VQE Ground State Energy: Transverse Field Ising Model

## Background The Variational Quantum Eigensolver (VQE) is the most widely studied algorithm for near-term quantum computers. It combines a parameterized quantum circuit (ansatz) with a classical optimizer to find the lowest eigenvalue of a Hamiltonian — the ground state energy of a quantum system. The Transverse Field Ising Model (TFIM) is a fundamental model in condensed matter physics that describes interacting quantum spins in an external magnetic field. It exhibits a quantum phase transition: at low field strengths the spins align (ferromagnetic phase), and at high field strengths they align with the external field (paramagnetic phase). The critical point occurs at h/J = 1.0 for the 1D chain. This challenge asks you to implement VQE to find the ground state energy of the 4-qubit TFIM across multiple magnetic field strengths. ## Problem Statement Given the Transverse Field Ising Model Hamiltonian for N=4 qubits with periodic boundary conditions: H = -J * Sum_i(Z_i * Z_{i+1}) - h * Sum_i(X_i) where J=1.0 (coupling constant), h varies (transverse field strength), and Z_4 = Z_0 (periodic boundary). Your task: Implement VQE to compute the ground state energy E_0(h) for each test case value of h. Design your own ansatz and choose your optimizer. ## Hamiltonian (Pauli String Format) The Hamiltonian is specified in framework-agnostic Pauli string notation. For N=4, J=1.0, h=1.0: H = -1.0 ZZII - 1.0 IZZI - 1.0 IIZZ - 1.0 ZIIZ - 1.0 XIII - 1.0 IXII - 1.0 IIXI - 1.0 IIIX For different h values, only the X-term coefficients change (multiply by h). ## Test Cases TC-1: h=0.5, Exact Energy=-4.253346, Phase=Near-critical (Public) TC-2: h=1.0, Exact Energy=-5.226252, Phase=Critical point (Public) TC-3: h=1.5, Exact Energy=-6.708204, Phase=Paramagnetic (Hidden) TC-4: h=2.0, Exact Energy=-8.472136, Phase=Deep paramagnetic (Hidden) TC-5: h=2.3, Exact Energy=-9.663825, Phase=Deep paramagnetic (Hidden) ## Scoring Accuracy (40%): Mean absolute error vs exact energy across all test cases Circuit Efficiency (25%): Normalized gate count and circuit depth Qubit Economy (15%): Number of qubits used (4 minimum) Reproducibility (10%): Coefficient of variation across 5 seeded runs Documentation (10%): Completeness of methodology metadata Tolerance: Absolute error <= 0.01 per test case for full accuracy credit. Partial credit for error <= 0.1. ## Constraints Frameworks: PennyLane, Qiskit, Cirq, Braket, CUDA-Q, or any gate-based simulator Hardware: Simulator only (no QPU required) Qubits: Minimum 4, additional ancillas allowed Seed: Use seed=42 for reproducibility runs Time limit: 10 minutes per test case on consumer hardware ## References Peruzzo et al., A variational eigenvalue solver on a photonic quantum processor (2014) Kandala et al., Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets (2017) PennyLane Ising Uprising Challenge: pennylane.ai/challenges/ising_uprising