Quantum machine learning has recently been one of the prominent areas in the field of quantum computing where quantum algorithms can be generalized to solve important problems. For specific problems like search and integer factoring, quantum algorithms can be generalized to important subroutines replacing linear algebra and machine learning. More accurately, the quantum search can be generalized to amplitude amplification allowing speedups for sampling and estimation tasks. Quantum factoring contains the phase estimation subroutine, which can be used for decomposing a matrix into eigenvalues and eigenvectors. Also, quantum kernel methods and quantum gradient computation are being actively researched for various machine learning applications.

Many quantum algorithms use heuristic methods similar to classical machine leaning. In the classical regime, it is difficult to obtain provable guarantees regarding the quality of learning and the run time. However, in quantum regimes, some algorithms retain the benefits of provable classical algorithms for learning and at the same time examine provable quantum speedups for machine learning. One such quantum algorithm that is inspired by its classical analogue is presented in this paper. It is called Alphatron and the classical algorithm is enhanced by quantum sub-routines. It is a gradient-descent-like algorithm for isotonic regression with the inclusion of kernel functions. This algorithm can be used to learn a kernelized, non-linear concept class of functions with a bounded noise term, hence can be exploited to learn a two-layer neural network, where one layer of activation functions feeds into a single activation function. In this work, the authors provide quantum speedups for the Alphatron and its application to non-linear classes of functions and two-layer neural networks.

The work considers pre-computation of the kernel matrix used in the algorithm and provides three sets of algorithms: Alphatron with pre-computation (Pre), approximate pre-computation (Approx Pre), and quantum estimation pre-computation (Q-Pre). Each algorithm offers improvement on the run-time complexity by pre-computation and quantum estimation, eventually improving the original Alphatron. The algorithms were tested against 'n' dimensional data, which is comparatively larger than the other parameters, thus showing relevance to many practical applications. With the aid of pre-computation, the complexity term which is quadratic in the total size of the dataset (m) and linear in "n" dimensions of data, loses a factor of T (number of iterations), therefore improving the performance. Furthermore, utilizing quantum amplitude estimation, a gain of quadratic speedup can be obtained along the dimension.

The work further exploits a setting where the samples are provided via quantum query access in order to harness quantum subroutines to estimate the entries of the kernel matrices used in the algorithm. This access can be provided by quantum random access memory (QRAM) via the GroverRudolph procedure. Such a device stores the data in (classical) memory cells but allows for superposition queries to the data. The cost of the resources is proportional to the length of the vector to set up, but then can provide a run-time of the superposition state that is logarithmic in the length of the vector.

The quantum subroutines used in this work are adaptations of the amplitude estimation algorithm. The results show that the learning guarantees can be preserved despite the erroneous estimation of the kernel matrices. Also, there are estimations inside the main loop of the algorithm which can be replaced with quantum subroutines while keeping the main loop intact. Moreover, in the case where data dimension 'n' is much larger than the other parameters, quantum pre-computation requires asymptotically more time than Alphatron with Kernel, since the same Alphatron with Kernel algorithm is used after the preparation of the kernel matrices. Therefore, optimizing Alphatron with Kernel is not cost-effective when the cost of preparing the data of size n is taken into account. Finally, further exploration of a potential quantum speedup is required for Alphatron with Kernel, especially for the case of pre-computation occurring prior to the algorithm.