Juq470 【2025】

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| Tip | Explanation | |-----|-------------| | Use generators | Keep data lazy; avoid converting to lists unless necessary. | | Set appropriate chunk_size | Larger chunks reduce I/O calls but increase memory; test with your dataset. | | Prefer parallel() early | Parallelism works best when applied before expensive operations. | | Profile with cProfile | Identify bottlenecks; juq470’s thin wrappers add negligible overhead. | To help you better, could you provide additional context


Input: Sparse matrix A (N×N), RHS vector b, tolerance ε, max. quantum subspace size K_max
Output: Approximate solution x̃ such that ||A x̃ – b|| / ||b|| < ε
1. Classical preconditioning: compute M⁻¹ ≈ A⁻¹ (e.g., AMG)
2. Initialise quantum subspace V = ∅
3. while residual > ε and |V| < K_max:
     a. Quantum Subspace Generation (QSG):
         i.  Prepare |b⟩ on quantum device (amplitude encoding via QRAM or iterative loading)
         ii. Apply a shallow ansatz U(θ) (hardware‑efficient) to generate candidate state |ψ⟩
         iii. Perform *Quantum Phase Estimation* (QPE) with low precision to extract dominant eigenvalues λ_k
         iv. Orthogonalise |ψ⟩ against V (via Gram‑Schmidt in Hilbert space) → |φ⟩
         v. Append |φ⟩ to V
     b. Classical Subspace Projection:
         i.  Estimate matrix elements A_ij = ⟨φ_i|A|φ_j⟩ via Hadamard‑test circuits
         ii. Form effective system A_eff y = b_eff, where b_eff_i = ⟨φ_i|b⟩
         iii. Solve for y (size |V|) classically (dense linear solve)
     c. Reconstruct approximate solution on quantum device:
         |x_q⟩ = Σ_i y_i |φ_i⟩
     d. Compute residual r = b – A x_q (classically using M⁻¹ as a surrogate)
     e. If ||r||/||b|| < ε → terminate
4. Return classical vector x̃ = M⁻¹ r + x_q (final refinement)