm{polylog}^s (\frac{mn}ε))$, where $κ$ is the condition number of a matrix that related to the original data set, $s$ is the number of iterations, $m$ is the number of data points and $ε$ is the desired precision of the output state. Since the time complexity has an exponential dependence on $s$, the quantum algorithm can only be beneficial for high dimensional problems with a small number of iterations $s$. To get a further speedup, we propose an improved quantum AOP algorithm with time complexity $O(\frac{s κ^6 \sqrt{k}}ε\mathrm{polylog}(\frac{nm}ε) + \frac{s^2 κ^4}ε\mathrm{polylog}(\frac{κk}ε))$ and space complexity $O(\log_2(nk/ε)+s)$. With space complexity slightly worse, our algorithm achieves at least a polynomial speedup compared to Duan \emph{et al}.'s algorithm. Also, our algorithm shows exponential speedups in $n$ and $m$ compared with the classical algorithm when both $κ$, $k$ and $1/ε$ are $O(\mathrm{polylog}(nm))$.