Superposition (Quantum Computing)

quantum-mechanics quantum-computing

Definition

Superposition (Quantum Mechanics)

A superposition is a quantum state that is a linear combination of basis states. In quantum computing, a qubit can exist in a superposition of the computational basis states $|0⟩$ and $|1⟩$:

$$|\psi ⟩=\alpha |0⟩+\beta |1⟩$$

where $\alpha ,\beta \in \mathbb{C}$ are complex amplitudes satisfying $|\alpha {|}^2+|\beta {|}^2=1$.

Distinction:

• Classical bit: Either $|0⟩$ or $|1⟩$ (mutually exclusive)
• Qubit: Can be $|0⟩$, $|1⟩$, or any superposition $\alpha |0⟩+\beta |1⟩$

Multi-Qubit Superposition

For $n$ qubits, the superposition spans all $2^n$ computational basis states:

$$|\psi ⟩=\sum _{i=0}^{2^n-1}{\alpha }_i|i⟩$$

with ${\sum }_{i=0}^{2^n-1}|{\alpha }_i{|}^2=1$.

Two-Qubit Superposition

For $n=2$ qubits, a general superposition is

$$|\psi ⟩={\alpha }_0|00⟩+{\alpha }_1|01⟩+{\alpha }_2|10⟩+{\alpha }_3|11⟩$$

with ${\sum }_{i=0}^3|{\alpha }_i{|}^2=1$.

Why does the sum of the coefficients squared have to be equal to 1?

The squared magnitudes $|{\alpha }_i{|}^2$ represent probabilities of measurement outcomes. When measuring a superposition, outcome $|i⟩$ occurs with probability $|{\alpha }_i{|}^2$. Since measurement must yield some outcome, the probabilities must sum to 1. Mathematically, this is the normalisation condition ensuring $|\psi ⟩$ is a unit vector in Hilbert space: $⟨\psi |\psi ⟩={\sum }_i|{\alpha }_i{|}^2=1$.

Quantum Parallelism

A quantum computer operating on $n$ qubits in superposition can process all $2^n$ basis states “in parallel” through a single application of a quantum operation. This is the source of quantum speedup in many algorithms.

Measurement

When a superposition state is measured in the computational basis:

• Outcome $|0⟩$ occurs with probability $|\alpha {|}^2$
• Outcome $|1⟩$ occurs with probability $|\beta {|}^2$
• The state collapses to the measured basis state