Secure delegated quantum algorithms for solving Mahalanobis distance

Distance metric is an important subroutine of some artificial intelligence algorithms such as classification and clustering algorithms. However, when a distance metric such as Mahalanobis distance is needed to calculate the distance between two points, the time required for classical algorithms to deal with Mahalanobis distance grows exponentially with the size of dataset and the dimensionality of the data. In this paper, a quantum algorithm for solving Mahalanobis distance (QAMD) is proposed. It can achieve an exponential speedup over the best classical counterpart if the covariance matrix in the sample distribution is a sparse Hermitian matrix. Especially, we propose a secure delegated QAMD protocol by combining the presented QAMD and quantum homomorphic encryption (QHE) which allows users with limited quantum capability to solve Mahalanobis distance with the help of a remote quantum server and still keep their data private. We also simulate a special case of the proposed delegated QAMD protocol on IBM's quantum cloud platform to show its feasibility.& COPY; 2023 Elsevier B.V. All rights reserved.