An N-particle quantum many-body state needs 2^N complex amplitudes to describe exactly on a classical computer — one number for every possible combination of up/down at every site. A real quantum simulator (trapped ions, cold atoms, superconducting qubits) has only N physical particles, so its own resources — and the cost of preparing or measuring it — scale linearly in N instead.
Classical state-vector size: 2^N complex128 amplitudes = 2^N x 16 bytes
Quantum simulator resource: N physical qubits/particles (linear)
N=20 → 16 MB N=30 → 17 GB N=40 → 17 TB N=50 → 18 PB
- N slider — grows the simulated chain. The 3D lattice always renders instantly because it tracks only a single-excitation transport amplitude per site (an O(N) observable a real analog simulator can measure directly) — never the full 2^N state.
- Run excitation — injects an excitation at the centre site and lets it spread through nearest-neighbour coupling, mirroring how a magnon or defect spreads in a real spin-chain simulator.
- Resource chart — the red curve is what an exact classical simulation of this same chain would cost in memory; the green line is what the quantum simulator itself needs. The red curve runs off the chart within a few dozen particles; the green line barely leaves the floor at N=60.
This is the core case for analog quantum simulation: build a controllable quantum system whose natural dynamics already obey the same equations as the system you want to study, and read the answer off it directly — instead of trying to classically track an exponentially large state vector.