Quantum Computation - Basics of Qubits

I'm still in a hiatus at the moment, but I may write the occasional blog post now and again (during the hiatus) like in this example. I will be giving a brief insight into the concept of Qubits which is the analogue to Classical Bits (1's and 0's). The only real difference between the two types, is that Qubits can be manipulated and controlled by the laws of Quantum Mechanics such as Superposition and Quantum Entanglement; the Spin states are also of great importance here since the control of the Spin will help create strings of information which are studied in Quantum Information Theory.

Before reading this post, I will assume you have some mathematical knowledge of Linear Algebra and Dirac Notation. Otherwise, I'll explain the concepts as a I write about the fundamentals of Qubits.

Firstly, let's look at the concept of Spin. Spin is a very important concept for Qubits. Spin is the angular momentum of a particle intrinsic (property of itself) to it's body. Spin is often regarded as a vector in three dimensional space, and being in the Up or Down state. The Down state has less energy in comparison to the Up state, and with Qubits, the Down state is usually set and then the Spin state will be changed to the Up state by electromagnetic radiation when necessary. The notation below represents a state vector called a Ket.

$$\vert A>$$

The Ket vector can be used to represent Spin states, and commonly denoted in the following form: 

<u|A> for an Up state and <d|A> for a Down state. The lowercase a and d actually represent the probability amplitudes of the particle being in those two quantum states. Of course, the particle can be in a superposition of those two quantum states, but under observation this coherence is usually broken and the particle is measured as being in one quantum spin state. The probability amplitudes are typically complex numbers, which means they have a real part and imaginary part.

To gather the probability of spin we square these probability amplitudes to give the following notation:

$$\lvert d^2 \rvert$$ and $$\lvert u^2 \rvert$$ which will need to equal 1.

More specifically, the two quantum states are called basis vectors, and can be represented as |0> and |1>. Thee basis is simply all the possible vectors for a particular vector space. I will assume you will know the formal definition of a basis.

Another two important points are the Bloch Sphere and Quantum Entanglement. Quantum Entanglement means if we change the state of one particle then it will change the state of a another particle which is entangled with the first particle. This is a fundamental idea to Quantum Computing. It enables us to compute a larger number of bits simultaneously. For example, a Qubit has two basis states: |0> and |1>, or more correctly as: |00> and |11> since a Qubit can be in superposition of two states at once. Remember that when we measure or observe the state, it will be in one state. If we change one Qubit to |00> then the other Qubit will change to |00> too. There is equal probability of the Qubit being in the superposition of |00> or |11>.

A important point to remember is that, for every additional Qubit you add, the computing power in comparison to a Classical Computer is doubled like so:

$$2^n$$
n is the number of Qubits. If we had 3 Qubits, then classically this will be equivalent to 8 Classical Bits.

Typically, Qubits are represented as a Bloch Sphere, since Qubits exist in a two-dimensional Hilbert Space where only the superposition of two quantum states is possible. In later posts I will attempt to explain the Bloch Sphere in more depth.

This post was only supposed to be very introductory, and not go into depth about Qubits and Quantum Computation. In the future, I will build upon this post and go into more depth about Quantum Computation and other theoretical concepts.