What is Trotterization?

Because the Ising coupling term and the transverse-field term don't commute, the exact evolution operator can't be built directly from simple gates. Trotterization approximates it by alternately applying each term's own easy gate for a short step δt, repeating many times — the same approach used on real digital quantum simulators.

Why does the fidelity bar fall over time?

Each Trotter step has an approximation error of order O(δt²). The fidelity bar compares the coarse-step state to a much finer-grained reference evolution of the same physical time, so coarser steps (fewer 'Trotter steps / unit time') make the error accumulate faster.

What do the shrinking arrows mean?

Each arrow is a spin's reduced Bloch vector, computed by tracing the full many-body statevector down to one site. As the ZZ coupling entangles neighboring spins, each individual spin's reduced state becomes mixed and its Bloch vector shrinks below the disc's edge — a direct visual signature of quantum entanglement.

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Trotterized Ising Chain (2D)

2D digital quantum simulation of a transverse-field Ising spin chain: a Trotter-Suzuki gate circuit evolves the full statevector while fidelity against a fine-grained reference is tracked live, and each spin's reduced Bloch vector shrinks visibly as entanglement builds.

Quantum Physics2DAdvanced60 FPS📱 Mobile-adapted⇄ 3D version

This 2D companion runs the exact same statevector Trotter-Suzuki engine as the 3D version — a chain of N spins under the transverse-field Ising Hamiltonian H = -J ΣZ_kZ_k+1 - h ΣX_k, evolved by alternating the diagonal ZZ phase gate with the single-qubit X rotation — but rendered as a flat row instead of an orbiting scene. Each spin's exact reduced Bloch vector is drawn as an arrow inside a unit disc, its length shrinking as nearest-neighbor entanglement grows from the |+…+⟩ quench, while a live strip chart plots the Trotter fidelity and the chain's mean magnetization over a rolling time window.

⚙ Under the hood

2D digital quantum simulation of a transverse-field Ising spin chain: a Trotter-Suzuki gate circuit evolves the full 2^N-dimensional statevector, tracking fidelity against a fine-grained reference and each spin's reduced Bloch vector as entanglement builds.

quantum simulationTrotter-SuzukiIsing modelBloch vectorentanglementstatevector

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install