Quantum Computing

The Bloch sphere — a geometric representation of a single qubit’s state.

Quantum computers exploit superposition and entanglement to perform computations that would be intractable for classical machines. Not faster at everything — specifically powerful for factoring, search, and simulating quantum systems.

Qubits

A classical bit is always 0 or 1. A qubit can be in superposition of both:

|ψ⟩ = α|0⟩ + β|1⟩

where |α|² + |β|² = 1. When measured, it collapses to |0⟩ with probability |α|² or |1⟩ with probability |β|².

Bell state in Qiskit

from qiskit import QuantumCircuit, transpile
from qiskit_aer import AerSimulator

qc = QuantumCircuit(2, 2)
qc.h(0)       # Hadamard: put qubit 0 in superposition
qc.cx(0, 1)   # CNOT: entangle qubit 0 and qubit 1
qc.measure([0, 1], [0, 1])

sim = AerSimulator()
job = sim.run(transpile(qc, sim), shots=1000)
counts = job.result().get_counts()
print(counts)
# {'00': 503, '11': 497}

The result is always 00 or 11 — never 01 or 10. Measuring one qubit instantly determines the other, regardless of the distance between them.

Quantum gates

Algorithms with quantum advantage

Shor’s algorithm (1994) — factors N-bit integers in O(n³) time. Classically the best known is sub-exponential. Breaks RSA if a large enough quantum computer is built.

Grover’s search (1996) — finds a marked item in an unsorted database of N items in O(√N) queries, vs O(N) classically. Quadratic speedup; important for cryptanalysis.

Variational Quantum Eigensolver (VQE) — hybrid classical-quantum for chemistry simulation. Calculates ground state energies of molecules too complex for classical methods. Active use case for near-term hardware.

Current hardware (2024–2025)

The current era is called NISQ — Noisy Intermediate-Scale Quantum. Fault-tolerant quantum computing likely requires millions of physical qubits to encode thousands of logical ones.