Qubit Fundamentals

What Is a Qubit?

A qubit (quantum bit) is the fundamental unit of quantum information. Unlike a classical bit that can only be 0 or 1, a qubit can exist in a superposition of both states simultaneously.

Mathematically, a qubit state is represented as a vector in a two-dimensional complex Hilbert space:

where and are complex probability amplitudes satisfying the normalization condition:

The states and form the computational basis:

Dirac (Bra-Ket) Notation

Paul Dirac introduced a compact notation for quantum states:

• Ket: denotes a column vector (state vector)
• Bra: denotes the conjugate transpose (row vector)
• Inner product: gives a complex scalar
• Outer product: gives an operator

The completeness relation for a single qubit is:

This identity operator ensures probability conservation during measurement.

Example: Bell State

The Bell state is written as:

The Bloch Sphere

Every pure qubit state can be visualized on the Bloch sphere. Using spherical coordinates , a general qubit is:

Key points on the Bloch sphere:

Any single-qubit gate corresponds to a rotation on the Bloch sphere.

Superposition

Superposition is the ability of a qubit to be in multiple states at once. When we apply a Hadamard gate to :

The state is an equal superposition of and . This is not a classical mixture — it is a fundamentally quantum coherence that enables quantum parallelism.

Superposition vs Classical Randomness

A key distinction: superposition involves interference between amplitudes, while classical randomness has no phase relationships. This interference is what powers quantum algorithms.

Measurement

Measurement collapses a qubit state to one of the basis states with probabilities given by the Born rule:

After measuring , the state becomes:

The expectation value of an observable is:

Quantum State Tomography

To fully reconstruct an unknown qubit state, we perform quantum state tomography by measuring in multiple bases. For a single qubit, we need measurements in the X, Y, and Z bases to reconstruct the density matrix:

where is the Bloch vector and are the Pauli matrices.

Python: Simulating a Qubit

import numpy as np

# Define computational basis
KET_0 = np.array([1, 0], dtype=complex)
KET_1 = np.array([0, 1], dtype=complex)

# Hadamard gate
H = (1 / np.sqrt(2)) * np.array([[1, 1],
                                   [1, -1]], dtype=complex)

# Apply H to |0>
psi = H @ KET_0
print(f"State after Hadamard: {psi}")
# Output: [0.707+0.j 0.707+0.j]

# Probabilities
probs = np.abs(psi) ** 2
print(f"P(|0>) = {probs[0]:.4f}, P(|1>) = {probs[1]:.4f}")

This demonstrates how a deterministic unitary operation creates a superposition with equal probabilities of measuring or .