Improved quantum algorithm for A-optimal projection

Dimensionality reduction algorithms, which reduce the dimensionality of a given data set whereas preserving the information of the 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 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 the algorithm of Duan et al. to O[κ⁴ˢ√kˢεˢpolylogˢ(mnε)], 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 $O[{s{{κ}}⁶√k}{{ε}}polylog({nm}{{ε}})+{{s}²{{κ}}⁴}{{ε}}polylog({{κ}k}{{ε}})]$ and space complexity $O[{log}₂(nk/{ε})+s]$. With space complexity slightly worse, our algorithm achieves, at least, a polynomial speedup compared to the algorithm of Duan et al.. Also, our algorithm shows exponential speedups in $n$ and $m$ compared with the classical algorithm when ${κ},{{0.28em}{0ex}}k$, and $1/{ε}$ are $O[polylog(nm)]$.