Harmonic sequence state-preparation

Abstract

We demonstrate an efficient circuit to prepare a quantum state with amplitudes proportional to a harmonic sequence. We do this by first preparing a large quantum state with linearly related amplitudes and then applying a quantum Fourier transform; this has a direct analogy to the fact that the Fourier coefficients of a sawtooth wave follow a harmonic sequence. We then consider an extension of this problem by block-encoding a matrix with a harmonic sequence along its diagonal. The cost of both circuits is dominated by the costs associated with the quantum Fourier transform.