Simulating a Small Anneal

A gate-model simulator like ours has no built-in "evolve under $H$ for time $dt$ " command — but we can build one. The trick is Trotterization: approximate the continuous-time evolution of an annealing Hamiltonian by a sequence of small, simple gate steps.

The problem with sums of non-commuting terms

The evolution operator over a small time $dt$ is $U(dt) = e^{-i\,dt\,H}$ (setting $\hbar = 1$ ). When $H = A + B$ with $[A, B] \neq 0$ — as for our $H(s) = -(1-s)X - sZ$ , since $X$ and $Z$ anticommute — we cannot simply factor $e^{-i\,dt(A+B)}$ into $e^{-i\,dt A} e^{-i\,dt B}$ . The two are equal only when $A$ and $B$ commute.

The first-order Trotter formula

For small $dt$ the Lie–Trotter formula gives a controlled approximation:

e^{-i\,dt\,(A + B)} = e^{-i\,dt\,A}\,e^{-i\,dt\,B} + O(dt^2).

The error per step is $O(dt^2)$ , set by the commutator $[A, B]$ . Splitting a total time $T$ into $N = T/dt$ steps gives total error $O(T\,dt) = O(T^2/N)$ , which shrinks as you use more, smaller steps. This is exactly how Lloyd's universal quantum simulator turns any local Hamiltonian into a gate sequence.

One step of the single-qubit anneal

Take $H(s) = -(1-s)X - sZ$ , so $A = -(1-s)X$ and $B = -sZ$ . A single Trotter step is

e^{-i\,dt\,H(s)} \approx e^{i\,dt(1-s)\,X}\;e^{i\,dt\,s\,Z}.

Now translate to gates. The simulator's rotations are

RX(\phi) = e^{-i\phi X/2}, \qquad RZ(\phi) = e^{-i\phi Z/2}.

Matching exponents, $e^{i\,dt(1-s)X} = RX\bigl(-2\,dt(1-s)\bigr)$ and $e^{i\,dt\,s\,Z} = RZ(-2\,dt\,s)$ . Because the rightmost operator acts first on a ket, apply the $Z$ rotation, then the $X$ rotation:

U_{\text{step}} = RX\bigl(-2\,dt(1-s)\bigr)\,RZ(-2\,dt\,s).

Try it

Build one first-order Trotter step of $e^{-i\,dt\,H(s)}$ at $s = 0.5$ , $dt = 0.1$ , using one $RZ$ and one $RX$ . The grader checks the resulting $2 \times 2$ unitary.

Run your code to see the quantum state.

The two rotation angles are both $-0.1$ here (since $s = 1-s = 0.5$ and $2\,dt = 0.2$ times each gives $-0.1$ ), producing a unitary very close to the identity for this small time step.