Introduction to quantum computing part -1 Representation of qubit using Bloch sphere
In quantum computing, a qubit or quantum bit is the basic unit of quantum information the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two state quantum mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include: the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states/levels simultaneously, a property which quantum mechanics is fundamental to quantum mechanics and quantum computing. In , the Bloch sphere is a geometrical representation of the pure state space of a two level quantum mechanical system(qubit)
Two qubit states |0⟩ and |1⟩ are represented by z and -z axes respectively. |0⟩ state denotes upward spin of electron and |1⟩ spin denotes downward spin of electron. Any point |ψ⟩ on this sphere is represented by equation |ψ⟩= α|0⟩+ β|1⟩ where α² is probability of of electron having upward spin and β² is probability of electron having downward spin. As these are probabilities we can also say that |α|²+|β|² =1.
The Bloch Sphere is is a generalization of the representation of a complex number z with |z|² = 1 as a point on the unit circle in the complex plane.
If z = x + iy, where x and y are real, then:
|z|² = z*z = (x − iy)(x + iy) = x² + y²
and x² + y² = 1 is the equation of a circle of radius one, centered on the origin.
And for polar co-ordinates we know that ,
However, the only measurable quantities are the probabilities |α|²
and |β|² , so multiplying the state by an arbitrary factor e^(iγ) (a global
phase) has no observable consequences, because:
and similarly for |β|². So, we are free to multiply our state by e^(−i α)
giving:
Switching back to cartesian representation for the coefficient of |1⟩, consider coefficient of |1⟩ in above equation as x+iy as it’s also complex number. So,
r_α² (I have given _ symbol for subscript r subscript α)+ |x + iy|²
= r_α² + (x − iy)*(x + iy)
= r_α² + x² + y² = 1
which is the equation of a unit sphere in real 3D space with cartesian
coordinates (x, y, r_α).
Cartesian coordinates are related to polar coordinates by:
x = r sinθcos ϕ
y = r sinθsin ϕ
z = r cos θ
so renaming r_α to z in equation 1.2 , and remembering that r = 1, we can write:
|ψ’ ⟩ = z|0⟩ + (x + iy)|1⟩
= cosθ|0⟩ + sinθ(cos ϕ+ i sin ϕ)|1⟩
= cosθ|0⟩ + e^(iϕ) sinθ|1⟩
But notice that for θ=0, |ψ⟩ = |0⟩and θ= 90(pie/2), |ψ⟩ = 1⟩. So,we don’t have to consider full angles till 180. Because we get all values in 0 to 90 only so halving θ in above equation will give final equation for qubit representation in Bloch sphere.
From above equation, we can verify that |α|²+|β|² =1.
