Quantum Counting: Algorithm and Error Distribution

Counting is one of the most basic procedures in mathematics and statistics. In statistics literature it is usually done via the proportion estimation method. In this article we manifest a radically different counting procedure first proposed in the late 1990’s based on the techniques of quantum computation. It combines two major tools in quantum computation, quantum Fourier transform and quantum amplitude amplification, and shares similar structure to the quantum part of the celebrated Shor’s factoring algorithm. We present complete details of this quantum counting algorithm and the analysis of its error distribution. Comparing it with the conventional proportion estimation method, we demonstrate that this quantum approach achieves much faster convergence rate than the classical approach.