Quantum Computing Research¶

Core ideas¶

Core ideas in Quantum Computing — what it is and when to use it.

• A qubit is a unit vector in a 2D complex space: α|0⟩ + β|1⟩ with |α|² + |β|² = 1.
• Measurement collapses it to 0 or 1 with probabilities |α|² and |β|².
• Gates are unitary matrices (Hadamard, Pauli-X, CNOT) that rotate the state.
• Entanglement correlates qubits so measuring one determines the other.

A pure-Python single-qubit simulator¶

A pure-Python single-qubit simulator — a key concept in Quantum Computing.

No libraries needed — a qubit is just two complex amplitudes, and a gate is a 2×2 matrix:

import cmath

# |0> state
state = [1 + 0j, 0 + 0j]

# Hadamard gate puts |0> into an equal superposition
h = 1 / cmath.sqrt(2)
H = [[h, h], [h, -h]]

def apply(gate, s):
    return [gate[0][0] * s[0] + gate[0][1] * s[1],
            gate[1][0] * s[0] + gate[1][1] * s[1]]

state = apply(H, state)
p0 = abs(state[0]) ** 2
p1 = abs(state[1]) ** 2
print(round(p0, 3), round(p1, 3))   # 0.5 0.5  (equal superposition)

This is exactly what a quantum SDK does under the hood, scaled to 2ⁿ amplitudes for n qubits.

Qiskit basics¶

Qiskit basics in Quantum Computing — what it is and when to use it.

With a real framework you build circuits declaratively and run them on a simulator or hardware:

# pip install qiskit qiskit-aer
from qiskit import QuantumCircuit
from qiskit_aer import AerSimulator

qc = QuantumCircuit(2, 2)
qc.h(0)              # Hadamard — superposition on qubit 0
qc.cx(0, 1)          # CNOT — entangle qubit 1 with qubit 0
qc.measure([0, 1], [0, 1])

sim = AerSimulator()
result = sim.run(qc, shots=1000).result()
print(result.get_counts())   # {'00': ~500, '11': ~500} — a Bell state

The ~50/50 split across 00 and 11 (never 01 or 10) is the signature of entanglement.

Where to go next¶

Gates, algorithms, error correction, and other SDKs to explore.

• Gates: Pauli-X/Y/Z, phase, Toffoli
• Algorithms: Grover's search, Deutsch–Jozsa, Shor's factoring
• Variational: VQE, QAOA for optimization
• Error correction: surface codes
• Other SDKs: Cirq (Google), PennyLane (quantum ML)

Practice exercises¶

1. Add a Pauli-X gate ([[0,1],[1,0]]) to the pure-Python simulator and confirm it flips |0⟩ to |1⟩.
2. Apply Hadamard twice and verify the state returns to |0⟩.
3. Compute the probabilities after applying H to the |1⟩ state.
4. Extend the simulator to two qubits using a length-4 amplitude vector.