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‎doc/pub/week10/ipynb/week10.ipynb‎

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‎doc/src/week10/programs/.ipynb_checkpoints/lipkinvqe_qpe-checkpoint.ipynb‎

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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# VQE for Lipkin Model: Complete Working Implementation\n",
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"# VQE and QPE for Lipkin Model\n",
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"\n",
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"This notebook implements **Variational Quantum Eigensolver (VQE)** for the Lipkin model using:\n",
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"We conclude by implementing the **Variational Quantum Eigensolver (VQE)** for the Lipkin model as well as the **Quantum Phase Estimation Algorithm (QPE)** using:\n",
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"- **3 qubits** to encode 5D quasi-spin space (j=2)\n",
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"- **Pauli decomposition** of the Hamiltonian\n",
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"- **Computational basis measurements** with proper basis rotations\n",
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"- **4-layer hardware-efficient ansatz** (24 parameters)\n",
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"\n",
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"\n",
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"- **Note well that this notebook does not have a VQE implementation with shots**"
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"- **Note well that this notebook does not have a VQE implementation with shots. Figures and analysis at the end was added using Claude Code**"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 1. Imports and Setup"
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"## Imports and Setup"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 2. Pauli Matrices and 3-Qubit Operators"
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"## Pauli Matrices and 3-Qubit Operators"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 3. Quantum Gates\n",
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"## Quantum Gates\n",
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"\n",
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"**Critical:** For Y measurements, we use **H·S†** rotation.\n",
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"\n",
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 4. State Encoding: j=2 Quasi-Spin to 3 Qubits\n",
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"## State Encoding: j=2 Quasi-Spin to 3 Qubits\n",
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"\n",
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"| Quasi-spin | Qubit State | Binary |\n",
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"|------------|-------------|--------|\n",
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 5. Lipkin Hamiltonian Construction\n",
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"## Lipkin Hamiltonian Construction\n",
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"\n",
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"$$H = \\frac{\\varepsilon}{2}\\left(J_z - \\frac{N}{2}\\right) + \\frac{V}{2}(J_+^2 + J_-^2) + \\frac{W}{2}J_z^2$$\n",
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"\n",
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 6. Exact Diagonalization"
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"## Exact Diagonalization"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 7. Pauli Decomposition"
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"## Pauli Decomposition"
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]
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},
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 8. Computational Basis Measurements\n",
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"## Computational Basis Measurements\n",
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"\n",
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"**CRITICAL FIX:** Use H·S† for Y measurements!\n",
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"\n",
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 9. Hardware-Efficient Ansatz (4 Layers)\n",
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"## Hardware-Efficient Ansatz (4 Layers)\n",
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"\n",
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"Circuit structure per layer:\n",
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"- Ry rotations on all 3 qubits\n",
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 10. VQE with Exact Evaluation\n",
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"## VQE with Exact Evaluation\n",
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"\n",
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"Now we test VQE without shot noise to verify everything works. Note that we do not implement the VQE with shots"
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"Now we test VQE without shot noise to verify everything works. Note that we do not implement the VQE with shots."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 11. QPE Parameters and Phase Encoding"
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"## QPE Parameters and Phase Encoding"
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]
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},
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 12. Quantum Fourier Transform and Controlled-$U^{2^k}$"
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"## Quantum Fourier Transform and Controlled-$U^{2^k}$"
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]
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},
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"cell_type": "markdown",
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"source": [
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"## 13. Full QPE Circuit"
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"## Full QPE Circuit"
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]
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 14. Run QPE — Three Trial States"
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"## Run QPE — Three Trial States"
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 15. QPE Results — Visualisation"
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"## QPE Results — Visualisation"
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]
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},
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 16. Quantum Fourier Transform — Detailed Verification"
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"## Quantum Fourier Transform — Detailed Verification"
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]
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},
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{

‎doc/src/week10/programs/.ipynb_checkpoints/qft_qpe_notebook-checkpoint.ipynb‎

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‎doc/src/week10/programs/.ipynb_checkpoints/week10-checkpoint.ipynb‎

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‎doc/src/week10/programs/lipkinvqe_qpe.ipynb‎

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"cell_type": "markdown",
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"metadata": {},
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"source": [
7-
"# VQE for Lipkin Model: Complete Working Implementation\n",
7+
"# VQE and QPE for Lipkin Model\n",
88
"\n",
9-
"This notebook implements **Variational Quantum Eigensolver (VQE)** for the Lipkin model using:\n",
9+
"We conclude by implementing the **Variational Quantum Eigensolver (VQE)** for the Lipkin model as well as the **Quantum Phase Estimation Algorithm (QPE)** using:\n",
1010
"- **3 qubits** to encode 5D quasi-spin space (j=2)\n",
1111
"- **Pauli decomposition** of the Hamiltonian\n",
1212
"- **Computational basis measurements** with proper basis rotations\n",
1313
"- **4-layer hardware-efficient ansatz** (24 parameters)\n",
1414
"\n",
1515
"\n",
16-
"- **Note well that this notebook does not have a VQE implementation with shots**"
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"- **Note well that this notebook does not have a VQE implementation with shots. Figures and analysis at the end was added using Claude Code**"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 1. Imports and Setup"
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"## Imports and Setup"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 2. Pauli Matrices and 3-Qubit Operators"
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"## Pauli Matrices and 3-Qubit Operators"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 3. Quantum Gates\n",
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"## Quantum Gates\n",
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"\n",
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"**Critical:** For Y measurements, we use **H·S†** rotation.\n",
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"\n",
@@ -156,7 +156,7 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 4. State Encoding: j=2 Quasi-Spin to 3 Qubits\n",
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"## State Encoding: j=2 Quasi-Spin to 3 Qubits\n",
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"\n",
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"| Quasi-spin | Qubit State | Binary |\n",
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"|------------|-------------|--------|\n",
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 5. Lipkin Hamiltonian Construction\n",
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"## Lipkin Hamiltonian Construction\n",
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"\n",
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"$$H = \\frac{\\varepsilon}{2}\\left(J_z - \\frac{N}{2}\\right) + \\frac{V}{2}(J_+^2 + J_-^2) + \\frac{W}{2}J_z^2$$\n",
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"\n",
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 6. Exact Diagonalization"
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"## Exact Diagonalization"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 7. Pauli Decomposition"
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"## Pauli Decomposition"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 8. Computational Basis Measurements\n",
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"## Computational Basis Measurements\n",
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"\n",
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"**CRITICAL FIX:** Use H·S† for Y measurements!\n",
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"\n",
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 9. Hardware-Efficient Ansatz (4 Layers)\n",
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"## Hardware-Efficient Ansatz (4 Layers)\n",
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"\n",
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"Circuit structure per layer:\n",
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"- Ry rotations on all 3 qubits\n",
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 10. VQE with Exact Evaluation\n",
580+
"## VQE with Exact Evaluation\n",
581581
"\n",
582-
"Now we test VQE without shot noise to verify everything works. Note that we do not implement the VQE with shots"
582+
"Now we test VQE without shot noise to verify everything works. Note that we do not implement the VQE with shots."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 11. QPE Parameters and Phase Encoding"
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"## QPE Parameters and Phase Encoding"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 12. Quantum Fourier Transform and Controlled-$U^{2^k}$"
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"## Quantum Fourier Transform and Controlled-$U^{2^k}$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 13. Full QPE Circuit"
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"## Full QPE Circuit"
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]
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 14. Run QPE — Three Trial States"
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"## Run QPE — Three Trial States"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 15. QPE Results — Visualisation"
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"## QPE Results — Visualisation"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 16. Quantum Fourier Transform — Detailed Verification"
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"## Quantum Fourier Transform — Detailed Verification"
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]
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},
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{

‎doc/src/week10/programs/qft_qpe_notebook.ipynb‎

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"| 6 | Pauli Decomposition of 4×4 Matrices |\n",
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"| 7 | Quantum Phase Estimation Algorithm |\n",
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"| 8 | QPE Example: Eigenvalues of a 4×4 Hamiltonian |\n",
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"| 9 | Summary & Key Formulae |\n"
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"| 9 | Summary & Key Formulae |\n",
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"\n",
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"At the end of this notebook, you will also find a detailed comparison between the VQE and the QPE for the Lipkin model of project 1.\n"
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]
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},
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{

‎doc/src/week10/programs/qpe_lipkin_results.png‎

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