# 🔢 WIA-QUA-002: Quantum Algorithm Standard

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> **Standard ID:** WIA-QUA-002
> **Version:** 1.0.0
> **Status:** Active
> **Category:** QUA (미래기술/양자/물리)
> **Color:** Indigo (#6366F1)

---

## 🌟 Overview

The WIA-QUA-002 standard defines the computational framework for quantum algorithms, including quantum gates, circuits, Shor's algorithm, Grover's search, variational quantum eigensolver (VQE), quantum approximate optimization algorithm (QAOA), and quantum error correction.

**弘익人間 (Benefit All Humanity)** - This standard aims to provide a comprehensive foundation for quantum algorithm development and implementation, making quantum computing accessible to all.

## 🎯 Key Features

- **Quantum Gates**: Complete library of single and multi-qubit gates (Hadamard, CNOT, Pauli, T-gate, etc.)
- **Quantum Circuits**: Build and simulate quantum circuits with arbitrary qubit configurations
- **Shor's Algorithm**: Integer factorization using quantum period finding
- **Grover's Algorithm**: Quantum search with O(√N) complexity
- **VQE**: Variational Quantum Eigensolver for molecular simulations
- **QAOA**: Quantum Approximate Optimization Algorithm for combinatorial problems
- **Error Correction**: Quantum error correction codes (Shor, Steane, Surface codes)
- **Quantum Supremacy**: Benchmarks and metrics for quantum advantage

## 📊 Core Concepts

### 1. Quantum Gates

#### Single-Qubit Gates
```
Hadamard (H): |0⟩ → (|0⟩ + |1⟩)/√2
Pauli-X: |0⟩ ↔ |1⟩
Pauli-Y: |0⟩ → i|1⟩, |1⟩ → -i|0⟩
Pauli-Z: |0⟩ → |0⟩, |1⟩ → -|1⟩
T-gate: |1⟩ → e^(iπ/4)|1⟩
S-gate: |1⟩ → i|1⟩
```

#### Multi-Qubit Gates
```
CNOT: Controlled-NOT (2-qubit)
Toffoli: Controlled-Controlled-NOT (3-qubit)
SWAP: Exchange qubit states
Controlled-Z: Phase flip on |11⟩
```

### 2. Shor's Algorithm

Factors large integers in polynomial time:
```
N = p × q  (find p and q)
Complexity: O((log N)³)
Classical: O(exp(√(log N)))
```

### 3. Grover's Algorithm

Quantum search with quadratic speedup:
```
Find x: f(x) = 1
Iterations: π/4 × √(N/M)
Speedup: O(√N) vs O(N)
```

## 🔧 Components

### TypeScript SDK

```typescript
import {
  QuantumCircuit,
  applyHadamard,
  applyCNOT,
  shorFactorize,
  groverSearch,
  runVQE,
  runQAOA
} from '@wia/qua-002';

// Create quantum circuit
const circuit = new QuantumCircuit(3); // 3 qubits

// Apply gates
circuit.hadamard(0);
circuit.cnot(0, 1);
circuit.pauliX(2);

// Measure
const result = circuit.measure();
console.log('Measurement:', result);

// Shor's algorithm
const factors = shorFactorize(15);
console.log('Factors of 15:', factors); // [3, 5]

// Grover's search
const oracle = (x: number) => x === 7;
const result = groverSearch(oracle, 8);
console.log('Found:', result); // 7
```

### CLI Tool

```bash
# Create quantum circuit
wia-qua-002 circuit create --qubits 3

# Apply gates
wia-qua-002 gate hadamard --qubit 0
wia-qua-002 gate cnot --control 0 --target 1

# Run Shor's algorithm
wia-qua-002 shor --number 15

# Run Grover's search
wia-qua-002 grover --size 1024 --target 42

# Run VQE simulation
wia-qua-002 vqe --molecule H2 --basis sto-3g

# Run QAOA
wia-qua-002 qaoa --graph maxcut.json --layers 3

# Simulate error correction
wia-qua-002 error-correct --code shor --error-rate 0.01
```

## 📚 Documentation

| Document | Description |
|----------|-------------|
| [WIA-QUA-002-v1.0.md](./spec/WIA-QUA-002-v1.0.md) | Complete specification with quantum theory |
| [TypeScript SDK](./api/typescript/) | Full TypeScript implementation |
| [CLI Tool](./cli/wia-qua-002.sh) | Command-line interface |

## 🚀 Quick Start

### Installation

```bash
# Clone repository
git clone https://github.com/WIA-Official/wia-standards.git
cd wia-standards/standards/quantum-algorithm

# Run installation script
./install.sh

# Verify installation
wia-qua-002 --version
```

### TypeScript Usage

```bash
# Install via npm
npm install @wia/qua-002

# Or yarn
yarn add @wia/qua-002
```

```typescript
import { QuantumCircuit, shorFactorize, groverSearch } from '@wia/qua-002';

// Factor a number
const factors = shorFactorize(21);
console.log(`21 = ${factors[0]} × ${factors[1]}`); // 21 = 3 × 7

// Quantum search
const found = groverSearch((x) => x === 42, 100);
console.log(`Found: ${found}`); // Found: 42

// Build circuit
const qc = new QuantumCircuit(2);
qc.hadamard(0);
qc.cnot(0, 1);
const state = qc.getState();
console.log('Entangled state:', state); // Bell state
```

## 🔬 Quantum Constants

| Constant | Symbol | Value | Description |
|----------|--------|-------|-------------|
| Planck constant | h | 6.626 × 10⁻³⁴ J·s | Quantum of action |
| Reduced Planck | ℏ | 1.055 × 10⁻³⁴ J·s | h/(2π) |
| Rydberg constant | R∞ | 1.097 × 10⁷ m⁻¹ | Atomic spectra |
| Fine structure | α | 1/137.036 | Electromagnetic coupling |

## ⚡ Quantum Algorithms

### Shor's Algorithm
- **Purpose**: Integer factorization
- **Complexity**: O((log N)³)
- **Speedup**: Exponential over classical
- **Applications**: Cryptography, RSA breaking

### Grover's Algorithm
- **Purpose**: Unstructured search
- **Complexity**: O(√N)
- **Speedup**: Quadratic over classical
- **Applications**: Database search, SAT solving

### VQE (Variational Quantum Eigensolver)
- **Purpose**: Find ground state energy
- **Complexity**: Hybrid quantum-classical
- **Applications**: Molecular simulation, chemistry

### QAOA (Quantum Approximate Optimization)
- **Purpose**: Combinatorial optimization
- **Complexity**: Parameterized circuit
- **Applications**: Max-Cut, TSP, portfolio optimization

## 🛡️ Error Correction

### Shor Code
- **Qubits**: 9 physical → 1 logical
- **Errors**: Corrects arbitrary single-qubit errors
- **Distance**: 3

### Steane Code
- **Qubits**: 7 physical → 1 logical
- **Errors**: Corrects single-qubit errors
- **Distance**: 3

### Surface Code
- **Qubits**: d² physical → 1 logical
- **Errors**: Threshold ~1%
- **Distance**: d

## 🌐 WIA Integration

This standard integrates with:
- **WIA-INTENT**: Intent-based quantum algorithm selection
- **WIA-OMNI-API**: Universal quantum API gateway
- **WIA-CLOUD**: Quantum cloud computing access
- **WIA-SECURITY**: Post-quantum cryptography

## 📖 Use Cases

1. **Cryptography**: RSA factorization, discrete logarithm
2. **Drug Discovery**: Molecular simulation with VQE
3. **Optimization**: Supply chain, logistics with QAOA
4. **Machine Learning**: Quantum neural networks
5. **Chemistry**: Electronic structure calculations
6. **Finance**: Portfolio optimization, risk analysis

## 🌍 Philosophy

**홍익인간 (弘益人間)** - Benefit All Humanity

This standard embodies the Korean philosophy of 弘益人間, aiming to benefit all humanity through innovation and standardization.

## 🤝 Contributing

Contributions welcome! Please see our [Contributing Guide](https://github.com/WIA-Official/wia-standards/CONTRIBUTING.md).

## 📄 License

MIT License - see [LICENSE](https://github.com/WIA-Official/wia-standards/LICENSE)

## 🔗 Links

- **Website**: [wiastandards.com](https://wiastandards.com)
- **GitHub**: [github.com/WIA-Official/wia-standards](https://github.com/WIA-Official/wia-standards)
- **Documentation**: [docs.wiastandards.com](https://docs.wiastandards.com)
- **Certification**: [cert.wiastandards.com](https://cert.wiastandards.com)

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**弘익人間 (홍익인간) · Benefit All Humanity**

*WIA - World Certification Industry Association*
*© 2025 SmileStory Inc. / WIA*
*MIT License*

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**홍익인간 (弘益人間) - Benefit All Humanity** 🌍