quantrs2-core 0.1.3

# QuantRS2-Core Implementation Plan

## Overview
This document outlines the phased implementation plan for enhancing the quantrs2-core module with advanced quantum computing features, focusing on SciRS2 integration and sophisticated gate decomposition algorithms.

## Phase 1: Foundation & SciRS2 Integration (High Priority)

### 1.1 Enhanced Matrix Operations Module
- **File**: `src/matrix_ops.rs`
- **Purpose**: Create a unified interface for matrix operations using SciRS2
- **Tasks**:
  - Define traits for quantum matrix operations
  - Implement dense/sparse matrix conversions
  - Add specialized quantum matrix operations (tensor products, partial traces)
  - Integrate SciRS2 BLAS/LAPACK routines

### 1.2 Controlled Gate Framework
- **File**: `src/controlled.rs`
- **Purpose**: Support for controlled versions of arbitrary gates
- **Tasks**:
  - Implement generic controlled gate wrapper
  - Add multi-controlled gate support
  - Optimize for common patterns (Toffoli, Fredkin)
  - Implement phase-controlled gates

### 1.3 Gate Synthesis from Unitaries
- **File**: `src/synthesis.rs`
- **Purpose**: Convert arbitrary unitary matrices to gate sequences
- **Tasks**:
  - Implement Cosine-Sine decomposition using SciRS2
  - Add KAK decomposition for two-qubit unitaries
  - Create optimal single-qubit decomposition
  - Implement general n-qubit synthesis

## Phase 2: Advanced Decomposition Algorithms (High Priority)

### 2.1 Solovay-Kitaev Algorithm
- **File**: `src/decomposition/solovay_kitaev.rs`
- **Purpose**: Approximate arbitrary gates using finite gate sets
- **Tasks**:
  - Implement basic SK algorithm structure
  - Add gate sequence database generation
  - Optimize using SciRS2 matrix norms
  - Create configurable precision levels

### 2.2 Clifford+T Decomposition
- **File**: `src/decomposition/clifford_t.rs`
- **Purpose**: Decompose gates into Clifford+T with optimal T-count
- **Tasks**:
  - Implement exact synthesis for special cases
  - Add approximate synthesis with T-count optimization
  - Create phase polynomial representation
  - Implement grid search optimization

### 2.3 Non-Unitary Operations
- **File**: `src/operations.rs`
- **Purpose**: Support measurements and quantum channels
- **Tasks**:
  - Define measurement operation trait
  - Implement projective measurements
  - Add POVM measurements
  - Create reset operations

## Phase 3: Advanced Gate Operations (Medium Priority)

### 3.1 Quantum Shannon Decomposition
- **File**: `src/decomposition/shannon.rs`
- **Purpose**: Decompose n-qubit gates using divide-and-conquer
- **Tasks**:
  - Implement recursive Shannon decomposition
  - Use SciRS2 SVD for optimal decomposition
  - Add multiplexed rotation gates
  - Optimize for specific gate patterns

### 3.2 Cartan Decomposition
- **File**: `src/decomposition/cartan.rs`
- **Purpose**: Canonical two-qubit gate decomposition
- **Tasks**:
  - Implement KAK decomposition using SciRS2 eigensolvers
  - Add canonical gate parameters extraction
  - Create optimal CNOT count decomposition
  - Support arbitrary two-qubit basis gates

### 3.3 Gate Fusion Optimization
- **File**: `src/optimization/fusion.rs`
- **Purpose**: Combine adjacent gates for efficiency
- **Tasks**:
  - Identify fusable gate patterns
  - Implement matrix multiplication fusion
  - Add commutation-based optimization
  - Create peephole optimization passes

## Phase 4: Quantum Channels & Advanced Features (Medium Priority)

### 4.1 Quantum Channel Representations
- **File**: `src/channels/mod.rs`
- **Purpose**: Support for non-unitary quantum operations
- **Tasks**:
  - Implement Kraus representation
  - Add Choi matrix representation
  - Create channel conversion utilities
  - Implement common quantum channels

### 4.2 ZX-Calculus Primitives
- **File**: `src/zx_calculus/mod.rs`
- **Purpose**: Graph-based quantum circuit optimization
- **Tasks**:
  - Define ZX-diagram data structures
  - Implement basic rewrite rules
  - Add diagram simplification algorithms
  - Create conversion to/from circuits

### 4.3 Eigenvalue-Based Gate Analysis
- **File**: `src/analysis/eigenvalue.rs`
- **Purpose**: Gate characterization using eigendecomposition
- **Tasks**:
  - Implement gate eigenvalue extraction
  - Add gate classification algorithms
  - Create rotation angle extraction
  - Support gate parameter learning

## Phase 5: Performance & Integration (Ongoing)

### 5.1 SIMD Optimization Extensions
- **File**: Update `src/simd_ops.rs`
- **Tasks**:
  - Extend SIMD operations for new gate types
  - Add AVX-512 support where available
  - Implement vectorized gate fusion
  - Create SIMD-optimized decomposition

### 5.2 Memory-Efficient Extensions
- **File**: Update `src/memory_efficient.rs`
- **Tasks**:
  - Add sparse gate representations
  - Implement gate caching with LRU
  - Create memory-mapped gate databases
  - Optimize for large gate sequences

### 5.3 Testing Framework
- **File**: Update `src/testing.rs`
- **Tasks**:
  - Add decomposition accuracy tests
  - Create synthesis benchmarks
  - Implement gate equivalence checking
  - Add property-based testing

## Implementation Guidelines

### Code Organization
- Each major feature gets its own module
- Use traits for extensibility
- Maintain backward compatibility
- Follow existing code style

### Performance Considerations
- Use SciRS2 operations wherever possible
- Implement both dense and sparse variants
- Add benchmarks for critical paths
- Profile memory usage

### Testing Strategy
- Unit tests for each algorithm
- Integration tests for decomposition chains
- Property tests for mathematical invariants
- Benchmark tests for performance

### Documentation Requirements
- Detailed module documentation
- Algorithm explanations with references
- Usage examples for each feature
- Performance characteristics

## Dependencies to Add
```toml
[dependencies]
scirs2-linalg = { workspace = true, features = ["sparse", "eigensolvers", "svd"] }
scirs2-optimize = { workspace = true, features = ["nonlinear"] }
smallvec = "1.13"  # For efficient small collections
rustc-hash = "2.1"  # For fast hashing in caches
```

## Success Metrics
- All high-priority features implemented
- Test coverage > 90%
- Performance benchmarks show improvement
- Documentation complete and reviewed
- Integration with other modules verified