Quantum Killer Apps: Quantum Fourier Transform and Search Algorithms

Abstract

We present a brief overview of the Fourier series, the Fourier and discrete Fourier transforms and their applications. We discuss a quantum algorithm that encodes the Fourier transform of the mapping f : {0, 1}ⁿ →{0, 1} in an n-qubit register. It’s shown how the quantum Fourier transform (QFT) gate is constructed from single-qubit phase and two-qubit control gates. Due to the collapse postulate, the quantum Fourier transform for f is not available in a register query, but it does allow efficient period estimation. We illustrate how the QFT is exploited in the Shor algorithm for factoring large numbers. On the average, search for an item in an unordered list of size N requires N∕2 queries. We show how the Grover quantum algorithm improves on this figure of merit as it requires resources that scale as $\sqrt {N}$.