111 lines
3.6 KiB
Markdown
111 lines
3.6 KiB
Markdown
# Quantum Algorithms for Sublinear Linear Systems
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## Executive Summary
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Quantum computing offers potential exponential speedups for linear system solving through algorithms that exploit quantum superposition and entanglement. This research plan explores the intersection of quantum algorithms with our sublinear-time classical solvers.
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## Core Research Areas
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### 1. HHL Algorithm (Harrow-Hassidim-Lloyd)
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- **Complexity**: O(log n · κ² · 1/ε) where κ is condition number
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- **Key insight**: Exponential speedup for sparse, well-conditioned matrices
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- **Challenge**: Quantum state preparation and measurement
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### 2. Quantum-Inspired Classical Algorithms
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Recent breakthroughs show classical algorithms can achieve similar speedups:
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- Tang 2018: Dequantized recommendation systems
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- Gilyén et al. 2018: Quantum-inspired sublinear algorithms
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- **Our opportunity**: Combine with diagonal dominance for enhanced performance
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### 3. Variational Quantum Linear Solver (VQLS)
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- Near-term quantum devices (NISQ era)
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- Hybrid quantum-classical approach
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- **Application**: Small subproblems in our solver hierarchy
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## Implementation Plan
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### Phase 1: Theoretical Foundation
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1. Map diagonal dominance to quantum advantage regimes
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2. Identify quantum speedup boundaries
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3. Develop hybrid quantum-classical protocols
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### Phase 2: Quantum-Inspired Classical
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1. Implement sampling-based linear solvers
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2. Use quantum-inspired techniques for:
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- Matrix inversion via sampling
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- Low-rank approximations
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- Spectral sparsification
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### Phase 3: Actual Quantum Implementation
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1. VQLS for small dense subproblems
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2. HHL for sparse components
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3. Error mitigation strategies
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## Key Papers
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1. **Harrow, Hassidim, Lloyd (2009)**: "Quantum algorithm for linear systems of equations"
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- Original HHL algorithm
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- arXiv:0811.3171
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2. **Tang (2018)**: "A quantum-inspired classical algorithm for recommendation systems"
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- Dequantization breakthrough
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- arXiv:1807.04271
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3. **Chakraborty et al. (2018)**: "The power of block-encoded matrix powers"
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- Block encoding techniques
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- arXiv:1804.01973
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4. **Bravo-Prieto et al. (2019)**: "Variational Quantum Linear Solver"
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- NISQ-friendly approach
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- arXiv:1909.05820
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5. **Childs et al. (2017)**: "Quantum algorithm for systems of linear equations with exponentially improved dependence on precision"
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- Improved HHL
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- arXiv:1511.02306
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## Performance Projections
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### Classical Sublinear (Current)
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- Complexity: O(poly(1/ε, 1/δ, log n))
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- 1000×1000 matrix: ~1ms
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### Quantum-Inspired (Projected)
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- Complexity: O(poly(log n, 1/ε))
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- 1000×1000 matrix: ~0.1ms
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- **10x improvement** over current
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### True Quantum (Future)
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- Complexity: O(log n · poly(κ, 1/ε))
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- 1000×1000 matrix: ~0.001ms
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- **1000x improvement** (with quantum hardware)
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## Integration Strategy
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```python
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class QuantumInspiredSolver:
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def __init__(self, matrix, epsilon=1e-6):
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self.matrix = matrix
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self.epsilon = epsilon
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def solve_via_sampling(self, b):
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"""
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Quantum-inspired sampling approach
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Based on Tang 2018 dequantization
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"""
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# 1. Approximate matrix via sampling
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rank = self.estimate_rank()
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samples = self.importance_sample(rank)
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# 2. Low-rank approximation
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U, S, V = self.randomized_svd(samples)
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# 3. Solve in low-rank space
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return self.low_rank_solve(U, S, V, b)
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```
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## Next Steps
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1. **Immediate**: Implement quantum-inspired sampling techniques
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2. **Q1 2025**: Develop VQLS prototype for GPU simulation
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3. **Q2 2025**: Test on IBM Quantum / Google Cirq
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4. **Q3 2025**: Benchmark vs classical on real quantum hardware |