Quantum Superposition and Qubits

A classical bit holds one of two values, 0 or 1; a qubit can be prepared in a quantum superposition of the states corresponding to these values. Superposition means that the qubit's state is described by probability amplitudes for both basis states, rather than by a single binary value. However, measuring the qubit in this basis yields only one outcome: 0 or 1.

How a qubit represents information

The two reference states of a qubit are written as |0⟩ and |1⟩ and called the computational basis. A general pure qubit state is written as |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex probability amplitudes satisfying |α|² + |β|² = 1.

• A superposition contains contributions from both basis states when both amplitudes are non-zero; |0⟩ and |1⟩ themselves are also valid qubit states.
• A classical bit whose value is unknown represents ordinary uncertainty, not quantum superposition.

Measurement and interference

Under measurement in the computational basis, the probabilities of obtaining 0 and 1 are |α|² and |β|² respectively. After an ideal measurement, the qubit is in the basis state corresponding to the observed result.

• An equal superposition, (|0⟩ + |1⟩)/√2, gives each outcome with probability one-half, but a single measurement does not return both values.
• The amplitudes have a relative phase, which affects how quantum states combine and interfere during subsequent operations.
• Interference distinguishes a coherent superposition from a classical random mixture with the same measurement probabilities.

Why the distinction matters for computing

Quantum algorithms manipulate amplitudes and use interference to enhance desired outcomes and suppress others. Superposition is therefore a computational resource, but it does not allow all possible answers to be read out simultaneously.

• An n-qubit pure state is described by amplitudes for 2ⁿ computational basis states; measuring the register in that basis produces one n-bit outcome.
• Quantum advantage depends on the algorithm and task, not merely on placing qubits in superposition.
• For cybersecurity, Shor's algorithm shows that a sufficiently capable quantum computer could undermine public-key cryptography based on integer factorisation or discrete logarithms; superposition alone does not break encryption.