# Understanding the Matrices of Quantum Logic Gates

• Floating Info
In summary, the conversation discusses the correspondence between Quantum Logic Gates and matrices, with 2x2 matrices corresponding to gates that interact with one qubit, and 4x4 matrices corresponding to gates that interact with two qubits. The problem is understanding what these matrices do and how they relate to the gates. The suggestion is to refer to lecture notes for a deeper understanding.
Floating Info
(I am not sure if this is the right section for this).

This question probably is extremely trivial and silly, but I haven't been able to find the answer to it despite going through quite a bit of The Internet.

So, it appears that each Quantum Logic Gate corresponds to a matrix. Ones that interact with one qubit are 2 x 2, two qubits 4 x 4. The problem is, I have no clue what on Earth these matrices do and how they correspond to the gates. Do you multiply them by something? If so, what is that something?

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Floating Info said:
(I am not sure if this is the right section for this).

This question probably is extremely trivial and silly, but I haven't been able to find the answer to it despite going through quite a bit of The Internet.

So, it appears that each Quantum Logic Gate corresponds to a matrix. Ones that interact with one qubit are 2 x 2, two qubits 4 x 4. The problem is, I have no clue what on Earth these matrices do and how they correspond to the gates. Do you multiply them by something? If so, what is that something?

I haven't delved into the subject but If you want to really understand them you should work your way through the following lecture notes:

http://www.theory.caltech.edu/people/preskill/ph229/#lecture

My impression is that any type of summary articles (such as Wikipedia) will be too brief to get much understanding.

## 1. What are quantum logic gates?

Quantum logic gates are fundamental building blocks of quantum computers, which are devices that use the principles of quantum mechanics to process and manipulate information. These gates are responsible for performing logical operations on quantum bits, or qubits, which are the basic units of information in a quantum computer.

## 2. How are quantum logic gates different from classical logic gates?

Classical logic gates operate on classical bits, which can only be in either a 0 or 1 state. Quantum logic gates, on the other hand, can operate on qubits, which can exist in multiple states simultaneously, known as superposition. This allows quantum logic gates to perform more complex operations and calculations compared to classical logic gates.

## 3. What is the role of matrices in quantum logic gates?

Matrices are used to represent the operations performed by quantum logic gates. Each gate has an associated matrix that describes how it transforms the input qubits into the output qubits. These matrices are crucial for understanding the behavior of quantum logic gates and designing quantum algorithms.

## 4. How do I interpret the matrices of quantum logic gates?

The matrices of quantum logic gates can be interpreted as a series of operations that are applied to the input qubits. These operations include rotations, flips, and phase shifts, and their combination determines the effect of the gate on the qubits. By analyzing the matrices, we can understand the behavior of the gate and how it affects the state of the qubits.

## 5. What are some common quantum logic gates and their corresponding matrices?

Some common quantum logic gates include the Hadamard gate, which puts a qubit in a superposition state, the CNOT gate, which performs a controlled NOT operation on two qubits, and the SWAP gate, which exchanges the state of two qubits. Their corresponding matrices are:

• Hadamard gate: 1/sqrt(2) * [1 1; 1 -1]
• CNOT gate: [1 0 0 0; 0 1 0 0; 0 0 0 1; 0 0 1 0]
• SWAP gate: [1 0 0 0; 0 0 1 0; 0 1 0 0; 0 0 0 1]

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