May 31, 2000

J. Mark Ettinger

## 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.

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