Qubits and Superposition
Now it's time to meet the quantum bit! Discover what makes qubits fundamentally different from classical bits and explore the mind-bending concept of superposition.
About This Lesson
- Understand what a qubit is and how it differs from a classical bit
- Grasp the concept of superposition—being in multiple states at once
- Learn about quantum measurement and collapse
- Master quantum notation (ket notation)
Prerequisites: Lessons 1-2. You should understand classical bits and binary.
Introducing the Qubit
A classical bit can only be 0 or 1. But a qubit (quantum bit) can be in a superposition of both 0 and 1 simultaneously!
The Coin Analogy
Imagine a coin spinning in the air. While it's spinning, is it heads or tails? Neither—and both! Only when you catch it does it "decide" to be heads or tails.
- Before measurement: The qubit is in both states simultaneously
- During measurement: The qubit "collapses" to one definite value
- After measurement: The qubit is now definitely 0 or 1
The Spinning Coin Analogy
Classical Bit
A coin sitting on a table
Definitely heads OR tails
Quantum Qubit
A coin spinning in the air
Both heads AND tails!
While the coin is spinning, is it heads or tails? It's neither—and both! Only when you catch it (measure it) does it "decide" to be heads or tails. A superposition is like that spinning coin: the qubit is in BOTH states at once!
Three Ways to Understand Superposition
🎭 Through Analogy
Think of a spinning coin in the air, or a room where the light switch is both ON and OFF until you look. Imagine Schrödinger's famous cat that's both alive and dead in a box—that's superposition! The qubit exists in this "in-between" state until you measure it.
📊 Through Visualization
Imagine a bag with 5 red marbles and 5 blue marbles. Without looking inside, you have a 50% chance of pulling red and 50% chance of pulling blue. But once you reach in and grab one, you get a definite color—either red or blue, not both. A qubit in superposition is like that bag before you reach in: both outcomes exist with their probabilities. Measurement is like pulling out one marble—you get one definite result, and the superposition collapses.
🔢 Through Logic
Step 1: A classical bit has a definite state: 0 OR 1 (one is true).
Step 2: A qubit in superposition has probability for each: it could be 0 AND could be 1 (both have non-zero probability).
Step 3: Measurement forces a choice, giving you one outcome randomly based on those probabilities.
Conclusion: Before measurement = both possibilities exist. After measurement = one definite answer.
The famous thought experiment: a cat in a box is in superposition—both alive AND dead—until we observe!
Classical Bit
Always ONE definite value: 0 OR 1
Quantum Qubit
Can be in BOTH 0 and 1 at once!
💡 Why This Matters
Real Application: Google's Sycamore quantum computer uses 53 qubits in superposition to perform calculations in 200 seconds that would take the world's fastest classical supercomputer 10,000 years!
❌ Wrong: "Superposition means the qubit is rapidly switching between 0 and 1, we just don't know which"
✓ Correct: The qubit is genuinely in BOTH states at once—not switching, not unknown—it's fundamentally both until measured!
🔬 For Advanced Learners: The Mathematics
Mathematically, a qubit in superposition is written as:
Where:
- |ψ⟩ (pronounced "ket sigh" or "ket psi") represents the quantum state
- α and β (pronounced "alpha" and "beta") are complex numbers called "amplitudes"
- |0⟩ and |1⟩ (pronounced "ket zero" and "ket one") are the basis states
The probabilities come from squaring the amplitudes:
- Probability of measuring 0: |α|²
- Probability of measuring 1: |β|²
- Total probability must equal 1: |α|² + |β|² = 1
Quantum Measurement
When you measure a qubit in superposition, you can't predict which result you'll get! It's truly random. But you CAN predict the probability of each outcome.
- Measurement is destructive: Once measured, superposition is destroyed
- Measurement is probabilistic: You get 0 or 1 randomly based on probabilities
- Measurement is irreversible: You can't "un-measure" a qubit
Setup: Qubit in equal superposition |+⟩ (50% chance each)
Quantum Notation (Ket Notation)
Quantum physicists use ket notation to represent quantum states:
- |0⟩ — "ket zero" — the qubit is definitely in state 0
- |1⟩ — "ket one" — the qubit is definitely in state 1
- |+⟩ — equal superposition of |0⟩ and |1⟩ (50% each)
Where α and β are complex numbers called amplitudes, and |α|² is the probability of measuring 0 and |β|² is the probability of measuring 1.
Don't confuse these two concepts!
Amplitude (α, β)
- Complex numbers
- Can be negative or imaginary
- Describe the quantum state
- Used in calculations
Probability (|α|², |β|²)
- Real numbers (always positive)
- Between 0 and 1
- What you actually measure
- Found by squaring: |α|²
Example: For |+⟩ = (1/√2)|0⟩ + (1/√2)|1⟩
- Amplitude of |0⟩: α = 1/√2 ≈ 0.707
- Probability of measuring 0: |α|² = (1/√2)² = 1/2 = 50%
Understanding Measurement
Measurement is one of the most fascinating and important aspects of quantum computing. When you measure a qubit in superposition, something dramatic happens...
What Happens During Measurement?
Before Measurement: Superposition
The qubit exists in a superposition: |ψ⟩ = α|0⟩ + β|1⟩
The qubit is "both 0 and 1" simultaneously
During Measurement: Collapse
The superposition collapses to either |0⟩ or |1⟩
This collapse is random, but follows probability rules
After Measurement: Classical Bit
You get a definite result: either 0 or 1
The superposition is destroyed—you can't "unmeasure"!
When measuring |ψ⟩ = α|0⟩ + β|1⟩:
- Probability of getting 0: |α|² (square the amplitude!)
- Probability of getting 1: |β|²
- These must sum to 1: |α|² + |β|² = 1 (normalization)
Example: For |+⟩ = (1/√2)|0⟩ + (1/√2)|1⟩
- P(0) = |(1/√2)|² = 1/2 = 50%
- P(1) = |(1/√2)|² = 1/2 = 50%
⚠️ Key Facts About Measurement
- Measurement is destructive: It destroys the superposition
- You can only measure once: Second measurement gives the same result
- Results are random: But follow probability distributions
- Information is lost: You can't recover the original amplitudes α and β
This is why quantum computing is tricky! We need to design algorithms that extract useful information through measurements while carefully preserving superposition where we need it.
Before moving on, can you:
- Explain what superposition means in your own words?
- Describe what happens when you measure a qubit?
- Explain the difference between amplitude and probability?
- Calculate |α|² given an amplitude α?
If you checked all boxes, excellent! You've mastered the core concepts. If not, review the sections above.
You now understand superposition and measurement—the core concepts of quantum computing! But how do we create superposition? How do we manipulate qubits to perform computations?
That's where quantum gates come in! Quantum gates are operations that transform qubit states:
- X gate: Flips |0⟩ ↔ |1⟩ (like classical NOT)
- H gate: Creates and destroys superposition
- And many more...
Ready to start building quantum circuits? Let's learn the gates!
📋 Quick Reference Card
Qubits & Superposition Cheat Sheet
Key Notation
| Notation | Pronunciation | Meaning |
|---|---|---|
| |0⟩ | ket zero | Definitely in state 0 |
| |1⟩ | ket one | Definitely in state 1 |
| |+⟩ | ket plus | Equal superposition (50% each) |
| |ψ⟩ | ket sigh (or psi) | General quantum state |
Amplitude vs Probability
| Concept | What It Is | Key Property |
|---|---|---|
| Amplitude (α, β) | Complex numbers | Describe the quantum state |
| Probability (|α|², |β|²) | Real numbers (0-1) | What you measure |
General Superposition Formula
Where:
- α, β = complex amplitudes
- |α|² = probability of measuring 0
- |β|² = probability of measuring 1
- |α|² + |β|² = 1 (normalization)
💡 Key Insight: Measurement collapses superposition to either |0⟩ or |1⟩. The probabilities are determined by the squared amplitudes (Born rule).
Glossary
- Qubit
- Pronunciation: KYOO-bit
Definition: A quantum bit—the basic unit of quantum information that can be in superposition - Superposition
- Pronunciation: soo-per-puh-ZI-shun
Definition: The ability of a quantum system to be in multiple states simultaneously - Ket Notation
- Pronunciation: ket
Definition: A mathematical notation using |⟩ to represent quantum states (e.g., |0⟩, |1⟩, |+⟩) - |0⟩
- Pronunciation: ket zero
Definition: The quantum state representing definitely 0 - |1⟩
- Pronunciation: ket one
Definition: The quantum state representing definitely 1 - |+⟩
- Pronunciation: ket plus
Definition: Equal superposition state: 50% chance of measuring 0 or 1 - |ψ⟩
- Pronunciation: ket sigh (or ket psi)
Definition: Symbol for a general quantum state - Amplitude
- Pronunciation: AM-plih-tood
Definition: A complex number (α or β) that determines the probability of measuring a state. The probability is |amplitude|² - Measurement
- Definition: The act of observing a qubit, which collapses superposition to either |0⟩ or |1⟩
- Collapse
- Definition: When measurement causes a superposition to become a definite state (|0⟩ or |1⟩)
- Born Rule
- Definition: The probability of measuring a state is the squared magnitude of its amplitude: P = |α|²
- Normalization
- Definition: The requirement that all probabilities sum to 1: |α|² + |β|² = 1
Test Your Understanding
Q1: What is superposition?
Q2: What happens when you measure a qubit in superposition?
Q3: If |+⟩ is measured 100 times, how many |0⟩ results?