Superposition in Quantum Computing

Superposition:

Superposition is a fundamental principle of quantum physics. It states that all states of a quantum system may be superimposed. It is combined like waves in classical physics to yield a coherent quantum state that is distinct from its component states. The state however collapses into a random state once it is measured.

For example, assume an electron as a qubit with a spin-up orientation representing state |0> and spin-down state |1>. However, unlike a classical bit that can only be in a single state at any time. A qubit can be in the state up, down or a combination of both states at the same time because of the wave-like characteristics of subatomic particles.

A qubit in superposition behaves as if it were in both |0> and |1> states simultaneously. This new state |Ψ > of the qubit can be written as:

| Ψ> = α| ↑> + β | ↓> = α|0> + β|1>

Where α and β are complex numbers and are known as probability amplitudes as indicated earlier, satisfying the relation:

α² + β² = 1

This indicates that a qubit has the probability α² of being in spin-up (classical 0-state). It also has probability β² of being in spin-up (classical 1-state). It can also be in a coherent superposition of both. Thus, |Ψ> and be considered as a vector in the two-dimensional complex vector space C² spanned by two basis states |0> and |1>.