Multiple Qubits
Discover how quantum computers achieve exponential power by combining multiple qubits into a single quantum system.
About this module
In this module you will:
- Understand exponential scaling with multiple qubits
- Learn how to represent two-qubit states
- Explore tensor products and state spaces
- See how H gates scale to multiple qubits
Prerequisites: Lessons 1-5. Understanding of single qubits and quantum gates.
6.1 The Power of Exponential Scaling
Classical computers scale linearly: 2 bits store 4 values (00, 01, 10, 11) but can only be in one at a time. Quantum computers scale exponentially: 2 qubits can be in a superposition of all 4 states simultaneously!
This is the key to quantum advantage. With n qubits, a quantum computer can represent 2n states at once. Just 50 qubits can represent more states than there are atoms in the solar system!
Key Insight: Exponential State Space
1 qubit → 2 states
2 qubits → 4 states
3 qubits → 8 states
n qubits → 2n states
With just 300 qubits, you have more states than atoms in the universe (2300 ≈ 1090)!
Interactive: Exponential Calculator
See how quickly the state space grows with more qubits.
Notice how even modest numbers of qubits create enormous state spaces!
6.2 Representing Two-Qubit States
Two qubits have four basis states: |00⟩, |01⟩, |10⟩, and |11⟩. Each represents a specific configuration where the first qubit is in state 0 or 1, and the second qubit is in state 0 or 1.
A general two-qubit state is a superposition: |ψ⟩ = α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩, where |α|² + |β|² + |γ|² + |δ|² = 1.
Interactive: Two-Qubit State Explorer
Explore different two-qubit basis states and see what they mean.
6.3 Gates on Multiple Qubits
Single-qubit gates can be applied to individual qubits in a multi-qubit system. For example, H ⊗ H (Hadamard on both qubits) creates a uniform superposition over all four basis states.
Starting with |00⟩, applying H to both qubits gives: (H ⊗ H)|00⟩ = (1/2)(|00⟩ + |01⟩ + |10⟩ + |11⟩) — equal superposition of all states!
Interactive: H ⊗ H Circuit
Apply Hadamard gates to both qubits and measure the results.
You should see roughly 25% of each outcome (|00⟩, |01⟩, |10⟩, |11⟩) — perfect uniform superposition!
6.4 Visualizing State Space Growth
As we add qubits, the computational space explodes. This exponential growth is why quantum computers can solve certain problems that classical computers cannot.
Comparison: Classical vs Quantum
See how classical and quantum systems scale differently.
Classical System
n bits store ONE of 2n values
Quantum System
n qubits in superposition of ALL 2n states!
This exponential advantage is the foundation of quantum computing's power!
Test Your Understanding
Question 1: How many states can 3 qubits represent simultaneously in superposition?
Question 2: What does (H ⊗ H)|00⟩ create?
Question 3: What is the main advantage of multiple qubits?
Question 4: In |01⟩, which qubit is in state |1⟩?