1. Introduction

The Fourier transform is an operator that expresses a time-dependent signal as a sum (or integral) of periodic signals. In other words, the Fourier transform changes a function of time s(t) into a function of frequency S(ω). If a signal is a function of time it said to be in the “time domain” and if it is a function of frequency it is said to be in the “frequency domain”.

For signals whose spectrum is changing in time, i.e. nonstationary signals, sometimes the best description is a mixture of the time and frequency components. Signal representations which mix the time and frequency domains are called, naturally enough, “time-frequency representations” and are often used to describe time-varying signals for which the pure frequency or Fourier representation is inadequate. A familiar example of a time-frequency representation is a musical score, which describes when (time) certain notes (frequency) are to be played.

We present efficient algorithms for quantum versions of the Zak and Weyl-Heisenberg transforms. Both these time-frequency transforms can be seen as generalizations of Fourier transforms and the quantum algorithms make heavy use of the Quantum Fourier Transform.

2. Zak Transforms

Yak transform is a function that mediates between the time domain and frequency domain depending on the subgroup B. We use the Quantum Zach Transform (QZT) to get desired results.

2.2. The Quantum Algorithm

We now show that the QZT is efficiently implementable. Applying the formula for the Zak transform yields F(a, a*) which can then be transformed into the quantum version.

3. Weyl-Heisenberg Transforms

We will use time-frequency translates to form orthonormal bases. We define the Quantum Weyl-Heisenberg Transform (QWHT) by |ψi 7→ Σ (b,b*)∈Δ hψ|g (b,b*)i|b, b*i. In other words, the QWHT expresses |ψi in the orthonormal basis of time-frequency translates of the window function.

3.2. The Quantum Algorithm

To compute WHT coefficients, we have to compute the Fourier coefficients of P=F/G. We use the phase kickback technique to implement the transformation on |ai. After this, we can extract the contents of the quantum register.