Quantum subroutine for variance estimation: algorithmic design and applications

Quantum computing sets the foundation for new ways of designing algorithms, thanks to the peculiar properties inherited by quantum mechanics. The exploration of this new paradigm faces new challenges concerning which field quantum speedup can be achieved. Toward finding solutions, looking for the design of quantum subroutines that are more efficient than their classical counterpart poses solid pillars to new powerful quantum algorithms. Herewith, we delve into a grounding subroutine, the computation of the variance, whose usefulness spaces across different fields of application, particularly the artificial intelligence (AI) one. Indeed, the finding of the quantum counterpart of these building blocks impacts vertically those algorithms that leverage this metric. In this work, we propose QVAR, a quantum subroutine, to compute the variance that exhibits a logarithmic complexity both in the circuit depth and width, excluding the state preparation cost. With the vision of showing the use of QVAR as a subroutine for new quantum algorithms, we tackle two tasks from the AI domain: feature selection and outlier detection. In particular, we showcase two AI hybrid quantum algorithms that leverage QVAR: the hybrid quantum feature selection (HQFS) algorithm and the quantum outlier detection algorithm (QODA). In this manuscript, we describe the implementation of QVAR, HQFS, and QODA, providing their correctness and complexities and showing the effectiveness of these hybrid quantum algorithms with respect to their classical counterpart.