1- TITLE: Lipkin Model Notebook (DocOnce Conversion)
2- AUTHOR:
3- DATE:
1+ TITLE: Lipkin Model Notebook
2+ AUTHOR: Morten Hjorth-Jensen
3+ DATE: Spring 2026
44
55
66# VQE for Lipkin Model: Complete Working Implementation
@@ -11,15 +11,6 @@ This notebook implements **Variational Quantum Eigensolver (VQE)** for the Lipki
1111- **Computational basis measurements** with proper basis rotations
1212- **4-layer hardware-efficient ansatz** (24 parameters)
1313
14- ## Key Corrections Applied:
15- 1. ✅ **Proper CNOT gates** (correct tensor products)
16- 2. ✅ **4-layer ansatz** (sufficient expressivity)
17- 3. ✅ **Correct Y measurement** (H·S†, not S†·H)
18-
19- ## Expected Results:
20- - Exact ground state: **E₀ = -3.0**
21- - VQE (exact evaluation): **-3.0000** (error < 10⁻¹⁰)
22- - VQE (with shots): **~-2.95 to -3.00** (depending on shots and optimization)
2314
2415
2516## 1. Imports and Setup
@@ -553,30 +544,3 @@ plt.show()
553544!ec
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555546
556- ## 13. Summary
557-
558- ### ✅ All Critical Bugs Fixed:
559-
560- 1. **CNOT Gates:** Proper 3-qubit gates using `kron(CNOT_2qubit, I)`
561- 2. **Ansatz Depth:** 4 layers (24 parameters) for sufficient expressivity
562- 3. **Y Measurement:** H·S† rotation (NOT S†·H, NOT H·S)
563-
564- ### 📊 Results:
565-
566- - **Exact energy:** -3.0000000000
567- - **VQE (exact eval):** Essentially exact (error < 10⁻⁹)
568- - **VQE (with shots):** Within shot noise of ground state
569-
570- ### 🔬 Physical System:
571-
572- - **j = 2** (N = 4 particles)
573- - **5D quasi-spin space** encoded in 3 qubits
574- - **Ground state:** Superposition of |000⟩, |010⟩, |100⟩
575- - **Correlation energy:** -1.0 (reference state at -2.0)
576-
577- ### ✓ Verification:
578-
579- - Hamiltonian is Hermitian ✓
580- - Pauli coefficients all real ✓
581- - Measurements preserve variational principle ✓
582- - VQE converges to ground state ✓
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