8 Using the quantum Fourier transform

 

This chapter covers

  • Introducing the discrete sinc distribution and phased discrete sinc quantum states
  • Using the IQFT to find the encoded frequency of periodic quantum states
  • Using the QFT to encode some trigonometric distributions in quantum states

Now that we understand what happens when the quantum Fourier transform (QFT) and inverse QFT (IQFT) are applied to a quantum state, let’s look at how we can use them. We will look at examples of two of the most common uses of the QFT: converting difference in phase to difference in magnitude and efficiently preparing some useful quantum states.

In the previous chapter, we saw how to encode a certain frequency into a quantum state in the form of a geometric sequence state. The encoded frequency is reflected in directions of the amplitudes of a geometric sequence state. In this chapter, we will use the IQFT to convert difference in phase to difference in magnitude. The magnitudes after applying the IQFT to a geometric sequence state match the values of the discrete sinc function. We will go deeper into understanding the significance of this pattern in both wave diffraction (single-slit experiment) and quantum states.

8.1 The single-slit experiment: Wave diffraction

8.1.1 Introducing the discrete sinc function

8.2 Encoding a periodic signal using discrete sinc quantum states

8.2.1 Phase-to-magnitude frequency encoding with the IQFT

8.2.2 Some useful numerical forms of the frequency encoding pattern

8.2.3 Reversed qubit implementation of phased discrete sinc quantum states

8.3 Discrete sinc as a sequence of coin flips

8.4 Encoding trigonometric distributions in a quantum state

8.4.1 Raised cosine

8.4.2 Other trigonometric functions

Summary