⚛️ Quantum Computing Playground

🧠 What is Quantum Superposition?

Superposition is one of the most fundamental principles in quantum mechanics. It means that a quantum system can exist in multiple states at the same time until it is measured.

🎯 Single Qubit Superposition

For a single qubit, we can write its state as:

Where α and β are complex numbers (amplitudes) that determine the probability of measuring |0⟩ or |1⟩. The probabilities are |α|² and |β|², and they must sum to 1.

🎲 Equal Superposition

In this experiment, we create an equal superposition using the Hadamard gate:

This means: 50% probability of measuring 0, and 50% probability of measuring 1.

🚀 Exponential Growth with Multiple Qubits

The real power comes when we have multiple qubits in superposition. With n qubits, we can represent 2ⁿ states simultaneously!

• 1 qubit: 2 states (|0⟩, |1⟩)
• 2 qubits: 4 states (|00⟩, |01⟩, |10⟩, |11⟩)
• 3 qubits: 8 states
• 4 qubits: 16 states
• 5 qubits: 32 states
• 6 qubits: 64 states - all at once!

🔧 How It Works

We apply the Hadamard gate (H) to each qubit:

• Each H gate puts one qubit into equal superposition
• When combined, they create a superposition of all possible bit strings
• All 2ⁿ states exist simultaneously until measurement

📊 What You'll Observe

After measurement, each of the 2ⁿ possible states should appear with approximately equal probability (1/2ⁿ or about 100/2ⁿ percent).

Try increasing the number of qubits and watch how the number of possible outcomes grows exponentially!

🧠 What is Grover's Algorithm?

Grover's algorithm, discovered by Lov Grover in 1996, is a quantum search algorithm that finds a specific item in an unsorted database. It provides a quadratic speedup over any classical algorithm.

📊 Speedup Comparison

See how the advantage grows with database size:

• 4 items: Classical: ~2 queries, Quantum: 1 query (2x faster)
• 16 items: Classical: ~8 queries, Quantum: 3 queries (2.7x faster)
• 64 items: Classical: ~32 queries, Quantum: 6 queries (5.3x faster)
• 1000 items: Classical: ~500 queries, Quantum: 25 queries (20x faster!)
• 1,000,000 items: Classical: ~500,000 queries, Quantum: 1,000 queries (500x faster!)

🎯 The Problem

Imagine you have an unsorted database (like a phone book in random order) and you need to find one specific entry. Classically, you'd have to check entries one by one until you find it.

⚡ The Quantum Solution

Grover's algorithm works through two main operations repeated in a loop:

1️⃣ The Oracle

A quantum circuit that "marks" the target item by flipping its phase. It doesn't tell us which item is the target, but it changes its quantum amplitude in a way we can exploit.

Where f(x) = 1 for the target item and 0 for all others.

2️⃣ The Diffusion Operator (Amplitude Amplification)

This operation amplifies the amplitude of the marked item while reducing the amplitudes of all other items. It's like making the target "brighter" and the rest "dimmer" in the quantum superposition.

🔄 Iteration Count

The optimal number of iterations is approximately π/4 × √N. Too few iterations and you won't find the target. Too many and the probability actually decreases!

• 4 items (2 qubits): 1 iteration → ~100% success
• 8 items (3 qubits): 2 iterations → ~95% success
• 16 items (4 qubits): 3 iterations → ~100% success

🔧 Circuit Structure

The algorithm follows these steps:

1. Initialize: Create equal superposition of all states using Hadamard gates
2. Repeat √N times:
   • Apply Oracle (mark the target)
   • Apply Diffusion operator (amplify the target)
3. Measure: The target state has very high probability!

📊 What You'll Observe

After running the circuit, you should see the target item measured with very high probability (~90-100%), while all other items appear much less frequently.

The green bar in the chart highlights the target item you're searching for.

🎓 Fun Fact

While √N speedup might not sound as impressive as the exponential speedups in some other quantum algorithms, it's actually provably optimal for unstructured search. No quantum algorithm can do better than Grover's!

🧠 What is Hash Breaking?

A preimage attack on a hash function means finding an input that produces a specific hash output. This is "reversing" or "breaking" the hash.

🎯 The Problem

Hash functions are designed to be one-way:

• Easy: Computing hash(x) is fast
• Hard: Finding x given hash(x) should be impossible

⚡ Quantum Attack with Grover's

Grover's algorithm can speed up the search for preimages:

• Classical brute force: Try N/2 inputs on average
• Grover's algorithm: Try √N inputs
• Speedup: Quadratic (not exponential!)

🔢 Example: 2-bit Hash

With a 2-bit hash, there are only 4 possible outputs (0, 1, 2, 3). Given target hash = 3:

• Search space: 4 possible inputs (with 2-bit inputs)
• Classical: Try ~2 inputs on average
• Grover's: 1 iteration to find the answer!

🛡️ Why SHA-256 Stays Safe

Real SHA-256 has 256 bits of output:

Even with quantum speedup, 10³⁸ operations is still:

• Billions of years on perfect quantum computer
• Requires billions of qubits (we have ~1,000 today)
• More energy than the sun produces

📊 Scaling Comparison

🎓 What You'll Learn

This demonstration shows:

• How to encode hash functions as quantum oracles
• Grover's algorithm applied to preimage attacks
• Why quadratic speedup isn't enough to break strong hashes
• The difference between theoretical and practical attacks

How Bitcoin Cryptography Works:

Private Key (256 bits, secret)
        ↓
    [ECDSA Elliptic Curve Math]
        ↓
Public Key (512 bits, derived from private)
        ↓
    [SHA-256 → RIPEMD-160]
        ↓
Bitcoin Address (160 bits, public)

Two Quantum Attack Vectors:

The Real Threat Timeline:

• 2025 (Current): ~1,000 noisy qubits - Bitcoin safe
• 2030-2035: Estimated 4,000+ error-corrected qubits - DANGER ZONE for signatures
• 2040: Large-scale quantum computers likely - Bitcoin must be upgraded

🔐 Minimum Bits Grover's Can Break:

🎯 What is DeFi Arbitrage?

DeFi arbitrage is the practice of profiting from price differences across decentralized exchanges (DEXs). A trader might swap ETH → USDC → DAI → WBTC → ETH, ending with more ETH than they started with!

⚛️ How Quantum Helps: QAOA

QAOA (Quantum Approximate Optimization Algorithm) is specifically designed for combinatorial optimization problems like finding the best arbitrage path.

How QAOA Works:

1. Encode the Problem: Convert the arbitrage problem into a quantum Hamiltonian (energy function)
2. Quantum Superposition: Create a superposition of ALL possible paths simultaneously
3. Optimize Parameters: Use alternating Cost and Mixer Hamiltonians to evolve toward optimal solutions
4. Measure Results: Collapse the quantum state to find high-profit paths

Where H_C encodes profitability (Cost Hamiltonian) and H_M allows exploration (Mixer Hamiltonian)

💼 Real-World Usage in Finance

📊 Why Quantum Wins for DeFi Optimization

Comparison: Bitcoin Breaking vs DeFi Optimization

🚀 The Future: Quantum DeFi Trading

By 2028-2030, quantum-enhanced DeFi bots may:

• Find arbitrage opportunities in milliseconds instead of seconds
• Optimize cross-chain paths (Ethereum ↔ Polygon ↔ Arbitrum)
• Calculate optimal MEV strategies (Maximal Extractable Value)
• Dynamically adjust to gas prices and slippage

Timeline Estimate: Quantum advantage for simple arbitrage is achievable with ~100 qubits (available by 2026). For complex multi-chain strategies, we'll need ~1000 qubits (estimated 2028-2030).