Simulating Heisenberg Spin Chains: Classical Methods vs. NISQ Hardware

One-dimensional Heisenberg spin chains are the standard proving ground for quantum simulation: analytically tractable enough to know the right answer, hard enough that classical cost grows quickly with chain length.

I implemented the classical side — exact diagonalisation, the density-matrix renormalisation group (DMRG), and quantum Monte Carlo — and the quantum side, a variational quantum eigensolver built in Qiskit and executed on IBM Quantum hardware, then compared accuracy and computational cost as a function of chain length.

The interesting result is not that hardware is noisy, but how the noise scales: quantifying the way gate error and decoherence degrade fidelity makes it possible to state the system sizes at which current devices remain genuinely competitive with classical simulation.