# The Postulates of Quantum Mechanics

1: Below is mainly taken from book of Nielsen and Chuang P80 — p96.

2. I mainly follow the lecture of Quantum Computing of University of Cambridge.

The Pauli Matrix:

Pauli — X:

# Basic Linear Algebra Required by Quantum Alg

1: Below is mainly taken from book of Nielsen and Chuang P60 — p79.

2. I mainly follow the lecture of Quantum Computing of University of Cambridge.

We requires complex number to define quantum phenomena:

The complex number is defined as:∀z ∈ℂ, z = a + ib, ョa,b∈R and i = /sqr(-1).

ℂ is the vector space of n-tuples of complex numbers [z1,..zn]^T, with addition and scalar multiplication.

A matrix is an array of ( in general) complex numbers with addition and scalar multiplication:

Matrix multiplication can be given by:

# Quantum Teleportation

Below is mainly taken from:

Alice wants to send quantum information to Bob. Specifically, suppose she wants to send the qubit state |ψ⟩=α|0⟩+β|1⟩. This entails passing on information about α and β to Bob.

However, according to the non-clone theorem, Alice can’t simply generate a copy of |ψ⟩ and give the copy to Bob. So we need to use Quantum Teleportation to build the tunnel.

A third party Telamon has been introduced in this protocol:

Step 1:

Alice needs to transmit |ψ⟩=α|0⟩+β|1⟩(a random qubit) to Bob. She doesn’t know the state of the qubit. For this, Alice and Bob take…

# Quantum Circuits

Unlike the classical circuits, quantum circuits don’t allow loops. And it doesn’t allow FANIN and FANOUT.

We can’t copy qubit, which can be illustrated using CNOT gate. The CNOT gate can be seen as below:

where the top line represents control qubit and the bottom line represents target qubit. It has priority that if the control qubit is set to 0, the target qubit is left alone. Otherwise, the target qubit is flipped.

Quantum Logic Gates

Below are mainly taken from wikipedia.

https://en.wikipedia.org/wiki/Quantum_logic_gate

Quantum gates are operators that transform quantum states. Unlike classical logic gates, quantum logic gates are reversible. Quantum gates are unitary operators, and are described as unitary matrices relative to some basis.

Quantum logic gates are represented by unitary matrices. A gate which acts on n qubits is represented by a 2^n* 2^n unitary matrix. The quantum states that the gates act upon are vectors in 2^n complex dimensions.

The basis vectors are the possible outcomes if measured, and a quantum state is a linear combination of these outcomes. The…

What’s Qubit and Qubit States?

Below is the summary of the section 1.1–1.2 of book “Quantum Computation and Quantum Information”

Quantum bits, also known as qubit, is the fundamental concept of Quantum Computation. Just like bits, Qubit has states, which can be defined as:

Where a and b is a vector in a two-dimensional complex vector space, and |0> and |1> are known as computational basis states. For measurement, we get the probability of each state instead of exact states. That is, we get state |0> with probability |a|² and |1> with probability |b|², where |a|² + |b|² = 1…

## Ran Liu

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