The Exercise

A quantum system evolves under a Hamiltonian $H = H_1 + H_2$ whose two parts do not commute. When evolution under each part alone is easier to carry out than evolution under $H$, a simulation replaces a short step of the true evolution by a product of steps under the two parts. With $A = -iH_1$ and $B = -iH_2$, in units with $\hbar = 1$, the true step over a time $t$ is $e^{t(A+B)}$.

The exercise compares this step with three products. The plainest is $e^{tA}e^{tB}$, one step of the Trotter product formula. Strang splitting uses the symmetric product $e^{tA/2}e^{tB}e^{tA/2}$, with the step under $A$ cut into two halves placed around the step under $B$. The third product, $e^{tA}e^{tB}e^{-t^2[A,B]/2}$, corrects the first by an exponential of the commutator $[A,B] = AB - BA$, the leading term of the formula of Baker, Campbell and Hausdorff. For each product the question is at what order in $t$ it departs from $e^{t(A+B)}$.

Exponentials in a Normed Algebra

The library states the estimates for two elements $A$ and $B$ of a normed algebra $\mathcal{A}$ over the real numbers, with $t$ real. The operators on a state space of finite dimension, with the operator norm, form such an algebra.

The exponential $e^X$ of an element $X$ of $\mathcal{A}$ is the sum of its power series, $$e^{X} = \sum_{n=0}^{\infty} \frac{X^n}{n!}.$$(1)

A function $f$ of a real variable with values in $\mathcal{A}$ is $O(t^k)$ near $t = 0$ when there are constants $C$ and $\delta > 0$ such that $$\|f(t)\| \le C\,|t|^k \quad \text{whenever } |t| < \delta.$$(2)

The Claim

What Is Checked

The claim is checked using the exponential (1), the commutator $[A,B]$, and the order of a function near $t = 0$ in the sense of (2).

About these constructions the library proves the three bounds (3), (4) and (5), for every such pair $A$, $B$, whether or not the two commute.