# On Strong Bounds for Trotter and Zeno Product Formulas with Bosonic
Applications

Tim Möbus1,2<EMAIL_ADDRESS>1 Department of Mathematics, Technical
University of Munich, München, Germany  2 Munich Center for Quantum Science
and Technology (MCQST), München, Germany

###### Abstract.

The Trotter product formula and the quantum Zeno effect are both indispensable
tools for constructing time-evolutions using experimentally feasible building
blocks. In this work, we discuss assumptions under which quantitative bounds
can be proven in the strong operator topology on Banach spaces and provide
natural bosonic examples. Specially, we assume the existence of a continuously
embedded Banach space, which relatively bounds the involved generators and
creates an invariant subspace of the limiting semigroup with a stable
restriction. The slightly stronger assumption of admissible subspaces is well-
recognized in the realm of hyperbolic evolution systems (time-dependent
semigroups), to which the results are extended. By assuming access to a
hierarchy of continuously embedded Banach spaces, Suzuki-higher-order bounds
can be demonstrated. In bosonic applications, these embedded Banach spaces
naturally arise through the number operator, leading to a diverse set of
examples encompassing notable instances such as the Bose-Hubbard model, the
Ornstein-Uhlenbeck semigroup, and multi-photon driven dissipation used in
bosonic error correction.

###### Table of Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Strongly continuous semigroups
    2. 2.2 Bosonic Sobolev preserving semigroups
  3. 3 Trotter-like product formulas
    1. 3.1 Trotter product formula
    2. 3.2 Trotter-Suzuki product formula
    3. 3.3 Time-dependent Trotter product formula
    4. 3.4 Application: Bosonic strongly continuous semigroups
  4. 4 Quantum Zeno effect
    1. 4.1 Projective Zeno effect
    2. 4.2 Time-dependent Zeno with general measurement
    3. 4.3 Application: The $l$-photon driven dissipation in error correction
  5. 5 Conclusion
  6. A Technical tools
  7. B Proof of Zeno improvement

##  1\. Introduction

The experimental motivation for operator product formulas, particularly the
(symmetrized) Trotter product formula and the quantum Zeno effect, is the
approximation of a time evolution by simpler building blocks. An instance from
quantum dynamics is the approximation of a global Hamiltonian evolution by
simpler locally defined Hamiltonians [38]. On an abstract structure-wise
level, we establish convergence rates for differences like

$\|\prod_{j=1}^{n}F_{j}(t,s)x-T(t,s)x\|\leq\,?,$

where ${F_{j}(t,s)}_{j=1}^{n}$ is a sequence of contractions, $T(t,s)$ a
contractive strongly continuous ($C_{0}$-) evolution system (time-dependent
$C_{0}$-semigroup), and $x$ an element of the underlying Banach space. In the
present paper, the building blocks ${F_{j}(t,s)}_{j=1}^{n}$ exhibit the
structure given by the Trotter product formula or the quantum Zeno effect. For
example, the former is given by $T(t,0)=e^{t(A+B)}$ and
$F_{j}=e^{t\frac{A}{n}}e^{t\frac{B}{n}}$ for two generators $A,B$ of
contractive $C_{0}$-semigroups, for which [52] proved that

$\Bigl{\|}\left(e^{t\frac{A}{n}}e^{t\frac{B}{n}}\right)^{n}x-e^{t(A+B)}x\Bigr{\|}\rightarrow
0\quad\text{for}\quad n\rightarrow\infty$ (1)

if the closure of $A+B$ generates a contractive $C_{0}$-semigroup. The books
[54, 41, Appx. D] provide a comprehensive overview of the extensive literature
on the topic, and we will only cover some closely related results.

The seminal work by Sophus Lie in 1875 proves Equation 1 as well as a
symmetrized product (see Equation 20) for arbitrary real matrices (Lie’s
product formula). Following the symmetrization, which improves the convergence
rate to $n^{-2}$, [47] investigated different families of product formulas
able to achieve arbitrary powers in the convergence rate [47, 49, 48, 20].
In the direction of the mentioned result by [52] for $C_{0}$-semigroups, [25]
proved a convergence result for non-negative self-adjoint operators on a
separable Hilbert space with weak assumptions on the common domain as well as
a generalization going beyond semigroups [25, 27].
In the direction of Trotter-Suzuki product formulas, the work [2] proves
explicit convergence rate for Hamiltonian evolutions on quantum lattices
systems in the infinite volume limit. Recently, the quantitative result [7]
addresses Trotter-Suzuki formulas in infinite closed quantum systems,
leveraging knowledge of the eigenstates of the limiting dynamics.
Combining the search for a good convergence rate and the generalization to
$C_{0}$-semigroups, many interesting works have investigated the interplay of
the domain of the generators with the convergence rate in the uniform operator
topology [42, 36, 22, 50, 37, 55]. Additionally, the extension to evolution
systems (time-dependent semigroups) can, for example, be found in [53, 35, 23,
45].

The structural relation to the quantum Zeno effect becomes clear at the
example of the projective Zeno effect in closed quantum systems introduced by
[5]. Here, $T(t,0)=e^{iPHP}P$ and $F_{j}=Pe^{t\frac{A}{n}}P$ for any Hermitian
matrix $H$ and projection $P$ represent the result by [31], who named the
effect after the greek philosopher Zeno of Elea,

$\|(Pe^{\frac{it}{n}H})^{n}-e^{it\,PHP}P\|\rightarrow 0\qquad\text{for}\qquad
n\rightarrow\infty\,.$ (2)

Since then, the Zeno effect was generalized in several different directions
(see Facchi.2008Schmidt.2004Itano.1990 for an overview).

Building upon the research contributions of
Burgarth.2020Mobus.2019Barankai.2018, the quantum Zeno effect has been
extended to encompass open and infinite-dimensional quantum systems, featuring
general quantum operations but uniformly continuous time evolutions.
Especially, we allowed for evolution systems in the work [34]. Going beyond
uniformly continuous semigroups, which are always generated by an bounded
operator, to $C_{0}$-semigroups, [13] have recently demonstrated the
convergence of the projective Zeno effect in the strong operator topology for
unbounded Hamiltonians under the weak assumption that $H^{1/2}P$ is densely
defined and $(H^{1/2}P)^{\dagger}H^{1/2}P$ self-adjoint [13]. In open quantum
systems, the recent work by [4] [4] and our follow up [33] show that the
optimal convergence rate can be achieved in the strong operator topology: Let
$(\mathcal{L},\mathcal{D}(\mathcal{L}))$ be the generator of a contractive
$C_{0}$-semigroup on a Banach space $\mathcal{X}$ and $M$ a contraction
satisfying the uniform power convergence, i.e. there is a projection $P$ such
that $\norm{M^{n}-P}_{\mathcal{X}\rightarrow\mathcal{X}}\leq\delta^{n}$. If
$P\mathcal{L}P$ is a generator of a contractive $C_{0}$-semigroup and the
asymptotic Zeno condition, i.e. there is a $b\geq 0$ such that
$\|Pe^{t\mathcal{L}}(\mathbbm{1}-P)\|_{\mathcal{X}\rightarrow\mathcal{X}},\|(\mathbbm{1}-P)e^{t\mathcal{L}}P\|_{\mathcal{X}\rightarrow\mathcal{X}}\leq
tb$ for all $t\geq 0$ holds true, then there is an explicit constant
$c(t,b)>0$ independent of $n$ so that for any $t\geq 0$ and all
$x\in\mathcal{D}((\mathcal{L}P)^{2})$

$\norm{\left(Me^{\frac{t}{n}\mathcal{L}}\right)^{n}x-e^{tP\mathcal{L}P}Px}\leq\frac{c(t,b)}{n}\left(\norm{x}+\norm{\mathcal{L}Px}+\norm{(\mathcal{L}P)^{2}x}\right)+\delta^{n}\norm{x}\,.$

Both works ([4, 33]) are proven in the setup of Banach spaces and can be
applied to open or closed quantum systems choosing the underlying space to be
an separable Hilbert space $\mathcal{H}$ or the space of trace-class operators
$\mathcal{T}_{1}(\mathcal{H})$.
As already mentioned, the projective quantum Zeno effect frequently measures
an evolving quantum state and thereby forces the evolution to remain within
the projective space. This concept is, for example, used to construct the
$Z(\theta)$-gate in bosonic error correction [30, 51, 19, 18]. Moreover,
certain measurements can even decouple the system from its environment
Facchi.2004Burgarth.2020 or protect the initial state from decoherence
Dominy.2013Erez.2004. Interestingly, it can even be applied in commercial
atomic magnetometers [28].

In this work, we generalize the Trotter product formula and the quantum Zeno
effect in the following directions. First, we prove quantitative bounds for
both by assuming that there exists a stronger Banach space $\mathcal{Y}$, on
which the reduction of the limiting $C_{0}$-semigroup is well-defined and
stable and relatively bounds all involved generators. This assumption is not
only motivated by hyperbolic $C_{0}$-evolution systems (time-dependent
$C_{0}$-semigroups) but also by so-called Sobolev-preserving semigroups [16],
which is the main example of the present paper. Nevertheless, we also extend
the product formulas to evolution systems and present a large class of Bosonic
semigroups, for which the results can be easily applied. Moreover, we manage
to prove higher-order approximations for the Trotter product formula.

For that, the paper is divided into three parts: First, we briefly recap the
necessary notation and statements in Section 2. Next, we discuss the results
and proofs for Trotter-like product formulas in Section 3. Here, we begin with
a bound for the Trotter product formula in the strong operator topology and
extend this to $C_{0}$-evolution systems and Suzuki’s higher-order
approximations. We then apply the Trotter-like results to Sobolev-preserving
semigroups. Finally, we generalize the quantum Zeno effect in Section 4,
starting with a bound for the projective Zeno effect, followed by an extension
to $C_{0}$-evolution systems and more general measurements. Once again, the
results are applied to Sobolev-preserving semigroups, particularly, the
$Z(\theta)$-gate in Bosonic error correction.

##  2\. Preliminaries

We shortly introduce Banach space theory, which is described in detail in [Ch.
III]Kato.1995[Ch. IV]Conway.1994[Ch. 2-3]Simon.2015[Ch. 2-3]Hille.1996. Let
$(\mathcal{X},\|\cdot\|_{\mathcal{X}}),(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$
denote Banach spaces and $\mathcal{B}(\mathcal{X},\mathcal{Y})$ the space of
bounded linear maps from $\mathcal{X}$ to $\mathcal{Y}$, which is, equipped
with the operator norm

$\|A\|_{\mathcal{X}\rightarrow\mathcal{Y}}\coloneqq\sup_{\|x\|_{\mathcal{X}}=1}\|A(x)\|_{\mathcal{Y}}\qquad\text{for}\,A\in\mathcal{B}(\mathcal{X},\mathcal{Y}),$
(3)

a Banach space again. We denote the identity map in
$\mathcal{B}(\mathcal{X},\mathcal{Y})$ by $\mathbbm{1}$ and the space of
bounded endomorphisms by $\mathcal{B}(\mathcal{X})$. More generally, an
unbounded operator $A:\mathcal{D}(A)\subset\mathcal{X}\rightarrow\mathcal{X}$
is a linear map defined on a domain $\mathcal{D}(A)$ and is called densely
defined if its domain is a dense subspace of $\mathcal{X}$. The addition and
concatenation of two unbounded operators $(A,\mathcal{D}(A))$ and
$(B,\mathcal{D}(B))$ is defined on
$\mathcal{D}(A+B)=\mathcal{D}(A)\cap\mathcal{D}(B)$ and
$\mathcal{D}(AB)=B^{-1}(\mathcal{D}(A))$. An operator $(A,\mathcal{D}(A))$ is
closed iff its graph $\\{(x,A(x))\,|\,x\in\mathcal{D}(A)\\}$ is a closed set
in the product space $\mathcal{X}\crossproduct\mathcal{X}$. By convention, we
extend all densely defined and bounded operators by the bounded linear
extension theorem to bounded operators on $\mathcal{X}$ [29, Thm. 2.7-11].
Here, $\overline{A}$ is an extension of $A$ if
$\mathcal{D}(A)\subset\mathcal{D}(\overline{A})$ and $Ax=\overline{A}x$ for
all $x\in\mathcal{D}(A)$. A densely defined operator is called closable if
there is a closed extension — its closure. Given operators111In this work, we
use the convention that an operator is a linear map on a Banach space.
$(A,\mathcal{D}(A)),(B,\mathcal{D}(B))$ on $\mathcal{X}$, the operator
$(B,\mathcal{D}(B))$ is relatively $A$-bounded if
$\mathcal{D}(B)\subseteq\mathcal{D}(A)$ and there are $a,b\geq 0$ for all
$x\in\mathcal{D}(A)$ such that

$\|B(x)\|_{\mathcal{X}}\leq a\|A(x)\|_{\mathcal{X}}+b\|x\|_{\mathcal{X}}\,.$
(4)

Similarly, we call $(A,\mathcal{D}(A))$ relatively $\mathcal{Y}$-bounded
w.r.t. a Banach space $\mathcal{Y}\subset\mathcal{D}(\mathcal{A})$ if
$(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ is continuously embedded, i.e. there
is a $b\geq 0$ such that $\|x\|_{\mathcal{X}}\leq b\|x\|_{\mathcal{Y}}$ for
all $x\in\mathcal{Y}$, and $\|A\|_{\mathcal{Y}\rightarrow\mathcal{X}}<\infty$.
Besides the convergence w.r.t. the operator norm (i.e. uniform convergence), a
sequence of operators $\\{A_{k}\\}_{k\in\mathbbm{N}}$ defined on a common
domain $\mathcal{D}(A)$ converges strongly to the operator
$(A,\mathcal{D}(A))$ if for all $x\in\mathcal{D}(A)$

$\lim_{k\rightarrow\infty}\|A_{k}x-Ax\|_{\mathcal{X}}=0\,.$

In the next step, we shortly introduce the theory of weighted Banach spaces,
which is important for the examples of this work. Let
$(\mathcal{W},\mathcal{D}(\mathcal{W}))$ be an injective operator, then
$\mathcal{D}(\mathcal{W})$ equipped with

$\|X\|_{\mathcal{W}}\coloneqq\|\mathcal{W}(X)\|_{\mathcal{X}}$ (5)

defines a normed space, which is additionally complete if
$(\mathcal{W},\mathcal{D}(\mathcal{W}))$ is surjective. This can be seen by
the closed Graph theorem, which shows that $\mathcal{W}$ is a closed operator
[16, Lem. 2.2].

In our examples, an operator,$\mathcal{W}$, will play the role of a reference
operator with specific relative bounds to the considered generators, and then
define the stable or admissible subspace required for our theorems.

###  2.1. Strongly continuous semigroups

A quantum mechanical evolution is often described by a differential equation
called the master equation, for which strongly continuous ($C_{0}$-)semigroups
and $(C_{0})$-evolution systems constitute an important toolbox for solution
theory. More details can for example be found in [11, Chap. II][26, Chap. 9]
or [21, Chap. X-XIII]. A $C_{0}$-semigroup is defined by a family of operators
$(T_{t})_{t\geq 0}\subset\mathcal{B}(\mathcal{X})$, which satisfies

$T_{t}T_{s}=T_{t+s},\quad
T_{0}=\mathbbm{1},\quad\text{and}\quad\lim\limits_{t\downarrow 0}T_{t}x=x$ (6)

for all $t,s\geq 0$ and $x\in\mathcal{X}$. Equivalently, every semigroup is
uniquely determined by its generator [11, Thm. II.1.4] — a possible unbounded,
densely defined, and closed operator defined by

$\mathcal{L}(x)=\lim\limits_{t\to 0^{+}}\frac{1}{t}(T_{t}(x)-x)\,,$ (7)

for any

$x\in\mathcal{D}(\mathcal{L})=\\{x\in\mathcal{X}:t\mapsto T_{t}(x)\text{
differentiable on }\mathbbm{R}_{\geq 0}\\}\,.$ (8)

This justifies the notation $T_{t}=e^{t\mathcal{L}}$. One can show that every
semigroup is stable [11, Prop. I.5.5], i.e

$\|T_{t}\|_{\mathcal{X}\rightarrow\mathcal{X}}\leq ce^{\omega t}$ (9)

for a $c\geq 0$ and $\omega\geq 0$, in particular, it is a contraction if
$c\leq 1$ and $\omega=0$. Since the semigroup leaves the domain
$\mathcal{D}(\mathcal{L})$ invariant and commutes with its generator, the
following master equation admits the unique solution $T_{t}(x(0))=x(t)$ [11,
Prop. II.6.2]

$\frac{d}{dt}x(t)=\mathcal{L}(x(t))\quad
x(0)\in\mathcal{D}(\mathcal{L})\quad\text{and}\quad t\geq 0\,.$ (10)

A possible extension of the above master equation is by assuming that the
generator is time-dependent. In this case, a two-parameter family of bounded
operators $(T_{t,s})_{(t,s)\in I_{T}}$ with $I_{T}\coloneqq\\{(t,s)\,|\,T\geq
t\geq s\geq 0\\}$ is called a $C_{0}$-evolution system (or time-dependent
$C_{0}$-semigroup) if

$T_{t,t}=\mathbbm{1}\,,\quad
T_{t,r}T_{r,s}=T_{t,s}\,,\quad\text{and}\quad(t,s)\mapsto T_{t,s}x\text{ is
strongly continuous}$ (11)

for all $0\leq s\leq r\leq t\leq T$. Clearly, every $C_{0}$-semigroup $T_{t}$
can be identified by an evolution system via $T_{t,s}=T_{t-s}$. However, a
crucial difference is that evolution systems are not necessarily
differentiable for any $x\neq 0$ [p. 478]Engel.2000. However, in the present
paper all considered evolution systems are defined to be differentiable in the
sense that there is a family of generators
$(\mathcal{L}_{s},\mathcal{D}(\mathcal{L}_{s}))_{0\leq s\leq T}$ such that for
all $x\in\mathcal{D}(\mathcal{L}_{s})$ and all $(t,s)\in I_{T}$,

$\displaystyle\frac{\partial}{\partial s}T_{t,s}x=-T_{t,s}\mathcal{L}_{s}x\,.$
(12)

We say that $(\mathcal{L}_{s},\mathcal{D}(\mathcal{L}_{s}))_{0\leq s\leq T}$
generates $T(t,s)$ if it is uniquely determined by the above relation. For a
continuously embedded Banach space $(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ in
$\mathcal{X}$, we call a continuously differentiable function $t\mapsto
x(t)\in\mathcal{Y}$ for $(t,s)\in I_{T}$ a $\mathcal{Y}$-solution if it
satisfies the differential equation

$\frac{\partial}{\partial t}x(t)=\mathcal{L}_{t}(x(t))\quad\text{and}\quad
x(s)=x_{s}\in\mathcal{Y}\quad\text{ for }0\leq s\leq t\leq T$ (13)

for a given family
$(\mathcal{L}_{t},\mathcal{Y}\subset\mathcal{D}(\mathcal{L}_{t}))$ with
$(t,s)\in I_{T}$. Conversely, we introduce sufficient assumptions on
$(\mathcal{L}_{s},\mathcal{D}(\mathcal{L}_{s}))$ such that the differential
Equation 13 admits a unique solution operator [Chap. 5]Pazy.1983[pp. 127
ff.]Giuseppe.1976. For that, we introduce admissible subspaces:

######  Definition 2.1 (Stable and admissible subspaces)

Let $(\mathcal{Y}\subset\mathcal{X},\|\cdot\|_{\mathcal{Y}})$ be a
continuously embedded subspace of $\mathcal{X}$. Then,
$(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ is $\mathcal{L}$-stable, where
$\mathcal{L}$ is the generator of an $C_{0}$-semigroup on $\mathcal{X}$, if it
is an invariant subspace, i.e. $e^{t\mathcal{L}}\mathcal{Y}\subset\mathcal{Y}$
for all $t\geq 0$, and the reduced operator-valued function
$e^{t\mathcal{L}}|_{\mathcal{Y}}:\mathcal{Y}\rightarrow\mathcal{Y}$ satisfies
$\|e^{t\mathcal{L}}\|_{\mathcal{Y}\rightarrow\mathcal{Y}}\leq ce^{t\omega}$
for appropriate $c,\omega\geq 0$. If the reduced function is additionally
continuous — defines a semigroup, we call
$(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ is $\mathcal{L}$-admissible. The same
definition translates to $C_{0}$-evolution systems.

Then, the sufficient coefficients, which refer to evolutions system of
hyperbolic type, are:

  1. (1)

For every $s\geq 0$, $(\mathcal{L}_{s},\mathcal{D}(\mathcal{L}_{s}))$
generates a $C_{0}$-semigroup ;

  2. (2)

For all $0\leq s\leq T$, there exists a $\mathcal{L}_{s}$-admissible subspace
$(\mathcal{Y}\subset\mathcal{X},\|\cdot\|_{\mathcal{Y}})$ ;

  3. (3)

Finally, the map
$s\mapsto\mathcal{L}_{s}\in\mathcal{B}((\mathcal{Y},\|.\|_{\mathcal{Y}}),(\mathcal{X},\|.\|_{\mathcal{X}}))$
is uniformly continuous.

To achieve a $\mathcal{Y}$-solution of the differential equation (13), one can
additionally assume

  1. (4)

$T_{t,s}\mathcal{Y}\subseteq\mathcal{Y}$ for $0\leq s\leq t\leq T$;

  2. (5)

$(s,t)\mapsto T_{s,t}$ is strongly continuous on
$(\mathcal{Y},\norm{\cdot}_{\mathcal{Y}})$ .

Under these assumptions, one can show:

######  Theorem 2.2 (Hyperbolic evolutions systems [39, Thm. 5.3.1/4.3])

Let $(\mathcal{L}_{s},\mathcal{D}(\mathcal{L}_{s}))_{s\geq 0}$ be a family of
operators satisfying assumptions $(1-3)$. Then, there is a unique hyperbolic
$C_{0}$-evolution system on $(\mathcal{X},\|\cdot\|_{\mathcal{X}})$ satisfying

$\|T_{t,s}\|_{\mathcal{X}\rightarrow\mathcal{X}}\leq
c\,e^{(t-s)\omega}\,,\quad\frac{\partial^{+}}{\partial
s}T_{t,s}x\biggl{|}_{t=s}=\mathcal{L}_{s}x\,,\quad\text{and}\quad\frac{\partial}{\partial
s}T_{t,s}x=-T_{t,s}\mathcal{L}_{s}x$

for all $(t,s)\in I_{T}$ and $x\in\mathcal{Y}$. If further $(4)$ and $(5)$
hold, then for every $x_{s}\in\mathcal{Y}$, $T_{t,s}x_{s}$ is a unique
solution for the initial value problem in (13).

###  2.2. Bosonic Sobolev preserving semigroups

One of the main examples in our work is the class of bosonic Sobolev
preserving semigroups, which are discussed in detail in [16]. For those, we
shortly introduce bosonic (or continuous variable) systems in the second
quantization formalism [6, Chap. 5.2]. Let $\mathcal{H}=L^{2}(\mathbb{R}^{m})$
be the Fock space of $m$-modes endowed with an orthonormal (Fock-)basis
$\\{\ket{\textbf{n}}\\}_{\textbf{n}\in\mathbbm{N}^{m}}$, where $n_{j}$ labels
the number of particles in the $j^{\text{th}}$-mode. Then, the annihilation
and creation operators are defined by

$a_{j}\ket{\textbf{n}}=\sqrt{n_{j}}\ket{n_{1},...,n_{j}-1,...,n_{m}},\quad
a^{\dagger}_{j}\ket{\textbf{n}}=\sqrt{n_{j}+1}\ket{n_{1},...,n_{j}+1,...,n_{m}}$

and satisfy

$[a_{i},a^{\dagger}_{j}]\ket{\textbf{n}}=\delta_{i,j}\ket{\textbf{n}}\,,\quad[a_{i},a_{j}]\ket{\textbf{n}}=0$

for all $\textbf{n}\in\mathbbm{N}^{m}$ and $i,j\in\\{1,...,m\\}$. Moreover,
the number operator is defined by

$N_{j}\ket{\textbf{n}}=a^{\dagger}_{j}a_{j}\ket{\textbf{n}}=n_{j}\ket{\textbf{n}}$
(14)

and counts the number of particles in the $j^{\text{th}}$-mode. Another
important class are the coherent states, which are eigenvectors of $a_{j}$,
i.e. $a_{j}\ket{\alpha}=\alpha_{j}\ket{\alpha}$ for all $j\in\\{1,...,m\\}$.
Here, the $j^{\text{th}}$-mode is defined by
$\ket{\alpha_{j}}=\exp(-\frac{|\alpha_{j}|^{2}}{2})\sum_{j=0}^{\infty}\frac{\alpha_{j}^{n}}{\sqrt{n!}}\ket{n_{j}}$
for any $\alpha_{j}\in\mathbbm{C}$. In most of our proofs, we use the finitely
supported subspace
$\mathcal{H}_{f}=\\{\ket{\psi}=\sum_{\textbf{n}=0}^{M}\lambda_{\textbf{n}}\ket{\textbf{n}}\,|\,M\in\mathbbm{N}\,,\,\textbf{n}\in\mathbbm{N}^{m}\,,\,\lambda_{\textbf{n}}\in\mathbbm{C}\\}$
as a domain for $a_{j}$, $a^{\dagger}_{j}$, and polynomials in both and extent
the bound afterwards by continuity in an appropriate topology.

For $A\in\mathcal{B}(\mathcal{H})$, the adjoint is uniquely defined by
$\braket{A\phi\,,\varphi}=\braket{\phi\,,A^{\dagger}\varphi}$ for all
$\phi,\varphi\in\mathcal{H}$ [44, Sec. 2.1] and the space of self-adjoint
operators denoted by
$\mathcal{B}_{\operatorname{sa}}(\mathcal{H})\coloneqq\\{A\in\mathcal{B}(\mathcal{H})\,|\,A=A^{\dagger}\\}$.
Then, we define the trace by
$\tr[A]=\sum_{\textbf{n}\in\mathbbm{N}^{m}}\bra{\textbf{n}}A\ket{\textbf{n}}$
for a positive $A\in\mathcal{B}_{\operatorname{sa}}(\mathcal{H})$, i.e. there
is a $B\in\mathcal{B}(\mathcal{H})$ such that $A=B^{\dagger}B$, and the trace
norm by $\|A\|_{1}=\tr[\sqrt{A^{\dagger}A}]$. Finally, the space of self-
adjoint trace-class operators is denoted by
$\mathcal{T}_{1,\operatorname{sa}}\coloneqq\\{A\in\mathcal{B}_{\operatorname{sa}}(\mathcal{H})\,|\,\|A\|_{1}<\infty\\}$.
Using the convention $\textbf{n}\leq M$ iff $n_{j}\leq M$ for all
$j\in\\{1,...,m\\}$, we define
$\mathcal{T}_{f}\coloneqq\\{A=\sum_{\textbf{n},\textbf{m}=0}^{M}a_{\textbf{n}\textbf{m}}\outerproduct{\textbf{n}}{\textbf{m}}\,|\,M\in\mathbbm{N}\,,\,\,\textbf{n},\textbf{m}\in\mathbbm{N}^{m}\,,\,\,a_{\textbf{n},\textbf{m}}=\bar{a}_{\textbf{m},\textbf{n}}\in\mathbbm{C}\\}$
as before, which is dense in $\mathcal{T}_{1,\operatorname{sa}}$ [44, Sec.
3.1].

Then, we define the bosonic Sobolev spaces by

$\mathcal{D}(\mathcal{W}^{\textbf{k}})=\mathcal{W}^{-\textbf{k}}(\mathcal{T}_{1,\,\operatorname{sa}})\quad\text{via}\quad\mathcal{W}^{-\textbf{k}}(x)\coloneqq\prod_{j=1}^{m}(\mathbbm{1}+N_{j})^{-k_{j}/4}\,\,x\,\prod_{j=1}^{m}(\mathbbm{1}+N_{j})^{-k_{j}/4}\,.$
(15)

Since $\mathcal{W}^{-\textbf{k}}$ is an injective bounded operator, its
inverse $\mathcal{W}^{\textbf{k}}$ is a closed operator on
$\mathcal{D}(\mathcal{W}^{\textbf{k}})$ by the closed graph theorem and
$(\mathcal{D}(\mathcal{W}^{\textbf{k}}),\|\cdot\|_{\mathcal{W}^{\textbf{k}}})$
a Banach space. For the sake of notation, we define

$W^{\textbf{k},1}\coloneqq\mathcal{D}(\mathcal{W}^{\textbf{k}})\quad\text{and}\quad\|\cdot\|_{W^{\textbf{k},1}}\coloneqq\|\cdot\|_{\mathcal{W}^{\textbf{k}}}$
(16)

with the special case
$(W^{0,1},\|\cdot\|_{W^{0,1}})=(\mathcal{T}_{1,\,\operatorname{sa}},\|\cdot\|_{1})$.
Interestingly, one can show that if $0\leq k_{j}<k_{j}^{\prime}$ for all
$j\in\\{1,...,m\\}$, $W^{\textbf{k}^{\prime},1}$ is compactly embedded in
$W^{\textbf{k},1}$ [16].

Next, we discuss a sufficient condition so that an operator
$(\mathcal{L},\mathcal{T}_{f})$ is a generator. First, we assume that the
operator is in $\operatorname{GKSL}$ form, i.e.

$\mathcal{L}:\mathcal{T}_{f}\to\mathcal{T}_{f}\quad
x\mapsto\mathcal{L}(x)=-i[H,x]+\sum\limits_{j=1}^{K}L_{j}xL_{j}^{\dagger}-\frac{1}{2}\\{L_{j}^{\dagger}L_{j},x\\}$
(17)

with $K\in\mathbb{N}$, $L_{j}\coloneqq p_{j}(a_{1},a^{\dagger}_{1},...)$
defined by polynomials of $a_{j}$ and $a^{\dagger}_{j}$ for all
$j\in\\{1,...,m\\}$222Here, we assume that the monomials in each mode are
ordered like $(a^{\dagger}_{j})^{k}a_{j}^{l}$., and symmetric $H\coloneqq
p_{H}(a_{1},a^{\dagger}_{1},...)$, i.e.
$\braket{A\psi,\phi}=\braket{\psi,A\phi}$ for all
$\psi,\phi\in\mathcal{H}_{f}$. Second, we assume that there are non-negative
sequences $(\textbf{k}_{r})_{r\in\mathbbm{N}}\subset\mathbbm{N}^{m}$ diverging
coordinate-vice to infinity and $(\omega_{\textbf{k}_{r}})_{r\in\mathbbm{N}}$
so that

$\tr[\mathcal{W}^{\textbf{k}_{r}}\mathcal{L}(x)]\leq\omega_{\textbf{k}_{r}}\tr[\mathcal{W}^{\textbf{k}_{r}}(x)]$
(18)

for all positive semi-definite $x\in\mathcal{T}_{f}$. Then, the following
theorem holds:

###### Theorem 2.3

Let $(\mathcal{L},\mathcal{T}_{f})$ be an operator of the form (17) satisfying
Equation 18, then the closure $\overline{\mathcal{L}}$ generates a positivity
preserving $C_{0}$-semigroup $(\mathcal{T}_{t})_{t\geq 0}$ on
$W^{\textbf{k}_{r},1}$ for $r\in\mathbbm{N}$ and

$\norm{\mathcal{T}_{t}}_{W^{\textbf{k}_{r},1}\to W^{\textbf{k}_{r},1}}\leq
e^{\omega_{\textbf{k}_{r}}t}$ (19)

for all $t\geq 0$. In the special case $\textbf{k}=0$, the semigroup is
contractive and trace-preserving.

The same holds true if the coefficients of the operator are time-dependent but
continuous and if the right-hand side in Equation 18 does not depend on $t$
[33, Thm. 4.5].

##  3\. Trotter-like product formulas

In this section, we discuss quantitative bounds for Trotter-like product
formulas. We begin with the fundamental Trotter product formula [52] and
enhance the convergence rate through the Suzuki approach [48]. Additionally,
we generalize the involved semigroups to evolution systems. Finally, we delve
into bosonic examples, in particular form bosonic Hamiltonian learning, as
well as error correction. Most of our proofs rely on a telescopic sum approach
known for example from proving the Lie-Trotter product formula [41, 54]. The
following generalization is proven in A.1:

###### Lemma 3.1

Let $I_{T}\ni(t,s)\mapsto F(t,s)\in\mathcal{B}(\mathcal{X})$ be an operator-
valued map with $\|F(t,s)\|_{\mathcal{X}\rightarrow\mathcal{X}}\leq 1$ and
$F(s,s)=\mathbbm{1}$ for all $(t,s)\in I_{T}\coloneqq\\{(t,s)\,|\,T\geq t\geq
s\geq 0\\}$. Moreover, let $T(t,s)$ be an evolution system, then for all
$(t,s)\in I_{T}$

$\|\prod_{j=1}^{n}F(s_{j},s_{j-1})x-T(t,s)x\|_{\mathcal{X}}\leq
n\max_{j\in\\{1,...,n\\}}\left\\{\|\left(F(s_{j},s_{j-1})-T(s_{j},s_{j-1})\right)T(s_{j-1},s_{0})x\|_{\mathcal{X}}\right\\}\,,$

for any increasing sequence $s=s_{0}<s_{1}<...<s_{n-1}<s_{n}=t$. Note that the
product over $F(s_{j},s_{j-1})$ is in decreasing order from the left to the
right.

###  3.1. Trotter product formula

In Trotter.1959, [52] generalized Lie’s product formula for matrices to
contractive $C_{0}$-semigroups on a Banach space $\mathcal{X}$ generated by
$(\mathcal{L},\mathcal{D}(\mathcal{L}))$ and
$(\mathcal{K},\mathcal{D}(\mathcal{K}))$. Under the assumption that the
closure of $\mathcal{L}+\mathcal{K}$ generates again a contractive
$C_{0}$-semigroup, he proved [52] that for all $x\in\mathcal{X}$

$\left(e^{\frac{\mathcal{L}}{n}}e^{\frac{\mathcal{K}}{n}}\right)^{n}x\rightarrow
e^{\mathcal{L}+\mathcal{K}}x\qquad\text{for}\qquad n\rightarrow\infty\,.$

In the following, we consider a set of assumptions on the generators
$(\mathcal{L},\mathcal{D}(\mathcal{L}))$ and
$(\mathcal{K},\mathcal{D}(\mathcal{K}))$ such that a quantitative error bound
can be found.

###### Theorem 3.2

Let $(\mathcal{L},\mathcal{D}(\mathcal{L}))$ and
$(\mathcal{K},\mathcal{D}(\mathcal{K}))$ be generators of two contractive
$C_{0}$-semigroups and
$(\mathcal{Y}\subset\mathcal{X},\|\cdot\|_{\mathcal{Y}})$ a Banach space such
that $\mathcal{Y}\subset\mathcal{D}(\mathcal{L}+\mathcal{K})$ and 333In the
following, we omit the assumption
$\mathcal{Y}\subset\mathcal{D}(\mathcal{L}+\mathcal{K})$ because it is clear
by the notation $(\mathcal{L}+\mathcal{K},\mathcal{Y})$ the closure of
$(\mathcal{L}+\mathcal{K},\mathcal{Y})$ generates a contractive
$C_{0}$-semigroup in $\mathcal{X}$. If additionally $\mathcal{K}$,
$\mathcal{L}$, $\mathcal{K}^{2}$, $\mathcal{L}\mathcal{K}$,
$\mathcal{K}\mathcal{L}$, $\mathcal{L}^{2}$ are relatively
$\mathcal{Y}$-bounded and $(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ is
$(\mathcal{L}+\mathcal{K})$-stable, in particular
$\|e^{t(\mathcal{L}+\mathcal{K})}\|_{\mathcal{Y}\rightarrow\mathcal{Y}}\leq\tilde{c}e^{t\omega}$,
then there is a $c\geq 0$ such that for all $x\in\mathcal{Y}$, $t\geq 0$, and
$n\in\mathbbm{N}$

$\displaystyle\norm{\left(e^{\frac{t}{n}\mathcal{L}}e^{\frac{t}{n}\mathcal{K}}\right)^{n}x-e^{t(\mathcal{L}+\mathcal{K})}x}_{\mathcal{X}}$
$\displaystyle\leq c\frac{t^{2}}{n}e^{t\omega}\|x\|_{\mathcal{Y}}\,.$

###### Proof.

By rescaling, assume $t\in[0,1]$. Then, we apply 3.1 to

$F(t,s)=e^{(t-s)\mathcal{L}}e^{(t-s)\mathcal{K}}\quad
T(t,0)=e^{(t-s)(\mathcal{L}+\mathcal{K})}$

w.r.t. the sequence $s_{0}=0<\frac{1}{n}<...<\frac{n-1}{n}<1$ so that

$\|\left(e^{\frac{\mathcal{L}}{n}}e^{\frac{\mathcal{K}}{n}}\right)^{n}x-e^{\mathcal{L}+\mathcal{K}}x\|_{\mathcal{X}}\leq
n\max_{s\in[0,1-1/n]}\|\left(e^{\frac{\mathcal{L}}{n}}e^{\frac{\mathcal{K}}{n}}-e^{\frac{\mathcal{L}+\mathcal{K}}{n}}\right)e^{s(\mathcal{L}+\mathcal{K})}x\|_{\mathcal{X}}\,.$

Then, the integral equation (40) shows

$\displaystyle
e^{\frac{\mathcal{L}}{n}}e^{\frac{\mathcal{K}}{n}}x_{s}-e^{\frac{\mathcal{L}+\mathcal{K}}{n}}x_{s}$
$\displaystyle=(e^{\frac{\mathcal{L}}{n}}-\mathbbm{1})x_{s}+e^{\frac{\mathcal{L}}{n}}(e^{\frac{\mathcal{K}}{n}}-\mathbbm{1})x_{s}-(e^{\frac{\mathcal{L}+\mathcal{K}}{n}}-\mathbbm{1})x_{s}$
$\displaystyle=\frac{1}{n}\int_{0}^{1}\left(e^{s_{1}\frac{\mathcal{L}}{n}}\mathcal{L}+e^{\frac{\mathcal{L}}{n}}e^{s_{1}\frac{\mathcal{K}}{n}}\mathcal{K}-e^{s_{1}\frac{\mathcal{L}+\mathcal{K}}{n}}(\mathcal{L}+\mathcal{K})\right)x_{s}ds_{1}$
$\displaystyle=\frac{1}{n^{2}}\iint_{0}^{1}\Bigl{(}s_{1}e^{s_{1}s_{2}\frac{\mathcal{L}}{n}}\mathcal{L}^{2}+e^{s_{2}\frac{\mathcal{L}}{n}}\mathcal{L}\mathcal{K}+s_{1}e^{\frac{\mathcal{L}}{n}}e^{s_{1}s_{2}\frac{\mathcal{K}}{n}}\mathcal{K}^{2}$
$\displaystyle\qquad\qquad\qquad-
s_{1}e^{s_{1}s_{2}\frac{\mathcal{L}+\mathcal{K}}{n}}(\mathcal{L}+\mathcal{K})^{2}\Bigr{)}x_{s}ds_{2}ds_{1}\,,$

where $x_{s}\coloneqq e^{s(\mathcal{L}+\mathcal{K})}x$. The integral on the
right-hand side, is well-defined due to the continuity of the integrand in
$s_{1}$ and $s_{2}$ (see [11, Lem. II.1.3]), which is given by the structure
above and [11, Lem. B.15]. Moreover, the assumptions show

$\|e^{s\frac{\mathcal{L}}{n}}\mathcal{L}^{2}x_{s}\|_{\mathcal{X}}\leq\|\mathcal{L}^{2}x_{s}\|_{\mathcal{X}}\leq
a_{\mathcal{L}^{2}}\|x_{s}\|_{\mathcal{Y}}\leq
a\tilde{c}e^{s\omega}\|x\|_{\mathcal{Y}}$

for a $a\geq 0$. Similarly, all other terms of the integrand in the last line
are bounded, which implies

$\displaystyle\|\left(e^{\frac{\mathcal{L}}{n}}e^{\frac{\mathcal{K}}{n}}\right)^{n}x-e^{\mathcal{L}+\mathcal{K}}x\|_{\mathcal{X}}$
$\displaystyle\leq\frac{1}{n}\biggl{(}a_{\mathcal{L}^{2}}+\frac{3}{2}a_{\mathcal{L}\mathcal{K}}+a_{\mathcal{K}^{2}}+a_{\mathcal{K}\mathcal{L}}\biggr{)}\tilde{c}e^{\omega}\|x\|_{\mathcal{Y}}\,.$

This finishes the proof of the theorem with
$c=\bigl{(}a_{\mathcal{L}^{2}}+\frac{3}{2}a_{\mathcal{L}\mathcal{K}}+a_{\mathcal{K}^{2}}+a_{\mathcal{K}\mathcal{L}}\bigr{)}\tilde{c}$.
∎

To make the abstract assumption of the existence of the Banach space
$(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ more tangible, we shortly go through a
simple theoretical example. Note also that in Section 3.4, we discuss a large
class of bosonic semigroups for which such a Banach space $\mathcal{Y}$ can
easily be defined.

######  Remark.

Note that in the above proof the assumption that
$(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ is a
$(\mathcal{K}+\mathcal{L}$)-stable Banach space can be weakened to a normed
space invariant under the $C_{0}$-semigroup $e^{t(\mathcal{L}+\mathcal{K})}$
satisfying
$\|e^{t(\mathcal{L}+\mathcal{K})}\|_{\mathcal{Y}\rightarrow\mathcal{Y}}\leq\tilde{c}e^{t\omega}$.
Together with the other assumptions, the same poof works.

Next, we discuss a possible choice of $\mathcal{Y}$ implied by certain
assumptions on $\mathcal{L}$ and $\mathcal{K}$.

######  Example 1.

Let $(\mathcal{L},\mathcal{D}(\mathcal{L}))$ and
$(\mathcal{K},\mathcal{D}(\mathcal{K}))$ be generators of two contractive
$C_{0}$-semigroups such that the closure of
$(\mathcal{L}+\mathcal{K},\mathcal{D}(\mathcal{L}+\mathcal{K}))$ defines a
contractive $C_{0}$-semigroup again and assume that $\mathcal{K}^{2}$,
$\mathcal{L}\mathcal{K}$, $\mathcal{K}\mathcal{L}$, and $\mathcal{L}^{2}$ are
relatively $(\overline{\mathcal{L}+\mathcal{K}})^{2}$-bounded. Then, we equip
$\mathcal{Y}=\mathcal{D}((\overline{\mathcal{L}+\mathcal{K}})^{2})$ with the
norm

$\|x\|_{\mathcal{Y}}\coloneqq\|x\|_{\mathcal{X}}+\|(\overline{\mathcal{L}+\mathcal{K}})x\|_{\mathcal{X}}+\|(\overline{\mathcal{L}+\mathcal{K}})^{2}x\|_{\mathcal{X}}$

for $x\in\mathcal{Y}$, which defines a Banach space by repeating the reasoning
why $\mathcal{D}(\mathcal{L})$ with the graph norm defines a Banach space
twice [26, Sec. 5.2]. Specially, it defines an
$(\mathcal{L}+\mathcal{K})$-admissible subspace, there satisfies the
assumptions of 3.2, with $\omega=0$ so that the bound does not depend
exponentially on $t$. Note that the above construction is equivalent to the
so-called Sobolev Tower of order $2$ [11, Sec. 5] if
$\overline{\mathcal{L}+\mathcal{K}}$ is invertible.

###  3.2. Trotter-Suzuki product formula

In the case of matrices $\mathcal{L},\mathcal{K}\in\mathcal{B}(\mathcal{X})$
for finite dimensional $\mathcal{X}$, a symmetrized version of Lie’s product
formula (for $C_{0}$-semigroups Trotter’s product formula) improves the
convergence rate to a power of two:

$\|\left(e^{\frac{\mathcal{L}}{2n}}e^{\frac{\mathcal{K}}{n}}e^{\frac{\mathcal{L}}{2n}}\right)^{n}-e^{\mathcal{L}+\mathcal{K}}\|_{\mathcal{X}\rightarrow\mathcal{X}}=\mathcal{O}\Bigl{(}\frac{1}{n^{2}}\Bigr{)}\,.$
(20)

As seen in the proof of 3.2, the question reduces to find an appropriate
$F(t)$ so that

$\|F^{(m)}\Bigl{(}\frac{1}{n}\Bigr{)}-e^{\frac{\mathcal{L}+\mathcal{K}}{n}}\|_{\mathcal{X}\rightarrow\mathcal{X}}=\mathcal{O}\Bigl{(}\frac{1}{n^{m+1}}\Bigr{)}\,.$

In the case of the symmetrized Lie product formula $F^{(2)}(t)\coloneqq
e^{t\frac{\mathcal{L}}{2}}e^{t\mathcal{K}}e^{t\frac{\mathcal{L}}{2}}$ (see
3.4). Following this idea, we assume the structure

$F^{(m)}(t)\coloneqq e^{tp_{1}A}e^{tp_{2}B}e^{tp_{3}A}\cdots
e^{tp_{M-1}A}e^{tp_{M}B}$

for $p_{1},...,p_{m}\in\mathbbm{R}$. For that, [47] proved different schemes
to achieve higher order convergence rates (see [20] for an overview) for
bounded generators. In the following, we consider a general framework and
explicitly demonstrate the symmetrized product formula afterwards.

###### Lemma 3.3

Let $(\mathcal{L},\mathcal{D}(\mathcal{L}))$ and
$(\mathcal{K},\mathcal{D}(\mathcal{K}))$ be generators of two contractive
$C_{0}$-semigroups, $m\in\mathbbm{N}_{\geq 1}$, and
$(\mathcal{Y}^{(m)}\subset\mathcal{X},\|\cdot\|_{\mathcal{Y}^{(m)}})$ a Banach
space such that the closure of $(\mathcal{L}+\mathcal{K},\mathcal{Y}^{(m)})$
generates a $C_{0}$-semigroup on $\mathcal{X}$. Moreover, we assume there
exists a contractive and strongly continuous map $t\mapsto F^{(m)}(t)$ for
$t\geq 0$ such that

$\|F^{(m)}(t)-e^{t(\mathcal{L}+\mathcal{K})}\|_{\mathcal{Y}^{(m)}\rightarrow\mathcal{X}}\leq
b^{(m)}t^{m+1}\,.$ (21)

If $(\mathcal{Y}^{(m)},\|\cdot\|_{\mathcal{Y}^{(m)}})$ is
$(\mathcal{L}+\mathcal{K})$-stable, in particular
$\|e^{t(\mathcal{L}+\mathcal{K})}\|_{\mathcal{Y}\rightarrow\mathcal{Y}}\leq\tilde{c}^{(m)}e^{t\omega^{(m)}}$,
then for all $x\in\mathcal{Y}^{(m)}$ and $t\geq 0$

$\norm{\left(F^{(m)}\Bigl{(}\frac{t}{n}\Bigr{)}\right)^{n}x-e^{t(\mathcal{L}+\mathcal{K})}x}_{\mathcal{X}}\leq\frac{t^{m+1}}{n^{m}}\tilde{c}^{(m)}b^{(m)}e^{t\omega^{(m)}}\|x\|_{\mathcal{Y}^{(m)}}\,.$

###### Proof.

Similar to the proof of 3.2, we apply 3.1 to

$F(t,0)=F^{(m)}(t)\quad T(t,0)=e^{t(\mathcal{L}+\mathcal{K})}$

w.r.t. the sequence $s_{0}=0<t\frac{1}{n}<...<t\frac{n-1}{n}<s_{n}=t$ so that

$\displaystyle\|\left(F^{(m)}\Bigl{(}\frac{t}{n}\Bigr{)}\right)^{n}x-e^{t(\mathcal{L}+\mathcal{K})}x\|_{\mathcal{X}}$
$\displaystyle\leq
n\max_{s\in[0,t(1-1/n)]}\|\left(F^{(m)}\Bigl{(}\frac{t}{n}\Bigr{)}-e^{\frac{t}{n}(\mathcal{L}+\mathcal{K})}\right)e^{s(\mathcal{L}+\mathcal{K})}x\|_{\mathcal{X}}$
$\displaystyle\leq\frac{t^{m+1}}{n^{m}}\tilde{c}^{(m)}b^{(m)}e^{t\omega^{(m)}}\|x\|_{\mathcal{Y}^{(m)}}$

where we used Assumption 21 and that $\mathcal{Y}^{(m)}$ is
$({\mathcal{L}+\mathcal{K}})$-stable in the last step. ∎

On quantum lattice system in the infinite volume limit, [2] shows that
Equation 21 is satisfied and the above convergence rate in the strong topology
is achieved. Next, we apply the above structure to the symmetrized Trotter
product formula (or second order Trotter-Suzuki product formula) and give
assumptions such that quadratic convergence is achieved.

###### Theorem 3.4

Let $(\mathcal{L},\mathcal{D}(\mathcal{L}))$ and
$(\mathcal{K},\mathcal{D}(\mathcal{K}))$ be generators of two contractive
$C_{0}$-semigroups and
$(\|\cdot\|_{\mathcal{Y}^{(2)}},\mathcal{Y}^{(2)}\subset\mathcal{X})$ a Banach
space such that the closure of $(\mathcal{Y}^{(2)},\mathcal{L}+\mathcal{K})$
generates a contractive $C_{0}$-semigroup in $\mathcal{X}$. If additionally
all products of $\mathcal{K}$ and $\mathcal{L}$ up to the third
power444$\mathcal{L}$, $\mathcal{K}$, $\mathcal{L}^{2}$, $\mathcal{K}^{2}$,
$\mathcal{L}\mathcal{K}$, $\mathcal{K}\mathcal{L}$, $\mathcal{K}^{3}$,
$\mathcal{K}^{2}\mathcal{L}$, $\mathcal{K}\mathcal{L}^{2}$, $\mathcal{L}^{3}$,
$\mathcal{L}^{2}\mathcal{K}$, $\mathcal{K}\mathcal{L}\mathcal{K}$,
$\mathcal{L}\mathcal{K}^{2}$, $\mathcal{L}\mathcal{K}\mathcal{L}$ are
relatively $\mathcal{Y}^{(2)}$-bounded and
$(\mathcal{Y}^{(2)},\|\cdot\|_{\mathcal{Y}^{(2)}})$ is
$(\mathcal{L}+\mathcal{K})$-stable, in particular
$\|e^{t(\mathcal{L}+\mathcal{K})}\|_{\mathcal{Y}\rightarrow\mathcal{Y}}\leq\tilde{c}^{(2)}e^{t\omega^{(2)}}$,
then for all $x\in\mathcal{Y}^{(2)}$ and $t\geq 0$ there is a $c\geq 0$ so
that

$\displaystyle\norm{\left(e^{\frac{t}{2n}\mathcal{K}}e^{\frac{t}{n}\mathcal{L}}e^{\frac{t}{2n}\mathcal{K}}\right)^{n}x-e^{t(\mathcal{L}+\mathcal{K})}x}_{\mathcal{X}}$
$\displaystyle\leq
c\frac{t^{3}}{n^{2}}e^{t\omega^{(2)}}\|x\|_{\mathcal{Y}^{(2)}}\,.$

###### Proof.

To apply 3.3, we prove a bound like Equation 21 with the help of the integral
equation (40): For $x\in\mathcal{Y}^{(2)}$

$\displaystyle e^{\frac{\mathcal{K}}{2n}}$ $\displaystyle
e^{\frac{\mathcal{L}}{n}}e^{\frac{\mathcal{K}}{2n}}x-e^{\frac{\mathcal{L}+\mathcal{K}}{n}}x$
$\displaystyle=\frac{1}{n}\int_{0}^{1}e^{s_{1}\frac{\mathcal{K}}{2n}}\frac{\mathcal{K}}{2}x+e^{\frac{\mathcal{K}}{2n}}e^{s_{1}\frac{\mathcal{L}}{n}}\mathcal{L}x+e^{\frac{\mathcal{K}}{2n}}e^{\frac{\mathcal{L}}{n}}e^{s_{1}\frac{\mathcal{K}}{2n}}\frac{\mathcal{K}}{2}x-e^{s_{1}\frac{\mathcal{L}+\mathcal{K}}{n}}(\mathcal{L}+\mathcal{K})xds_{1}$
$\displaystyle=\frac{1}{n^{2}}\iint_{0}^{1}s_{1}e^{s_{1}s_{2}\frac{\mathcal{K}}{2n}}\frac{\mathcal{K}^{2}}{4}x+e^{s_{2}\frac{\mathcal{K}}{2n}}\frac{\mathcal{K}}{2}\mathcal{L}x+s_{1}e^{\frac{\mathcal{K}}{2n}}e^{s_{1}s_{2}\frac{\mathcal{L}}{n}}\mathcal{L}^{2}x-s_{1}e^{s_{1}s_{2}\frac{\mathcal{L}+\mathcal{K}}{n}}(\mathcal{L}+\mathcal{K})^{2}x$
$\displaystyle\qquad\qquad+e^{s_{2}\frac{\mathcal{K}}{2n}}\frac{\mathcal{K}^{2}}{4}x+e^{\frac{\mathcal{K}}{2n}}e^{s_{2}\frac{\mathcal{L}}{n}}\mathcal{L}\frac{\mathcal{K}}{2}x+s_{1}e^{\frac{\mathcal{K}}{2n}}e^{\frac{\mathcal{L}}{n}}e^{s_{1}s_{2}\frac{\mathcal{K}}{2n}}\frac{\mathcal{K}^{2}}{4}xds_{2}ds_{1}$
$\displaystyle=\frac{1}{n^{3}}\iiint_{0}^{1}s_{1}^{2}s_{2}e^{s_{1}s_{2}s_{3}\frac{\mathcal{K}}{2n}}\frac{\mathcal{K}^{3}}{8}x+s_{2}e^{s_{2}s_{3}\frac{\mathcal{K}}{2n}}\frac{\mathcal{K}^{2}}{4}\mathcal{L}x+s_{1}e^{s_{3}\frac{\mathcal{K}}{2n}}\frac{\mathcal{K}}{2}\mathcal{L}^{2}x+s_{1}^{2}s_{2}e^{\frac{\mathcal{K}}{2n}}e^{s_{1}s_{2}s_{3}\frac{\mathcal{L}}{n}}\mathcal{L}^{3}x$
$\displaystyle\qquad\qquad+s_{2}e^{s_{2}s_{3}\frac{\mathcal{K}}{2n}}\frac{\mathcal{K}^{3}}{8}x+e^{s_{3}\frac{\mathcal{K}}{2n}}\frac{\mathcal{K}}{2}\mathcal{L}\frac{\mathcal{K}}{2}x+s_{2}e^{\frac{\mathcal{K}}{2n}}e^{s_{2}s_{3}\frac{\mathcal{L}}{n}}\mathcal{L}^{2}\frac{\mathcal{K}}{2}x$
$\displaystyle\qquad\qquad+s_{1}e^{s_{3}\frac{\mathcal{K}}{2n}}\frac{\mathcal{K}^{3}}{8}x+s_{1}e^{\frac{\mathcal{K}}{2n}}e^{s_{3}\frac{\mathcal{L}}{n}}\mathcal{L}\frac{\mathcal{K}^{2}}{4}x+s_{1}^{2}s_{2}e^{\frac{\mathcal{K}}{2n}}e^{\frac{\mathcal{L}}{n}}e^{s_{1}s_{2}s_{3}\frac{\mathcal{K}}{2n}}\frac{\mathcal{K}^{3}}{8}x$
$\displaystyle\qquad\qquad-
s_{1}^{2}s_{2}e^{s_{1}s_{2}s_{3}\frac{\mathcal{L}+\mathcal{K}}{n}}(\mathcal{L}+\mathcal{K})^{3}x\,ds_{3}ds_{2}ds_{1}\,.$

Similar to the proof of 3.2, the integral is well-defined (see [11, Lem.
II.1.3]) and the relative boundedness shows

$\|\int_{0}^{1}s_{1}^{2}s_{2}e^{s_{1}s_{2}s_{3}\frac{\mathcal{K}}{2n}}\frac{\mathcal{K}^{3}}{8}xds_{1}ds_{2}\|_{\mathcal{X}}\leq\iint_{0}^{1}\frac{s_{1}^{2}s_{2}}{8}ds_{1}ds_{2}\|x\|_{\mathcal{Y}^{(2)}}=\frac{1}{48}\|x\|_{\mathcal{Y}^{(2)}}$

for a $b\geq 0$. Applying the relative bound to all terms of the integrand in
a similar manner, we achieve

$\|e^{\frac{\mathcal{K}}{2n}}e^{\frac{\mathcal{L}}{n}}e^{\frac{\mathcal{K}}{2n}}x-e^{\frac{\mathcal{L}+\mathcal{K}}{n}}x\|_{\mathcal{X}}\leq\frac{19}{6}\|x\|_{\mathcal{Y}^{(2)}}\,,$

which finishes the proof of the statement by applying 3.3. ∎

One observation in the proof is that higher-order convergence rates need
higher order derivatives and thereby a higher regularity of the input state.
Good examples for this observation are given in the bosonic setup Section 3.4.

######  Remark (Higher order approximation).

Note that the above proof idea can be utilized to achieve higher order
approximation by following the proofs of [49] [49]. However, one has to keep
in mind that the evolution needs to be time-reversible for $m\geq 3$ as shown
in [46].

###  3.3. Time-dependent Trotter product formula

For the time-dependent case the ideas are roughly the same by exploiting the
assumption of a subspace on which the evolution defines a stable function.
Here, we consider the limit

$\prod_{j=1}^{n}U(s_{j+1},s_{j})V(s_{j+1},s_{j})\rightarrow
T(t,s)\qquad\text{for}\qquad n\rightarrow\infty\,,$ (22)

where $U(t,s)$ and $V(t,s)$ are two $C_{0}$-evolution systems with generator
families $(\mathcal{L}_{s},\mathcal{D}(\mathcal{L}_{s}))$ and
$(\mathcal{K}_{s},\mathcal{D}(\mathcal{K}_{s}))$, and appropriate assumptions.

###### Theorem 3.5

Let $U(t,s)$ and $V(t,s)$ be contractive $C_{0}$-evolution system for
$(t,s)\in I_{T}$ and with generator families
$(\mathcal{L}_{t},\mathcal{D}(\mathcal{L}_{t}))_{t\in[0,T]}$ and
$(\mathcal{K}_{t},\mathcal{D}(\mathcal{K}_{t}))_{t\in[0,T]}$. Moreover, assume
that there is a Banach space
$(\mathcal{Y}\subset\mathcal{X},\|\cdot\|_{\mathcal{Y}})$ so that the closure
of $(\mathcal{L}_{t}+\mathcal{K}_{t},\mathcal{Y})_{t\in[0,T]}$ generates a
unique contractive $C_{0}$-evolution systems $T(t,s)$ on $\mathcal{X}$ for
$(t,s)\in I_{T}$. If $\mathcal{L}_{s}$, $\mathcal{K}_{s}$,
$\mathcal{L}_{s}\mathcal{L}_{t}$, $\mathcal{K}_{s}\mathcal{L}_{t}$,
$\mathcal{L}_{s}\mathcal{K}_{t}$, $\mathcal{K}_{s}\mathcal{K}_{t}$ are well-
defined on $\mathcal{Y}$ and relatively $\mathcal{Y}$-bounded uniformly for
all $(s,t)\in I_{T}$ and $(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ is
$(\mathcal{L}+\mathcal{K})$-stable, in particular
$\|e^{t(\mathcal{L}+\mathcal{K})}\|_{\mathcal{Y}\rightarrow\mathcal{Y}}\leq\tilde{c}e^{t\omega}$,
then there is a $c\geq 0$ so that

$\displaystyle\norm{\prod_{j=1}^{n}U(s_{j},s_{j-1})V(s_{j},s_{j-1})-T(t,0)}_{\mathcal{X}}$
$\displaystyle\leq c\frac{t^{2}}{n}e^{t\omega}\|x\|_{\mathcal{Y}}$

for all $x\in\mathcal{Y}$, $t\in[0,T]$, and
$s_{0}=0<\frac{t}{n}<...<\frac{(n-1)t}{n}<t=s_{n}$.

###### Proof.

By applying 3.1 to

$F(t,s)=U(t,s)V(t,s)$

and the sequence $s_{0}=0<\frac{1}{n}<...<\frac{n-1}{n}<1$, we achieve

$\displaystyle\|\prod_{j=1}^{n}U(s_{j},s_{j-1})V(s_{j},s_{j-1})$
$\displaystyle-T(t,s)x\|_{\mathcal{X}}$ $\displaystyle\leq
n\max_{j\in\\{1,...,n\\}}\|\Bigl{(}U(s_{j},s_{j-1})V(s_{j},s_{j-1})-T(s_{j},s_{j-1})\Bigr{)}T(s_{j-1},s_{0})x\|_{\mathcal{X}}\,.$

Then, the integral equation (40) as well as A.2 show with
$s_{j}(\tau)=s_{j}+\frac{1-\tau}{n}$

$\displaystyle U(s_{j}$
$\displaystyle,s_{j-1})V(s_{j},s_{j-1})x-T(s_{j},s_{j-1})\tilde{x}$ (23)
$\displaystyle=\frac{1}{n}\int_{0}^{1}\frac{\partial}{\partial\tau_{1}}\Bigl{(}U(s_{j},s_{j-1}(\tau_{1}))\tilde{x}+U(s_{j},s_{j-1})V(s_{j},s_{j-1}(\tau_{1}))\tilde{x}-T(s_{j},s_{j-1}(\tau_{1}))\tilde{x}\Bigr{)}d\tau_{1}$
$\displaystyle=\frac{1}{n}\int_{0}^{1}\biggl{(}U(s_{j},s_{j-1}(\tau_{1}))\mathcal{L}_{s_{j-1}(\tau_{1})}\tilde{x}+U(s_{j},s_{j-1})V(s_{j},s_{j-1}(\tau_{1}))\mathcal{K}_{s_{j-1}(\tau_{1})}\tilde{x}$
$\displaystyle\qquad\qquad\qquad-T(s_{j},s_{j-1}(\tau_{1}))(\mathcal{L}_{s_{j-1}(\tau_{1})}+\mathcal{K}_{s_{j-1}(\tau_{1})})\tilde{x}\Bigr{)}d\tau_{1}$
$\displaystyle=\frac{1}{n^{2}}\iint_{0}^{1}\Bigl{(}\tau_{1}U(s_{j},s_{j-1}(\tau_{1}\tau_{2}))\mathcal{L}_{s_{j-1}(\tau_{1}\tau_{2})}\mathcal{L}_{s_{j-1}(\tau_{1})}\tilde{x}+U(s_{j},s_{j-1}(\tau_{2}))\mathcal{L}_{s_{j-1}(\tau_{2})}\mathcal{K}_{s_{j-1}(\tau_{1})}\tilde{x}$
$\displaystyle\qquad\qquad\qquad+\tau_{1}U(s_{j},s_{j-1})V(s_{j},s_{j-1}(\tau_{1}\tau_{2}))\mathcal{K}_{s_{j-1}(\tau_{1}\tau_{2})}\mathcal{K}_{s_{j-1}(\tau_{1})}\tilde{x}$
$\displaystyle\qquad\qquad\qquad-\tau_{1}T(s_{j},s_{j-1}(\tau_{1}\tau_{2}))(\mathcal{L}_{s_{j-1}(\tau_{1}\tau_{2})}+\mathcal{K}_{s_{j-1}(\tau_{1}\tau_{2})})(\mathcal{L}_{s_{j-1}(\tau_{1})}+\mathcal{K}_{s_{j-1}(\tau_{1})})\tilde{x}\biggr{)}d\tau_{2}d\tau_{1}\,,$

where $\tilde{x}\coloneqq T(s_{j-1},s_{0})x$. Next, we use the relative
boundedness assumptions as well as the property that
$(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ defines an
$(\mathcal{L}_{s}+\mathcal{K}_{s})$-stable subspace to show

$\displaystyle\|\mathcal{L}_{s_{j-1}(\tau_{1})}\tilde{x}\|_{\mathcal{X}}$
$\displaystyle\leq b\|T(s_{j-1},s_{0})x\|_{\mathcal{Y}}\leq
b\,\tilde{c}\,e^{(s_{j-1}-s_{0})\omega}\|x\|_{\mathcal{Y}}$

and

$\displaystyle\|U(s_{j},s_{j-1}(\tau_{1}\tau_{2}))\mathcal{L}_{s_{j-1}(\tau_{1}\tau_{2})}\mathcal{L}_{s_{j-1}(\tau_{1})}\tilde{x}\|_{\mathcal{X}}$
$\displaystyle\leq\|\mathcal{L}_{s_{j-1}(\tau_{1}\tau_{2})}\mathcal{L}_{s_{j-1}(\tau_{1})}T(s_{j-1},s_{0})x\|_{\mathcal{X}}$
$\displaystyle\leq b\|T(s_{j-1},s_{0})x\|_{\mathcal{Y}}$ $\displaystyle\leq
b\,\tilde{c}\,e^{(s_{j-1}-s_{0})\omega}\|x\|_{\mathcal{Y}}$

as well as similar bounds for all the other terms coming up in Equation 23.
These two bounds finish the proof of the error bound of the statement. ∎

######  Remark (Variation of time-steps).

Note that the above proofs above hold even for time-steps that are not equally
distributed, thanks to the generality of 3.1 and the semigroup property of the
limiting semigroup. In this context, it suffices for the largest time-step to
converge to zero faster than order $\frac{1}{\sqrt{n}}$ in the strong
topology.

###  3.4. Application: Bosonic strongly continuous semigroups

In this section, we focus on Bosonic $C_{0}$-semigroups defined on the space
of self-adjoint trace-class operators on the Fock space (see Section 2.2).
Specially, we consider generators admitting the following GKSL structure

$\mathcal{L}:\mathcal{T}_{f}\to\mathcal{T}_{f}\quad
x\mapsto\mathcal{L}(x)=-i[H,x]+\sum\limits_{j=1}^{K}L_{j}xL_{j}^{\dagger}-\frac{1}{2}\\{L_{j}^{\dagger}L_{j},x\\}$
(24)

with $K\in\mathbb{N}$, $L_{j}\coloneqq p_{j}(a_{1},a^{\dagger}_{1},...)$ for
all $j\in\\{1,...,m\\}$, and a symmetric $H\coloneqq
p_{H}(a_{1},a^{\dagger}_{1},...)$. Note that in the time-dependent case the
coefficients would additionally depend continuously on time. Assuming Equation
18 not only shows that the closure of $(\mathcal{L},\mathcal{T}_{f})$
generates a quantum dynamical $C_{0}$-semigroup, but also that it is Sobolev
preserving, i.e.

$\|\mathcal{T}_{t}\|_{W^{\textbf{k}_{r},1}\to W^{\textbf{k}_{r},1}}\leq
e^{\omega_{\textbf{k}_{r}}t}$ (25)

for an interpolated sequence $\textbf{k}_{r}\subset\mathbbm{N}^{m}$ converging
to infinity in each mode (for more details [16]) and all $t\geq 0$. Note that
$W^{\textbf{k}_{r},1}$ is a $\mathcal{L}$-admissible subspace and plays
exactly the role of the Banach space $\mathcal{Y}$ discussed in the statements
before. Examples for Sobolev-preserving semigroups are besides Gaussian
semigroups, the Bose-Hubbard model, and the Ornstein-Uhlenbeck semigroup, the
$l$-photon driven dissipative semigroup used in bosonic error correction
codes. Furthermore, the finite degree assumption on $\mathcal{L}$ in Equation
24 implies by [16, Lem. E3] the existence of a $\textbf{k}\in\mathbbm{N}^{m}$
and $b\geq 0$ such that

$\|\mathcal{L}(x)\|_{1}\leq b\|x\|_{W^{\textbf{k},1}}\,.$ (26)

With the help of these two observations, we can apply the above theorems to
bosonic $C_{0}$-semigroups. For that, let
$(\mathcal{L},\mathcal{D}(\mathcal{L}))$ and
$(\mathcal{K},\mathcal{D}(\mathcal{K}))$ be generators of contractive
$C_{0}$-semigroups on $\mathcal{T}_{1,\operatorname{sa}}$ satisfying Equation
24. Additionally, assume that the closure of
$(\mathcal{L}+\mathcal{K},\mathcal{T}_{f})$ generates a Sobolev
preserving555Directly follows from [16] if Equation 18 is satisfied., which
satisfies Equation 25 in particular. Then, there is a k and $b\geq 0$ such
that

$\displaystyle\norm{\left(e^{\frac{t}{n}\mathcal{L}}e^{\frac{t}{n}\mathcal{K}}\right)^{n}x-e^{t(\mathcal{L}+\mathcal{K})}x}_{\mathcal{X}}$
$\displaystyle\leq
b\frac{t^{2}}{n}e^{t\omega_{2\textbf{k}}}\|x\|_{\mathcal{W}^{2\textbf{k},1}}$
$\displaystyle\norm{\left(e^{\frac{t}{2n}\mathcal{K}}e^{\frac{t}{n}\mathcal{L}}e^{\frac{t}{2n}\mathcal{K}}\right)^{n}x-e^{t(\mathcal{L}+\mathcal{K})}x}_{\mathcal{X}}$
$\displaystyle\leq
b\frac{t^{3}}{n^{2}}e^{t\omega_{3\textbf{k}}}\|x\|_{\mathcal{W}^{3\textbf{k},1}}\,.$

Here, k can be understood as the component-wise maximum of k’s satisfying
Equation 26 for $\mathcal{L}$ and $\mathcal{K}$. Then, concatenations of
$\mathcal{L}$ and $\mathcal{K}$ simply accumulate in the regularity k of the
Sobolev space — we get $2\textbf{k}$ or $3\textbf{k}$.

An interesting observation is that only the limiting semigroup needs to be
Sobolev preserving, which enables the scheme to be used as a ”regularization”
method in the sense of Sobolev preserving semigroups. More explicitly, many
semigroups even satisfy a stronger assumption than Equation 18, which is

$\tr[\mathcal{W}^{\textbf{k}_{r}}\mathcal{L}(x)]\leq-\mu_{\textbf{k}_{r}}\tr[\mathcal{W}^{\textbf{k}_{r}+\textbf{l}}(x)]+c_{\textbf{k}_{r}}$
(27)

for a sequence $\textbf{k}_{r}$ converging component-wise to infinity,
$\textbf{l}\in\mathbbm{N}^{m}$ depending highly on the structure of the
generator $\mathcal{L}$, and constants
$\mu_{\textbf{k}_{r}},c_{\textbf{k}_{r}}\geq 0$. This stronger assumption not
only allows us to improve the bound Equation 25 for states to a constant
scaling in $t$ (see [16, Prop. 5.1]), but also to bound all other terms which
are not of leading order by a constant. For example, we can show that for any
Hamiltonian defined by
$H=p_{H}(a_{1},a^{\dagger}_{1},...,a_{m},a^{\dagger}_{m})$ there is a
$l\in\mathbbm{N}$ so that

$-i[H,\cdot]+\mathcal{L}[a^{l}-\alpha^{l}](\cdot)$ (28)

generates a Sobolev preserving semigroup and, in particular, it satisfies
Equation 27. This idea is for example used in the recent bosonic Hamiltonian
Learning scheme [32], in which a similar method is used to derive Lieb-
Robinson bounds.

The same reasoning translates to the time-dependent case. Here, the
coefficients of the polynomials in Equation 24 depend continuously on time
$t\in[0,T]$. Then, a similar statement as 2.3 can be proven if Equation 18 is
satisfied with a time-independent right-hand side (see [16, Thm. 4.5]).
Assuming that all involved evolution systems are well-defined and of GKSL form
(24), the assumption that the Trotter-limit is again Sobolev preserving
implies

$\displaystyle\norm{\prod_{j=1}^{n}U(s_{j+1},s_{j})V(s_{j+1},s_{j})x-T(t,s)x}_{\mathcal{X}}\leq
c\frac{t^{2}}{n}e^{t\omega}\|x\|_{W^{2\textbf{k},1}}$

for a $c,\omega\geq 0$, $\textbf{k}\in\mathbbm{N}^{m}$ and all $t\in[0,T]$,
$x\in W^{2\textbf{k},1}$ by 3.5.

##  4\. Quantum Zeno effect

Similar to the Trotter product formula, the quantum Zeno effect describes an
operator product of a contractive projection $P\mathcal{B}(\mathcal{X})$, i.e.
$P^{2}=P$, and a $C_{0}$-semigroup $e^{t\mathcal{L}}$:

$(Pe^{\frac{\mathcal{L}}{n}})^{n}\rightarrow
e^{P\mathcal{L}P}P\qquad\text{for}\quad n\rightarrow\infty\,.$ (29)

Under certain assumptions the convergence as well as the convergence rate can
be achieved in the strong operator topology [33, 4]. Similarly to the bounded
generator case [34], we extend the result in [33] to evolution systems. We
start with the projective Zeno effect without assuming the asymptotic Zeno
condition [33, Eq. 6]. Then, we extend the result to more general contractions
$M$. Finally, we apply the abstract theory again to Sobolev preserving
semigroups, in particular, to an example from bosonic continuous error
correction.

###  4.1. Projective Zeno effect

We start with the simplest case presented in Equation 29 where a
$C_{0}$-semigroup is interrupted by a contractive projection $P$.

###### Proposition 4.1

Let $(\mathcal{L},\mathcal{D}(\mathcal{L}))$ be a generator of a contractive
$C_{0}$-semigroup and $P$ a projective contraction. Assume there is a Banach
space $(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ invariant under $P$ such that
the closure of $(P\mathcal{L}P,\mathcal{Y})$ defines a contractive
$C_{0}$-semigroup, $\mathcal{L}P$ and $\mathcal{L}^{2}P$ are relatively
$\mathcal{Y}$-bounded, and $(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ is a
$P\mathcal{L}P$-stable subspace, in particular
$\|e^{tP\mathcal{L}P}\|_{\mathcal{Y}\rightarrow\mathcal{Y}}\leq\tilde{c}e^{t\omega}$.
Then, there is a $c_{1}\geq 0$ such that for all $t\geq 0$, $x\in\mathcal{Y}$,
and $n\in\mathbbm{N}$

$\displaystyle\|\left(Pe^{t\frac{\mathcal{L}}{n}}P\right)^{n}Px-e^{tP\mathcal{L}P}Px\|_{\mathcal{X}}$
$\displaystyle\leq\frac{c_{1}t^{2}}{n}\left(e^{t\omega}\|Px\|_{\mathcal{Y}}+\|(P\mathcal{L}P)^{2}x\|_{\mathcal{X}}\right).$

If we additionally assume that $(P\mathcal{L}P)^{2}$ is relatively
$\mathcal{Y}$-bounded, then there is a $c_{2}$ such that

$\displaystyle\|\left(Pe^{\frac{\mathcal{L}}{n}}P\right)^{n}Px-e^{P\mathcal{L}P}Px\|_{\mathcal{X}}$
$\displaystyle\leq\frac{c_{2}(1+e^{t\omega})}{n}\|Px\|_{\mathcal{Y}}\,.$

###### Proof.

By rescaling the generator $\mathcal{L}$, we assume $t\in[0,1]$. Then, we
apply 3.1 to

$F(t,s)=Pe^{(t-s)\mathcal{L}}P\quad T(t,s)=e^{(t-s)P\mathcal{L}P}$

w.r.t. the sequence $s_{0}=0<\frac{1}{n}<...<\frac{n-1}{n}<1$ so that

$\|\left(Pe^{\frac{\mathcal{L}}{n}}P\right)^{n}Px-e^{P\mathcal{L}P}Px\|_{\mathcal{X}}\leq
n\max_{s\in[0,1-1/n]}\|\left(Pe^{\frac{\mathcal{L}}{n}}P-e^{\frac{P\mathcal{L}P}{n}}\right)e^{sP\mathcal{L}P}Px\|_{\mathcal{X}}\,.$

Then, we use $e^{P\mathcal{L}P}Px=Pe^{P\mathcal{L}P}Px$ and the integral
equation (40) shows

$\displaystyle Pe^{\frac{\mathcal{L}}{n}}P-e^{\frac{P\mathcal{L}P}{n}}Px_{s}$
$\displaystyle=\frac{1}{n^{2}}\iint_{0}^{1}\left(s_{1}Pe^{s_{1}s_{2}\frac{\mathcal{L}}{n}}\mathcal{L}^{2}P-s_{1}e^{s_{1}s_{2}P\frac{P\mathcal{L}P}{n}}(P\mathcal{L}P)^{2}\right)x_{s}ds_{2}ds_{1}\,,$

where $x_{s}\coloneqq e^{sP\mathcal{L}P}x$. By the assumptions that
$\mathcal{L}^{2}P$ is relatively $\mathcal{Y}$-bounded, i.e.
$\|\mathcal{L}^{2}P\|_{\mathcal{Y}\rightarrow\mathcal{X}}<a_{1}$ for an
$a_{1}\geq 0$, and $\mathcal{Y}$ is a $P\mathcal{L}P$-stable subspace, in
particular $\|e^{t\mathcal{L}}\|_{\mathcal{Y}\rightarrow\mathcal{Y}}\leq
b_{1}e^{t\omega}$ for a $b_{1},\omega\geq 0$, we can show

$\displaystyle\|\iint_{0}^{1}s_{1}Pe^{s_{1}s_{2}\frac{\mathcal{L}}{n}}\mathcal{L}^{2}Px_{s}ds_{2}ds_{1}\|_{\mathcal{X}}$
$\displaystyle\leq\frac{a_{1}}{2}\|e^{sP\mathcal{L}P}Px\|_{\mathcal{Y}}\leq\frac{a_{1}b_{1}}{2}e^{\omega}\|Px\|_{\mathcal{Y}}\,.$
(30)

Therefore, $c_{1}=\frac{a_{1}b_{1}}{2}$ and rescaling of $t$ implies

$\displaystyle\|\left(Pe^{t\frac{\mathcal{L}}{n}}P\right)^{n}Px-e^{tP\mathcal{L}P}Px\|_{\mathcal{X}}$
$\displaystyle\leq\frac{c_{1}t^{2}}{n}\left(e^{t\omega}\|Px\|_{\mathcal{Y}}+\|(P\mathcal{L}P)^{2}x\|_{\mathcal{X}}\right).$

If $(P\mathcal{L}P)^{2}$ is additionally relatively $\mathcal{Y}$-bounded with
constants $a_{2}$, we directly achieve

$\displaystyle\|\left(Pe^{\frac{\mathcal{L}}{n}}P\right)^{n}Px-e^{P\mathcal{L}P}Px\|_{\mathcal{X}}$
$\displaystyle\leq\frac{c_{2}(1+e^{t\omega})}{n}\|Px\|_{\mathcal{Y}}\,,$

with $c_{2}=\max\\{c_{1},\frac{a_{2}}{2}\\}$. This finishes the proof of the
theorem. ∎

######  Remark 2.

To get a better understanding of the above result, we compare it to Theorem I
in [33]. Assume that $(P\mathcal{L}P,\mathcal{D}(P\mathcal{L}P))$ generates a
contractive $C_{0}$-semigroup. Then,
$\mathcal{Y}=\mathcal{D}((P\mathcal{L}P)^{2})\cap\mathcal{D}(P\mathcal{L}P)$
defines a Banach space with the norm
$\|x\|_{\mathcal{Y}}\coloneqq\|x\|_{\mathcal{X}}+\|P\mathcal{L}Px\|_{\mathcal{X}}+\|(P\mathcal{L}P)^{2}x\|_{\mathcal{X}}$.
Clearly, $\mathcal{Y}$ is invariant under $P$ and an even
$P\mathcal{L}P$-admissible subspace, in particular,
$\|e^{tP\mathcal{L}P}\|_{\mathcal{X}\rightarrow\mathcal{Y}}\leq 1$. Next, the
asymptotic Zeno condition could be used for the following relative bound

$\displaystyle\frac{1}{n^{2}}\|\iint_{0}^{1}s_{1}Pe^{s_{1}s_{2}\frac{\mathcal{L}}{n}}\mathcal{L}^{2}Px_{s}ds_{2}ds_{1}\|_{\mathcal{X}}$
$\displaystyle=\frac{1}{n}\|\int_{0}^{1}Pe^{s_{1}\frac{\mathcal{L}}{n}}(P+\mathbbm{1}-P)\mathcal{L}Px_{s}ds_{1}-P\mathcal{L}Px_{s}\|_{\mathcal{X}}$
$\displaystyle\leq\frac{1}{n}\|\int_{0}^{1}Pe^{s_{1}\frac{\mathcal{L}}{n}}(\mathbbm{1}-P)\mathcal{L}Px_{s}ds_{1}+\frac{1}{n}Pe^{s_{1}\frac{\mathcal{L}}{n}}\mathcal{L}P\mathcal{L}Px_{s}ds_{1}\|_{\mathcal{X}}$
$\displaystyle\leq\frac{1}{n^{2}}\left(b^{2}\|x_{s}\|_{\mathcal{X}}+b\|P\mathcal{L}Px_{s}\|_{\mathcal{X}}+\|(P\mathcal{L}P)^{2}x_{s}\|_{\mathcal{X}}\right)$
$\displaystyle\leq\frac{1}{n^{2}}\max\\{b^{2},b,1\\}\|x_{s}\|_{\mathcal{Y}}\,,$

which is slightly different from the relative bound used in Equation 30. Here,
$b$ is defined in Theorem I in [33].

In the next step, we generalize the above result to evolution systems:

###### Theorem 4.2

Let $I_{T}\ni(t,s)\mapsto V(t,s)\in\mathcal{B}(\mathcal{X})$ be a
$C_{0}$-evolution system with generator family
$(\mathcal{L}_{t},\mathcal{D}(\mathcal{L}_{t}))_{t\in[0,T]}$ and
$P\in\mathcal{B}(\mathcal{X})$ a projective contraction. Assume that there is
a Banach space $(\mathcal{Y}\subset\mathcal{X},\|\cdot\|_{\mathcal{Y}})$
invariant under $P$ so that the closure of
$(P\mathcal{L}_{t}P,\mathcal{Y})_{t\in[0,T]}$ generates a unique contractive
$C_{0}$-evolution systems $T(t,s)$ for $(t,s)\in I_{T}$ commuting with $P$.
Moreover, assume that $\mathcal{L}_{t}P$, $\mathcal{L}_{t}\mathcal{L}_{s}P$,
$P\mathcal{L}_{t}P$, and $P\mathcal{L}_{t}P\mathcal{L}_{s}P$ are well-defined
on $\mathcal{Y}$ and relatively $\mathcal{Y}$-bounded uniformly for all
$(t,s)\in I_{T}$, and $(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ is a
$P\mathcal{L}_{t}P$-stable subspace, in particular
$\|T(t,s)\|_{\mathcal{Y}\rightarrow\mathcal{Y}}\leq\tilde{c}e^{(t-s)\omega}$.
Then, there is a $c\geq 0$ so that

$\displaystyle\Bigl{\|}\prod_{j=1}^{n}PV(s_{j},s_{j-1})Px-T(t,0)Px\Bigr{\|}_{\mathcal{X}}\leq
c\frac{t^{2}}{n}e^{t\omega}\|Px\|_{\mathcal{Y}}$

for all $t\geq 0$, $s_{0}=0<\frac{t}{n}<...<\frac{t(n-1)}{n}<t=s_{n}$,
$x\in\mathcal{Y}$, and $n\in\mathbbm{N}$.

Note that the assumption about the existence of the Banach space $\mathcal{Y}$
is motivated by hyperbolic evolution systems for which by definition an
admissible subspace exists 2.2.

###### Proof.

As before, we rescale $t\in[0,1]$ and apply 3.1 to

$F(t,s)=PV(t,s)P$

w.r.t. the sequence $s_{0}=0<\frac{1}{n}<...<\frac{n-1}{n}<1$ so that

$\displaystyle\Bigl{\|}\prod_{j=1}^{n}PV(s_{j},s_{j-1})$ $\displaystyle
Px-T(t,0)Px\Bigr{\|}_{\mathcal{X}}$ $\displaystyle\leq
n\max_{j\in\\{1,...,n\\}}\|\left(PV(s_{j},s_{j-1})P-T(s_{j},s_{j-1})\right)T(s_{j-1},s_{0})Px\|_{\mathcal{X}}\,.$

Then, we use $T(s_{j-1},s_{0})Px=PT(s_{j-1},s_{0})Px$ and the integral
equation (40) to show with $s_{j}(\tau)=s_{j}+\frac{1-\tau_{1}}{n}$

$\displaystyle P$ $\displaystyle
V(s_{j},s_{j-1})Px_{j}-T(s_{j},s_{j-1})Px_{j}$
$\displaystyle=\frac{1}{n}\int_{0}^{1}\Biggl{(}PV(s_{j},s_{j-1}(\tau_{1}))\mathcal{L}_{s_{j-1}(\tau_{1})}P-T(s_{j},s_{j-1}(\tau_{1}))P\mathcal{L}_{s_{j-1}(\tau_{1})}P\Biggr{)}x_{j}d\tau_{1}$
$\displaystyle=\frac{1}{n^{2}}\iint_{0}^{1}\Biggl{(}\tau_{1}PV(s_{j},s_{j-1}(\tau_{1}\tau_{2}))\mathcal{L}_{s_{j-1}(\tau_{1}\tau_{2})}\mathcal{L}_{s_{j-1}(\tau_{1})}P$
$\displaystyle\qquad\qquad\qquad-\tau_{1}T(s_{j},s_{j-1}(\tau_{1}\tau_{2}))P\mathcal{L}_{s_{j-1}(\tau_{1}\tau_{2})}P\mathcal{L}_{s_{j-1}(\tau_{1})}P\Biggr{)}x_{j}d\tau_{2}d\tau_{1}\,,$

where $x_{j}\coloneqq PT(s_{j-1},s_{0})Px$. Note that the integral above is
again well-defined by A.2 as well as [11, Lem. B.15]. As in the proof of 4.1,
we use the assumption to show

$\displaystyle\|\iint_{0}^{1}\tau_{1}PV(s_{j},s_{j-1}(\tau_{1}))\mathcal{L}_{s_{j-1}(\tau_{1}\tau_{2})}\mathcal{L}_{s_{j-1}(\tau_{1})}Px_{j}d\tau_{2}d\tau_{1}\|_{\mathcal{X}}$
$\displaystyle\leq\frac{a_{1}}{2}\|T(s_{j-1},s_{0})Px\|_{\mathcal{Y}}\leq
c_{1}e^{\omega}\|Px\|_{\mathcal{Y}}$

for $c_{1}=\frac{a_{1},b_{1}}{2}$ with $a_{1},b_{1}\geq 0$ and similarly

$\displaystyle\|\iint_{0}^{1}\tau_{1}T(s_{j},s_{j-1}(\tau_{1}))P\mathcal{L}_{s_{j-1}(\tau_{1}\tau_{2})}P\mathcal{L}_{s_{j-1}(\tau_{1})}Px_{j}d\tau_{2}d\tau_{1}\|_{\mathcal{X}}$
$\displaystyle\leq c_{2}e^{\omega}\|x_{j}\|_{\mathcal{Y}}$

for $c_{2}=\frac{a_{2},b_{2}}{2}$ with $a_{2},b_{2}\geq 0$. Rescaling and
combining both bounds show

$\displaystyle\Bigl{\|}\prod_{j=1}^{n}PV(s_{j},s_{j-1})$ $\displaystyle
Px-T(t,0)Px\Bigr{\|}_{\mathcal{X}}\leq\frac{ct^{2}}{n}e^{t\omega}\|Px\|_{\mathcal{Y}}$

with $c=2\max\\{c_{1},c_{2}\\}$. ∎

###  4.2. Time-dependent Zeno with general measurement

In the next step, we generalize Theorem I in [33] to $C_{0}$-evolution
systems. For that, we use again the help of a reference subspace
$\mathcal{Y}$:

###### Theorem 4.3

Let $I_{T}\ni(t,s)\mapsto V(t,s)\in\mathcal{B}(\mathcal{X})$ be a
$C_{0}$-evolution system with generator family
$(\mathcal{L}_{t},\mathcal{D}(\mathcal{L}_{t}))_{t\in[0,T]}$,
$M\in\mathcal{B}(\mathcal{X})$ a contraction, and $P$ a projection satisfying

$\norm{M^{n}-P}_{\mathcal{X}\rightarrow\mathcal{X}}\leq\delta^{n}$ (31)

for $\delta\in(0,1)$ and all $n\in\mathbbm{N}$ and

$\norm{PV(t,s)(\mathbbm{1}-P)}_{\mathcal{X}\rightarrow\mathcal{X}}\leq(t-s)b\quad\text{and}\quad\norm{(\mathbbm{1}-P)V(t,s)P}_{\mathcal{X}\rightarrow\mathcal{X}}\leq(t-s)b$
(32)

for $b\geq 0$ and all $(t,s)\in I_{T}$. Furthermore, assume there is a Banach
space $(\mathcal{Y}\subset\mathcal{X},\|\cdot\|_{\mathcal{Y}})$ invariant
under $P$ so that the closure of $(P\mathcal{L}_{t}P,\mathcal{Y})_{t\in[0,T]}$
generates a unique contractive $C_{0}$-evolution systems $T(t,s)$ for
$(t,s)\in I_{T}$ commuting with $P$. Moreover, assume that $\mathcal{L}_{t}P$,
$\mathcal{L}_{t}\mathcal{L}_{s}P$, $P\mathcal{L}_{t}P$, and
$P\mathcal{L}_{t}P\mathcal{L}_{s}P$ ar well-defined on $\mathcal{L}$ and
relatively $\mathcal{Y}$-bounded uniformly for all $(t,s)\in I_{T}$, and
$(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ is a $P\mathcal{L}_{t}P$-stable
subspace, in particular
$\|T(t,s)\|_{\mathcal{Y}\rightarrow\mathcal{Y}}\leq\tilde{c}e^{(t-s)\omega}$.
Then, there is a $c\geq 0$ so that

$\displaystyle\Bigl{\|}\prod_{j=1}^{n}MV(s_{j},s_{j-1})x-T(t,0)Px\Bigr{\|}_{\mathcal{X}}\leq
c\frac{t^{2}}{n}\left(\|x\|_{\mathcal{X}}+e^{t\omega}\|Px\|_{\mathcal{Y}}\right)$

for all $t\geq 0$, $s_{0}=0<\frac{t}{n}<...<\frac{t(n-1)}{n}<t=s_{n}$,
$x\in\mathcal{Y}$, and $n\in\mathbbm{N}$.

###### Proof.

As in the proof of [33, Thm. 5.1], we approximate $M$ by $P$ in the first
step. To do so, we follow the proof strategy of Lemma 5.2 in [33]. By just
changing the roles of $A$ to $A_{j}\coloneqq
M(\mathbbm{1}-P)V(s_{j},s_{j-1})$, $B$ to $B_{j}\coloneqq PV(s_{j},s_{j-1})$,
and, additionally, using the fact that the bounds only dependent on the step
size, the same proof shows

$\norm{\prod_{j=1}^{n}MV(s_{j},s_{j-1})x-\prod_{j=1}^{n}PV(s_{j},s_{j-1})Px}_{\mathcal{X}}\leq\left(\delta^{n}+\frac{b}{n}+\frac{1}{n}\frac{b(2+b)(\delta-\delta^{n})}{1-\delta}e^{2b}\right)\norm{x}_{\mathcal{Y}}.$
(33)

for all $x\in\mathcal{X}$. If we assume $\mathcal{L}_{t}\mathcal{L}_{s}P$
(instead of $P\mathcal{L}_{t}P$) is relatively $\mathcal{Y}$-bounded for all
$(t,s)\in I_{T}$, then, the proof directly follows by 4.2 after Equation 33. ∎

######  Remark.

In Appendix B, we give an extended proof of 4.3, which improves the above
statement slightly in the sense that the relative $\mathcal{Y}$-boundedness of
$\mathcal{L}_{t}\mathcal{L}_{s}P$ is not needed anymore. This is interesting
because then Theorem B.1 reduces directly to Theorem I in [33].

###  4.3. Application: The $l$-photon driven dissipation in error correction

We continue with the example in Section 3.4 about time-(in)dependent
$C_{0}$-semigroups generated by operators in GKSL form (24), in particular,
Sobolev preserving semigroups induced by Equation 18. In the following, we
shortly repeat the proof of these two properties for the $l$-photon driven
dissipation for simplicity in one-mode [16, Lem. 5.3] and give two specific
examples for projections satisfying the assumptions of the statements before
(4.1, 4.2, 4.3). The $l$-photon driven dissipation (28),

$\mathcal{L}[a^{l}-\alpha^{l}][x]=(a^{l}-\alpha^{l})x((a^{\dagger})^{l}-\bar{\alpha}^{l})-\frac{1}{2}\left\\{((a^{\dagger})^{l}-\bar{\alpha}^{l})(a^{l}-\alpha^{l}),x\right\\}$
(34)

is the fundamental dynamics for the bosonic error correction code introduced
in [30, 19, 18]. Using the CCR and the simple commutation relation
$af(N+j\mathbbm{1})=f(N+(j+1)\mathbbm{1})a$ for any real-valued function
$f:\mathbbm{R}\rightarrow\mathbbm{R}$, we have for
$f(x)=(x+1)^{k/2}1_{x\geq-1}$ and $g_{l}(x)\coloneqq f(x)-f(x-l)$:

$\displaystyle\mathcal{L}[a_{1}^{l}-\alpha^{l}]^{\dagger}(f(N))$
$\displaystyle=-(N-(l-1)\mathbbm{1})\cdots
Ng_{l}(N)+\frac{1}{2}\left(\overline{\alpha}^{l}a^{l}g_{l}(N)+\alpha^{l}g_{l}(N)(a^{\dagger})^{l}\right)\,.$

Note that the first term contributes a diagonal and the others, two off-
diagonals with respect to the Fock basis. The operator can be decomposed in
blocks defined by

$\displaystyle\frac{1}{2}$
$\displaystyle\Bigl{(}-a\ket{n}\bra{n}-b\ket{n+l}\bra{n+l}+c^{*}\ket{n}\bra{n+l}+c\ket{n+l}\bra{n}\Bigr{)}$
$\displaystyle\coloneqq\frac{1}{2}\Bigl{(}-h_{l}(n)g_{l}(n)\ket{n}\bra{n}-h_{l}(n+l)g_{l}(n+l)\ket{n+l}\bra{n+l}$
$\displaystyle\qquad\qquad\qquad+\sqrt{h_{l}(n+l)}\left(\bar{\alpha}^{l}g_{l}(n+l)\ket{n}\bra{n+l}+\alpha^{l}g_{l}(n+l)\ket{n+l}\bra{n}\right)\Bigr{)}$

for $n\in\mathbbm{N}$ and the real valued function
$h_{l}:\mathbbm{R}\rightarrow\mathbbm{R},h(x)\coloneqq(x-(l-1)\mathbbm{1})\cdots
x1_{x\geq l}$. By considering the eigenvalues of the above operator, it can be
upper bounded by

$\displaystyle\frac{1}{2}$
$\displaystyle\Bigl{(}-a\ket{n}\bra{n}-b\ket{n+l}\bra{n+l}+c^{*}\ket{n}\bra{n+l}+c\ket{n+l}\bra{n}\Bigr{)}$
$\displaystyle\leq\frac{1}{4}\left(-a-b+\sqrt{(a-b)^{2}+4|c|^{2}}\right)\Bigl{(}\ket{n}\bra{n}+\ket{n+l}\bra{n+l}\Bigr{)}$
$\displaystyle\leq\frac{1}{2}\left(-\min\\{a,b\\}+|c|\right)\Bigl{(}\ket{n}\bra{n}+\ket{n+l}\bra{n+l}\Bigr{)}$
$\displaystyle=\frac{1}{2}\left(-h_{l}(n)g_{l}(n)+\sqrt{h_{l}(n+l)}|\alpha|^{l}g_{l}(n+l)\right)\Bigl{(}\ket{n}\bra{n}+\ket{n+l}\bra{n+l}\Bigr{)}$
$\displaystyle\leq\frac{1}{2}\left(-(n+1)^{k/2-1+l}+c_{k}^{(l)}\right)\Bigl{(}\ket{n}\bra{n}+\ket{n+l}\bra{n+l}\Bigr{)}\,,$

where the constant $c_{k}^{(l)}\geq 0$ exists because the first term is of
negative leading order (a more detailed discussion can be found in [16, Lem.
5.3]). All together, we achieve

$\mathcal{L}[a^{l}-\alpha^{l}]^{\dagger}(f(N))\leq-\frac{l}{2}(N+\mathbbm{1})^{k/2-1+l}+c_{k}^{(l)}\mathbbm{1}\,,$

which implies Assumption 18 and a better Sobolev preserving constant for
states [16, Prop. 5.1]:

$\displaystyle\tr\big[\mathcal{L}[a^{l}-\alpha^{l}](\rho)(N+\mathbbm{1})^{k/2}\big{]}$
$\displaystyle\leq-\frac{l}{2}\tr[\rho\,(N+\mathbbm{1})^{k/2}\big{]}+c_{k}^{(l)}\,.$

However, we are, in particular, interested in the Sobolev preserving property
of $(P\mathcal{L}P,\mathcal{D}(P\mathcal{L}P))$ because this is one way to
find the right Banach space $\mathcal{Y}=W^{k,1}$ in the statements above.
Here, the idea is to choose $k$ large enough such that the relative
boundedness assumption is achieved (for comparison see Equation 26). However,
this strongly depends on $P$. To analyze this, we will consider two cases:
First, $P(\cdot)=K\cdot K^{\perp}$ for a projection $K$ diagonal in the Fock
basis. Then, the same reasoning as above proves the well-definedness as well
as the Sobolev preserving property of the semigroup defined by
$P\mathcal{L}P$. The same holds true for the time-dependent analogous
generated by (eq. 13). Another example is the $Z(\theta)$ gate of the bosonic
error correction code discussed in [30, 19, 18]. The physical codespace to
which the evolution is exponentially converging (see eq. 38) and [1]) is
defined by

$\mathcal{C}_{l}\coloneqq\textrm{span}\left\\{\outerproduct{\alpha_{1}}{\alpha_{2}}\,:\,\alpha_{1},\alpha_{2}\in\left\\{\alpha
e^{\frac{i2\pi j}{l}}\,|\,j\in\\{0,...,l-1\\}\right\\}\right\\}\,,$

where $\ket{\alpha}$ denotes the coherent state. In particular, we are
interested in the case $l=2$. Next, we define the Schrödinger CAT-states,
which represent the logical qubits:

$\ket{CAT_{\alpha}^{+}}\coloneqq\frac{1}{\sqrt{2(1+e^{2|\alpha|^{2}})}}\left(\ket{\alpha}+\ket{-\alpha}\right)\qquad\text{and}\qquad\ket{CAT_{\alpha}^{-}}\coloneqq\frac{1}{\sqrt{2(1-e^{2|\alpha|^{2}})}}\left(\ket{\alpha}-\ket{-\alpha}\right)\,.$

Note that the above CAT-states are orthonormal. Then, a possible rotation
around the $z$-axis is described in [30] and experimentally realized in [51].
The gate is defined by

$Z_{\theta}=\cos(\frac{\theta}{2})(P^{+}_{\alpha}+P^{-}_{\alpha})+i\sin(\frac{\theta}{2})Z_{\alpha}\,,$
(35)

where $P^{+}=\ket{CAT^{+}_{\alpha}}\bra{CAT^{+}_{\alpha}}$,
$P^{-}=\ket{CAT^{-}_{\alpha}}\bra{CAT^{-}_{\alpha}}$ are the projection of the
CAT-states and
$Z_{\alpha}=\ket{CAT^{+}_{\alpha}}\bra{CAT^{-}_{\alpha}}+\ket{CAT^{-}_{\alpha}}\bra{CAT^{+}_{\alpha}}$.
Specially, here we discuss a construction of a $Z(\theta)$-gate by a discrete
projective Zeno effect w.r.t. the projection
$P_{\alpha}=P_{\alpha}^{+}+P_{\alpha}^{-}$ and the driving Hamiltonian
$H=a+a^{\dagger}$. Then, the Zeno limit describes Equation 35:

$P_{\alpha}HP_{\alpha}=(\alpha+\alpha^{\dagger})Z_{\alpha}\,.$ (36)

Since $P_{\alpha}HP_{\alpha}$ acts only non-trivially on a 2-dimensional
subspace, the Zeno semigroup is

$e^{tiP_{\alpha}HP_{\alpha}}P_{\alpha}=\cos(t(\alpha+\alpha^{\dagger}))P_{\alpha}+i\sin(t(\alpha+\alpha^{\dagger}))Z_{\alpha}$
(37)

and constructs the rotation (35) for appropriate $t$. The convergence

$\|\left(P_{\alpha}e^{i\frac{t}{n}[H,\cdot]}\right)^{n}x-e^{ti[P_{\alpha}HP_{\alpha},P_{\alpha}\cdot
P_{\alpha}]}P_{\alpha}x\|_{1\rightarrow
1}=\mathcal{O}\Bigl{(}\frac{t^{2}}{n}\Bigr{)}$

is a consequence of 4.1 or a simple case of 4.3. Here, $\mathcal{Y}$ is a
certain Sobolev space or 2. By the boundedness of $P_{\alpha}H$ and
$HP_{\alpha}$ all assumptions are satisfied and the bound holds true in the
uniform topology. The same would hold true for more complex and time-dependent
$H$ defined by a polynomial in annihilation and creation operator.

######  Remark 3.

In practice the projection is implemented by the $l$-photon driven dissipation
semigroup, which converges in the following sense to $P_{\alpha}$ [1, Theorem
2]

$\tr[(a^{l}-\alpha^{l})e^{t\mathcal{L}_{l}}(\rho)(a^{l}-\alpha^{l})^{\dagger}]\leq
e^{-l!t}\tr[(a^{l}-\alpha^{l})\rho(a^{l}-\alpha^{l})^{\dagger}]\,.$ (38)

This convergence does not satisfy 4.3. Therefore, a future line of research
could be to improve Equation 38 or Assumption 31 in 4.3 in the direction of
Theorem 2 in [4].

######  Remark 4.

The case $l=4$ described in [30] is not necessarily Sobolev preserving anymore
so that 4.1 does not apply. Nevertheless, 4.3 applies as discussed before.

##  5\. Conclusion

To summarize, we have seen a range of quantitative error bounds for
generalizations of Trotter’s product formula and the quantum Zeno effect in
the case of $C_{0}$-semigroups. Our primary contribution lies not only in the
extension of the Trotter product formula and the quantum Zeno effect to
evolution systems (time-dependent semigroups) but also in providing Suzuki-
higher order error bounds for the Trotter-Suzuki product formula in the strong
operator topology and a large class of bosonic examples. Moreover, we
discussed some examples to explore possible definitions of the abstract
reference space $\mathcal{Y}$.

For applications, a desirable line of research would entail a detailed
discussion of possible choices of $\mathcal{Y}$ as well as examples which does
not satisfy the assumption. For the quantum Zeno effect, a generalization of
the convergence assumption of the contraction $M$, as discussed in 3, would be
fruitful. Here, an interesting interplay of the convergence assumption as well
as the regularity of the semigroup quantified by the stable subspace could
come up.

_Acknowledgments:_ I would like to thank Paul Gondolf, Robert Salzmann, Simon
Becker, Lauritz van Luijk, Niklas Galke, Stefan Teufel, Marius Lemm, and Yu-
Jie Liu for valuable feedback and discussions on the topic. I also acknowledge
the support of the Munich Center for Quantum Sciences and Technology and of
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – TRR
352 – Project-ID 470903074. Furthermore, this work was supported by the
University of Nottingham and the University of Tübingen’s funding as part of
the Excellence Strategy of the German Federal and State Governments, in close
collaboration with the University of Nottingham.

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##  Appendix A Technical tools

First, we generalize the telescopic sum approach known for example from
proving the Lie-Trotter product formula [41, 54]:

###### Lemma A.1

Let $I_{T}\ni(t,s)\mapsto F(t,s)\in\mathcal{B}(\mathcal{X})$ be an operator-
valued map with $\|F(t,s)\|_{\mathcal{X}\rightarrow\mathcal{X}}\leq 1$ and
$F(s,s)=\mathbbm{1}$ for all $(t,s)\in I_{T}\coloneqq\\{(t,s)\,|\,T\geq t\geq
s\geq 0\\}$. Moreover, let $T(t,s)$ be an evolution system, then for all
$(t,s)\in I_{T}$

$\|\prod_{j=1}^{n}F(s_{j},s_{j-1})x-T(t,s)x\|_{\mathcal{X}}\leq
n\max_{j\in\\{1,...,n\\}}\left\\{\|\left(F(s_{j},s_{j-1})-T(s_{j},s_{j-1})\right)T(s_{j-1},s_{0})x\|_{\mathcal{X}}\right\\}\,,$

for any increasing sequence $s=s_{0}<s_{1}<...<s_{n-1}<s_{n}=t$. Note that the
product is in decreasing order from the left to the right.

###### Proof.

The statement directly follows by the following telescopic sum [41, 54]:

$\displaystyle\prod_{j=1}^{n}F_{j}x-\prod_{j=1}^{n}T_{j}x$
$\displaystyle=\sum_{k=1}^{n}\left(\prod_{j=k+1}^{n}F_{j}\right)(F_{k}-T_{k})\left(\prod_{j=1}^{k-1}T_{j}\right)x\,,$
(39)

where the products are in decreasing order from the left to the right. Then,
the definition $T_{j}=T(s_{j},s_{j-1})$, the semigroup property, i.e.

$T(t,s)=T(s_{n},s_{n-1})\circ T(s_{n-1},s_{n-2})\circ\cdots\circ
T(s_{1},s_{0})\eqqcolon\prod_{j=1}^{n}T_{j}\,,$

and the contractivity of $F(t,s)$ combined with the submultiplicativity of the
norm finish the proof. ∎

In the considered cases related to the Trotter product formula and the quantum
Zeno effect, we are interested in an upper bound on the right-hand side of
A.1. Here, the following integral equation plays an important role, i.e.

$G(1)x-G(0)x=\int_{0}^{1}\frac{\partial}{\partial s}G(s)xds\,,$ (40)

which is well-defined if the operator-valued function $[0,1]\ni s\mapsto
G(s)\in\mathcal{B}(\mathcal{X})$ is strongly continuously differentiable [17].
In the cases discussed in the paper (see for example 3.2, 4.3) $G$ is given by
a semigroup our evolution system and $G(0)=\mathbbm{1}$. The continuity in the
former case is directly clear (see [11, Lem. 2.1.3]). In the latter, we repeat
the proof of [11, Lem. B.16].

###### Lemma A.2

Let $\mathcal{Y}\subset\mathcal{X}$ be a continuously embedded subspace of
$\mathcal{X}$, $[0,1]\ni t\mapsto G_{1}(t)\in\mathcal{B}(\mathcal{X})$ and
$[0,1]\ni t\mapsto(G_{2}(t),\mathcal{Y})$ be two strongly continuous vector-
valued maps. Then

$[0,1]\ni t\mapsto(G_{1}(t)G_{2}(t),\mathcal{Y})$

is again a strongly continuous vector-valued map on $\mathcal{Y}$.

###### Proof.

The proof follows the proof of [11, Lem. B.15]. Let $x\in\mathcal{Y}$,
$t\in[0,1]$, and $(t_{n})_{n\in\mathbbm{N}}$ a sequence converging to $t$.
Then, there is an uniform bound on
$\|G_{1}(t_{n})\|_{\mathcal{X}\rightarrow\mathcal{X}}$ by the uniform
boundedness theorem. Therefore,

$\displaystyle\|G_{1}(t_{n})G_{2}(t_{n})x-G_{1}(t)G_{2}(t)x\|_{\mathcal{X}}$
$\displaystyle=\|G_{1}(t_{n})\|_{\mathcal{X}\rightarrow\mathcal{X}}\,\|G_{2}(t_{n})x-G_{2}(t)x\|_{\mathcal{X}}$
$\displaystyle\qquad+\|(G_{1}(t_{n})-G_{1}(t))G_{2}(t)x\|_{\mathcal{X}}$

finishes the proof. ∎

##  Appendix B Proof of Zeno improvement

Here, we shortly discuss a slight improvement of 4.3, where the assumption
that $\mathcal{L}_{t}\mathcal{L}_{s}P$ is relatively $\mathcal{Y}$-bounded is
omitted. Therefore, Theorem I in [33] follows directly from the result below.

###### Theorem B.1

Let $I_{T}\ni(t,s)\mapsto V(t,s)\in\mathcal{B}(\mathcal{X})$ be a
$C_{0}$-evolution system with generator family
$(\mathcal{L}_{t},\mathcal{D}(\mathcal{L}_{t}))_{t\in[0,T]}$,
$M\in\mathcal{B}(\mathcal{X})$ a contraction, and $P$ a projection satisfying

$\norm{M^{n}-P}_{\mathcal{X}\rightarrow\mathcal{X}}\leq\delta^{n}$ (41)

for $\delta\in(0,1)$ and all $n\in\mathbbm{N}$. Moreover, we assume the
asymptomatic Zeno condition:

$\norm{PV(t,s)(\mathbbm{1}-P)}_{\mathcal{X}\rightarrow\mathcal{X}}\leq(t-s)b\quad\text{and}\quad\norm{(\mathbbm{1}-P)V(t,s)P}_{\mathcal{X}\rightarrow\mathcal{X}}\leq(t-s)b$
(42)

for $b\geq 0$ and all $t\in[0,T]$. Furthermore, there is a Banach space
$(\mathcal{Y}\subset\mathcal{X},\|\cdot\|_{\mathcal{Y}})$ invariant under $P$
so that the closure of $(P\mathcal{L}_{t}P,\mathcal{Y})_{t\in[0,T]}$ generates
a unique contractive $C_{0}$-evolution system $T(t,s)$ for $(t,s)\in I_{T}$
commuting with $P$. Moreover, assume that $\mathcal{L}_{t}P$,
$P\mathcal{L}_{t}P$, and $P\mathcal{L}_{t}P\mathcal{L}_{s}P$ are well-defined
on $\mathcal{Y}$ and relatively $\mathcal{Y}$-bounded for all $(t,s)\in
I_{T}$, and $(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ is a
$P\mathcal{L}_{t}P$-stable subspace, in particular
$\|T(t,s)\|_{\mathcal{Y}\rightarrow\mathcal{Y}}\leq\tilde{c}e^{(t-s)\omega}$.
Then, there is a $c\geq 0$ so that

$\displaystyle\Bigl{\|}\prod_{j=1}^{n}MV(s_{j},s_{j-1})x-T(t,0)Px\Bigr{\|}_{\mathcal{X}}\leq
c\frac{t^{2}}{n}\left(\|x\|_{\mathcal{X}}+e^{t\omega}\|Px\|_{\mathcal{Y}}\right)$

for all $t\geq 0$, $s_{0}=0<\frac{t}{n}<...<\frac{t(n-1)}{n}<t=s_{n}$,
$x\in\mathcal{Y}$ and $n\in\mathbbm{N}$.

###### Proof.

We shortly repeat the first step in the proof of 4.3, where we approximate $M$
by $P$ in the first step. To do so, we follow the proof strategy of Lemma 5.3
in [33]. By just changing the roles of $A$ to $A_{j}\coloneqq
M(\mathbbm{1}-P)V(s_{j},s_{j-1})$, $B$ to $B_{j}\coloneqq PV(s_{j},s_{j-1})$,
and, additionally, using the fact that the bounds only depend on the step
size, the same proof shows

$\norm{\prod_{j=1}^{n}MV(s_{j},s_{j-1})x-\prod_{j=1}^{n}PV(s_{j},s_{j-1})Px}_{\mathcal{X}}\leq\left(\delta^{n}+\frac{b}{n}+\frac{1}{n}\frac{b(2+b)(\delta-\delta^{n})}{1-\delta}e^{2b}\right)\norm{x}_{\mathcal{X}}.$
(43)

for all $x\in\mathcal{X}$. Then, we cannot apply 4.1 due to the missing
relative $\mathcal{Y}$-boundedness of $\mathcal{L}_{t}\mathcal{L}_{s}P$.
Instead, we follow the proof of 4.2 and apply A.1 to

$F(t,s)=PV(t,s)P$

w.r.t. the sequence $s_{0}=0<\frac{t}{n}<...<\frac{t(n-1)}{n}<t=s_{n}$ so that

$\displaystyle\Bigl{\|}\prod_{j=1}^{n}PV(s_{j},s_{j-1})$ $\displaystyle
Px-T(t,0)Px\Bigr{\|}_{\mathcal{X}}$ $\displaystyle\leq
n\max_{j\in\\{1,...,n\\}}\|\left(PV(s_{j},s_{j-1})P-T(s_{j},s_{j-1})\right)T(s_{j-1},s_{0})Px\|_{\mathcal{X}}\,.$

Then, we use $T(s_{j-1},s_{0})Px=PT(s_{j-1},s_{0})Px$ and the integral
equation (40) to show with $s_{j}(\tau)=s_{j}+\frac{1-\tau_{1}}{n}$

$\displaystyle P$ $\displaystyle
V(s_{j},s_{j-1})Px_{j}-T(s_{j},s_{j-1})Px_{j}$
$\displaystyle=\frac{1}{n}\int_{0}^{1}\Biggl{(}PV(s_{j},s_{j-1}(\tau_{1}))\mathcal{L}_{s_{j-1}(\tau_{1})}P-T(s_{j},s_{j-1}(\tau_{1}))P\mathcal{L}_{s_{j-1}(\tau_{1})}P\Biggr{)}x_{j}d\tau_{1}$
$\displaystyle=\frac{1}{n}\int_{0}^{1}\Biggl{(}PV(s_{j},s_{j-1}(\tau_{1}))(\mathbbm{1}-P)\mathcal{L}_{s_{j-1}(\tau_{1})}P$
$\displaystyle\qquad\qquad\qquad+PV(s_{j},s_{j-1}(\tau_{1}))P\mathcal{L}_{s_{j-1}(\tau_{1})}P-T(s_{j},s_{j-1}(\tau_{1}))P\mathcal{L}_{s_{j-1}(\tau_{1})}P\Biggr{)}x_{j}d\tau_{1}$

and for the last two terms

$\displaystyle\frac{1}{n}\int_{0}^{1}\Biggl{(}$ $\displaystyle
PV(s_{j},s_{j-1}(\tau_{1}))P\mathcal{L}_{s_{j-1}(\tau_{1})}P-T(s_{j},s_{j-1}(\tau_{1}))P\mathcal{L}_{s_{j-1}(\tau_{1})}P\Biggr{)}x_{j}d\tau_{1}$
$\displaystyle=\frac{1}{n^{2}}\int_{0}^{1}\Biggl{(}\tau_{1}PV(s_{j},s_{j-1}(\tau_{1}\tau_{2}))(P+\mathbbm{1}-P)\mathcal{L}_{s_{j-1}(\tau_{1}\tau_{2})}P\mathcal{L}_{s_{j-1}(\tau_{1})}P$
$\displaystyle\qquad\qquad\qquad-\tau_{1}T(s_{j},s_{j-1}(\tau_{1}\tau_{2}))P\mathcal{L}_{s_{j-1}(\tau_{1}\tau_{2})}P\mathcal{L}_{s_{j-1}(\tau_{1})}P\Biggr{)}x_{j}d\tau_{2}d\tau_{1}\,,$

where $x_{j}\coloneqq PT(s_{j-1},s_{0})Px$. Note that the integral above is
again well-defined by A.2 or [11, Lem. B.15]. Next, we use the relative
$\mathcal{Y}$-boundedness as in 4.1, but also the asymptotic Zeno condition,
which implies boundedness for all $t\in[0,T]$

$\|P\mathcal{L}_{t}(\mathbbm{1}-P)\|_{\mathcal{X}\rightarrow\mathcal{X}},\|(\mathbbm{1}-P)\mathcal{L}_{t}P\|_{\mathcal{X}\rightarrow\mathcal{X}}\leq
b\,.$

Then, $\|P\mathcal{L}_{t}Px\|_{\mathcal{X}}\leq a_{1}\|x\|_{\mathcal{Y}}$ and
$\|P\mathcal{L}_{t}P\mathcal{L}_{s}Px\|_{\mathcal{X}}\leq
a_{2}\|x\|_{\mathcal{Y}}$ for all $(t,s)\in I_{T}$ show

$\displaystyle\|PV(s_{j},s_{j-1})Px_{j}-T(s_{j},s_{j-1})Px_{j}\|_{\mathcal{X}}$
$\displaystyle\leq\frac{1}{n^{2}}\left(b^{2}\|x_{j}\|_{\mathcal{X}}+\left(\frac{a_{1}b}{2}+a_{2}\right)\|x_{j}\|_{\mathcal{Y}}\right)\,.$

Applying $\|PT(s_{j-1},s_{0})Px\|_{\mathcal{Y}}\leq
a_{3}e^{\omega}\|Px\|_{\mathcal{Y}}$ to the above bound shows

$\displaystyle\|PV(s_{j},s_{j-1})Px_{j}-T(s_{j},s_{j-1})Px_{j}\|_{\mathcal{X}}\leq\frac{1}{n^{2}}\left(b^{2}\|Px\|_{\mathcal{X}}+\left(\frac{a_{1}b}{2}+a_{2}\right)e^{\omega}\|Px\|_{\mathcal{Y}}\right)\,.$

Rescaling the time finishes the proof of the second case. ∎