Project: Hamiltonian Simulation Study

Simulating quantum dynamics — evolving a state under $e^{-iHt}$ — was the original motivation for quantum computers. There is no single best method: product formulas (Trotter), linear combination of unitaries, qubitization, and classical tensor networks each win in different regimes. This capstone implements the workhorse Trotter step on a concrete model so you can study the accuracy–cost trade-off that distinguishes the methods.

The model

We simulate the two-site transverse-field Ising model

The two pieces $H_{ZZ} = J Z_0 Z_1$ and $H_X = h(X_0 + X_1)$ do not commute, so $e^{-iHt} \neq e^{-iH_{ZZ}t}e^{-iH_X t}$ in general — which is exactly why a method is needed.

First-order Trotter

The Lie–Trotter formula splits the evolution into a product of the individually-easy exponentials:

Each step of size $\delta t = t/n$ is a short circuit. The $ZZ$ factor compiles to $\texttt{CX};\ \texttt{RZ}(2J\delta t);\ \texttt{CX}$ and the field factor to two $\texttt{RX}(2h\delta t)$ rotations, using $R_Z(\theta) = e^{-i\theta Z/2}$ (hence the factor of $2$).

Comparing methods

The first-order error per step scales as $\mathcal{O}(\delta t^2)$, so total error is $\mathcal{O}(t^2/n)$ — halving the step size halves the error but doubles the gate count. A second-order (symmetric) Trotter step costs roughly $1.5\times$ as many gates but cuts the error to $\mathcal{O}(t^3/n^2)$, usually a better deal. Post-Trotter methods (qubitization, LCU) reach error $\varepsilon$ with cost scaling as $\log(1/\varepsilon)$ instead of $\text{poly}(1/\varepsilon)$ — asymptotically far better, but with larger constant overhead. Studying which wins for a given $(t, \varepsilon)$ on a given $H$ is precisely a "Hamiltonian simulation study."

Try it

Build one first-order Trotter step for $J = 1$, $h = 0.5$, $\delta t = 0.3$ starting from $|00\rangle$. The grader checks the resulting statevector.

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