feat: vendor midstream and sublinear-time-solver libraries

Add ruvnet/midstream (AIMDS real-time inference) and ruvnet/sublinear-time-solver (sublinear optimization algorithms) as vendored dependencies under vendor/. Co-Authored-By: claude-flow <ruv@ruv.net>
2026-08-04 19:31:42 +00:00 · 2026-03-02 23:32:45 -05:00
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+# Quantum Algorithms for Sublinear Linear Systems
+
+## Executive Summary
+
+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.
+
+## Core Research Areas
+
+### 1. HHL Algorithm (Harrow-Hassidim-Lloyd)
+- **Complexity**: O(log n · κ² · 1/ε) where κ is condition number
+- **Key insight**: Exponential speedup for sparse, well-conditioned matrices
+- **Challenge**: Quantum state preparation and measurement
+
+### 2. Quantum-Inspired Classical Algorithms
+Recent breakthroughs show classical algorithms can achieve similar speedups:
+- Tang 2018: Dequantized recommendation systems
+- Gilyén et al. 2018: Quantum-inspired sublinear algorithms
+- **Our opportunity**: Combine with diagonal dominance for enhanced performance
+
+### 3. Variational Quantum Linear Solver (VQLS)
+- Near-term quantum devices (NISQ era)
+- Hybrid quantum-classical approach
+- **Application**: Small subproblems in our solver hierarchy
+
+## Implementation Plan
+
+### Phase 1: Theoretical Foundation
+1. Map diagonal dominance to quantum advantage regimes
+2. Identify quantum speedup boundaries
+3. Develop hybrid quantum-classical protocols
+
+### Phase 2: Quantum-Inspired Classical
+1. Implement sampling-based linear solvers
+2. Use quantum-inspired techniques for:
+   - Matrix inversion via sampling
+   - Low-rank approximations
+   - Spectral sparsification
+
+### Phase 3: Actual Quantum Implementation
+1. VQLS for small dense subproblems
+2. HHL for sparse components
+3. Error mitigation strategies
+
+## Key Papers
+
+1. **Harrow, Hassidim, Lloyd (2009)**: "Quantum algorithm for linear systems of equations"
+   - Original HHL algorithm
+   - arXiv:0811.3171
+
+2. **Tang (2018)**: "A quantum-inspired classical algorithm for recommendation systems"
+   - Dequantization breakthrough
+   - arXiv:1807.04271
+
+3. **Chakraborty et al. (2018)**: "The power of block-encoded matrix powers"
+   - Block encoding techniques
+   - arXiv:1804.01973
+
+4. **Bravo-Prieto et al. (2019)**: "Variational Quantum Linear Solver"
+   - NISQ-friendly approach
+   - arXiv:1909.05820
+
+5. **Childs et al. (2017)**: "Quantum algorithm for systems of linear equations with exponentially improved dependence on precision"
+   - Improved HHL
+   - arXiv:1511.02306
+
+## Performance Projections
+
+### Classical Sublinear (Current)
+- Complexity: O(poly(1/ε, 1/δ, log n))
+- 1000×1000 matrix: ~1ms
+
+### Quantum-Inspired (Projected)
+- Complexity: O(poly(log n, 1/ε))
+- 1000×1000 matrix: ~0.1ms
+- **10x improvement** over current
+
+### True Quantum (Future)
+- Complexity: O(log n · poly(κ, 1/ε))
+- 1000×1000 matrix: ~0.001ms
+- **1000x improvement** (with quantum hardware)
+
+## Integration Strategy
+
+```python
+class QuantumInspiredSolver:
+    def __init__(self, matrix, epsilon=1e-6):
+        self.matrix = matrix
+        self.epsilon = epsilon
+
+    def solve_via_sampling(self, b):
+        """
+        Quantum-inspired sampling approach
+        Based on Tang 2018 dequantization
+        """
+        # 1. Approximate matrix via sampling
+        rank = self.estimate_rank()
+        samples = self.importance_sample(rank)
+
+        # 2. Low-rank approximation
+        U, S, V = self.randomized_svd(samples)
+
+        # 3. Solve in low-rank space
+        return self.low_rank_solve(U, S, V, b)
+```
+
+## Next Steps
+
+1. **Immediate**: Implement quantum-inspired sampling techniques
+2. **Q1 2025**: Develop VQLS prototype for GPU simulation
+3. **Q2 2025**: Test on IBM Quantum / Google Cirq
+4. **Q3 2025**: Benchmark vs classical on real quantum hardware