Introduction: Computing with Quantum Mechanics
Quantum computing exploits superposition and entanglement to process information in fundamentally new ways. A quantum computer uses qubits instead of classical bits.
The Qubit
\[|\psi\rangle = \alpha|0\rangle + \beta|1\rangle, \quad |\alpha|^2 + |\beta|^2 = 1\]Unlike a bit (0 or 1), a qubit can be in superposition of both. On measurement, it collapses to 0 with probability \(|\alpha|^2\) or 1 with probability \(|\beta|^2\).
Quantum Gates
Hadamard: \(H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\) — creates superposition
Pauli-X: \(X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) — quantum NOT gate
CNOT: Flips target qubit if control is 1 — creates entanglement
Quantum Parallelism
With \(n\) qubits, we can represent \(2^n\) states simultaneously:
\[|\psi\rangle = \sum_{x=0}^{2^n-1} \alpha_x |x\rangle\]A quantum operation acts on all these states at once!
The Quantum Connection
Quantum computers aren't just faster classical computers—they compute differently. Certain problems (factoring, database search, simulation) can be solved exponentially faster. The challenge: maintaining coherence while computing, as any interaction with the environment causes decoherence.