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Units, Constants and Useful Formulas
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Last modified: January 3, 2022 ✓
Qubit
-----
The bit is the basic unit of information used by
computers to represent information. The bit can
take the value of 0 or 1. The Qubit is similar to
the bit but can also be a superposition of both the
0 and 1 states. Thus,
|ψ> = α|0> + β|1>
where α and β are probability amplitudes that
can be complex numbers, i.e. α = eiθα.
The Qubit can also be represented in terms of the
BLOCH SPHERE.
|ψ> = α|0> + β|1>
= rαeiθα|0> + rβeiθβ|1>
e-iθα|ψ> = rα|0> + rβei(θβ - θα)|1>
= cos(θ/2)|0> + eiφsin(θ/2)|1>
Where rα2 + rβ2 = r2 = 1, rα = cos(θ/2), rβ = sin(θ/2)
∴ θ = 2cos-1rα = 2sin-1rβ.
A pure Qubit state can be represented by any point
on the surface.
- -
|ψ> = (1/√2)|0> + (1/√2)|1> => ρ = | 1/2 1/2 |
| 1/2 1/2 |
- -
= cos(θ/2)|0> + exp(iφ)sin(θ/2)|1>
θ = 0 => |ψ> = |0>
θ = π => |ψ> = |1>
θ = π/2, φ = 0 => |ψ> = (1/√2)|0> + (1/√2)|1>
ρ can also be also be written in the Pauli basis as:
ρ = aI + bσx + cσy + dσz}
- - - - - - - -
= a| 1 0 | + rx| 0 1 | + ry| 0 -i | + rz| 1 0 |
| 0 1 | | 1 0 | | i 0 | | 0 -1 |
- - - - - - - -
Where rx2 + ry2 + rz2 = 1
- -
= (1/2)| 1 + rz rx - iry |
| rx + iry 1 - rz |
- -
Therefore, rx = 1, ry = rz = 0
- -
ρ = | 1/2 1/2 |
| 1/2 1/2 |
- -
It is possible to put the Qubit in a mixed state,
a statistical combination of different pure states.
Mixed states can be represented by points inside
the Bloch sphere. A mixed Qubit state has three
degrees of freedom: the angles φ and θ, as well
as the length, r, of the vector that represents
the mixed state. In this case rx2 + ry2 + rz2 ≤ 1
and,
|ψ> = r|0> + (1 - r)|1>