An Intuitive Guide to Quantum Superposition & Qubits
Demystifying the foundational physics behind quantum computing, explaining how qubits differ from classical bits, and writing a basic circuit in Qiskit.
Quantum computing is often discussed with an air of mystique, but at its heart, it is a mathematical expansion of classical information theory. Instead of storing discrete binary states, quantum processors manipulate vector spaces.
In this introductory guide, we will explore the concept of a qubit, the physics of superposition, and write our first quantum circuit using Qiskit.
The Classical Bit vs. The Qubit
A classical bit represents a switch that is either off (0) or on (1). Mathematically, these can be represented as two orthogonal vectors in a 2D space:
$$|0\rangle = \begin{pmatrix} 1 \ 0 \end{pmatrix}, \quad |1\rangle = \begin{pmatrix} 0 \ 1 \end{pmatrix}$$
A quantum bit, or qubit, is a physical system that can exist in a linear combination of both states. This state $|\psi\rangle$ is written as:
$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$$
where $\alpha$ and $\beta$ are complex numbers representing probability amplitudes. When we measure the qubit, the probability of obtaining state $|0\rangle$ is $|\alpha|^2$, and the probability of obtaining state $|1\rangle$ is $|\beta|^2$. Since the total probability must equal 1, these coefficients must satisfy the normalization condition:
$$|\alpha|^2 + |\beta|^2 = 1$$
Creating Superposition with the Hadamard Gate
To transition a qubit from a definite state (like $|0\rangle$) into a uniform superposition, we apply a Hadamard gate ($H$). Geometrically, this performs a rotation on the Bloch Sphere. Mathematically, the Hadamard transformation is defined by the matrix:
$$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix}$$
When applied to $|0\rangle$:
$$H|0\rangle = \frac{1}{\sqrt{2}} (|0\rangle + |1\rangle)$$
This puts the qubit in a state where measuring it yields a 50% chance of being 0 and a 50% chance of being 1.
Implementing with Qiskit
Let's write a simple Python script to create a quantum circuit, apply a Hadamard gate, and measure the output using Qiskit:
from qiskit import QuantumCircuit
from qiskit_aer import AerSimulator
from qiskit.visualization import plot_histogram
# 1. Create a quantum circuit with 1 qubit and 1 classical bit
qc = QuantumCircuit(1, 1)
# 2. Apply a Hadamard gate to qubit 0
qc.h(0)
# 3. Measure qubit 0 and store the result in classical bit 0
qc.measure(0, 0)
# 4. Simulate the circuit execution
simulator = AerSimulator()
job = simulator.run(qc, shots=1000)
result = job.result()
# 5. Extract counts
counts = result.get_counts(qc)
print("Measurement counts:", counts)
Running this simulation returns counts distributed roughly equally between 0 and 1 (e.g., {'0': 508, '1': 492}), confirming that the qubit was in a superposition of both states prior to measurement.
Summary
Superposition is the entry point into quantum algorithms. In upcoming posts, we will explore Quantum Entanglement and how CNOT gates allow qubits to share information in ways classical channels cannot replicate.