# State-dependent Trotter Limits and their approximations

Daniel Burgarth  Center for Engineered Quantum Systems, Macquarie University,
2109 NSW, Australia  Niklas Galke  Física Teòrica: Informació i Fenòmens
Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193
Bellaterra, Spain  Alexander Hahn  Center for Engineered Quantum Systems,
Macquarie University, 2109 NSW, Australia  Lauritz van Luijk  Institut für
Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167
Hannover, Germany

###### Abstract

The Trotter product formula is a key instrument in numerical simulations of
quantum systems. However, computers cannot deal with continuous degrees of
freedom, such as the position of particles in molecules, or the amplitude of
electromagnetic fields. It is therefore necessary to discretize these
variables to make them amenable to digital simulations. Here, we give
sufficient conditions to conclude the validity of this approximate discretized
physics. Essentially, it depends on the state-dependent Trotter error, for
which we establish explicit bounds that are also of independent interest.

_Introduction._ —The Trotter product formula is a central tool in the study of
quantum dynamics with a vast amount of applications, such as quantum computing
[1, 2, 3, 4], quantum field theory [5, 6, 7, 8] and its path integral
formulation [9], quantum control [10, 11, 12], open quantum systems [13],
quantum optics [14], Floquet dynamics [15, 16, 17] and quantum many body
systems [18, 19, 20, 21]. It allows to split the time-evolution of a quantum
system, that is hard or cost-expensive to implement, into simpler components.
Due to this feature, it has become the standard contraption for numerically
simulating the dynamics of large quantum systems. Nevertheless, two important
questions regarding these simulations stay open. On one hand, it is not clear
how the error due to the Trotter approximation depends on the input state of
the system. For example, one would expect that the approximation becomes
better when only low-energy states are considered. On the other hand, digital
computers – no matter if classical or quantum – can only simulate a finite
number of levels. However, many relevant physical systems are actually
represented by continuous degrees of freedom, i.e. infinite dimensions.
Therefore, the common practice is to truncate the Hilbert space on some finite
level [22, 23, 17, 24] and deal with the finite-dimensional truncated
operators. These truncations, however, are only approximations to the full
system. Hence in total, this simulation procedure involves two simultaneous
approximations: First, approximating the time-evolution of the full quantum
system by some finite-dimensional truncations; second, approximating the full
time-evolution of the quantum system by the Trotter product formula. We
summarize the interplay of the two approximations in Fig. 1.

$H^{(1)}$, $H^{(2)}$continuousHamiltonians$H=H^{(1)}+H^{(2)}$quantum
simulation$H_{d}^{(1)}=P_{d}H^{(1)}P_{d}$$H_{d}^{(2)}=P_{d}H^{(2)}P_{d}$$H_{d}=H_{d}^{(1)}+H_{d}^{(2)}$Trottermathematically
hardnot digitally simulableTruncationmathematically easydigitally
simulablemathematically easydigitally simulableTrotterthis paper? Figure 1: In
the simulation of a quantum system one starts with two simple continuous-
degree Hamiltonians $H^{(1)}$ and $H^{(2)}$. Then, the goal is to simulate the
dynamics under $H=H^{(1)}+H^{(2)}$ through the Trotter product formula (blue
path). However, due to its continuous nature, it is not possible to implement
this strategy on digital simulators such as computers. Therefore, it is common
practice to follow the green path instead. This is, one first truncates
$H^{(1)}$ and $H^{(2)}$ to a finite level $d$. Afterwards, one implements the
dynamics under the sum of the truncated Hamiltonians
$H_{d}=H_{d}^{(1)}+H_{d}^{(2)}$ via the Trotter product formula.
Unfortunately, $H_{d}$ does not necessarily cover the physics of the full
model $H$ correctly, so that this method can lead to deceptive results. Here,
we establish a method to close this gap, which is indicated by the dashed
green arrow.

A priori, it is not self-evident whether they are compatible with each other.
In fact, we will discuss a simple and well-known example, where it seems that
they are not. In this context, the main obstacle is the question of
convergence, i.e.: Is the infinite-dimensional Trotter product formula a valid
approximation? Whilst it always converges in finite dimensions [4, 25], it
only converges under certain conditions in the infinite-dimensional case [26,
27]. Therefore, a rigorous numerical treatment of infinite-dimensional
truncated operators would need to additionally deliver a reasoning why the
Trotterization and truncation procedures are nonconflicting. Such a discussion
about the validity of the simulations is missing in the literature, making
numerical results not fully conclusive.

In this paper, we address both of the aforementioned questions and show that
they are closely related. That is, we first establish state-dependent bounds
for the Trotter error. Afterwards, we use them to provide sufficient
conditions on the truncations, which ensure the validity of simultaneous
Trotterization. Firstly, as a technical condition, the Hamiltonians of
consideration have to be jointly approximable by the _same_ truncation scheme.
Secondly, as one would expect intuitively, the state-dependent error of the
finite-dimensional Trotter approximation has to eventually saturate in the
truncation dimension. In this case, our method allows to infer bounds for the
Trotter error in infinite dimensions from the state-dependent ones of their
finite-dimensional truncations. These results have applications to a wide
class of physical models. Not only do even the simplest textbook examples such
as the quantum harmonic oscillator and the particle in a box have continuous
degrees. But also e.g. in quantum optics, the Rabi-model [28, 29] and the
spin-boson model [30], which describe the interaction of matter with light,
are infinite-dimensional Hamiltonians. In quantum chemistry, atoms and
molecules have continuous positions and momenta. The simulation of their
chemical behaviours has been proposed as an impactful application of the
emerging field of quantum simulation [1, 2]. In addition, modern quantum
information makes heavy use of infinite-dimensional systems, such as GKP
qubits [31] or superconducting transmon qubits [32, 33], which were the
platform for the “quantum supremacy” experiment [34].

Additionally, our state-dependent Trotter bound is of independent and broader
interest in the field of quantum information and computation beyond the scope
of this paper. Recently, such state-dependent Trotter error bounds enjoy a
considerable interest in particular in the area of Hamiltonian simulation. For
instance in [35, 36], the asymptotic scaling of the finite-dimensional Trotter
error is derived when the input state is only supported in a (low-energy)
subspace. These results rely on strong assumptions on the respective
Hamiltonians in order to bound the leakage from this (low-energy) subspace.
The authors in [37] study energy-constrained distances between time evolutions
generated by Hamiltonians with continuous degrees of freedom. Their results
have applications in the realm of Trotterization but only hold for special
Hamiltonians generating a particular type of dynamics. In addition, in [38,
39] state-dependent Trotter error bounds are derived for certain (time-
dependent) Hamiltonians. These bounds rely on restrictive assumptions on the
structure of the commutator applied to the input state. The advantage of our
bound, compared to the existing bounds in the literature, is its validity _for
all_ finite-dimensional Hamiltonians. Furthermore, it is an _explicit_ bound,
which shows a particularly simple dependence on the input state. By taking
linear combinations of input vectors, we are immediately able to bound the
Trotter convergence speed in arbitrary subspaces.

_Setting._ —Before turning to our main results, let us briefly summarise the
setting, in which the Trotter product formula and its truncated versions are
studied. A more elaborate introduction is given in the Supplementary Material,
Sec. A. We will consider two Hamiltonians $H^{(1)}$ and $H^{(2)}$ acting on an
infinite-dimensional Hilbert space $\mathcal{H}$. This is, two self-adjoint
operators, which are defined on a dense domain of input states
$\mathcal{D}(H^{(1)})$, $\mathcal{D}(H^{(2)})$, respectively. The domains
encode important features of the physical system, such as boundary conditions.
It often suffices to look at a core, which is a particular subspace of the
domain that already carries all the relevant information. See the
Supplementary Material, Sec. A for details. The time-evolution under each
Hamiltonian is given by a unitary operator
$U^{(1)}(t)=\mathrm{e}^{-\mathrm{i}tH^{(1)}}$,
$U^{(2)}(t)=\mathrm{e}^{-\mathrm{i}tH^{(2)}}$, respectively. Now, the Trotter
product formula, _if_ converging, approximates the dynamics
$U(t)=\mathrm{e}^{-\mathrm{i}tH}$ of a self-adjoint operator $H$ by the
$n$-fold product of the individual unitaries
$X(t/n)^{n}\equiv[U^{(1)}(t/n)U^{(2)}(t/n)]^{n}$. On the common core $H$
restricts to the sum of $H^{(1)}$ and $H^{(2)}$, see the Supplementary
Material, Sec. C. In order to construct finite-dimensional approximations of
the above operators, we will first fix a basis
$\\{|j\rangle:j\in\mathbb{N}_{0}\\}$ of the Hilbert space $\mathcal{H}$. Then,
we obtain a truncated, finite-dimensional Hilbert space
$V_{d}=\mathrm{span}\\{|0\rangle,\dots,|d-1\rangle\\}$ of dimension $d<\infty$
simply by gathering the first $d$ basis vectors. We can project any vector in
$\mathcal{H}$ onto $V_{d}$ with $P_{d}=\sum_{j=0}^{d-1}|j\rangle\langle j|$.
Similarly, for our Hamiltonians we receive their truncated versions by
$H_{d}^{(1)}=P_{d}H^{(1)}P_{d}$, $H_{d}^{(2)}=P_{d}H^{(2)}P_{d}$ and
$H_{d}=P_{d}HP_{d}$, if existing. All the truncated Hamiltonians only act on
the finite-dimensional $V_{d}$ and can, therefore, be represented by Hermitian
$d\times d$ matrices. They generate finite-dimensional unitary dynamics on
$V_{d}$ via $U_{d}^{(1)}(t)=\mathrm{e}^{-\mathrm{i}tH_{d}^{(1)}}$,
$U_{d}^{(2)}(t)=\mathrm{e}^{-\mathrm{i}tH_{d}^{(2)}}$ and
$U_{d}(t)=\mathrm{e}^{-\mathrm{i}tH_{d}}$, respectively. One goal of this
paper is to give conditions on the finite-dimensional truncated Trotter
product formula which imply the convergence of the full infinite-dimensional
case. We will explain in the Supplementary Material, Sec. A that it is
necessary to include the dependence on an input state (also see [40, Sec. 3]
and [41, Sec. VIII.8]). That is, by studying
$[U_{d}^{(1)}(t/n)U_{d}^{(2)}(t/n)]^{n}|\psi_{d}\rangle\xrightarrow{n\to\infty}U_{d}(t)|\psi_{d}\rangle$,
we would like to draw conclusions about
$[U^{(1)}(t/n)U^{(2)}(t/n)]^{n}|\psi\rangle\xrightarrow{n\to\infty}U(t)|\psi\rangle$,
where $|\psi_{d}\rangle=P_{d}|\psi\rangle$. To this end, we introduce the
state-dependent Trotter error in the finite-dimensional case

$\displaystyle b_{d}^{(n)}(|\psi\rangle;t):=$
$\displaystyle\big{\|}\big{(}[U_{d}^{(1)}(t/n)U_{d}^{(2)}(t/n)]^{n}-U_{d}(t)\big{)}|\psi_{d}\rangle\big{\|},$
(1)

from which we can inherit a dimension-independent Trotter error by taking the
limit $d\to\infty$

$b^{(n)}(|\psi\rangle;t):=\limsup_{d\to\infty}b_{d}^{\scaleobj{0.8}{(n)}}(|\psi\rangle;t).$
(2)

Notice that $b^{(n)}(|\psi\rangle;t)$ might reach the maximum distance of $2$
even though $b_{d}^{(n)}(|\psi\rangle;t)$ is considerably smaller for a
relatively large $d$.

_Main Results._ —Our first main result is an explicit state-dependent Trotter
error bound for arbitrary _finite-dimensional_ systems. Usually, when
computing the norm Trotter error, one would reduce the difference
$X(t/n)^{n}-U(t)$ to one period $X(t/n)-U(t/n)$ by using a telescope sum
identity, see for instance [25, 4]. This approach introduces additional
unitaries, which can be removed through sub-multiplicativity of the matrix
norm. However in the state-dependent case, these unitaries lead to a non-
trivial rotation of the input state, which hinges on the total evolution time
$t$. For this reason, we need to follow a conceptionally different approach
than all other proofs of Trotter convergence in the literature. Our proof is
inspired by [42, Lemma 1]. We treat the generator of the Trotterized evolution
$X(t/n)^{n}$ as a $\frac{2t}{n}$-periodic time-dependent Hamiltonian
$\tilde{H}(s)$, which is repeatedly switching between $H^{(1)}$ and $H^{(2)}$
after a time $s=\frac{t}{n}$. Then, the distance between the Trotterized and
the target evolution is related to an integral action
$S_{21}(\tau)=\int_{0}^{\tau}\frac{1}{2}H-\tilde{H}(s)\mathrm{d}s$, which
vanishes after a full cycle, $S_{21}(2t)=0$. Thus, applied to an eigenstate of
$H$, the integral action can essentially be bounded by the maximum of the
distance between $H^{(1)}$ or $H^{(2)}$ and the average generator
$\frac{1}{2}H$. See the Supplementary Material, Sec. B. In total, we obtain
the following bound.

######  Main Result 1 (Finite-dimensional state-dependent Trotter error
bound).

Consider the case of the _finite-dimensional_ Trotter formula, i.e. the two
Hamiltonians $H^{(1)}$ and $H^{(2)}$ are acting on a finite-dimensional
Hilbert space $\mathcal{H}$. If $|\varphi\rangle\in\mathcal{H}$ is an
eigenstate of $H^{(1)}+H^{(2)}$ according to the eigenvalue equation
$(H^{(1)}+H^{(2)})|\varphi\rangle=h|\varphi\rangle$, then the Trotter error is
bounded by

$\displaystyle\left\|\left([U^{(1)}(t/n)U^{(2)}(t/n)]^{n}-U(t)\right)|\varphi\rangle\right\|$
$\displaystyle\leq\frac{2t^{2}}{n}\mathrm{max}\left(\Big{\|}\Big{[}H^{(1)}-\frac{h}{2}\Big{]}^{2}|\varphi\rangle\Big{\|},\Big{\|}\Big{[}H^{(2)}-\frac{h}{2}\Big{]}^{2}|\varphi\rangle\Big{\|}\right).$
(3)

The usual error bounds in a matrix norm essentially quantify a worst-case
scenario. Instead, this state-dependent bound still accounts for the
relationship between the particular input state $|\varphi\rangle$ and the
fidelity of Trotterization. Therefore, it can lead to a tighter scaling for
certain $|\varphi\rangle$. For instance, this is crucial when considering
infinite-dimensional Trotter problems. Approximating those with increasing
cutoff dimensions $d$ adds more and more states to the Hilbert space $V_{d}$.
In turn, the worst-case Trotter error grows with the truncation dimension.
Since one is ultimately interested in the limit $d\to\infty$, this is a rather
unfavourable property and state-dependent error bounds are mandatory. We study
these truncated product formulas in our next main result. It manifests a
method to check convergence of the Trotter product formula which is
particularly well-suited to numerical simulations. Intuitively, if one seeks
out that the state-dependent Trotter error $b_{d}^{(n)}(|\psi\rangle;t)$
saturates with increasing $d$, one would expect that the infinite-dimensional
analogue converges for this particular $|\psi\rangle$. Indeed, this is what we
find.

######  Main Result 2 (Truncation of an infinite-dimensional Trotter product).

The _infinite-dimensional_ Trotter product formula converges for all
$|\psi\rangle\in\mathcal{H}$,

$\left\lparen
U^{\scaleobj{0.8}{(1)}}(t/n)U^{\scaleobj{0.8}{(2)}}(t/n)\right\rparen^{n}|\psi\rangle\xrightarrow{n\to\infty}U(t)|\psi\rangle,$
(4)

if the following two conditions are satisfied:

  1. (i)

$H^{(1)}$ and $H^{(2)}$ can be simultaneously approximated with $H_{d}^{(1)}$
and $H_{d}^{(2)}$ by the same truncation scheme. That is,
$\mathcal{V}=\bigcup_{d}V_{d}$ is a common core of $H^{(1)}$ and $H^{(2)}$.

  2. (ii)

For all total evolution times $t\in\mathbb{R}$, the dimension-independent
Trotter error goes to zero, $b^{(n)}(|\psi\rangle;t)\to 0$, as $n\to\infty$.

In this case, $U(t)$ gives a unitary dynamics, whose generator $H$ is self-
adjoint and agrees with $(H^{\scaleobj{0.8}{(1)}}+H^{\scaleobj{0.8}{(2)}})$,
wherever both are defined. A generalized version to non-unitary dynamics can
be found in the Supplementary Material, Sec. C, see Thm. 7.

In order to prove this, we first show that $U_{d}(t)|\psi\rangle$ has a limit
as $d\to\infty$, i.e. the finite-dimensional approximations applied to a state
$|\psi\rangle$ converge to a well-defined $|\phi(t)\rangle$. Since the Trotter
limit $n\to\infty$ is always well-defined in finite dimensions [25], the
$U_{d}(t)$ can be obtained by Trotterization. Then, we prove that the
infinite-dimensional Trotter formula,
$X(t/n)^{n}|\psi\rangle\to|\phi(t)\rangle$ as $n\to\infty$, converges to the
same limit $|\phi(t)\rangle$. The last step is to show that this limit is
indeed governed by a unitary time-evolution, whose generating Hamiltonian
agrees with $H^{(1)}+H^{(2)}$ on $\mathcal{V}$ These ramifications are derived
using a modification of the _Trotter-Kato approximation theorems_ [43, Thm.
4.8, Thm. 4.9] inspired by [44, Thm. 2.3]. Notice that we only have
$\mathcal{V}\subsetneq\mathcal{H}$, thus in particular
$\mathcal{V}\neq\mathcal{H}$. Furthermore, two Hamiltonians may have a non-
overlapping domains, see e.g. [11, Example 3.8]. This situation is excluded by
our common core assumption.

Our truncation main result establishes a way to obtain explicit error bounds
for the infinite-dimensional Trotter product formula once the finite-
dimensional case is under control: A simple calculation shows that (see the
Supplementary Material, Sec. C)

$\displaystyle
b^{(n)}(|\psi\rangle;t)\leq\bigg{\lparen}\sum_{j=1}^{d}\big{|}\langle
j|\psi\rangle\big{|}^{2}b^{(n)}(|j\rangle;t)^{2}\bigg{\rparen}^{1/2}.$ (5)

Thus, in order to conclude that the Trotter product formula converges for all
$|\psi\rangle\in\mathcal{H}$, it suffices to require
$b^{(n)}\left\lparen|j\rangle;t\right\rparen\to 0$ as $n\to\infty$ for all
basis vectors $|j\rangle$. This is because if $|\psi\rangle\in\mathcal{V}$ –
and hence $|\psi\rangle\in V_{d}$ for sufficiently large $d$ – then Eq. (5)
also goes to zero.

_Examples_.—Let us numerically study an example of a convergent and a non-
convergent Trotter scenario. We denote the position operator with $Q$ and the
momentum operator with $P$. The truncation is done in the Fock basis
$\\{|m\rangle\\}$, which is a common core of all the considered operators.
That is, we project with $P_{d}=\sum_{m=0}^{d-1}|m\rangle\langle m|$ onto the
finite-dimensional Hilbert space $V_{d}$. Fig. 2 shows the case of
$H^{(1)}=\frac{1}{2}\left(Q^{2}+P^{2}\right)$, i.e. the quantum harmonic
oscillator, and $H^{(2)}=\frac{1}{2}\left(QP+PQ\right)$, i.e. the Hamiltonian
which generates the squeezing transformation. It is known that this infinite-
dimensional Trotter problem converges for all $|\psi\rangle\in\mathcal{H}$,
see for instance [27]. Indeed, we numerically find that the Trotter error on
the finite-dimensional truncations $V_{d}$ is bounded independent of the
dimension of truncation for all considered Fock basis states $|m\rangle$. For
a non-convergent Trotter scenario, we look at the two operators
$H^{(1)}=Q^{3}$ and $H^{(2)}=P^{2}$ in Fig. 3. The corresponding sum
$H^{(1)}+H^{(2)}=P^{2}+Q^{3}$ describes a particle in a $Q^{3}$ potential.
This Trotter problem does not converge as even in the classical version of
this Hamiltonian, the particle would escape to infinity at finite times [45].
Our numerical study also reveals that the Trotter error on the finite-
dimensional truncations $V_{d}$ does not saturate in the truncation dimension
$d$. Instead, we observe a phase transition to a chaotic behaviour similar to
[17].

Through our state-dependent error bound, we can also perform an analytical
examination in certain cases. For this, consider $H^{(1)}=\frac{1}{2}Q^{2}$
and $H^{(2)}=\frac{1}{2}P^{2}$ and truncate at level $d$ in the Fock basis
$\\{|m\rangle\\}$ as before. We then find the analytic bound

$\displaystyle b_{d}^{(n)}$ $\displaystyle(|m\rangle;t)$
$\displaystyle\leq\frac{t^{2}}{2n}\sqrt{\frac{3}{8}\left(m(m+1)\left(m^{2}+m+14\right)+10\right)},$
(6)

see the Supplementary Material, Sec. D. This bound does not depend on the
cutoff dimension $d$. Hence, by our truncation main result the full continuous
Trotter problem converges and its Trotter error can be quantified for Fock
states explicitly by Eq. (6). For a numerical comparison, see Fig. 4.

$0$$100$$200$$300$$0$$2$$4$$6$$\cdot 10^{-2}$Dimension of truncation
$d$Trotter error
$b_{d}^{(n)}(|m\rangle;t)$$|0\rangle$$|1\rangle$$|2\rangle$$|3\rangle$$|4\rangle$
Figure 2: State-dependent Trotter error for the operators
$H^{(1)}=\frac{1}{2}\left(Q^{2}+P^{2}\right)$ and
$H^{(2)}=\frac{1}{2}\left(QP+PQ\right)$. We consider the first five Fock-basis
states $\\{|m\rangle\\}=\\{|0\rangle,\dots,|4\rangle\\}$ for different
dimensions of truncations $n=1,\dots,300$. The total evolution time is fixed
to $t=2$ and the number of Trotter steps is $N=1000$. The saturation of the
error indicates the convergence of this Trotter problem.
$0$$100$$200$$300$$0$$0.2$$0.4$Dimension of truncation $d$Trotter error
$b_{d}^{(n)}(|m\rangle;t)$$|0\rangle$$|1\rangle$$|2\rangle$$|3\rangle$$|4\rangle$
Figure 3: State-dependent error for the operators $H^{(1)}=Q^{3}$ and
$H^{(2)}=P^{2}$. We consider the first five Fock-basis states
$\\{|m\rangle\\}=\\{|0\rangle,\dots,|4\rangle\\}$ for different dimensions of
truncations $n=1,\dots,300$. The total evolution time is fixed to $t=1$ and
the number of Trotter steps is $N=1000$. The Trotter error does not saturate,
which shows that this Trotter problem does not converge.
$0$$10$$20$$30$$40$$50$$0$$2$$4$$6$$8$$\cdot 10^{-3}$Dimension of truncation
$d$Trotter error
$b_{d}^{(n)}(|m\rangle;t)$$|0\rangle$$|1\rangle$$|2\rangle$$|3\rangle$$|4\rangle$
Figure 4: State-dependent error for the operators $H^{(1)}=\frac{1}{2}Q^{2}$
and $H^{(2)}=\frac{1}{2}P^{2}$. We consider the first five Fock-basis states
$\\{|m\rangle\\}=\\{|0\rangle,\dots,|4\rangle\\}$ for different dimensions of
truncations $n=1,\dots,50$. The total evolution time is fixed to $t=1$ and the
number of Trotter steps is $N=1000$. The dots are a numerical simulation,
whereas the lines show the explicit error bounds from Eq. (6). Since the
Trotter error can be bounded independently of the truncation dimension, this
Trotter problem converges.

_Conclusion._ —From a practical perspective, our truncation main result
reduces the complexity of determining Trotter convergence and finding
infinite-dimensional error bounds to just computing finite-dimensional error
bounds, which saturate in the truncation dimension. The latter may be obtained
through our state-dependent error bounds from the first main result. This
makes the common practice of numerical Trotter simulations rigorous. From a
fundamental perspective, it would be great to have a generic method in order
to analytically check the convergence of the infinite-dimensional Trotter
product formula in a general setting. This is a hard problem and only very
little literature in the mathematical physics community exists on this topic.
Our result can extend to such a method in certain cases. For this, one needs
that there are eigenstates of the infinite-dimensional target Hamiltonian that
stay eigenstates of the truncated operators. Of course, generalizing our
finite-dimensional state-dependent error bounds to arbitrary input states
would make this assumption obsolete. Both of our main results naturally
generalize to Trotter products of more than two operators.

DB thanks Paolo Facchi and Dominic Berry for interesting discussions. AH would
like to thank Cahit Kargı for a helpful exchange on the numerical simulations.
LvL thanks Reinhard F. Werner for helpful discussions and suggestions. DB
acknowledges funding by the Australian Research Council (project numbers
FT190100106, DP210101367, CE170100009). NG acknowledges financial support from
the Spanish Agencia Estatal de Investigación, project PID2019-107609GB-I00,
and the Spanish Plan de Recuperación, Transformación y Resiliencia financed by
the European Union-NextGenerationEU. AH was supported by the Sydney Quantum
Academy. LvL acknowledges support by the Quantum Valley Lower Saxony.

## References

  * [1] R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).
  * [2] S. Lloyd, Universal Quantum Simulators, Science 273, 1073 (1996).
  * [3] D. Poulin, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doherty, and M. Troyer, The Trotter step size required for accurate quantum simulation of quantum chemistry, Quantum Inf. Comput. 361 (2015).
  * [4] A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Theory of Trotter Error with Commutator Scaling, Phys. Rev. X 11, 011020 (2021).
  * [5] R. D. Klauber, Student Friendly Quantum Field Theory (Sandtrove Press, Fairfield (Iowa), USA, 2013).
  * [6] B. Simon, Functional integration and quantum physics, 2nd ed. (AMS Chelsea Pub, Providence (Rhode Island), USA, 2005).
  * [7] F. Nicola and S. I. Trapasso, On the Pointwise Convergence of the Integral Kernels in the Feynman-Trotter Formula, Commun. Math. Phys. 376, 2277 (2019).
  * [8] B. Gaveau, E. Mihóková, M. Roncadelli, and L. S. Schulman, Path integral in a magnetic field using the Trotter product formula, Am. J. Phys. 72, 385 (2004).
  * [9] G. W. Johnson and M. L. Lapidus, The Feynman Integral and Feynman’s Operational Calculus, Oxford Mathematical Monographs (Oxford University Press, New York, USA, 2002).
  * [10] B. Misra and E. C. G. Sudarshan, The Zeno’s paradox in quantum theory, J. Math. Phys. 18, 756 (1977).
  * [11] C. Arenz, D. Burgarth, P. Facchi, and R. Hillier, Dynamical decoupling of unbounded Hamiltonians, J. Math. Phys. 59, 032203 (2018).
  * [12] C. Arenz, D. Burgarth, and R. Hillier, Dynamical decoupling and homogenization of continuous variable systems, J. Phys. A: Math. Theor. 50, 135303 (2017).
  * [13] Ángel Rivas and S. F. Huelga, Open Quantum Systems (Springer, Berlin, Heidelberg, Germany, 2012).
  * [14] F. Fitzek, J.-N. Siemß, S. Seckmeyer, H. Ahlers, E. M. Rasel, K. Hammerer, and N. Gaaloul, Universal atom interferometer simulation of elastic scattering processes, Nat. Sci. Rep. 10, 22120 (2020).
  * [15] L. M. Sieberer, T. Olsacher, A. Elben, M. Heyl, P. Hauke, F. Haake, and P. Zoller, Digital quantum simulation, Trotter errors, and quantum chaos of the kicked top, npj Quantum Inf. 5, (2019).
  * [16] K. Chinni, M. H. Muñoz-Arias, I. H. Deutsch, and P. M. Poggi, Trotter Errors from Dynamical Structural Instabilities of Floquet Maps in Quantum Simulation, PRX Quantum 3, 010351 (2022).
  * [17] C. Kargi, J. P. Dehollain, F. Henriques, L. M. Sieberer, T. Olsacher, P. Hauke, M. Heyl, P. Zoller, and N. K. Langford, Quantum Chaos and Universal Trotterisation Behaviours in Digital Quantum Simulations,  (2021).
  * [18] F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, Matrix Product Density Operators: Simulation of Finite-Temperature and Dissipative Systems, Phys. Rev. Lett. 93, 207204 (2004).
  * [19] M. Zwolak and G. Vidal, Mixed-State Dynamics in One-Dimensional Quantum Lattice Systems: A Time-Dependent Superoperator Renormalization Algorithm, Phys. Rev. Lett. 93, 207205 (2004).
  * [20] G. Vidal, Efficient Simulation of One-Dimensional Quantum Many-Body Systems, Phys. Rev. Lett. 93, 040502 (2004).
  * [21] L. Cincio, J. Dziarmaga, and M. M. Rams, Multiscale Entanglement Renormalization Ansatz in Two Dimensions: Quantum Ising Model, Phys. Rev. Lett. 100, 240603 (2008).
  * [22] N. Lambert, S. Ahmed, M. Cirio, and F. Nori, Modelling the ultra-strongly coupled spin-boson model with unphysical modes, Nat. Comm. 10, (2019).
  * [23] E. Jeckelmann and S. R. White, Density-matrix renormalization-group study of the polaron problem in the Holstein model, Physical Review B 57, 6376 (1998).
  * [24] G. Vidal, Classical Simulation of Infinite-Size Quantum Lattice Systems in One Spatial Dimension, Phys. Rev. Lett. 98, 070201 (2007).
  * [25] M. Suzuki, Decomposition formulas of exponential operators and Lie exponentials with some applications to quantum mechanics and statistical physics, J. Math. Phys. 26, 601 (1985).
  * [26] T. Kato, in Topics in Functional Analysis. Ad. Math. Suppl. Studies, edited by I. Gohberg and M. Kac (Academic Press, New York, USA, 1978), Vol. 3, pp. 185–195.
  * [27] M. L. Lapidus, Generalization of the Trotter-Lie formula, Integral Equ. Oper. Theory 4, 366 (1981).
  * [28] I. I. Rabi, On the Process of Space Quantization, Phys. Rev. 49, 324 (1936).
  * [29] I. I. Rabi, Space Quantization in a Gyrating Magnetic Field, Phys. Rev. 51, 652 (1937).
  * [30] E. T. Jaynes and F. W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc. IEEE 51, 89 (1963).
  * [31] D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001).
  * [32] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design derived from the Cooper pair box, Phys. Rev. A 76, 042319 (2007).
  * [33] J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R. Johnson, J. M. Chow, J. M. Gambetta, J. Majer, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Suppressing charge noise decoherence in superconducting charge qubits, Phys. Rev. B 77, 180502 (2008).
  * [34] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrà, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  * [35] B. Şahinoğlu and R. D. Somma, Hamiltonian simulation in the low-energy subspace, npj Quantum Inf. 7, (2021).
  * [36] C. Yi and E. Crosson, Spectral analysis of product formulas for quantum simulation, npj Quantum Inf. 8, (2022).
  * [37] S. Becker, N. Datta, L. Lami, and C. Rouzé, Energy-Constrained Discrimination of Unitaries, Quantum Speed Limits, and a Gaussian Solovay-Kitaev Theorem, Phys. Rev. Lett. 126, 190504 (2021).
  * [38] T. Jahnke and C. Lubich, Error Bounds for Exponential Operator Splittings, BIT 40, 735 (2000).
  * [39] D. An, D. Fang, and L. Lin, Time-dependent unbounded Hamiltonian simulation with vector norm scaling, Quantum 5, 459 (2021).
  * [40] T. Ichinose and H. Tamura, Note on the Norm Convergence of the Unitary Trotter Product Formula, Lett. Math. Phys. 70, 65 (2004).
  * [41] M. Reed and B. Simon, I: Functional Analysis, revisited and enlarged ed. (Academic Press Inc., San Diego (California), USA, 1981).
  * [42] D. Burgarth, P. Facchi, G. Gramegna, and K. Yuasa, One bound to rule them all: from Adiabatic to Zeno, Quantum 6, 737 (2022).
  * [43] K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Semigroup Forum 63, 278 (2001).
  * [44] N. G. Duffield and R. F. Werner, Mean-field dynamical semigroups on $C^{*}$-algebras, Rev. Math. Phys. 04, 383 (1992).
  * [45] C. Zhu and J. R. Klauder, Classical symptoms of quantum illnesses, Am. J. Phys. 61, 605 (1993).

Supplementary Material

##  I A. Mathematical Preliminaries

In this section, we introduce all the mathematical concepts and the notation
needed for the proof of our main results. We also elaborate more on the
mathematical setting and motivate why we have to consider state-dependent
error bounds instead of norm error bounds.

Consider a separable infinite-dimensional Hilbert space $\mathcal{H}$ with an
orthonormal basis $\\{|j\rangle\\}$. A corresponding sequence of truncated
finite-dimensional Hilbert spaces of increasing dimension $d<\infty$ is then
given by
$V_{d}=\mathrm{span}\\{|0\rangle,\dots,|d-1\rangle\\}\subset\mathcal{H}$.
Denote the orthogonal projection onto $V_{d}$ by
$P_{d}=\sum_{j=0}^{d-1}|j\rangle\langle j|$. We denote the union of the
truncated Hilbert spaces by $\mathcal{V}:=\bigcup_{d}V_{d}$. Since
$\\{|j\rangle\\}$ is a basis of $\mathcal{H}$, $\mathcal{V}$ lies dense in
$\mathcal{H}$, i.e. $\mathcal{V}^{\perp}=\\{0\\}$. The scalar product of
$\mathcal{H}$ is denoted by $\langle{\hskip
1.0pt\,{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt\,}|{\hskip
1.0pt\,{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt\,}\rangle$ and its induced norm by
$\|\psi\|=\sqrt{\langle\psi|\psi\rangle}$, $\psi\in\mathcal{H}$.

A _strongly continuous contraction semigroup_ is a set of linear operators
$T(t)$ on $\mathcal{H}$, $t\geq 0$, with the following properties

  1. 1.

semigroup: for all $t,s\in\mathbb{R}_{\geq 0}$, $T(t)T(s)=T(t+s)$ and
$T(0)=\operatorname{id}$,

  2. 2.

contraction: $\|T(t)|\psi\rangle\|\leq\||\psi\rangle\|$, for all
$|\psi\rangle\in\mathcal{H}$,

  3. 3.

strong continuity: For all $|\psi\rangle\in\mathcal{H}$, $\lim_{t\downarrow
0}\|T(t)|\psi\rangle-|\psi\rangle\|=0$.

Every strongly continuous contraction semigroup defines an unbounded operator
$K:\mathcal{D}(K)\to\mathcal{H}$ by setting $\mathcal{D}(K)$ to be the set
such that $K\psi:=\lim_{t\downarrow 0}\tfrac{\mathrm{i}}{t}(T(t)\psi-\psi)$
exists [43]. As is typical for unbounded operators $\mathcal{D}(K)$ will
usually be a proper subset of $\mathcal{H}$. In fact,
$\mathcal{D}(K)=\mathcal{H}$ if and only if the semigroup is uniformly
continuous. Here, _uniform_ refers to the topology induced by the operator
norm $\|A\|_{\infty}=\sup_{\||\psi\rangle\|=1}\|A|\psi\rangle\|$. In finite
dimensions, the uniform and the strong operator topology coincide, but in
infinite dimensions this is wrong and the uniform topology is stronger. The
operator $K$ is then called the _generator_ of the semigroup and (in
principle) encodes all information necessary to compute $T(t)$. Self-adjoint
(unbounded) operators $H$ on $\mathcal{H}$ are precisely the generators of
unitary groups $U(t)=\mathrm{e}^{-\mathrm{i}tH}$. We will prove a version of
Main Result 2 for strongly continuous contraction semigroups. The unitary case
as considered in the main text then follows from that. See Thm. 7 in Sec. C of
this Supplementary Material.

Next, we recall the Trotter product formula. This discussion will motivate the
use of state-dependent error bounds. For this purpose, let us focus on the
finite-dimensional unitary case first: Let $\dim(\mathcal{H})<\infty$ and let
$H^{(1)}:\mathcal{H}\to\mathcal{H}$, $H^{(2)}:\mathcal{H}\to\mathcal{H}$ be
self-adjoint, hence generators of two continuous unitary groups. Then the
_uniform Trotter error_ satisfies the bound [4, 25]

$\displaystyle\Big{\|}\left(\mathrm{e}^{-\mathrm{i}\frac{t}{n}H^{(1)}}\mathrm{e}^{-\mathrm{i}\frac{t}{n}H^{(2)}}\right)^{n}-\mathrm{e}^{-\mathrm{i}t(H^{(1)}+H^{(2)})}\Big{\|}_{\infty}\leq\frac{t^{2}}{2n}\big{\|}[H^{(1)},H^{(2)}]\big{\|}_{\infty}.$
(7)

Due to the explicit bound with $\mathcal{O}(1/N)$ scaling, we can conclude
from Eq. (7) that the finite-dimensional Trotter product formula _always_
converges uniformly, i.e. the uniform Trotter error goes to zero as
$n\to\infty$. But what does Eq. (7) imply in the situation where the
Hamiltonians are finite-dimensional truncations of infinite-dimensional
operators? Let $H^{(1)}:\mathcal{D}(H^{(1)})\to\mathcal{H}$ and
$H^{(2)}:\mathcal{D}(H^{(2)})\to\mathcal{H}$ be two self-adjoint operators
acting on an infinite-dimensional Hilbert space $\mathcal{H}$. Denote their
finite-dimensional approximations as $H_{d}^{(i)}:=P_{d}H^{(i)}P_{d}:V_{d}\to
V_{d}$ with $i=1,2$. Obviously, Eq. (7) holds for $H^{(1)}_{n}$ and
$H^{(2)}_{n}$, so that

$\displaystyle\beta_{d}^{(n)}(t):=\Big{\|}\left(\mathrm{e}^{-\mathrm{i}\frac{t}{n}H_{d}^{(1)}}\mathrm{e}^{-\mathrm{i}\frac{t}{n}H_{d}^{(2)}}\right)^{n}-\mathrm{e}^{-\mathrm{i}t(H_{d}^{(1)}+H_{d}^{(2)})}\Big{\|}_{\infty}\leq\frac{t^{2}}{2n}\big{\|}[H_{d}^{(1)},H_{d}^{(2)}]\big{\|}_{\infty}.$
(8)

In this case, for a fixed total evolution time $t$ and a fixed number of
Trotter steps $n$, the commutator error
$\|[H_{d}^{(1)},H_{d}^{(2)}]\|_{\infty}$ will typically diverge with the
dimension $d$ of truncation. This is due to the fact that the operator norm
$\|{\hskip 1.0pt\,{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt\,}\|_{\infty}$ gives a worst-case error by taking the supremum over all
normalised $|\psi\rangle\in V_{d}$. By increasing $d$, new vectors will be
added so that the Trotter error in Eq. (8) becomes larger. Consequently, the
Trotter approximants might fail to converge uniformly. In fact:

######  Proposition 3.

If the uniform Trotter error $\beta_{d}^{(n)}(t)$ is bounded independently of
$d$ then the infinite-dimensional Trotter product converges uniformly.

###### Proof.

A proof will be given in the next section. ∎

This will, of course, not hold for arbitrary Hamiltonians. For an example, see
[40, Sec. 3]. Indeed, this can also be observed numerically: As an example, we
study the operators $H^{(1)}=\frac{1}{2}\left(Q^{2}+P^{2}\right)$ and
$H^{(2)}=\frac{1}{2}\left(QP+PQ\right)$ with $Q$ denoting the position
operator and $P$ the momentum operator. Note, that $H^{(1)}$ is the
Hamiltonian of the quantum harmonic oscillator and that $H^{(2)}$ generates
squeezing transformations. As can be seen from Fig. 5, the Trotter error
$\beta_{d}^{(n)}(t)$ increases with the truncation dimension $d$ for this
example and eventually reaches $2$, which is the maximum of the operator norm
distance of two unitaries.

$0$$50$$100$$150$$0$$1$$2$$H^{(1)}=\frac{1}{2}\left(Q^{2}+P^{2}\right)$$H^{(2)}=\frac{1}{2}\left(QP+PQ\right)$Dimension
of truncation $d$Uniform Trotter error $\beta_{d}^{(n)}(t)$ Figure 5:  Uniform
Trotter error for $H^{(1)}=\frac{1}{2}\left(Q^{2}+P^{2}\right)$ and
$H^{(2)}=\frac{1}{2}\left(QP+PQ\right)$ with $Q$ the position and $P$ the
momentum operator. Parameters: $n=10$ Trotter steps, total evolution time of
$t=2$. The error reaches the maximal norm distance of $2$ showing that the
Trotter formula does not converge in norm.

For this reason, we cannot expect uniform convergence in the infinite-
dimensional setting. Rather, we need a fine-tuned error measure. This can be
achieved by considering the state-dependent, or _strong_ , Trotter error

$b_{d}^{(n)}(|\psi\rangle;t):=\Big{\|}\left(\mathrm{e}^{\mathrm{i}\frac{t}{n}H_{d}^{(1)}}\mathrm{e}^{\mathrm{i}\frac{t}{n}H_{d}^{(2)}}\right)^{n}|\psi\rangle-\mathrm{e}^{\mathrm{i}t(H_{d}^{(1)}+H_{d}^{(2)})}|\psi\rangle\Big{\|}$
(9)

instead. If $b_{d}^{(n)}(|\psi\rangle;t)\to 0$ for $n\to\infty$ for all
$|\psi\rangle$, we say that the Trotter product formula _converges strongly_.
See [41, Sec. VI.1] for details regarding strong convergence. In the main
text, we used Eq. (9) to define

$b^{(n)}(|\psi\rangle;t):=\limsup_{d\to\infty}b_{d}^{(n)}(|\psi\rangle;t).$
(10)

Lastly, consider a linear operator $H:\mathcal{D}(H)\to\mathcal{H}$ acting on
an infinite-dimensional separable Hilbert space $\mathcal{H}$. Then the graph
norm $\|{\hskip
1.0pt\,{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt\,}\|_{H}$ of $H$ is defined for $|\psi\rangle\in\mathcal{D}(H)$ by

$\||\psi\rangle\|_{H}^{2}=\||\psi\rangle\|^{2}+\|H|\psi\rangle\|^{2}.$ (11)

$H$ is called _closed_ if $\mathcal{D}(H)$ is complete with respect to the
graph norm $\lVert{\hskip
1.0pt\,{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt\,}\rVert_{H}$. If $H_{1}$ is another operator, then we write $H\subset
H_{1}$ and call $H_{1}$ an _extension_ of $H$ if
$\mathcal{D}(H)\subset\mathcal{D}(H_{1})$ and if
$H_{1}|\psi\rangle=H|\psi\rangle$ for $|\psi\rangle\in\mathcal{D}(H)$. $H$ is
called _densely defined_ if $\mathcal{D}(H)$ is dense in $\mathcal{H}$. In
that case, its adjoint $H^{*}$ is a well-defined closed operator. If $H$ is
_symmetric_ , i.e. $\langle\psi,H\varphi\rangle=\langle H\psi,\varphi\rangle$,
then $H^{**}$ is the smallest closed extension of $H$, called the _closure_.

We say that a subspace $\mathcal{W}\subset\mathcal{D}(H)$ is a _core_ for $H$
if $\mathcal{W}$ is dense in $\mathcal{D}(H)$ with respect to the graph norm
of $H$.

##  II B. Proof of the first main result

This section demonstrates the proof of our Main Result 1. Throughout, we will
assume that $\mathcal{{H}}$ is a finite-dimensional Hilbert space. We then
consider two Hamiltonians on $\mathcal{H}$, i.e. self-adjoint operators
$H^{(1)}:\mathcal{H}\to\mathcal{H}$ and $H^{(2)}:\mathcal{H}\to\mathcal{H}$,
whose Trotter product is examinated. Denote

$U(t)=\mathrm{e}^{-\mathrm{i}(H^{(1)}+H^{(2)})t}$ (12)

and

$W^{(n)}(t)=\left(\mathrm{e}^{-\mathrm{i}H^{(1)}\frac{t}{n}}\mathrm{e}^{-\mathrm{i}H^{(2)}\frac{t}{n}}\right)^{n}.$
(13)

We now recapitulate [42, Lemma 1], in particular Eq. (2.4) therein. This will
be needed in order to prove our Main Result 1.

######  Lemma 4.

Let $\tilde{H}_{1}(t)$, $\tilde{H}_{2}(t)$ be two families of time-dependent,
self-adjoint and locally integrable operators. Define the corresponding
unitaries they generate

$\tilde{U}_{j}(t)=\mathcal{T}\exp\left(-\mathrm{i}\int_{0}^{t}\tilde{H}_{j}(s)\mathrm{d}s\right),\quad
j=1,2,$ (14)

where $\mathcal{T}$ denotes time-ordering. Furthermore, define the integral
action $S_{21}(t)$ as

$S_{21}(t)=\int_{0}^{t}\left[\tilde{H}_{2}(s)-\tilde{H}_{1}(s)\right]\mathrm{d}s.$
(15)

Then for all $t\in\mathbb{R}_{\geq 0}$,

$\tilde{U}_{2}(t)-\tilde{U}_{1}(t)=-\mathrm{i}S_{21}(t)\tilde{U}_{2}(t)-\int_{0}^{t}\tilde{U}_{1}(t)\tilde{U}_{1}(s)^{\dagger}\left[\tilde{H}_{1}(s)S_{21}(s)-S_{21}(s)\tilde{H}_{2}(s)\right]\tilde{U}_{2}(s)\mathrm{d}s.$
(16)

###### Proof.

This is proved in [42, Lemma 1]. ∎

Let us restate our Main Result 1 in a slightly generalized version.

######  First Main Result.

Let $\mathcal{H}$ be a finite-dimensional Hilbert space and let
$H^{(1)}:\mathcal{H}\to\mathcal{H}$ and $H^{(2)}:\mathcal{H}\to\mathcal{H}$ be
self-adjoint operators on $\mathcal{H}$. Define $U(t)$ and $W^{(n)}(t)$ as in
Eq. (12) and Eq. (13), respectively. If $|\varphi\rangle$ is an eigenvector of
the self-adjoint operator $H^{(1)}+H^{(2)}$ accoring to the eigenvalue
equation $(H^{(1)}+H^{(2)})|\varphi\rangle=h|\varphi\rangle$, then we have for
all $t\in\mathbb{R}_{\geq 0}$ and $n\in\mathbb{N}$,

$\left\|\left(U(t)-W^{(n)}(t)\right)|\varphi\rangle\right\|\leq\frac{2t^{2}}{n}\inf_{\alpha\in\mathbb{R}}\mathrm{max}\left(\Big{\|}\Big{[}H^{(1)}-\alpha
h\Big{]}^{2}|\varphi\rangle\Big{\|},\Big{\|}\Big{[}H^{(2)}-(1-\alpha)h\Big{]}^{2}|\varphi\rangle\Big{\|}\right).$
(17)

###### Proof.

We start by applying Lemma 4. For this, take $\tilde{U}_{1}(2t)=W^{(n)}(t)$
and $\tilde{U}_{2}(2t)=U(t)$. Notice that $\tilde{U}_{1}$ is generated by a
piece-wise constant Hamiltonian $\tilde{H}_{1}(s)$ over the time interval
$[0,2t].$ This Hamiltonian is defined by

$\tilde{H}_{1}(s)\equiv\begin{cases}H^{(2)},\quad&t\in\left[0,\frac{t}{n}\right)\\\
H^{(1)},\quad&t\in\left[\frac{t}{n},\frac{2t}{n}\right)\end{cases}$ (18)

and then periodically extended by
$\tilde{H}_{1}(s+\frac{2t}{n})=\tilde{H}_{1}(s)$. The unitary $\tilde{U}_{2}$
is simply generated by the average constant Hamiltonian
$\tilde{H}_{2}(s)=\frac{H^{(1)}+H^{(2)}}{2}$. Since both Hamiltonians are
locally integrable, Lemma 4 can be applied. We want to estimate the error on
an eigenvector $\left(H^{(1)}+H^{(2)}\right)|\varphi\rangle=h|\varphi\rangle:$

$\left(U(t)-W^{(n)}(t)\right)|\varphi\rangle=-\mathrm{i}\mathrm{e}^{-\mathrm{i}ht}S_{21}(2t)|\varphi\rangle-\int_{0}^{2t}\mathrm{e}^{-\mathrm{i}\frac{h}{2}s}\tilde{U}_{1}(2t)\tilde{U}_{1}^{\dagger}(s)\left[\tilde{H}_{1}(s)-\frac{h}{2}\right]S_{21}(s)|\varphi\rangle\mathrm{d}s.$
(19)

The action reads

$\displaystyle S_{21}(s)$
$\displaystyle=\int_{0}^{s}\left(\frac{H^{(1)}+H^{(2)}}{2}-\tilde{H}_{1}(\tau)\right)\mathrm{d}\tau$
$\displaystyle=\int_{0}^{\left\lfloor
s/\frac{2t}{n}\right\rfloor\frac{2t}{n}}\left(\frac{H^{(1)}+H^{(2)}}{2}-\tilde{H}_{1}(\tau)\right)\mathrm{d}\tau+\int_{\left\lfloor
s/\frac{2t}{n}\right\rfloor\frac{2t}{n}}^{s}\left(\frac{H^{(1)}+H^{(2)}}{2}-\tilde{H}_{1}(\tau)\right)\mathrm{d}\tau$
$\displaystyle=\int_{\left\lfloor
s/\frac{2t}{n}\right\rfloor\frac{2t}{n}}^{s}\left(\frac{H^{(1)}+H^{(2)}}{2}-\tilde{H}_{1}(\tau)\right)\mathrm{d}\tau$
$\displaystyle=\int_{0}^{\left\\{s/\frac{2t}{n}\right\\}}\left(\frac{H^{(1)}+H^{(2)}}{2}-\tilde{H}_{1}(\tau)\right)\mathrm{d}\tau,$
(20)

where we split the integral into full periods and a remainder and used the
periodicity of $\tilde{H}_{1}(s).$ In particular, $S_{21}(2t)=0$ and we can
bound

$\displaystyle\left\|\left(U(t)-W^{(n)}(t)\right)|\varphi\rangle\right\|$
$\displaystyle\leq
2t\sup_{s\in[0,2t]}\left\|\left[\tilde{H}_{1}(s)-\frac{h}{2}\right]S_{21}(s)|\varphi\rangle\right\|$
$\displaystyle=2t\sup_{s\in[0,\frac{2t}{n}]}\left\|\left[\tilde{H}_{1}(s)-\frac{h}{2}\right]S_{21}(s)|\varphi\rangle\right\|$
(21)

where we used the periodicity of $\tilde{H}_{1}(s)$ and of $S_{21}(s)$ in
order to reduce the range of the supremum. For
$s\in\left[0,\frac{t}{n}\right)$, we have

$S_{21}(s)=\int_{0}^{s}\left(\frac{H^{(1)}+H^{(2)}}{2}-H^{(2)}\right)\mathrm{d}\tau=s\frac{H^{(1)}-H^{(2)}}{2}.$
(22)

Analogously for $s\in\left[\frac{t}{n},\frac{2t}{n}\right)$, we obtain

$S_{21}(s)=\frac{t}{n}\frac{H^{(1)}-H^{(2)}}{2}+\int_{\frac{t}{n}}^{s}\left(\frac{H^{(1)}+H^{(2)}}{2}-H^{(1)}\right)\mathrm{d}\tau=\left(\frac{2t}{n}-s\right)\frac{H^{(1)}-H^{(2)}}{2}.$
(23)

Therefore,

$\displaystyle\left\|\left(U(t)-W^{(n)}(t)\right)|\varphi\rangle\right\|\leq$
$\displaystyle
2t\max\left(\sup_{s\in[0,\frac{t}{n}]}\left\|\left[H^{(2)}-\frac{h}{2}\right]s\frac{H^{(1)}-H^{(2)}}{2}|\varphi\rangle\right\|\right.,$
$\displaystyle\left.\sup_{s\in[\frac{t}{n},\frac{2t}{n}]}\left\|\left[H^{(1)}-\frac{h}{2}\right]\left(\frac{2t}{n}-s\right)\frac{H^{(1)}-H^{(2)}}{2}|\varphi\rangle\right\|\right)$
$\displaystyle=$
$\displaystyle\frac{t^{2}}{n}\max\left(\left\|\left[H^{(2)}-\frac{h}{2}\right]\left(H^{(1)}-H^{(2)}\right)|\varphi\rangle\right\|,\left\|\left[H^{(1)}-\frac{h}{2}\right]\left(H^{(1)}-H^{(2)}\right)|\varphi\rangle\right\|\right).$
(24)

We expand

$\displaystyle\left\|\left[H^{(2)}-\frac{h}{2}\right]\left(H^{(1)}-H^{(2)}\right)|\varphi\rangle\right\|$
$\displaystyle=\left\|\left[H^{(2)}-\frac{h}{2}\right]\left(H^{(1)}+H^{(2)}-2H^{(2)}\right)|\varphi\rangle\right\|$
$\displaystyle=\left\|\left[H^{(2)}-\frac{h}{2}\right]\left(h-2H^{(2)}\right)|\varphi\rangle\right\|$
$\displaystyle=2\left\|\left[H^{(2)}-\frac{h}{2}\right]^{2}|\varphi\rangle\right\|$
(25)

and simliar for the second term

$\left\|\left[H^{(1)}-\frac{h}{2}\right]\left(H^{(1)}-H^{(2)}\right)|\varphi\rangle\right\|=2\left\|\left[H^{(1)}-\frac{h}{2}\right]^{2}|\varphi\rangle\right\|.$
(26)

Therefore, in total we receive

$\left\|\left(U(t)-W^{(n)}(t)\right)|\varphi\rangle\right\|\leq\frac{2t^{2}}{n}\mathrm{max}\left(\Big{\|}\Big{[}H^{(1)}-\frac{h}{2}\Big{]}^{2}|\varphi\rangle\Big{\|},\Big{\|}\Big{[}H^{(2)}-\frac{h}{2}\Big{]}^{2}|\varphi\rangle\Big{\|}\right),$
(27)

which is the bound from our Main Result 1 in the main text. In order to show
the slightly generalized version of this bound stated in Eq. (17), notice that
we can always rescale the Hamiltonians $H^{(1)}$ and $H^{(2)}$ by a multiple
of the identity $\mathds{1}$ without changing the Trotter error. That is, by
rescaling $H^{(1)}\rightarrow H^{(1)}+a\mathds{1}$ and $H^{(2)}\rightarrow
H^{(2)}+b\mathds{1}$ we will not change
$\|\left(U(t)-W^{(n)}(t)\right)|\varphi\rangle\|$ since $\mathds{1}$ commutes
with any Hamiltonian. This procedure leads to a rescaling $h\rightarrow h+a+b$
and Eq. (27) becomes

$\left\|\left(U(t)-W^{(n)}(t)\right)|\varphi\rangle\right\|\leq\frac{2t^{2}}{n}\mathrm{max}\left(\Big{\|}\Big{[}H^{(1)}-\frac{h+b-a}{2}\Big{]}^{2}|\varphi\rangle\Big{\|},\Big{\|}\Big{[}H^{(2)}-\frac{h+a-b}{2}\Big{]}^{2}|\varphi\rangle\Big{\|}\right),$
(28)

which is true for any $a,b\in\mathbb{R}$. The assertion then follows by
defining $\alpha=(h+b-a)/(2h)$ and taking the infimum over $\alpha$. The
latter is possible since the inequality holds for all $\alpha\in\mathbb{R}$. ∎

##  III C. Proof of the second main result

In this section, we present the proof of Main Result 2. For this, we will use
the notation introduced in Sec. A of this Supplementary Material. We will
first prove a more general result, which holds for strongly continuous
contraction semigroups. This is given by Thm. 7. Our Main Result 2 will then
follow as a special case. In the proofs of the theorems, we will use the
following consequence of the _Trotter-Kato approximation theorem_ [43, Sec.
4]:

######  Theorem 5.

For each $n\in\mathbb{N}$, let $T_{d}(t)$ be a contraction semigroup on
$V_{d}$. Let $H_{d}$ be the sequence of generators. The following statements
are equivalent

  1. (a)

The strong limit $T(t)=\operatorname*{s-lim}_{d\to\infty}T_{d}(t)$ exists for
all times $t>0$ and defines a strongly continuous contraction semigroup.

  2. (b)

The operator $H:\mathcal{D}(H)\to\mathcal{H}$, defined by

$\displaystyle|\psi\rangle\in\mathcal{D}(H)\iff$ $\displaystyle\text{there
exists a sequence }(|\psi_{d}\rangle)_{d\in\mathbb{N}}\in\mathscr{C},\text{
such that }|\psi_{d}\rangle\to|\psi\rangle\in\mathcal{H},$
$\displaystyle\text{and
}H|\psi\rangle:=\lim_{d\to\infty}H_{d}|\psi_{d}\rangle\text{ exists},$

is actually well-defined in the sense that $H|\psi\rangle$ does not depend on
the chosen sequence $(|\psi_{d}\rangle)$. Additionally, $H$ generates a
contraction semigroup.

Furthermore, if these equivalent statements hold then $H$ coincides with the
generator of $T(t)$.

For the proof, we extend $T_{d}(t)$ to act as the identity on the orthogonal
complement $V_{d}^{\perp}$ of $V_{d}$. This turns $T_{d}(t)$ into a strongly
continuous contraction semigroup on $\mathcal{H}$, which would be wrong for
the extension by $0$. Notice that this does not affect strong convergence: The
difference of the extension by zero and the extension by the identity is the
projection $P_{d}^{\perp}$ onto $V_{d}^{\perp}$ which vanishes strongly as
$d\to\infty$. The proof is inspired by [44, Thm. 2.3]:

###### Proof.

For the proof, choose $z\in\mathbb{C}$ with $\mathrm{Im}\,z>0$.

a $\Rightarrow$ b: Our setting is a special case of the _second Trotter-Kato
approximation theorem_ [43, Thm. 4.9]. Therefore, a is equivalent to the
statement that the resolvents $R(H_{d};z)=(z-H_{d})^{-1}$ are strongly
convergent with $\operatorname*{s-lim}_{d\to\infty}R(H_{d};z)$ having dense
range. For a sequence $(|\psi_{d}\rangle)_{d}$ with $|\psi_{d}\rangle\in
V_{d}$, we write $(|\psi_{d}\rangle)_{d}\in\mathscr{C}$ if it converges in
$\mathcal{H}$, i.e.,

$\mathscr{C}=\left\\{(|\psi_{d}\rangle)_{d}\subset\mathcal{H}\nonscript\
\middle|\allowbreak\nonscript\ \mathopen{}|\psi_{d}\rangle\in
V_{d},\,\exists\,|\psi\rangle\in\mathcal{H}:\lim_{d\to\infty}|\psi_{d}\rangle=|\psi\rangle\right\\}.$
(29)

Furthermore, we write

$\mathscr{D}:=\big{\\{}(|\psi_{d}\rangle)_{d}\in\mathscr{C}\nonscript\
\big{|}\allowbreak\nonscript\
\mathopen{}(H_{d}|\psi_{d}\rangle)_{d}\in\mathscr{C}\big{\\}}.$ (30)

Consider the resolvent operator $\mathscr{R}(z)$ which maps a sequence
$(|\psi_{d}\rangle)_{d}\subset\mathcal{H}$ to the sequence
$(R(H_{d};z)|\psi_{d}\rangle)_{d}$. On one hand, by definition of
$\mathscr{D}$, it holds that $((z-H_{d})|\psi_{d}\rangle)_{d}$ is a sequence
in $\mathscr{C}$ if $(|\psi_{d}\rangle)_{d}\in\mathscr{D}$, so

$\mathscr{D}\subset\mathscr{R}(z)\mathscr{C}=\left\\{*{R(H_{d};z)|\psi_{d}\rangle}_{d}\nonscript\
\middle|\allowbreak\nonscript\
\mathopen{}(|\psi_{d}\rangle)_{d}\in\mathscr{C}\right\\}.$ (31)

On the other hand, since the $R(H_{d};z)$ are uniformly bounded [43, Thm.
II.1.10] and strongly convergent in $d$, we have that
$*{R(H_{d};z)|\psi_{d}\rangle}_{d}\in\mathscr{C}$ if
$(|\psi_{d}\rangle)_{d}\in\mathscr{C}$. But then
$(|\psi_{d}\rangle)_{d}\in\mathscr{D}$ due to $(z-H_{d})R(H_{d};z)=\mathds{1}$
or, equivalently, $H_{d}R(H_{d};z)=zR(H_{d};z)-\mathds{1}$. This shows

$\mathscr{D}=\mathscr{R}(z)\mathscr{C}.$ (32)

For now, we denote the generator of the strong limit $T(t)$ by $A$ to avoid
confusion with the operator $H$ defined in b. Of course, we will show later
that the two coincide. Note that by the Trotter-Kato Theorem, we have
$\lim_{d}R(H_{d};z)=R(A;z)$. For $(|\psi_{d}\rangle)_{d}\in\mathscr{C}$ with
$\lim_{d}|\psi_{d}\rangle=|\psi\rangle$, we have

$\displaystyle\lim_{d\to\infty}H_{d}R(H_{d};z)|\psi_{d}\rangle$
$\displaystyle=\lim_{d\to\infty}[\big{]}{zR(H_{d};z)|\psi_{d}\rangle-|\psi_{d}\rangle}$
$\displaystyle=zR(A;z)|\psi\rangle-|\psi\rangle$
$\displaystyle=AR(A;z)|\psi\rangle.$ (33)

Since $\mathscr{D}=\mathscr{R}(z)\mathscr{C}$, this proves that
$\lim_{d}H_{d}|\phi_{d}\rangle=A[\big{]}{\lim_{d}|\phi_{d}\rangle}$ for all
$(|\phi_{d}\rangle)_{d}\in\mathscr{D}$. In particular, this shows the well-
definedness of $H$ and that $A$ is an extension of $H$. Finally, we prove
$H=A$ via

$\mathcal{D}(H)=\left\\{\lim_{d\to\infty}|\psi_{d}\rangle\nonscript\
\middle|\allowbreak\nonscript\
\mathopen{}(|\psi_{d}\rangle)_{d}\in\mathscr{D}\right\\}=\left\\{\lim_{d}R(H_{d};z)|\phi_{d}\rangle\nonscript\
\middle|\allowbreak\nonscript\
\mathopen{}(|\phi_{d}\rangle)_{d}\in\mathscr{C}\right\\}=R(A;z)\mathcal{H}=\mathcal{D}(A),$
(34)

where the third equality uses that
$\lim_{d}R(H_{d};z)|\phi_{d}\rangle=R(A;z)(\lim_{d}|\phi_{d}\rangle)$ and that
any $|\psi\rangle\in\mathcal{H}$ is the limit of a sequence in $\mathscr{C}$.

b $\Rightarrow$ a: Due to uniform boundedness, we only have to show strong
convergence on the dense subset $(z-H)\mathcal{D}(H)\subset\mathcal{H}$. Let
$|\phi\rangle\in\mathcal{D}(H)$ and put
$|\psi\rangle=(z-H)|\phi\rangle\in\mathcal{H}$. We know that
$|\psi\rangle=\lim_{d\to\infty}(z-H_{d})|\phi\rangle$. This implies that

$\displaystyle\|R(H_{d};z)|\psi\rangle-$ $\displaystyle R(H;z)|\psi\rangle\|$
$\displaystyle\leq\lVert
R(H_{d};z)(|\psi\rangle-(z-H_{d})|\phi\rangle)\rVert+\lVert
R(H_{d};z)(z-H_{d})|\phi\rangle-|\phi\rangle\rVert$ (35)

goes to zero as $d\to\infty$: The first summand on the right-hand side is
dominated by $\lvert
z\rvert^{-1}\lVert|\psi\rangle-(z-H_{d})|\phi\rangle\rVert\to 0$. For the
second summand, $R(H_{d};z)(z-H_{d})|\phi\rangle=|\phi\rangle$ for
sufficiently large $d$. This shows that the resolvents $R(H_{d};z)$ are
strongly convergent. By construction, their limit
$R(z)=\operatorname*{s-lim}_{d\to\infty}R(H_{d};z)$ has dense range. However,
as already pointed out before, this is equivalent to a due to the second
Trotter-Kato approximation theorem [43, Thm. 4.9]. ∎

In Thm. 5, we started with a sequence of generators $H_{d}$ and defined $H$ as
a certain limit. However, for the proof of Thm. 7, we also need to look at the
reversed setting, i.e. start from $H$ and define the $H_{d}$ as its finite-
dimensional approximations. This scenario is elaborated in the following
lemma.

######  Lemma 6.

If $\mathcal{V}\subset\mathcal{D}(H)$ is a core of $H$, then it holds that
$P_{d}HP_{d}=:H_{d}\to H$ strongly on $\mathcal{V}$. Furthermore, we have
$T_{d}(t)\to T(t)$ strongly on all of $\mathcal{H}$.

###### Proof.

Let $|\psi\rangle\in\mathcal{V}$. Then there exists an
$\tilde{d}\in\mathbb{N}$ such that $|\psi\rangle\in V_{\tilde{d}}$. Hence, for
$d\geq\tilde{d}$ it holds that
$|\psi_{d}\rangle=P_{d}|\psi\rangle=|\psi\rangle$ and, thus,
$H_{d}|\psi_{d}\rangle=P_{d}H|\psi\rangle\to H|\psi\rangle$. By the _first
Trotter-Kato approximation theorem_ [43, Thm. 4.8], the assertion follows. ∎

Before we turn towards strong convergence of the Trotter product, we provide
the proof of Prop. 3. It also introduces some of the notation and techniques
that we will employ later on. For the proof, recall Eq. (8).

###### Proof of Prop. 3.

Let $H^{(i)}:\mathcal{D}(H^{(i)})\to\mathcal{H}$, $i=1,2$, be Hamiltonians
such that
$\mathcal{V}=\bigcup_{d}V_{d}\subset\mathcal{D}(H^{(1)})\cap\mathcal{D}(H^{(2)})$.
As before, $H^{\scaleobj{0.8}{(i)}}_{d}=P_{d}H^{\scaleobj{0.8}{(i)}}P_{d}$.
Denote
$X_{d}(t)=\mathrm{e}^{-\mathrm{i}tH_{d}^{(1)}}\mathrm{e}^{-\mathrm{i}tH_{d}^{(2)}}$
and $U_{d}(t)=\mathrm{e}^{-\mathrm{i}t(H_{d}^{(1)}+H_{d}^{(2)})}$ and by
$X(t)$ and $U(t)$ the analogous operators on the infinite dimensional space.
Assume that there exists $c_{n}=c_{n}(t)>0$ such that
$\beta_{d}^{(n)}(t)<c_{n}$ with $c_{n}\to 0$ if $n\to\infty$. Let
$\psi\in\mathcal{V}$, w.l.o.g. $\psi\in V_{d}$, and let $\tilde{d}>d$. Then
$P_{\tilde{d}}\psi=\psi$, so

$\displaystyle\lVert(X(t/n)^{n}-U(t))\psi\rVert$
$\displaystyle\leq\lVert(X_{\tilde{d}}(t/n)^{n}-U_{\tilde{d}}(t))\psi\rVert+\lVert(\mathds{1}-P_{\tilde{d}})(X(t/n)^{n}-U(t))\psi\rVert$
$\displaystyle\leq
c_{n}\lVert\psi\rVert+\lVert(\mathds{1}-P_{\tilde{d}})(X(t/n)^{n}-U(t))\psi\rVert.$
(36)

Taking $\tilde{d}\to\infty$, we thus obtain
$\lVert(X(t/n)^{n}-U(t))\psi\rVert\leq c_{n}\lVert\psi\rVert$. This shows
uniform convergence on $\mathcal{V}$. Approximating $\psi\in\mathcal{H}$ by
elements in $\mathcal{V}$ and using the triangle inequality and unitarity of
$X(t)$ and $U(t)$ then shows $\lVert X(t/n)^{n}-U(t)\rVert_{\infty}<c_{n}\to
0$. ∎

Recall the definitions in (9) and (10). We are now ready to present and prove:

######  Theorem 7.

Let $H^{(i)}:\mathcal{D}(H^{(i)})\to\mathcal{H}$, $i=1,2$, be generators of
two strongly continuous contraction semigroups $T^{(i)}(t)$, $i=1,2$. Assume
that
$\mathcal{V}=\bigcup_{d}V_{d}\subset\mathcal{D}(H^{(1)})\cap\mathcal{D}(H^{(2)})$
is a common core of $H^{(1)}$ and $H^{(2)}$. Denote by $T_{d}(t)$ the strongly
continuous contraction semigroup on $V_{d}$ generated by
$H^{\scaleobj{0.8}{(1)}}_{d}+H^{\scaleobj{0.8}{(2)}}_{d}$. Then

  1. (1)

If for some $|\psi\rangle\in\mathcal{V}$, $t>0$ one has
$b^{(n)}(|\psi\rangle;t)\to 0$ as $n\to\infty$, then both limits
$\lim_{d\to\infty}T_{d}(t)|\psi\rangle$ and
$\lim_{n\to\infty}*{T^{\scaleobj{0.8}{(1)}}(t/n)T^{\scaleobj{0.8}{(2)}}(t/n)}^{n}|\psi\rangle$
exist in $\mathcal{H}$ and coincide.

  2. (2)

If for all $t\geq 0$ and all $|\psi\rangle\in\mathcal{V}$ one has
$b^{(n)}(|\psi\rangle;t)\to 0$, then there is a strongly continuous
contraction semigroup $T(t)$ on $\mathcal{H}$ with
$T(t)\psi=\lim_{d}T_{d}(t)|\psi\rangle$, $|\psi\rangle\in\mathcal{V}$. The
generator $H$ of $T(t)$ is an extension of
$H^{\scaleobj{0.8}{(1)}}|_{\mathcal{V}}+H^{\scaleobj{0.8}{(2)}}|_{\mathcal{V}}$.
The Trotter product of $T^{\scaleobj{0.8}{(1)}}$ and $T^{\scaleobj{0.8}{(2)}}$
converges strongly to $T(t)$.

###### Proof.

We use the notations
$X_{d}(t)=T^{\scaleobj{0.8}{(1)}}_{d}(t)T^{\scaleobj{0.8}{(2)}}_{d}(t)$ and
$X(t)=T^{\scaleobj{0.8}{(1)}}(t)T^{\scaleobj{0.8}{(2)}}(t)$.

1: We have to prove two statements. First, we prove that for the series of
finite-dimensional approximations, $T_{d}(t)|\psi\rangle$ converges to a well-
defined vector $|\phi\rangle$ as the level of truncation goes to infinity,
$d\to\infty$. Second, we show that for the Trotter product,
$X(t/n)^{n}|\psi\rangle$ converges to exactly this vector $|\phi\rangle$ in
the Trotter limit $n\to\infty$. Let us start with the first assertion assuming
$|\psi\rangle\in\mathcal{V}$. By the triangle inequality, it holds that

$\left\lVert*{T_{d}(t)-T_{\tilde{d}}(t)}|\psi\rangle\right\rVert\leq
b_{d}^{\scaleobj{0.8}{(n)}}(|\psi\rangle;t)+\left\lVert*{X_{d}(t/n)^{n}-X_{\tilde{d}}(t/n)^{n}}|\psi\rangle\right\rVert+b_{\tilde{d}}^{\scaleobj{0.8}{(n)}}(|\psi\rangle;t),$
(37)

Due to Lemma 6, we have
$\lim_{\tilde{d}\to\infty}\lim_{d\to\infty}\left\lVert*{X_{d}(t/n)^{n}-X_{\tilde{d}}(t/n)^{n}}|\psi\rangle\right\rVert=0$,
and therefore

$\limsup_{d\to\infty}\limsup_{\tilde{d}\to\infty}\left\lVert*{T_{d}(t)-T_{\tilde{d}}(t)}|\psi\rangle\right\rVert\leq
2b^{(n)}(|\psi\rangle;t)\xrightarrow{n\to\infty}0.$ (38)

Hence, $T_{d}(t)|\psi\rangle$ forms a Cauchy sequence and thus converges to
some $|\phi\rangle\in\mathcal{H}$. For the second assertion, let
$|\psi\rangle\in\mathcal{V}$ and use the triangle inequality again in order to
receive

$\displaystyle\left\lVert X(t/n)^{n}|\psi\rangle-|\phi\rangle\right\rVert\leq$
$\displaystyle\left\lVert*{X(t/n)^{n}-X_{d}(t/n)^{n}}|\psi\rangle\right\rVert$
$\displaystyle+b_{d}^{\scaleobj{0.8}{(n)}}(|\psi\rangle;t)$
$\displaystyle+\left\lVert T_{d}(t)|\psi\rangle-|\phi\rangle\right\rVert.$
(39)

By convergence of $T_{d}(t)|\psi\rangle$ and, therefore,
$X_{d}(t)|\psi\rangle$ and by taking the $\limsup_{d\to\infty}$, we obtain

$\left\lVert X(t/n)^{n}|\psi\rangle-|\phi\rangle\right\rVert\leq
b^{(n)}(|\psi\rangle;t)\xrightarrow{n\to\infty}0,$ (40)

which proves the claim.

2: Again, two claims are to be proved. For the first claim, we have to show
that $T(t)$ is a strongly continuous contraction semigroup on $\mathcal{H}$.
For the second claim, we have to show that the generator $H$ of $T(t)$ is an
extension of
$H^{\scaleobj{0.8}{(1)}}|_{\mathcal{V}}+H^{\scaleobj{0.8}{(2)}}|_{\mathcal{V}}$,
i.e. the sum of the individual generators of $T^{(1)}(t)$ and $T^{(2)}(t)$.
For the first claim, notice that by (1), $T(t)$ is the strong limit of the
semigroups $T_{d}(t)$. Hence, it is a semigroup itself. For strong continuity,
let $|\psi\rangle\in\mathcal{V}$. Then for $d$ large enough we have that
$|\psi\rangle\in V_{d}$ and it holds that [43, Lemma 1.3]

$\displaystyle\|T_{d}(t)|\psi\rangle-|\psi\rangle\|$
$\displaystyle=\left\lVert\int_{0}^{t}T_{d}(s)(H^{\scaleobj{0.8}{(1)}}_{d}+H^{\scaleobj{0.8}{(2)}}_{d})|\psi\rangle\
\textrm{d}s\right\rVert$
$\displaystyle\leq\int_{0}^{t}\|(H_{d}^{\scaleobj{0.8}{(1)}}+H_{d}^{\scaleobj{0.8}{(2)}})|\psi\rangle\|\
\textrm{d}s$
$\displaystyle=t\|P_{d}(H^{\scaleobj{0.8}{(1)}}+H^{\scaleobj{0.8}{(2)}})|\psi\rangle\|.$
(41)

By continuity of $\lVert{\hskip
1.0pt\,{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt\,}\rVert$ we have

$\|T(t)|\psi\rangle-|\psi\rangle\|=\lim_{d\to\infty}\|T_{d}(t)|\psi\rangle-|\psi\rangle\|\leq
t\|(H^{\scaleobj{0.8}{(1)}}+H^{\scaleobj{0.8}{(2)}})|\psi\rangle\|\xrightarrow{t\to
0}0.$ (42)

By density of $\mathcal{V}$ in $\mathcal{H}$ and, again, uniform continuity,
strong continuity of $T(t)$ follows on all of $\mathcal{H}$. Now, let
$|\psi\rangle\in\mathcal{V}\subset\mathcal{D}(H^{\scaleobj{0.8}{(1)}})\cap\mathcal{D}(H^{\scaleobj{0.8}{(2)}})$
and $|\psi_{d}\rangle=P_{d}|\psi\rangle$. Then as $d\to\infty$,
$|\psi_{d}\rangle\to|\psi\rangle$ and
$H^{\scaleobj{0.8}{(i)}}_{d}|\psi_{d}\rangle\to
H^{\scaleobj{0.8}{(i)}}|\psi\rangle$, since for $d$ large enough
$|\psi_{d}\rangle=|\psi\rangle$. Hence,
$(H_{d}^{\scaleobj{0.8}{(1)}}+H_{d}^{\scaleobj{0.8}{(2)}})|\psi_{d}\rangle\to(H^{\scaleobj{0.8}{(1)}}+H^{\scaleobj{0.8}{(2)}})|\psi\rangle$,
so $|\psi\rangle\in\mathcal{D}(H)$ and
$H|\psi\rangle=(H^{\scaleobj{0.8}{(1)}}+H^{\scaleobj{0.8}{(2)}})|\psi\rangle$
by Thm. 5 b. Thus
$\overline{H^{\scaleobj{0.8}{(1)}}_{|\mathcal{V}}+H^{\scaleobj{0.8}{(2)}}_{|\mathcal{V}}}\subset
H$. ∎

Now, our Main Result 2 follows as a special case of Thm. 7:

######  Second Main Result.

Let $H^{(1)}:\mathcal{D}(A^{(1)})\to\mathcal{H}$ and
$H^{(2)}:\mathcal{D}(H^{(2)})\to\mathcal{H}$ be self-adjoint, hence generators
of two strongly continuous unitary groups $T^{(i)}(t)$, $i=1,2$. Assume that
$\mathcal{V}=\bigcup_{d}V_{d}\subset\mathcal{D}(H^{(1)})\cap\mathcal{D}(H^{(2)})$
is a common core of $H^{(1)}$ and $H^{(2)}$. If for all $t\in\mathbb{R}$,
$b^{(n)}(|\psi\rangle;t)\to 0$ as $n\to\infty$, then

$*{T^{\scaleobj{0.8}{(1)}}(t/n)T^{\scaleobj{0.8}{(2)}}(t/n)}^{n}|\psi\rangle\xrightarrow{n\to\infty}T(t)|\psi\rangle,$
(43)

where $T(t)$ is a strongly continuous unitary group, whose generator $H$ is a
self-adjoint and agrees with $(H_{1}+H_{2})$ wherever both are defined.

###### Proof.

Since $H^{\scaleobj{0.8}{(1)}}$ and $H^{\scaleobj{0.8}{(2)}}$ are self-
adjoint, so are $H^{\scaleobj{0.8}{(i)}}_{d},i=1,2$ and $H_{d}$. All these
operators generate unitary groups that we will denote by
$U^{\scaleobj{0.8}{(i)}}$, $U^{\scaleobj{0.8}{(i)}}_{d}$ and $U_{d}$,
respectively. Since $b^{(n)}(|\psi\rangle;t)\to 0$ as $n\to\infty$ for $t\geq
0$, we obtain a contraction semigroup $U(t)$, $t\geq 0$, generated by $H$.
Now, $U(t)$ is isometric: $\|U(t)|\psi\rangle\|=\lim_{d\to\infty}\left\lVert
U_{d}(t)|\psi\rangle\right\rVert=\||\psi\rangle\|.$ Hence, $H$ is symmetric as
the generator of a strongly continuous isometry semigroup. To see this, let
$|\psi\rangle,|\phi\rangle\in\mathcal{D}(H)$ and differentiate
$\langle\psi|\phi\rangle=\langle U(t)\psi|U(t)\phi\rangle$ at $t=0$. The same
argument can be applied for $t\leq 0$, in which case we obtain a semigroup
generated by $-H^{*}$. Hence, $H^{*}$ is symmetric as well. Notice that an
operator $A$ is symmetric if and only if (i) $A\subset A^{*}$, and (ii)
$A\subset B$ implies $B^{*}\subset A^{*}$ [41]. Applying this to $H$ and
$H^{*}$ gives $H\subset H^{*}\subset H^{**}=H$, which shows that $H$ is self-
adjoint. In order to see that $H$ is equal to
$(H^{\scaleobj{0.8}{(1)}}+H^{\scaleobj{0.8}{(2)}})$ wherever both are defined,
we consider the difference: Let $Y$ be the closure of the symmetric operator
$(H-H^{\scaleobj{0.8}{(1)}}+H^{\scaleobj{0.8}{(2)}})$, defined on the
intersection of the domains. We know from Thm. 7 2 that $Y|_{\mathcal{V}}=0$.
But a closed operator which vanishes on a dense subset must vanish everywhere.
To see this, note that since $Y$ is closed, we have that $Y$ is an extension
of $\overline{(Y|_{\mathcal{V}})}$. But $Y|_{\mathcal{V}}=0|_{\mathcal{V}}$
and since the zero operator is bounded this means that
$\overline{(Y|_{\mathcal{V}})}=0$. As an extension of the zero operator, $Y$
is itself zero, i.e., $H=H^{\scaleobj{0.8}{(1)}}+H^{\scaleobj{0.8}{(2)}}$
wherever all are defined. ∎

It remains to prove Eq. (5).

###### Proof of Eq. (5).

By the triangle inequality,
$b^{(n)}(|\psi\rangle;t)^{2}\leq\limsup_{d\to\infty}\sum_{j=1}^{d}\big{|}\langle
j|\psi\rangle\big{|}^{2}b_{d}^{(n)}(|j\rangle;t)^{2}$. If
$|\psi\rangle\in\mathcal{V}$, then there exists an $\tilde{d}\in\mathbb{N}$,
such that $|\psi\rangle\in V_{\tilde{d}}$ and therefore
$|\psi\rangle=\sum_{j=1}^{\tilde{d}}\langle j|\psi\rangle|j\rangle$. As a
consequence for $d\geq\tilde{d}$,
$b_{d}^{(n)}(|\psi\rangle;t)^{2}\leq\sum_{j=1}^{\tilde{d}}\big{|}\langle
j|\psi\rangle\big{|}^{2}b_{d}^{(n)}(|j\rangle;t)^{2}$, and therefore,

$\displaystyle
b^{(n)}(|\psi\rangle;t)\leq[\bigg{]}{\sum_{j=1}^{\tilde{d}}\big{|}\langle
j|\psi\rangle\big{|}^{2}b^{(n)}(|j\rangle;t)^{2}}^{1/2},$ (44)

which proves the statement. ∎

##  IV D. Example: The quantum harmonic oscillator

The bounds from our Main Result 1 hold for arbitrary Hamiltonians acting on a
finite-dimensional Hilbert space. In particular, we can use them for bounding
strong Trotter errors for the finite-dimensional truncated operators, in which
we are interested. For instance, consider the case where
$H^{(1)}=\frac{1}{2}Q^{2}$ and $H^{(2)}=\frac{1}{2}P^{2}$, where where
$Q:\mathcal{{D}}(Q)\rightarrow L^{2}(\mathbb{{R}})$ is the position operator
and $P:\mathcal{{D}}(P)\rightarrow L^{2}(\mathbb{{R}})$ is the momentum
operator. We can rewrite $Q$ and $P$ in terms of the creation operator
$a^{\dagger}$ and annihilation operator $a$ as follows

$\displaystyle Q$
$\displaystyle=\frac{1}{\sqrt{2}}\left(a+a^{\dagger}\right),$ (45)
$\displaystyle P$
$\displaystyle=\frac{\mathrm{{i}}}{\sqrt{2}}\left(a^{\dagger}-a\right).$ (46)

The operators $a$ and $a^{\dagger}$ satisfy the commutation relation
$\left[a,a^{\dagger}\right]=\mathds{1}$. In this particular Trotter scenario,
we would like to implement the target Hamiltonian
$H^{\mathrm{osc}}=\frac{1}{2}\left(Q^{2}+P^{2}\right)$ by switching between
the time evolutions generated by $H^{(1)}=\frac{1}{2}Q^{2}$ and
$H^{(2)}=\frac{1}{2}P^{2}$, respectively. $H^{\mathrm{osc}}$ is the quantum
harmonic oscillator and is related to the _number operator_ $N=a^{\dagger}a$
by $H^{\mathrm{osc}}=N+\frac{1}{2}$. The eigenstates $\left|m\right\rangle$ of
$N$ according to the eigenvalue equation
$N\left|m\right\rangle=m\left|m\right\rangle$ are the Fock states. In turn,
the $|m\rangle$ are eigenstates of $H^{\mathrm{osc}}$ as well and satisfy the
eigenvalue equation
$H^{\mathrm{osc}}\left|m\right\rangle=\left(m+\frac{1}{2}\right)\left|m\right\rangle$.
Let us use our Main Result 1 to compute a strong error bound for this Trotter
problem for the corresponding finite-dimensional approximations. Obviously, we
have

$\displaystyle Q^{2}$
$\displaystyle=\frac{1}{2}\left(2N+aa+a^{\dagger}a^{\dagger}+1\right),$ (47)
$\displaystyle P^{2}$
$\displaystyle=\frac{1}{2}\left(2N-aa-a^{\dagger}a^{\dagger}+1\right)$ (48)

and the Fock states are a common core of $H^{(1)}=\frac{1}{2}Q^{2}$ and
$H^{(2)}=\frac{1}{2}P^{2}$. Define $R^{\pm}=\frac{1}{2}\left(2N\pm aa\pm
a^{\dagger}a^{\dagger}+1\right)$, so that $R^{+}=Q^{2}$ and $R^{-}=P^{2}$.
Then the action of $R^{\pm}$ onto a Fock state $|m\rangle$ with $m\geq 2$
computes to

$R^{\pm}\left|m\right\rangle=\left(m+\frac{1}{2}\right)\left|m\right\rangle\pm\frac{1}{2}\sqrt{m\left(m-1\right)}\left|m-2\right\rangle\pm\frac{1}{2}\sqrt{\left(m+1\right)\left(m+2\right)}\left|m+2\right\rangle.$
(49)

Analogously, for $\left(R^{\pm}\right)^{2}$ we obtain

$\displaystyle\left(R^{\pm}\right)^{2}=$ $\displaystyle\frac{1}{4}\left(2N\pm
aa\pm a^{\dagger}a^{\dagger}+1\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(4N^{2}\pm 2Naa\pm
2Na^{\dagger}a^{\dagger}+4N\pm 2aaN+aaaa+aaa^{\dagger}a^{\dagger}\pm
2aa\right.$ $\displaystyle\left.\pm
2a^{\dagger}a^{\dagger}N+a^{\dagger}a^{\dagger}aa+a^{\dagger}a^{\dagger}a^{\dagger}a^{\dagger}\pm
2a^{\dagger}a^{\dagger}+1\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(4N^{2}\pm 4aaN\pm
4a^{\dagger}a^{\dagger}N+2aaa^{\dagger}a^{\dagger}+aaaa+a^{\dagger}a^{\dagger}a^{\dagger}a^{\dagger}\mp
2aa\pm 6a^{\dagger}a^{\dagger}-1\right),$ (50)

where we have used $\left[N,aa\right]=-2aa$,
$\left[N,a^{\dagger}a^{\dagger}\right]=2a^{\dagger}a^{\dagger}$ and
$\left[aa,a^{\dagger}a^{\dagger}\right]=4N+2$. Hence, by acting on a Fock
state $|m\rangle$ with $m\geq 4$,

$\displaystyle\left(R^{\pm}\right)^{2}\left|m\right\rangle=$
$\displaystyle\left(N^{2}\pm aaN\pm
a^{\dagger}a^{\dagger}N+\frac{1}{2}aaa^{\dagger}a^{\dagger}+\frac{1}{4}aaaa+\frac{1}{4}a^{\dagger}a^{\dagger}a^{\dagger}a^{\dagger}\mp\frac{1}{2}aa\pm\frac{3}{2}a^{\dagger}a^{\dagger}-\frac{1}{4}\right)\left|m\right\rangle$
$\displaystyle=$ $\displaystyle m^{2}\left|m\right\rangle\pm
m\sqrt{m\left(m-1\right)}\left|m-2\right\rangle\pm
m\sqrt{\left(m+1\right)\left(m+2\right)}\left|m+2\right\rangle$
$\displaystyle+\frac{1}{2}\left(m+1\right)\left(m+2\right)\left|m\right\rangle+\frac{1}{4}\sqrt{m\left(m-1\right)\left(m-2\right)\left(m-3\right)}\left|m-4\right\rangle$
$\displaystyle+\frac{1}{4}\sqrt{\left(m+1\right)\left(m+2\right)\left(m+3\right)\left(m+4\right)}\left|m+4\right\rangle\mp\frac{1}{2}\sqrt{m\left(m-1\right)}\left|m-2\right\rangle$
$\displaystyle\pm\frac{3}{2}\sqrt{\left(m+1\right)\left(m+2\right)}\left|m+2\right\rangle-\frac{1}{4}\left|m\right\rangle$
$\displaystyle=$
$\displaystyle\frac{3}{4}\left(2m\left(m+1\right)+1\right)\left|m\right\rangle\pm\frac{1}{2}\left(2m-1\right)\sqrt{m\left(m-1\right)}\left|m-2\right\rangle$
$\displaystyle\pm\frac{1}{2}\left(2m+3\right)\sqrt{\left(m+1\right)\left(m+2\right)}\left|m+2\right\rangle+\frac{1}{4}\sqrt{m\left(m-1\right)\left(m-2\right)\left(m-3\right)}\left|m-4\right\rangle$
$\displaystyle+\frac{1}{4}\sqrt{\left(m+1\right)\left(m+2\right)\left(m+3\right)\left(m+4\right)}\left|m+4\right\rangle.$
(51)

In order to apply our Main Result 1, we have to compute the norm of

$\displaystyle\left(\left(R^{\pm}\right)^{2}-2\left(m+\frac{1}{2}\right)R^{\pm}+\left(m+\frac{1}{2}\right)^{2}\right)\left|m\right\rangle=$
$\displaystyle\frac{3}{4}\left(2m\left(m+1\right)+1\right)\left|m\right\rangle\pm\frac{1}{2}\left(2m-1\right)\sqrt{m\left(m-1\right)}\left|m-2\right\rangle$
$\displaystyle\pm\frac{1}{2}\left(2m+3\right)\sqrt{\left(m+1\right)\left(m+2\right)}\left|m+2\right\rangle$
$\displaystyle+\frac{1}{4}\sqrt{m\left(m-1\right)\left(m-2\right)\left(m-3\right)}\left|m-4\right\rangle$
$\displaystyle+\frac{1}{4}\sqrt{\left(m+1\right)\left(m+2\right)\left(m+3\right)\left(m+4\right)}\left|m+4\right\rangle$
$\displaystyle-\left(m+\frac{1}{2}\right)^{2}\left|m\right\rangle\mp\left(m+\frac{1}{2}\right)\sqrt{m\left(m-1\right)}\left|m-2\right\rangle$
$\displaystyle\mp\left(m+\frac{1}{2}\right)\sqrt{\left(m+1\right)\left(m+2\right)}\left|m+2\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(m^{2}+m+1\right)\left|m\right\rangle\mp\sqrt{m\left(m-1\right)}\left|m-2\right\rangle$
$\displaystyle\pm\sqrt{\left(m+1\right)\left(m+2\right)}\left|m+2\right\rangle+\frac{1}{4}\sqrt{m\left(m-1\right)\left(m-2\right)\left(m-3\right)}\left|m-4\right\rangle$
$\displaystyle+\frac{1}{4}\sqrt{\left(m+1\right)\left(m+2\right)\left(m+3\right)\left(m+4\right)}\left|m+4\right\rangle.$
(52)

Since the $\left\\{\left|m\right\rangle\right\\}$ form an orthonormal basis,

$\displaystyle\left\|\left(\left(R^{\pm}\right)^{2}-2\left(m+\frac{1}{2}\right)R^{\pm}+\left(m+\frac{1}{2}\right)^{2}\right)\left|m\right\rangle\right\|^{2}=$
$\displaystyle\frac{3}{8}\left(m(m+1)\left(m^{2}+m+14\right)+10\right).$ (53)

We now truncate the operators at dimension $d$ in the Fock basis. That is, we
use the orthogonal projector
$P_{d}=\sum_{m=0}^{d-1}\left|m\right\rangle\left\langle m\right|$ as a
truncation scheme. Denote the truncated vectors and operators with a subscript
$d$ and assume that $d\geq m+4$. Then

$\left\|\left(U_{d}(t)-W_{d}^{(n)}(t)\right)\left|m_{d}\right\rangle\right\|\leq\frac{t^{2}}{2n}\sqrt{\frac{3}{8}\left(m(m+1)\left(m^{2}+m+14\right)+10\right)},$
(54)

which is Eq. (6) of the main text. The assumption $d\geq m+4$ is made in order
to ensure that $(R_{d}^{\pm})^{2}$ (and $R_{d}^{\pm}$) acts correctly on the
truncated Fock states $\left|m_{d}\right\rangle$: Since $(R^{\pm})^{2}$
involves terms up to fourth order in $a^{\dagger}$, we have to make sure that
all vectors in the linear combination of $(R^{\pm})^{2}\left|m\right\rangle$
in Eq. (52) are actually again vectors in the truncated Hilbert space $V_{d}$.
If we drop this assumption and consider $m\leq d<m+4$ instead, all terms in
$(R^{\pm})^{2}\left|m\right\rangle$, which give Fock states
$\left|m+j\right\rangle$ with $j\geq d-m+1$, will not contribute to
$(R_{d}^{\pm})^{2}\left|m_{d}\right\rangle$. In this case, Eq. (53) will
become smaller and Eq. (54) will still give an upper bound. Similarly, for the
validity of Eq. (49)–(53), we made the assumption $m\geq 4$. However, our
bound in Eq. (54) stays valid even for Fock states $|m\rangle$ with $m<4$. In
this case, all terms for which $m-4<0$ vanish in the derivation of the bound.
Nevertheless, adding these additional terms only increases the norm so that
Eq. (54) is still an upper bound. This bound is independent of the truncation
dimension $d$, which shows that the full infinite-dimensional Trotter problem
$H^{(1)}=\frac{1}{2}Q^{2}$, $H^{(2)}=\frac{1}{2}P^{2}$ converges strongly to
$H^{\mathrm{osc}}=\frac{1}{2}\left(X^{2}+P^{2}\right)$. Furthermore, since
$Q^{2}$ and $P^{2}$ are self-adjoint operators when acting on the Fock space
$\mathcal{F}=\overline{\mathrm{span}\left\\{\left|m\right\rangle\right\\}}$,
$H^{\mathrm{osc}}=\frac{1}{2}\left(Q^{2}+P^{2}\right)$ is a self-adjoint
operator on the Fock space.