Multiple Qubits
Discover how quantum computers achieve exponential power by combining multiple qubits into a single quantum system.
About This Lesson
- 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.
The Power of Exponential Scaling
Classical computers scale linearly: 2 bits can be in one of 4 states at a time. Quantum computers scale exponentially: 2 qubits can be in a superposition of all 4 states simultaneously!
N qubits = 2N states in superposition. 1 qubit → 2 states | 2 qubits → 4 states | 3 qubits → 8 states | n qubits → 2n states
With 300 qubits: 2300 ≈ 1090 states (more than atoms in the universe!)
Representing Two-Qubit States
Two qubits have four basis states: |00⟩, |01⟩, |10⟩, and |11⟩. A general state is a superposition of all four.
🔬 For Advanced Learners: Tensor Products and Hilbert Spaces
When combining quantum systems, we use the tensor product (⊗) to build the combined state space. This mathematical operation is what gives quantum computing its exponential power.
Tensor Product Basics
For two qubits, each qubit lives in a 2-dimensional space. The combined system lives in a 2 ⊗ 2 = 4-dimensional space:
|0⟩ ⊗ |1⟩ = |01⟩
|1⟩ ⊗ |0⟩ = |10⟩
|1⟩ ⊗ |1⟩ = |11⟩
For n qubits, the state space has dimensions 2ⁿ. This is why 300 qubits would have more states than atoms in the universe (2³⁰⁰ ≈ 10⁹⁰)!
Hilbert Spaces
The Hilbert space is the mathematical framework where quantum states live. For n qubits:
- Dimension: 2ⁿ (exponential growth!)
- Basis states: All possible bit strings of length n
- General state: Complex linear combination of basis states
- Inner product: Allows calculating probabilities and overlaps
Why it matters: Understanding tensor products explains both the power (exponential state space) and the challenge (exponentially growing complexity) of quantum computing. Classical computers cannot efficiently simulate large quantum systems because they'd need to track 2ⁿ complex numbers!
Gates on Multiple Qubits
H ⊗ H (Hadamard on both qubits) creates uniform superposition: (1/2)(|00⟩ + |01⟩ + |10⟩ + |11⟩)
Classical vs Quantum Comparison
Classical System
n bits store ONE of 2n values
Quantum System
n qubits in ALL 2n states!
So far, we've applied single-qubit gates (H, X, Z) to multiple qubits independently. Each qubit was manipulated separately.
But what if we want qubits to interact with each other? What if one qubit should influence another?
That's where multi-qubit gates come in! The most important is the CNOT gate—a gate that lets one qubit control another. This interaction is the key to creating entanglement, the most powerful quantum phenomenon.
Before moving on, can you:
- Calculate how many states N qubits can represent?
- Write a 2-qubit state in notation (e.g., |01⟩)?
- Understand why exponential scaling is powerful?
Ready to make qubits interact with CNOT?
📋 Quick Reference Card
Multiple Qubits Cheat Sheet
State Space Scaling
| # Qubits | Possible States | Example States |
|---|---|---|
| 1 | 2 | |0⟩, |1⟩ |
| 2 | 4 | |00⟩, |01⟩, |10⟩, |11⟩ |
| n | 2n | Exponential growth! |
💡 Key Insight: N qubits can be in superposition of ALL 2N states simultaneously—this exponential scaling is quantum computing's power!
Glossary
- Multi-Qubit System
- Definition: A quantum system with 2 or more qubits
- Computational Basis
- Definition: The standard basis states: |00⟩, |01⟩, |10⟩, |11⟩ for 2 qubits
- Tensor Product
- Symbol: ⊗
Definition: Mathematical operation combining quantum states: |0⟩ ⊗ |1⟩ = |01⟩ - Exponential Scaling
- Definition: N qubits can represent 2N states simultaneously—grows exponentially!
- State Space
- Definition: The set of all possible states a quantum system can be in
- Product State
- Definition: A multi-qubit state where each qubit can be described independently
- Hilbert Space
- Pronunciation: HIL-bert
Definition: The mathematical space where quantum states live
Test Your Understanding
Q1: How many states can 3 qubits represent in superposition?
Q2: What does (H ⊗ H)|00⟩ create?
Q3: What is the main advantage of multiple qubits?