The Exercise

The Hamiltonian $H$ of a quantum system is written as a sum $H = H_1 + \dots + H_L$ of Hermitian terms that need not commute, and the evolution it generates is to be built, in units with $\hbar = 1$, from the evolutions $e^{-iH_k\Delta t}$ of the terms alone. Over a time step $\Delta t$ the terms act one at a time, $H_1$ first and $H_L$ last, and then once more in the reverse order, from $H_L$ back to $H_1$. Each term acts for a time $2\Delta t$ in all, so the step is compared with the true evolution $e^{-2iH\Delta t}$. The exercise is to show that the two differ by an error of order $\Delta t^3$, and that $m$ steps in succession differ from $e^{-2imH\Delta t}$ by at most $m\alpha\,\Delta t^3$ for a constant $\alpha$.

A Quantum System

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ An observable $A$ of $S$ is a self adjoint operator on the state space of $S$, $A^\dagger = A$, whose real spectrum is the set of its possible measured values; the Hamiltonian $H$ and its terms $H_k$ are observables of $S$.^✦ A unitary $U$ of $S$ is a closed system evolution of $S$: an operator on the state space of $S$ with $$U^\dagger U = U U^\dagger = 1.$$(1)^✦

The propagator of an observable $H$ of $S$, with Planck’s constant $\hbar$, from time $t_1$ to time $t_2$ is the unitary $$U(t_1, t_2) = \exp\bigl[-iH(t_2 - t_1)/\hbar\bigr],$$(2)^✦ which solves the Schrödinger equation, $|\psi(t_2)\rangle = U(t_1, t_2)|\psi(t_1)\rangle$. With $\hbar = 1$ and $t_1 = 0$ it is $e^{-iHt_2}$, the form of every evolution in the exercise.

The gate error $E(U, V)$ between two unitaries $U$ and $V$ of $S$ is the operator norm of their difference, $$E(U, V) = \|U - V\| = \max_{\langle\psi|\psi\rangle = 1} \bigl\|(U - V)|\psi\rangle\bigr\|.$$(3)^✦

The Symmetric Trotter Step

The symmetric Trotter step $U_{\Delta t}$ of observables $H_1, \dots, H_L$ of $S$ with time step $\Delta t$ is the unitary $$U_{\Delta t} = e^{-iH_1\Delta t} e^{-iH_2\Delta t} \cdots e^{-iH_L\Delta t}\, e^{-iH_L\Delta t} \cdots e^{-iH_2\Delta t} e^{-iH_1\Delta t},$$(4)^✦ where each factor $e^{-iH_k\Delta t}$ is the propagator (2) of $H_k$.

The Claim

What Is Checked

The claim is stated over the symmetric Trotter step (4) of the terms $H_k$, the propagator (2) of their sum $H$ over the times $2\Delta t$ and $2m\Delta t$, and the gate error (3) between two unitaries of the quantum system $S$.