QUANTUM COMPUTATION: QUANTUM FOURIER TRANSMORMATION [PART 1]

This brief article intends to clarify in the most simple way (if it can be made simpler please let me know I will be happy to learn the other way!!!) what the Quantum Fourier Transformation is, how is derived and expressed in tensor form. Future articles will focus on how the QFT is correlated to quantum computational Gates.

The article is written under the assumption that whomever reads it shall have a basic knowledge of what a qubit is and how it can be represented through computational basis and Dirac notation. I tried to eliminate as much as I could Tensor Calculus, albeit lacking formalism and generalisation to favour simplicity, on the presupposition and understanding that not everybody is interested to reason at higher levels of abstraction.

On personal basis this constitutes an attempt to share knowledge with the collective from a "generative/first principle perspective" (i.e the essence and the meaning behind any formula or concept) deemed the only approach to really develop a deep mastery (theoretical and practical) over the subject to derive innovative practical solutions capable of changing and redesign entire markets and trends.

When time allows further Articles will be shared with the aim to disseminate knowledge useful to new ideas generations for those who truly have a genuine passion for Relentless Innovation and "Obsessive" Research.

P.S if anyone in linked in and all over the world has the possibility to let me access the Hardware of one of these machines to study the cooling systems and make some experiments I will be grateful.

WHAT THE QUANTUM FOURIER TRANSFORM IS

The Quantum Fourier Transformation, namely QFT, is a key component of quantum computing and is important for a number of reasons including:

• Speed: QFT can exponentially speed up the computation of Fourier transforms, which are essential for analysing signals. This allows for faster data compression, pattern recognition, and solving differential equations
• Parallelism and superposition: QFT's parallelism and superposition properties allow it to handle large datasets and complex computations faster than classical methods
• Quantum algorithms: QFT is a key part of many quantum algorithms, including Shor's algorithm for factoring integers and quantum phase estimation
• Cryptography and optimization: QFT is an important tool in cryptography and optimization
• Quantum chemistry: QFT facilitates quantum phase evaluation and state preparation in quantum chemistry.?
• Industrial applications: QFT can be used to analyze the mechanical and thermal properties of materials

The QFT is the quantum version of the discrete Fourier transform, which is a key tool in digital signal processing.

MATHEMATICAL DERIVATION AND TENSORIAL EXPRESSION

Fourier Transformation, Laplace Transformation, Hankel Transformation etc, are called integral Transformations.

Question:

Why to use these transformation?

For example, let's assume that we are working in the time domain but in this domain is very difficult to solve our problem. Then, what we can do is to choose the frequency domain, for example. In other words we will transfer the function from the time domain to the frequency domain. This transfer is accomplished through an INTEGRAL TRANSFORMATION.

What is the Quantum Fourier Transform (QFT)?

Is a Transformation that Changes Quantum computational basis to Fourier Basis

Let's discuss Briefly about the computational basis (here we assume as minimum requirement a basic knowledge around Hilbert Spaces and bases (a single q bit can be mathematically represented as a two-dimensional Hilbert Space C2) and Dirac Notation.

In the quantum domain, we can represent a q bit with two computational basis as illustrated in Figure 1

Let's now review some basics about Fourier Basis

What are Fourier Basis?

We can represent the Fourier basis as illustrated in FIGURE 2:

The Quantum Fourier Transform will transform the 0, 1 basis into the + and - basis as illustrated in FIGUR3 3

The Hadamard Gate is the quantum gate capable of this transformation as illustrated in Figure 4 below:

Now that the relationship between Computational basis and Fourier basis via the Hadamard Transformation is clarified, let's come back to the Quantum Fourier Transformation to generalise its derivation from an intuitive perspective.

How Can we represent the Quantum Fourier Transformation as an integral Yransformation?Well, we have, mathematically speaking, an {x} basis which will be transformed in a {y} basis through an Integral transformation which we will indicate with IT. We will also represent the transformation through a Kernel K(x, y) function (this applies to all the transformation; in other words Laplace Transformation will have its Kernel function, Hankel its kernel and Fourier its kernel, for the sake of generalising).

Therefore we can write the transformation generically as below:

IT[x] = K(x, y) {y}

In the QFT case we will have:

Let's now express the Kernel as function of X and Y and the number :

Here N represent the number of the computational basis states (example: for 3 qubits N = 2 ** 3 = 8)

We can then rewrite the QFT using the discrete notation as a sum over all the computational basis states as follows where the square root of N is introduced for mathematical convenience as normalisation factor:

Let's now calculate the QFT for 1 qubit.

According to QFT formula in FIGURE 7, for 1 q bit N=2 ** 1 =2.

The QFT can be calculated as illustrated below:

Let's now consider another example: a linear combination of the states | 0 > and | 1 > and calculate the QFT:

Let's start articulating some reasoning on the QFT equation expressed in FIGURE 7. To be in the position to make an initial inference and derive quantum gates based circuits from this equation, or at least to develop some intuition around it, we need to express the Y in terms of X (X are the inputs and we work under the assumption that they are known.

What is y?

We can immediately say that y represents a decimal value and can be expressed in binary form as follows:

The equivalence in FIGURE 10 tells us that if we have n qubits, y is nothing but a state and y1....yn represent the binary values used to express that state.

For Example, if we consider n=3 qubits we can write that Y = 5 = | 101 ?

Let's develop the equivalence of FIGURE 10 by expressing the binary terms in terms of powers of 2 and then let's generalise the expression:

Let's now substitute the expression found for y in FIGURE 10 in the QFT expression (FIGURE 7)

Looking at the QFT equation, and applying the property in red type, we have that the exponential summation can be expressed as a product.

Let's re-write the QFT equation then:

Let's now look at the summation made over y (from 0 to n-1 which is omitted in the expression) in the equation of FIGURE 12.

This summation can be expressed as the product of n summations, being n the dimension of y | y1, y2, y3, ……, yn ?. We can then express the summation as follows:

Let's then re-arrange the QFT expression according to what observed in FIGURE 13 and observing that the sum of a product is commutative (to express the QFT as Tensor product on an n-dimensional Hilbert Space)

Let's now develop the Tensor product: