Quantum computing maths

The mathematical principles of superposition form the foundation of quantum computing, representing one of the most significant technological applications of quantum mechanics. Classical computers process information using bits that exist in definite states of 0 or 1. Quantum computers, however, utilize qubits that can exist in superposition states described by:

|ψ⟩ = α|0⟩ + β|1⟩

Where |α|² + |β|² = 1. This seemingly simple mathematical expression enables quantum computers to process exponentially more information than classical computers for certain types of problems.

Consider an n-qubit quantum computer. While a classical n-bit system can represent only one of 2ⁿ possible states at any given time, a quantum system can exist in a superposition of all 2ⁿ states simultaneously:

|ψ⟩ = ∑_{i=0}^{2^n-1} cᵢ|i⟩

This exponential scaling is what gives quantum computers their potential advantage. Algorithms like Shor's algorithm for factoring large numbers and Grover's algorithm for searching unsorted databases exploit superposition and interference to achieve speedups that are impossible with classical computation.