Qubit

Quantum Bits also known as qubits or qbits are basic unit of Quantum computing, just like Binary Bits for the classical computing.

A classical bit can be in one of the two states 0 or 1, where as a qubit can be in Superposition (in-between) of both of these classical states.

A pure qubit, say|ψ⟩, is represented as the linear combination of two basis states 0 & 1 (for now lets represent them by |0⟩ & |1⟩ respectively)

|ψ⟩ = α |0⟩ + β |1⟩

where α, β ∈ [0,1], such that |α|² + |β|² = 1

We can see α & β can take infinite number of values between [0,1] (basically all points on the unit circle, because of constraint) and because of this, the above qubit can take infinite number of values, this is where it becomes important to switch to Vector Spaces (Linear combinations of basis vectors) to discuss qubits.

Now here comes the interesting parts
1. when we initialize a qubit it starts in one of the pure |0⟩ or |1⟩ state & using a quantum gate/ operation, we put it into superposition (combination of these two states).
2. We then do all of our computations in this pure quantum (superposition states)
3. Finally Once all the computations are done, we measure them back into one of the original basic states of step 1

Simple Qubit Life Cycle

The 3rd step above is where we need a new concept call measurement of qubit, now here when the qubit is measured it can land up in one of the possible basis (non-superposition) vector states, with probability equal to the square of the coefficient of each of the state, as shown above.

At the final measurement,
P(|ψ⟩ = |0⟩) =γ² , &
P(|ψ⟩ = |1⟩) =δ²

Computational Basis

As we want to define a quantum state which can be both 0 & 1 at the same time, we want to define a vector space where 0 & 1 are the basis vectors and a qubit can then be any linear combination of these basic vectors.

Now to distinguish 0 as a bit vs 0 as a vector of this space, we would use |0⟩ to represent a pure 0 quantum state. So a qubit is a vector in the vector space defined/ spanned by |0⟩ & |1⟩ basis vectors

Note: |0⟩ is read as ket-0, |1⟩ is read as ket-1 & in general |x⟩ is read as ket-x

|0⟩ & |1⟩ are called the computation basis of the two dimensional vector state they form, and are represented as below:

Computational Basis for single qubit

Now a qubit is basically an element of this vector space, and a qubit |ψ⟩ can now be denoted as:

Single Qubit

You can verify |0⟩ & |1⟩ are the orthogonal vectors & hence they can be referred as Orthogonal (computation) basis.

This post is first part of the “Quantum Computing 101” Series, check the next post of the series, Classical Bit vs Qubit .