An improved quantum algorithm for A-optimal projection

Dimensionality reduction (DR) algorithms, which reduce the dimensionality of a given data set while preserving the information of original data set as well as possible, play an important role in machine learning and data mining. Duan et al. proposed a quantum version of 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 complexity of Duan et al.'s algorithm to $O(\frac{\kappa^{4s}\sqrt{k^s}} {\epsilon^{s}}\mathrm{polylog}^s (\frac{mn}{\epsilon}))$, where $\kappa$ 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 $\epsilon$ is the desired precision of the output state. Since the 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 proposed an improved quantum AOP algorithm with complexity $O(\frac{s \kappa^6 \sqrt{k}}{\epsilon}\mathrm{polylog}(\frac{nm}{\epsilon}) + \frac{s^2 \kappa^4}{\epsilon}\mathrm{polylog}(\frac{\kappa k}{\epsilon}))$. Our algorithm achieves a nearly exponential speedup if $s$ is large and a polynomial speedup even if $s$ is a small constant compared with Duan et al.'s algorithm. Also, our algorithm shows exponential speedups in $n$ and $m$ compared with the classical algorithm when both $\kappa$, $k$ and $1/\epsilon$ are $O(\mathrm{polylog}(nm))$.