$\displaystyle\left\|\left[O_{X}(\tau)-O_{X}(H_{X[\ell]},\tau)\right]\rho_{0}(t_{1})\right\|_{1}$
(778)

for $t_{1}\leq t-\tau$. However, we cannot directly apply the subtheorem
because we encounter the following difficulty. In high dimensional cases, when
the radius $\ell$ becomes large, the subset size $|L_{0}^{\ast}|$ in Eq. (561)
is roughly given by $|L_{0}^{\ast}|=|X[\ell]\setminus X|=\Theta\left[({\rm
diam}(X)+\ell)^{D}\right]$, and hence the number of bosons in the region
$L_{0}^{\ast}$, denoted by $Q$, is proportional to $|L_{0}^{\ast}|$, i.e.,
$Q\propto\Theta\left[({\rm diam}(X)+\ell)^{D}\right]$. Therefore, in applying
the inequality (563) to an operator $O_{X}$ supported on a large subset $X$
such that ${\rm diam}(X)\gtrsim\ell$, we have to consider a significantly
short time such that

$\displaystyle\frac{\tau Q\log(Q)}{\ell^{2}}\propto\tau\left(\frac{{\rm
diam}(X)}{\ell}\right)^{2}{\rm diam}(X)^{D-2}\log[{\rm
diam}(X)]\lesssim\frac{1}{e},$ (779)

which is necessary to make the inequality (563) meaningful. Unfortunately,
under the above condition, we can no longer derive the optimal Lieb-Robinson
bound. To avoid the constraint (779), before applying the subtheorem to the
derivation of the Lieb-Robinson bound, we first reduce the problem to the time
evolution of local operators (see Fig. 23 below).

For this purpose, we first define the projection $\bar{\Pi}_{Y,r,\leq Q}$ as
follows:

$\displaystyle\bar{\Pi}_{Y,r,\leq Q}:=\prod_{i\in Y}\Pi_{i[r],\leq Q}.$ (780)

From the projection, for any site $i\in Y$, the number of bosons in the ball
region $i[r]$ is smaller than or equal to $Q$. We also define the following
subsets (see Fig. 23):

$\displaystyle X_{1}:=X[2\ell_{0}],\quad X_{2}:=X[2\ell_{0}+k],\quad
X_{3}:=X[3\ell_{0}+k],\quad\tilde{X}:=X_{2}\setminus
X_{1}=X[2\ell_{0}+k]\setminus X[2\ell_{0}],$ (781)

where $\ell_{0}$ has been defined in the main statement, i.e.,
$\ell_{0}=(\ell-k)/3$. We then adopt the projection of

$\displaystyle\bar{\Pi}^{\ast}:=\bar{\Pi}_{\tilde{X},\ell_{0},\leq
Q}=\prod_{i\in\tilde{X}}\Pi_{i[\ell_{0}],\leq Q}.$ (782)

We next consider the decomposition of

$\displaystyle H=H_{1}+H_{2}+\partial H_{0}$ (783)

with

$\displaystyle H_{1}=H_{0,X_{1}}+V_{X_{2}},\quad H_{2}=H_{0,X_{2}^{\rm
c}}+V_{X_{2}^{\rm c}}+\sum_{Z\in\mathcal{S}_{X_{2}}}v_{Z},$
$\displaystyle\partial H_{0}=H-H_{1}-H_{2}=H_{0,X_{2}\setminus X_{1}}+\partial
h_{X_{1}}+\partial h_{X_{2}}=\sum_{\begin{subarray}{c}\left\langle
i,j\right\rangle\\\
\\{i,j\\}\cap\tilde{X}\neq\emptyset\end{subarray}}J_{i,j}(b_{i}b_{j}^{\dagger}+{\rm
h.c.}),$ (784)

where we use the notations of Eq. (28), and $\mathcal{S}_{X_{2}}$ has been
defined in Eq. (547), i.e., $\mathcal{S}_{X_{2}}:=\\{Z\subseteq\Lambda|Z\cap
X_{2}\neq\emptyset,Z\cap X_{2}^{\rm c}\neq\emptyset\\}$. Because the width of
the region $X_{2}\setminus X_{1}$ is equal to $k$ (see Fig. 23), we ensure
that the support of $v_{Z}$ ($Z\in\mathcal{S}_{X_{2}}$) has no overlap with
$X_{1}$, and hence

$\displaystyle[H_{1},H_{2}]=\sum_{Z\in\mathcal{S}_{X_{2}}}[H_{0,X_{1}},v_{Z}]=0.$
(785)

By using the above notations, we consider

$\displaystyle
e^{-iH\tau}=e^{-i\left(H_{1}+H_{2}\right)\tau}\mathcal{T}e^{-i\int_{0}^{\tau}\partial
H_{0,x}dx},$ (786)

with

$\displaystyle\partial H_{0,x}:=\partial
H_{0}\left(H_{1}+H_{2},x\right)=e^{i\left(H_{1}+H_{2}\right)x}\partial
H_{0}e^{-i\left(H_{1}+H_{2}\right)x}.$ (787)

Under the above setup, we will take the following strategy to prove
Proposition 45. From Eq. (786), we reduce the target time evolution
$O_{X}(\tau)$ to

$\displaystyle O_{X}(\tau)$
$\displaystyle=\left(\mathcal{T}e^{-i\int_{0}^{\tau}\partial
H_{0,x}dx}\right)^{\dagger}O_{X}(H_{1}+H_{2},\tau)\mathcal{T}e^{-i\int_{0}^{\tau}\partial
H_{0,x}dx}$ $\displaystyle=:O_{X_{2},\tau}(\partial H_{0,x},0\to\tau),$ (788)

where we denote $O_{X}(H_{1}+H_{2},\tau)=O_{X}(H_{1},\tau)=:O_{X_{2},\tau}$.
Note that from Eq. (785) and $[O_{X},H_{2}]=0$, we have

$e^{i(H_{1}+H_{2})\tau}O_{X}e^{-i(H_{1}+H_{2})\tau}=e^{iH_{1}\tau}e^{iH_{2}\tau}O_{X}e^{-iH_{2}\tau}e^{-iH_{1}\tau}=e^{iH_{1}\tau}O_{X}e^{-iH_{1}\tau}.$

We then aim to approximate the time evolution (Supplementary Note 9.3) by
using the effective Hamiltonian as

$\displaystyle\partial\tilde{H}^{\ast}_{0,x}:=\bar{\Pi}^{\ast}\partial
H_{0,x}\bar{\Pi}^{\ast}=\bar{\Pi}^{\ast}e^{i\left(H_{1}+H_{2}\right)x}\partial
H_{0}e^{-i\left(H_{1}+H_{2}\right)x}\bar{\Pi}^{\ast},$ (789)

where the second equation results from Eq. (787). As a first task, we need to
estimate the approximation error of

$\displaystyle O_{X_{2},\tau}(\partial H_{0,x},0\to\tau)\approx
O_{X_{2},\tau}(\bar{\Pi}^{\ast}\partial H_{0,x}\bar{\Pi}^{\ast},0\to\tau),$
(790)

and then our second task is to approximate $\bar{\Pi}^{\ast}\partial
H_{0,x}\bar{\Pi}^{\ast}$ onto $X_{3}$ ($=X[\ell]$). Because $\partial H_{0,x}$
is constructed from $b_{i}b_{j}^{\dagger}$ with
$\\{i,j\\}\cap\tilde{X}\neq\emptyset$, we need to obtain the approximation of

$\displaystyle\bar{\Pi}^{\ast}e^{i\left(H_{1}+H_{2}\right)x}(b_{i}b_{j}^{\dagger})e^{-i\left(H_{1}+H_{2}\right)x}\bar{\Pi}^{\ast}\approx\bar{\Pi}^{\ast}e^{i\left(H_{1,i[\ell_{0}]}+H_{2,i[\ell_{0}]}\right)x}(b_{i}b_{j}^{\dagger})e^{-i\left(H_{1,i[\ell_{0}]}+H_{2,i[\ell_{0}]}\right)x}\bar{\Pi}^{\ast}.$
(791)

Because of $\\{i,j\\}\cap\tilde{X}\neq\emptyset$, the subset Hamiltonian
$H_{1,i[\ell_{0}]}+H_{2,i[\ell_{0}]}$ is still supported on $X_{3}$. To the
above approximation, we can utilize Subtheorem 2. Combining the approximations
(790) and (791) leads to the main inequality (768).

Supplementary Figure 23: Schematic picture of the setup for the proof. We
first define the regions $X_{1}$, $X_{2}$ and $X_{3}$ as in Eq. (781). The
shaded region $\tilde{X}$ is defined as $X_{2}\setminus X_{1}$. The
Hamiltonian is decomposed to $H_{1}+H_{2}+\partial H_{0}$ [see Eq.
(Supplementary Note 9.3)]. Roughly speaking, the Hamiltonians $H_{1}$ and
$H_{2}$ acts on the region $X_{1}$ and $X_{2}^{\rm c}$, respectively, and the
Hamiltonian $\partial H_{0}$ describes the interactions in the region
$\tilde{X}$. By these choices, the Hamiltonians $H_{1}$ and $H_{2}$ are
decoupled in the sense that $[H_{1}+H_{2}]=0$ as in Eq. (785). Then, the time
evolution $e^{-iHt}$ is decomposed to $e^{-i\left(H_{1}+H_{2}\right)\tau}$ and
$\mathcal{T}e^{-i\int_{0}^{\tau}\partial H_{0,x}dx}$ as in Eq. (786), where
$\partial H_{0,x}=\partial H_{0}\left(H_{1}+H_{2},x\right)$. The time
evolution $e^{-i\left(H_{1}+H_{2}\right)\tau}=e^{-iH_{1}\tau}e^{-iH_{2}\tau}$
keeps the operator $O_{X}$ in the region $X_{2}$, whereas the time evolution
$\mathcal{T}e^{-i\int_{0}^{\tau}\partial H_{0,x}dx}$ may induce the operator
spreading outside the region $X_{3}$. In order to ensure the quasi-locality of
the operator $\partial H_{0,x}=\partial H_{0}(H_{1}+H_{2},x)$, we first
approximate $\partial H_{0,x}$ by the effective Hamiltonian
$\bar{\Pi}^{\ast}\partial H_{0,x}\bar{\Pi}^{\ast}$ (Lemma 46), where
$\bar{\Pi}^{\ast}$ is defined in Eq. (782). Afterward, by using Subtheorem 2,
we approximate the effective Hamiltonian $\bar{\Pi}^{\ast}\partial
H_{0,x}\bar{\Pi}^{\ast}$ onto the region $X_{3}$ [Corollary 47 and the
inequality (813)].

First of all, for the approximation (790), we prove the following lemma:

######  Lemma 46.

For the time-evolved quantum state $\rho_{0}(t_{1})$ with an arbitrary $t_{1}$
($\leq t-\tau$), we upper-bound the norm of

$\displaystyle\left\|\left[O_{X}(\tau)-O_{X_{2},\tau}(\partial\tilde{H}^{\ast}_{0,x},0\to\tau)\right]\rho_{0}(t_{1})\right\|_{1}$
$\displaystyle=\left\|\left[O_{X_{2},\tau}(\partial
H_{0,x},0\to\tau)-O_{X_{2},\tau}(\partial\tilde{H}^{\ast}_{0,x},0\to\tau)\right]\rho_{0}(t_{1})\right\|_{1}\leq\delta_{\ell_{0},Q}^{(1)},$
(792)

where $\delta_{\ell_{0},Q}^{(1)}$ is defined as follows:

$\displaystyle\delta_{\ell_{0},Q}^{(1)}:=4\gamma
R^{D}(1+2\tau\gamma^{2}\bar{J}N_{0}R^{D})\zeta
e^{-\left[Q/(4b_{0}\ell_{0}^{D})\right]^{1/\kappa}/2}=\Theta(N_{0}R^{2D})e^{-\Theta\left(Q/\ell_{0}^{D}\right)^{1/\kappa}}.$
(793)

####  Supplementary Note 9.3.1 Proof of Lemma 46

For the proof, we utilize Corollary 27. We here denote

$\displaystyle U_{\tau^{\prime}}=U_{\partial H_{0,x},0\to\tau^{\prime}}.$
(794)

From the inequality (27), in estimating the norm error in (792), we have to
upper-bound

$\displaystyle\left\|\bar{\Pi}^{\ast{\rm
c}}U_{\tau_{1}}U_{\tau}^{\dagger}O_{X_{2},\tau}U_{\tau}\sqrt{\rho_{0}(t_{1})}\right\|_{F},$
(795)

and

$\displaystyle\left\|\bar{\Pi}^{\ast}\partial
H_{0,\tau-\tau_{1}}\bar{\Pi}^{\ast{\rm
c}}U_{\tau_{1}}U_{\tau}^{\dagger}O_{X_{2},\tau}U_{\tau}\sqrt{\rho_{0}(t_{1})}\right\|_{F}.$
(796)

Note that all the terms in LHS of (27) reduce to either of the above forms by
letting $\tau_{1}\to(0\ {\rm or}\ \tau)$ and $O_{X_{2}}\to(\hat{1}\ {\rm or}\
O_{X_{2}})$

First, from the definition (787), we aim to upper-bound

$\displaystyle\left\|\bar{\Pi}^{\ast}\partial
H_{0,\tau-\tau_{1}}\bar{\Pi}^{\ast{\rm
c}}\right\|\leq\sum_{\begin{subarray}{c}\left\langle i,j\right\rangle\\\
\\{i,j\\}\cap\tilde{X}\neq\emptyset\end{subarray}}\bar{J}\left\|\bar{\Pi}^{\ast}\left(b_{i}b_{j}^{\dagger}+{\rm
h.c.}\right)\right\|\leq\sum_{\begin{subarray}{c}\left\langle
i,j\right\rangle\\\
\\{i,j\\}\cap\tilde{X}\neq\emptyset\end{subarray}}2\bar{J}\left\|\bar{\Pi}^{\ast}\left(\hat{n}_{i}+\hat{n}_{j}\right)\right\|,$
(797)

where we use (459) in the last inequality. Because the Hilbert space is
restricted by $\tilde{\Pi}_{(i_{0}[R],N_{0})}$ as in (756), we can upper-bound
as $\left\|\hat{n}_{i}\right\|\leq N_{0}$ for $i\in i_{0}[R]$, which yields

$\displaystyle\left\|\bar{\Pi}^{\ast}\partial
H_{0,\tau-\tau_{1}}\bar{\Pi}^{\ast{\rm
c}}\right\|\leq\sum_{\begin{subarray}{c}\left\langle i,j\right\rangle\\\
\\{i,j\\}\cap\tilde{X}\neq\emptyset\end{subarray}}4\bar{J}N_{0}\leq
4\gamma\bar{J}N_{0}|X_{2}\setminus X_{1}|\leq 4\gamma\bar{J}N_{0}|X_{2}|,$
(798)

where we use $\tilde{X}:=X_{2}\setminus X_{1}$ as in Eq. (781). We hence
upper-bound the norm (796) as

$\displaystyle\left\|\bar{\Pi}^{\ast}\partial
H_{0,\tau-\tau_{1}}\bar{\Pi}^{\ast{\rm
c}}U_{\tau_{1}}U_{\tau}^{\dagger}O_{X_{2},\tau}U_{\tau}\sqrt{\rho_{0}(t_{1})}\right\|_{F}\leq
4\gamma\bar{J}N_{0}|X_{2}|\left\|\bar{\Pi}^{\ast{\rm
c}}U_{\tau_{1}}U_{\tau}^{\dagger}O_{X_{2},\tau}U_{\tau}\sqrt{\rho_{0}(t_{1})}\right\|_{F}.$
(799)

Thus, we only have to estimate the norm (795).

We first reduce the norm (795) to

$\displaystyle\left\|\bar{\Pi}^{\ast{\rm
c}}U_{\tau_{1}}U_{\tau}^{\dagger}O_{X_{2},\tau}U_{\tau}\sqrt{\rho_{0}(t_{1})}\right\|_{F}=$
$\displaystyle\left\|\bar{\Pi}^{\ast{\rm
c}}e^{i\left(H_{1}+H_{2}\right)\tau_{1}}e^{-iH\tau_{1}}e^{iH\tau}e^{-i\left(H_{1}+H_{2}\right)\tau}O_{X_{2},\tau}e^{i\left(H_{1}+H_{2}\right)\tau}e^{-iH\tau}\sqrt{\rho_{0}(t_{1})}\right\|_{F}$
$\displaystyle=$ $\displaystyle\left\|\bar{\Pi}^{\ast{\rm
c}}e^{i\left(H_{1}+H_{2}\right)\tau_{1}}e^{iH(\tau-\tau_{1})}O_{X}e^{-iH\tau}\sqrt{\rho_{0}(t_{1})}\right\|_{F},$
(800)

where in the first equation, we use from Eqs. (786) and (794)

$\displaystyle
U_{\tau^{\prime}}=\mathcal{T}e^{-i\int_{0}^{\tau^{\prime}}\partial
H_{0}\left(H_{1}+H_{2},x\right)dx}=e^{i\left(H_{1}+H_{2}\right)\tau^{\prime}}e^{-iH\tau^{\prime}},\quad
H=H_{1}+H_{2}+\partial H_{0},$ (801)

and the second equation is obtained from the definition of
$O_{X}(H_{1}+H_{2},\tau)=O_{X}(H_{1},\tau)=:O_{X_{2},\tau}$. By applying the
definition (782) to Eq. (Supplementary Note 9.3.1), we have

$\displaystyle\left\|\bar{\Pi}^{\ast{\rm
c}}e^{i\left(H_{1}+H_{2}\right)\tau_{1}}e^{iH(\tau-\tau_{1})}O_{X}e^{-iH\tau}\sqrt{\rho_{0}(t_{1})}\right\|_{F}$
$\displaystyle\leq\sum_{i\in\tilde{X}}\left\|\Pi_{i[\ell_{0}],>Q}e^{i\left(H_{1}+H_{2}\right)\tau_{1}}e^{iH(\tau-\tau_{1})}O_{X}e^{-iH\tau}\sqrt{\rho_{0}(t_{1})}\right\|_{F}.$
(802)

For the estimation of the norm of

$\left\|\Pi_{i[\ell_{0}],>Q}e^{i\left(H_{1}+H_{2}\right)\tau_{1}}e^{iH(\tau-\tau_{1})}O_{X}e^{-iH\tau}\sqrt{\rho_{0}(t_{1})}\right\|_{F}=\left\|\Pi_{i[\ell_{0}],>Q}e^{i\left(H_{1}+H_{2}\right)\tau_{1}}e^{iH(\tau-\tau_{1})}O_{X}e^{-iH\tau}e^{-iHt_{1}}\sqrt{\rho_{0}}\right\|_{F},$

we can apply the inequality (434) in Lemma 28 from $t_{1}+\tau\leq t$. In the
lemma, the boson creation by the operator $O_{X_{0}}$ is assumed to be smaller
than or equal to $q_{0}$ (see (49) for the formulation), which also holds for
$O_{X}$ from the assumption of the main statement in Proposition 45. Moreover,
from the assumption of $\ell_{0}\geq\bar{\ell}_{t,R}$ and $d_{i,X}\geq
2\ell_{0}$ for $i\in\tilde{X}\ (=X[2\ell_{0}+k]\setminus X[2\ell_{0}])$, we
have $i[\ell_{0}]\subset X[\ell_{0}]^{\rm c}\subseteq X[\bar{\ell}_{t,R}]^{\rm
c}$. Therefore, by using the inequality (434) with $p=0$ obtain

$\displaystyle\left\|\Pi_{i[\ell_{0}],>Q}e^{i\left(H_{1}+H_{2}\right)\tau_{1}}e^{iH(\tau-\tau_{1})}O_{X}e^{-iH\tau}\sqrt{\rho_{0}(t_{1})}\right\|_{F}^{2}\leq\zeta^{2}e^{-\left[Q/(4b_{0}\ell_{0}^{D})\right]^{1/\kappa}}.$
(803)

By applying the inequality (803) to (Supplementary Note 9.3.1) and
(Supplementary Note 9.3.1), we obtain

$\displaystyle\left\|\bar{\Pi}^{\ast{\rm
c}}U_{\tau_{1}}U_{\tau}^{\dagger}O_{X_{2},\tau}U_{\tau}\sqrt{\rho_{0}(t_{1})}\right\|_{F}=\left\|\bar{\Pi}^{\ast{\rm
c}}e^{i\left(H_{1}+H_{2}\right)\tau_{1}}e^{iH(\tau-\tau_{1})}O_{X}e^{-iH\tau}\sqrt{\rho_{0}(t_{1})}\right\|_{F}\leq|\tilde{X}|\zeta
e^{-\left[Q/(4b_{0}\ell_{0}^{D})\right]^{1/\kappa}/2},$ (804)

which also yields from (799)

$\displaystyle\left\|\bar{\Pi}^{\ast}\partial
H_{0,\tau_{1}}\bar{\Pi}^{\ast{\rm
c}}U_{\tau-\tau_{1}}^{\dagger}O_{X_{2},\tau}U_{\tau}\sqrt{\rho_{0}(t_{1})}\right\|_{F}\leq
4\gamma\bar{J}N_{0}|X_{2}|\cdot|\tilde{X}|\zeta
e^{-\left[Q/(4b_{0}\ell_{0}^{D})\right]^{1/\kappa}/2}.$ (805)

By combining the above two inequalities with Corollary 27, we obtain

$\displaystyle\left\|\left[O_{X_{2},\tau}(\partial
H_{0,x},0\to\tau)-O_{X_{2},\tau}(\partial\tilde{H}^{\ast}_{0,x},0\to\tau)\right]\rho_{0}(t_{1})\right\|_{1}$
$\displaystyle\leq$ $\displaystyle 4|\tilde{X}|\zeta
e^{-\left[Q/(4b_{0}\ell_{0}^{D})\right]^{1/\kappa}/2}+8\gamma\bar{J}N_{0}|X_{2}|\tau\cdot|\tilde{X}|\zeta
e^{-\left[Q/(4b_{0}\ell_{0}^{D})\right]^{1/\kappa}/2}$ $\displaystyle\leq$
$\displaystyle 4\gamma R^{D}(1+2\tau\gamma^{2}\bar{J}N_{0}R^{D})\zeta
e^{-\left[Q/(4b_{0}\ell_{0}^{D})\right]^{1/\kappa}/2},$ (806)

where we use $\tilde{X},X_{2}\subset i_{0}[R]$, which yields
$|\tilde{X}|,|X_{2}|\leq|i_{0}[R]|\leq\gamma R^{D}$. We thus prove the main
inequality (792). This completes the proof. $\square$

[ End of Proof of Lemma 46]

In the following, we aim to estimate the error to approximate
$\partial\tilde{H}^{\ast}_{0,x}$ onto the subset $X_{3}$ ($=X[3\ell_{0}+k]$).
We here denote

$\displaystyle h_{i,j}=J_{i,j}(b_{i}b_{j}^{\dagger}+{\rm
h.c.}),\quad\tilde{h}^{\ast}_{i,j}\left(H_{1}+H_{2},x\right)=\bar{\Pi}^{\ast}h_{i,j}\left(H_{1}+H_{2},x\right)\bar{\Pi}^{\ast}.$
(807)

We then consider each of the interaction terms
$\tilde{h}^{\ast}_{i,j}\left(H_{1}+H_{2},x\right)$ that is included in
$\partial\tilde{H}^{\ast}_{0,x}$, i.e.,

$\displaystyle\partial\tilde{H}^{\ast}_{0,x}=\bar{\Pi}^{\ast}e^{i\left(H_{1}+H_{2}\right)x}\partial
H_{0}e^{-i\left(H_{1}+H_{2}\right)x}\bar{\Pi}^{\ast}=\sum_{\left\langle
i,j\right\rangle:\\{i,j\\}\cap\tilde{X}\neq\emptyset}\tilde{h}^{\ast}_{i,j}\left(H_{1}+H_{2},x\right),$
(808)

where we use Eqs. (789) and (Supplementary Note 9.3). For each of
$\tilde{h}^{\ast}_{i,j}\left(H_{1}+H_{2},x\right)$, we aim to estimate the
approximation error of

$\displaystyle\bar{\Pi}^{\ast}h_{i,j}\left(H_{1}+H_{2},x\right)\bar{\Pi}^{\ast}\approx\bar{\Pi}^{\ast}h_{i,j\to
X_{3},x}\bar{\Pi}^{\ast}=\tilde{h}^{\ast}_{i,j\to X_{3},x},$ (809)

where the operator $\tilde{h}_{i,j\to X_{3},x}$ is an appropriate operator
supported on $X_{3}$. Here, from the definitions in Eq. (781), we can ensure
(see also Fig. 23)

$\\{i,j\\}[\ell_{0}-1]\subset
X_{3}\quad\textrm{for}\quad\\{i,j\\}\cap\tilde{X}\neq\emptyset\quad(d_{i,j}=1).$

By applying Subtheorem 2, we immediately obtain the following corollary:

######  Corollary 47.

Under the constraints on the Hilbert space as in (756) and the projection
(782), there exists an appropriate operator $\tilde{h}_{i,j\to X_{3},x}$ such
that

$\displaystyle\left\|\left[h_{i,j}\left(H_{1}+H_{2},x\right)-\tilde{h}^{\ast}_{i,j\to
X_{3},x}\right]\bar{\Pi}^{\ast}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}\right\|\leq\left\|h_{i,j}\right\|\delta_{\ell_{0},Q}^{(2)}\quad\textrm{for}\quad
x\leq\tau,$ (810)

where $\ell_{0}\geq\log[\Theta(N_{0})]$ and $\delta_{\ell_{0},Q}$ is defined
as

$\displaystyle\delta_{\ell_{0},Q}^{(2)}:=\Theta(Q^{2})\left\\{e^{-\Theta\left(\ell_{0}\right)}+e^{\Theta(\ell_{0})}\left[\Theta(\tau)+\Theta\left(\frac{\tau
Q\log(Q)}{\ell_{0}^{2}}\right)\right]^{\Theta(\ell_{0})}\right\\}.$ (811)

The statement is proved by letting $O_{Z_{0}}\to h_{i,j}$ ($r_{1}=1,\
q_{Z_{0}}=1)$ in Subtheorem 2.

Therefore, by using the notation of

$\displaystyle\partial\tilde{H}^{\ast}_{0,X_{3},x}=\sum_{\begin{subarray}{c}\left\langle
i,j\right\rangle\\\
\\{i,j\\}\cap\tilde{X}\neq\emptyset\end{subarray}}\tilde{h}^{\ast}_{i,j\to
X_{3},x},$ (812)

we obtain

$\displaystyle\left\|\mathcal{T}e^{-i\int_{0}^{\tau}\partial\tilde{H}^{\ast}_{0,x}dx}-\mathcal{T}e^{-i\int_{0}^{\tau}\partial\tilde{H}^{\ast}_{0,X_{3},x}dx}\right\|\leq\int_{0}^{\tau}\left\|\partial\tilde{H}^{\ast}_{0,x}-\partial\tilde{H}^{\ast}_{0,X_{3},x}\right\|dx$
$\displaystyle\leq\sum_{\begin{subarray}{c}\left\langle i,j\right\rangle\\\
\\{i,j\\}\cap\tilde{X}\neq\emptyset\end{subarray}}\int_{0}^{\tau}\left\|\tilde{h}^{\ast}_{i,j}\left(H_{1}+H_{2},x\right)-\tilde{h}^{\ast}_{i,j\to
X_{3},x}\right\|dx$ $\displaystyle=\sum_{\begin{subarray}{c}\left\langle
i,j\right\rangle\\\
\\{i,j\\}\cap\tilde{X}\neq\emptyset\end{subarray}}\int_{0}^{\tau}\left\|\bar{\Pi}^{\ast}\left[h_{i,j}\left(H_{1}+H_{2},x\right)-h_{i,j\to
X_{3},x}\right]\bar{\Pi}^{\ast}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}\right\|dx$
$\displaystyle\leq\sum_{\begin{subarray}{c}\left\langle i,j\right\rangle\\\
\\{i,j\\}\cap\tilde{X}\neq\emptyset\end{subarray}}\tau\left\|h_{i,j}\right\|\delta_{\ell_{0},Q}^{(2)}\leq
4\tau\gamma\bar{J}N_{0}|X_{2}|\delta_{\ell_{0},Q}^{(2)},$ (813)

where the equation is a consequence from the constraints of the Hilbert space
on $\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}$ as in (756), and we use the same
analysis as (798) in the last inequality.

By using the above inequality, we obtain the approximation of

$\displaystyle O_{X_{2},\tau}(\partial\tilde{H}^{\ast}_{0,x},0\to\tau)\approx
O_{X_{2},\tau}(\partial\tilde{H}^{\ast}_{0,X_{3},x},0\to\tau).$ (814)

From the inequality of

$\displaystyle\left\|U^{\dagger}OU-\tilde{U}^{\dagger}O\tilde{U}\right\|=\left\|U^{\dagger}OU-U^{\dagger}O\tilde{U}+U^{\dagger}O\tilde{U}-\tilde{U}^{\dagger}O\tilde{U}\right\|\leq
2\left\|O\right\|\cdot\left\|U-\tilde{U}\right\|,$ (815)

we obtain the approximation error of (814) by using the inequality (813) as

$\displaystyle\left\|O_{X_{2},\tau}(\partial\tilde{H}^{\ast}_{0,x},0\to\tau)-O_{X_{2},\tau}(\partial\tilde{H}^{\ast}_{0,X_{3},x},0\to\tau)\right\|$
$\displaystyle\leq
2\left\|O_{X}\right\|\cdot\left\|\mathcal{T}e^{-i\int_{0}^{\tau}\partial\tilde{H}^{\ast}_{0,x}dx}-\mathcal{T}e^{-i\int_{0}^{\tau}\partial\tilde{H}^{\ast}_{0,X_{3},x}dx}\right\|$
$\displaystyle\leq
8\zeta\tau\gamma\bar{J}N_{0}|X_{2}|\delta_{\ell_{0},Q}^{(2)}\leq
8\zeta\tau\gamma^{2}\bar{J}N_{0}R^{D}\delta_{\ell_{0},Q}^{(2)},$ (816)

where we use $|X_{2}|\leq i_{0}[R]\leq\gamma R^{D}$.

By combining the inequality (792) in Lemma 46 with (Supplementary Note 9.3.1),
we finally arrive at the inequality of

$\displaystyle\left\|\left[O_{X}(\tau)-O_{X_{2},\tau}(\partial\tilde{H}^{\ast}_{0,X_{3},x},0\to\tau)\right]\rho_{0}(t_{1})\right\|_{1}$
$\displaystyle\leq\delta_{\ell_{0},Q}^{(1)}+8\zeta\tau\gamma^{2}\bar{J}N_{0}R^{D}\delta_{\ell_{0},Q}^{(2)}$
$\displaystyle=\Theta(N_{0}R^{2D})e^{-\Theta\left(Q/\ell_{0}^{D}\right)^{1/\kappa}}+\zeta\Theta(N_{0}R^{D})\cdot\Theta(Q^{2})\left\\{e^{-\Theta\left(\ell_{0}\right)}+e^{\Theta(\ell_{0})}\left[\Theta(\tau)+\Theta\left(\frac{\tau
Q\log(Q)}{\ell_{0}^{2}}\right)\right]^{\Theta(\ell_{0})}\right\\}$
$\displaystyle=\zeta\Theta(N_{0}R^{2D}Q^{2})\left\\{e^{-\Theta\left(Q/\ell_{0}^{D}\right)^{1/\kappa}}+e^{-\Theta\left(\ell_{0}\right)}+e^{\Theta(\ell_{0})}\left[\Theta(\tau)+\Theta\left(\frac{\tau
Q\log(Q)}{\ell_{0}^{2}}\right)\right]^{\Theta(\ell_{0})}\right\\}.$ (817)

Therefore, we can choose the approximate operator for $O_{X}(\tau)$ as
$O_{X_{2},\tau}(\partial\tilde{H}^{\ast}_{0,X_{3},x},0\to\tau)$. Here, the
operator $O_{X_{2},\tau}(\partial\tilde{H}^{\ast}_{0,X_{3},x},0\to\tau)$ does
not depend on the interaction terms outside $X_{3}$, and hence we can adopt
the operator $O_{X_{2},\tau}(\partial\tilde{H}^{\ast}_{0,X_{3},x},0\to\tau)$
for the approximation of $O_{X}(H_{X_{3}},\tau)$. Thus, by using the same
analysis as (Supplementary Note 7), we prove the main inequality (768). This
completes the proof of Proposition 45. $\square$

##  Supplementary Note 10 Lieb-Robinson bound in one dimensional systems

In this section, we prove the optimal Lieb-Robinson bound in which the Lieb-
Robinson light cone is almost linear as $\tilde{\mathcal{O}}(t)$. The main
statement is given as follows:

######  Theorem 4.

We here adopt the same setups as in Theorems 2 and 3. Then, under the
condition of

$\displaystyle
R^{\prime\prime}:=R-r_{0}-\bar{\ell}_{t,R}-1\geq\Theta\left[t\log^{2}(q_{0}r_{0}t)\right],$
(818)

we obtain the Lieb-Robinson bound as

$\displaystyle\left\|\left[O_{X_{0}}(t)-O_{X_{0}}\left(H_{i_{0}[R]},t\right)\right]\rho_{0}\right\|_{1}\leq\left\|O_{X_{0}}\right\|\exp\left\\{-\Theta\left(\frac{R^{\prime\prime}}{t\log(R)}\right)^{1/\kappa}+\Theta[\log(R)]\right\\}.$
(819)

Remark. In the case where $r_{0}=\mathcal{O}(R)$ and $q_{0}=\mathcal{O}(R)$,
the RHS of (819) decays with $R$ as long as

$\displaystyle R\geq\Theta[t\log^{1+\kappa}(t)].$ (820)

Therefore, the light cone becomes almost linear as $\tilde{\mathcal{O}}(t)$ in
one-dimensional systems.

As a related result, Yin and Lucas have proved [10] that in one-dimensional
systems

$\displaystyle{\rm
tr}\left(\left[O_{X}(t),O_{Y}\right]\rho_{0}\right)\leq\left(\frac{\Theta(t)}{d_{X,Y}}\right)^{\Theta(d_{X,Y})}$
(821)

under a similar assumption to Assumption 3 on the initial state $\rho_{0}$.
The considered quantity is rather weaker than
$\left\|\left[O_{X}(t),O_{Y}\right]\rho_{0}\right\|_{1}$*13*13*13For example,
we cannot discuss the gate complexity of the digital quantum simulation only
from the bound (821)., but the upper bound is stronger in the sense that the
commutator decays for $R\geq\Theta(t)$, which means the strictly linear light
cone. Moreover, the provided proof there is much simpler than ours. Thus, it
leaves a hope to improve the present statement in Theorem 4 to the better form
(821) with a more straightforward proof.

Supplementary Figure 24: Schematic picture of the setup for the proof of
Theorem 4. We adopt the projections $\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}$
and $\Pi_{\tilde{X}_{0},\leq Q}$, where $\tilde{X}_{0}$ is defined as
$\tilde{X}_{0}:=i_{0}[R]\setminus X_{0}[\bar{\ell}_{t,R}+1]$. We restrict
ourselves to the Hilbert space spanned by these projections, and the error is
estimated in the inequality (Supplementary Note 10.1). In the main proof in
Sec. Supplementary Note 10.3, we decompose $\tilde{X}_{0}$ into $\bar{s}$
subspaces and refine the connection of short-time unitary evolutions in
(Supplementary Note 9.2).

###  Supplementary Note 10.1 Preparation

As in Sec. Supplementary Note 9.1, we restrict the Hilbert space which is
spanned by the Fock state on $\mathcal{S}(i_{0}[R],1/2,N_{0})$
($X_{0}=i_{0}[r_{0}]$, see also Fig. 24). In addition, we adopt the projection
$\Pi_{\tilde{X}_{0},\leq Q}$ onto the region $\tilde{X}_{0}$, where
$\tilde{X}_{0}$ is defined as

$\displaystyle\tilde{X}_{0}:=i_{0}[R]\setminus X_{0}[\bar{\ell}_{t,R}+1].$
(822)

In Proposition 32, we set

$\displaystyle L\to i_{0}[R],\quad N\to N_{0},\quad
L^{\prime}\to\tilde{X}_{0},\quad
Q\to\check{q}R^{\prime\prime}\quad(R^{\prime\prime}=R-r_{0}-\bar{\ell}_{t,R}-1).$
(823)

Note that the definitition (822) implies

$\displaystyle d_{X_{0},L^{\prime}}=\bar{\ell}_{t,R}+1,$ (824)

and hence the condition (493) is satisfied. Here, we have
$|\tilde{X}_{0}|=R^{\prime\prime}$, and we can replace $\ell_{1}$ in Eq. (493)
by $R^{\prime\prime}$, i.e., $\ell_{1}\to R^{\prime\prime}$. Then, from the
inequality (32), we obtain

$\displaystyle\left\|O_{X_{0}}(H,t)\rho_{0}-O_{X_{0}}\left(\tilde{H}_{(i_{0}[R],N_{0},\tilde{X}_{0},Q)},t\right)\tilde{\Pi}_{(i_{0}[R],N_{0},\tilde{X}_{0},Q)}\rho_{0}\tilde{\Pi}_{(i_{0}[R],N_{0},\tilde{X}_{0},Q)}\right\|_{1}$
$\displaystyle\leq\zeta_{0}e^{-R}+\zeta_{0}\Theta\left(tQ\right)e^{-\Theta\left(Q/R^{\prime\prime}\right)^{1/\kappa}}=\zeta_{0}e^{-R}+\zeta_{0}\Theta\left(tQ\right)e^{-\Theta\left(\check{q}\right)^{1/\kappa}},$
(825)

where the effective Hamiltonian $\tilde{H}_{(i_{0}[R],N_{0},\tilde{X}_{0},Q)}$
has been defined in Eq. (494), and we choose $N_{0}$ as in Eq. (755), i.e.,

$\displaystyle
N_{0}=\Theta\left(q_{0}^{3},R^{2D+1+\kappa}\right)=\Theta\left(q_{0}^{3},R^{\kappa+3}\right)\quad\quad\textrm{for}\quad
D=1.$ (826)

We recall that the projection $\tilde{\Pi}_{(i_{0}[R],N_{0},\tilde{X}_{0},Q)}$
has been defined in Eq. (502), i.e.,

$\displaystyle\tilde{\Pi}_{(i_{0}[R],N_{0},\tilde{X}_{0},Q)}=\Pi_{\tilde{X}_{0},\leq
Q}\tilde{\Pi}_{(i_{0}[R],N_{0})}=\Pi_{\tilde{X}_{0},\leq
Q}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}.$ (827)

Also, as in Eq. (756), we adopt the simplified notation of

$\displaystyle\tilde{H}_{(i_{0}[R],N_{0},\tilde{X}_{0},Q)}\to
H,\quad\tilde{\Pi}_{(i_{0}[R],N_{0},\tilde{X}_{0},Q)}\rho_{0}\tilde{\Pi}_{(i_{0}[R],N_{0},\tilde{X}_{0},Q)}\to\rho_{0}.$
(828)

###  Supplementary Note 10.2 Main proposition and the proof

The proof of Theorem 4 relies on the following proposition:

######  Proposition 48.

Let us choose $\check{q}$ as

$\displaystyle\check{q}=\Theta\left(\frac{R^{\prime\prime}}{t\log(R)}\right).$
(829)

Then, under the notations in Eq. (828), we can obtain the following
approximation error

$\displaystyle\left\|\left[O_{X_{0}}(t)-\bar{O}_{i_{0}[R]}\right]\rho_{0}\right\|_{1}$
$\displaystyle\leq\zeta_{0}e^{\Theta\left[t\log(N_{0})\right]-\Theta(R^{\prime\prime})}\quad\textrm{for}\quad
R^{\prime\prime}\geq\Theta[t\log^{1+\kappa}(N_{0})],$ (830)

where we can choose $\bar{O}_{i_{0}[R]}$ as $O_{X_{0}}(H_{i_{0}[R]},t)$ by
following the same discussion as in Subtheorem 2 [see also (Supplementary Note
8)].

By using Proposition 48 with (Supplementary Note 10.1), we obtain

$\displaystyle\left\|\left[O_{X_{0}}(t)-O_{X_{0}}(H_{i_{0}[R]},t)\right]\rho_{0}\right\|_{1}$
$\displaystyle\leq\zeta_{0}e^{-R}+\zeta_{0}\Theta\left(tQ\right)\exp\left[-\Theta\left(\frac{R^{\prime\prime}}{t\log(R)}\right)^{1/\kappa}\right]+\zeta_{0}e^{\Theta\left[t\log(R)\right]-\Theta(R^{\prime\prime})}.$
(831)

Because the second term in the RHS is dominant, we prove the main inequality
(819). Finally, because $N_{0}$ is given by Eq. (826), the condition
$R^{\prime\prime}\geq\Theta[t\log^{1+\kappa}(N_{0})]$ reduces to (818). This
completes the proof of Theorem 4. $\square$

###  Supplementary Note 10.3 Proof of Proposition 48: refined decomposition of
the time evolution

In the case of one-dimensional systems, if we use the same approach in Sec.
Supplementary Note 9.2, we cannot obtain the optimal Lieb-Robinson light cone
as $R=\tilde{\mathcal{O}}(t)$ (see also Remark in Theorem 3). To refine the
bound, we take a similar but different approach. In this section, we adopt the
notations $\tau$, $X_{m}$ and $\bar{m}$ similar to Eqs. (757) and (758) (see
also Fig. 24):

$\displaystyle t:=\bar{m}\tau,$ $\displaystyle
X_{m}:=i_{0}[r_{0}+\bar{\ell}_{t,R}+1+m\Delta r],$ (832) $\displaystyle\Delta
r=\left\lfloor\frac{R-(r_{0}+\bar{\ell}_{t,R}+1)}{\bar{s}}\right\rfloor=\left\lfloor\frac{R^{\prime\prime}}{\bar{s}}\right\rfloor\quad(\bar{s}\geq\bar{m}),$
(833) $\displaystyle R^{\prime\prime}:=R-r_{0}-\bar{\ell}_{t,R}-1,$ (834)

where $X_{\bar{s}}\subseteq i_{0}[R]$ and $\bar{s}$ is appropriately chosen
[see (910) below].

Here, we choose $\tau$ as an $\mathcal{O}(1)$ constant [$\tau\leq
1/(4\gamma\bar{J})$], which also implies

$\displaystyle\bar{m}:=t/\tau=\Theta(t).$ (835)

On the other hand, we redefine the parameter $\Delta r$ as follows:

$\displaystyle\Delta
r=\left\lfloor\frac{R^{\prime\prime}}{\bar{s}}\right\rfloor\quad(\bar{s}\geq\bar{m}),$
(836)

where By using $\Delta r$ and $R$, we choose $\check{q}$ such that

$\displaystyle\check{q}=\Theta\left(\frac{\Delta r}{\log(R)}\right),$ (837)

which reduces to the form of Eq. (829) from $\Delta
r=\Theta(R^{\prime\prime}/\bar{s})$ and $\bar{s}=\Theta(t)$. From the
inequality (Supplementary Note 10.1), $\check{q}$ should be at least larger
than $\log^{\kappa}(R)$, or else the upper bound in (Supplementary Note 10.1)
is worse than the trivial bound. Thus, from $\kappa\geq 1$, we add the
following additional condition to $\Delta r$ as

$\displaystyle\frac{\Delta r}{\log(R)}\geq\Theta[\log^{\kappa}(R)],$ (838)

which leads to the choice of $\Delta r$ as

$\displaystyle\Delta
r=\Theta[\log^{1+\kappa}(N_{0})]\geq\Theta[\log^{1+\kappa}(R)].$ (839)

The above condition yields the condition
$R^{\prime\prime}\geq\Theta[t\log^{1+\kappa}(N_{0})]$ in the main inequality
(830).

In the following, we refine the previous decomposition, which has been given
in Eq. (Supplementary Note 9.2) (see Fig. 25 for the schematic picture of the
present decomposition). Our purpose here is to derive the equation (850) and
the inequality (852). We start from the decomposition of the time-evolved
operator $O_{X_{0}}(\tau)$ as follows:

$\displaystyle O_{X_{0}}(\tau)$
$\displaystyle=\sum_{s_{1}=1}^{\bar{s}+1}\Delta O_{s_{1}},$ (840)

where we define $\Delta O_{s_{1}}^{(1)}$ as

$\displaystyle\Delta O_{1}:=U_{1}^{\dagger}O_{X_{0}}U_{1},$
$\displaystyle\Delta
O_{s_{1}}:=U_{{s_{1}}}^{\dagger}O_{X_{0}}U_{{s_{1}}}-U_{{s_{1}-1}}^{\dagger}O_{X_{0}}U_{{s_{1}-1}}\quad(2\leq
s_{1}\leq\bar{s}),$ $\displaystyle\Delta
O_{\bar{s}+1}:=O_{X_{0}}(\tau)-U_{{\bar{s}}}^{\dagger}O_{X_{0}}U_{{\bar{s}}},$
(841)

where, for an arbitrary $s$, we denote $U_{s}$ as the time evolution in the
region $X_{s}$, i.e.,

$\displaystyle U_{s}:=e^{-iH_{X_{s}}\tau}.$ (842)

In the next step, we consider the decomposition of $\Delta O_{s_{1}}(\tau)$ as
follows:

$\displaystyle\Delta
O_{s_{1}}(\tau)=\sum_{s_{2}=\min(\bar{s}+1,s_{1}+1)}^{\bar{s}+1}\Delta
O_{s_{1},s_{2}},$ (843)

where $\Delta O_{{s_{1},s_{2}}}$ ($s_{1}\leq\bar{s}$) is defined as

$\displaystyle\Delta O_{s_{1},s_{1}+1}:=U_{{s_{1}+1}}^{\dagger}\Delta
O_{s_{1}}U_{{s_{1}+1}},$ $\displaystyle\Delta
O_{s_{1},s_{2}}:=U_{{s_{2}}}^{\dagger}\Delta
O_{s_{1}}U_{{s_{2}}}-U_{{s_{2}-1}}^{\dagger}\Delta
O_{s_{1}}U_{{s_{2}-1}}\quad(s_{1}+2\leq s_{2}\leq\bar{s}),$
$\displaystyle\Delta O_{s_{1},\bar{s}+1}:=\Delta
O_{s_{1}}(\tau)-U_{{\bar{s}}}^{\dagger}\Delta O_{s_{1}}U_{{\bar{s}}}.$ (844)

Note that we adopt the same definition for $U_{s}$ as in Eq. (842). In the
case of $s_{1}=\bar{s}+1$, we define $\Delta O_{{s_{1},s_{2}}}$ as

$\displaystyle\Delta O_{{\bar{s}+1,\bar{s}+1}}:=\Delta
O_{\bar{s}+1}(\tau)\quad\textrm{for}\quad s_{1}=\bar{s}+1.$ (845)

By combining Eqs. (840) and (843), we obtain

$\displaystyle O_{X_{0}}(2\tau)$
$\displaystyle=\sum_{s_{1}=1}^{\bar{s}+1}\sum_{s_{2}=\min(\bar{s}+1,s_{1}+1)}^{\bar{s}+1}\Delta
O_{s_{1},s_{2}}.$ (846)

Supplementary Figure 25: Schematic picture of the new decomposition (850). We
start from the time evolution of $O_{X_{0}}(\tau)$, which is decomposed to the
summation of $\\{\Delta O_{s_{1}}\\}_{s_{1}=1}^{\bar{s}+1}$ as in Eq. (840).
Here, $\Delta O_{s_{1}}$ is defined by using the unitary operator $U_{s_{1}}$
[see Eq. (842)] and is supported on the subset $X_{s_{1}}$. In the next step,
the time evolution of the decomposed operator, i.e., $\Delta O_{s_{1}}(\tau)$,
is also decomposed to the summation of $\\{\Delta
O_{{s_{1},s_{2}}}\\}_{s_{2}=s_{1}+1}^{\bar{s}+1}$ as in Eq. (843). In the same
way, the time evolution of $\Delta O_{{s_{1},s_{2}}}(\tau)$ is further
decomposed to $\\{\Delta
O_{{s_{1},s_{2},s_{3}}}\\}_{s_{3}=s_{2}+1}^{\bar{s}+1}$. By connecting the
decompositions, we obtain the full decomposition as in Eq. (850). By
truncating the summation for $s_{1},s_{2},\ldots,s_{\bar{m}}$ up to $\bar{s}$,
all the decomposed operators $\Delta O_{\mathbb{s}_{\bar{m}}}$ are supported
on the subset $X_{\bar{s}}\subseteq i_{0}[R]$, and hence we can obtain the
refined approximation for the time evolution $O_{X_{0}}(t)$ [see Eq. (851)].

By repeating this process, we recursively define $\Delta O_{{\mathbb{s}_{m}}}$
($\mathbb{s}_{m}:=\\{s_{1},\ldots,s_{m-1},s_{m}\\}$, $s_{m-1}\leq\bar{s}$) as

$\displaystyle\Delta O_{\mathbb{s}_{m}}:=U_{{s_{m-1}+1}}^{\dagger}\Delta
O_{\mathbb{s}_{m-1}}U_{{s_{m-1}+1}}\quad(s_{m}=s_{m-1}+1),$
$\displaystyle\Delta O_{\mathbb{s}_{m}}:=U_{{s_{m}}}^{\dagger}\Delta
O_{\mathbb{s}_{m-1}}U_{{s_{m}}}-U_{{s_{m}-1}}^{\dagger}\Delta
O_{\mathbb{s}_{m-1}}U_{{s_{m}-1}}\quad(s_{m-1}+2\leq s_{m}\leq\bar{s}),$
$\displaystyle\Delta O_{\mathbb{s}_{m}}:=\Delta
O_{\mathbb{s}_{m-1}}(\tau)-U_{{\bar{s}}}^{\dagger}\Delta
O_{\mathbb{s}_{m-1}}U_{{\bar{s}}}\quad(s_{m}=\bar{s}+1).$ (847)

In the case of $s_{m-1}=\bar{s}+1$, we define $\Delta O_{\mathbb{s}_{m}}$ as

$\displaystyle\Delta O_{\mathbb{s}_{m}}:=\Delta
O_{\mathbb{s}_{m-1}}(\tau)\quad\textrm{for}\quad s_{m-1}=s_{m}=\bar{s}+1.$
(848)

By using the above notations, we obtain

$\displaystyle\Delta
O_{\mathbb{s}_{m-1}}(\tau)=\sum_{s_{m}=\min(\bar{s}+1,s_{m-1}+1)}^{\bar{s}+1}\Delta
O_{\mathbb{s}_{m}}.$ (849)

By applying Eq. (849) iteratively, we have the decomposition of
$O_{X_{0}}(\bar{m}\tau)=O_{X_{0}}(t)$ as follows:

$\displaystyle O_{X_{0}}(\bar{m}\tau)=O_{X_{0}}(t)$
$\displaystyle=\sum_{s_{1}=1}^{\bar{s}+1}\sum_{s_{2}=\min(\bar{s}+1,s_{1}+1)}^{\bar{s}+1}\cdots\sum_{s_{\bar{m}}=\min(\bar{s}+1,s_{\bar{m}-1}+1)}^{\bar{s}+1}\Delta
O_{\mathbb{s}_{\bar{m}}}.$ (850)

Then, by truncating the summation for each of $\\{s_{m}\\}_{m=1}^{\bar{m}}$ up
to $\bar{s}$, we define $\bar{O}_{X_{\bar{s}}}$ as

$\displaystyle\bar{O}_{X_{\bar{s}}}:=\sum_{s_{1}=1}^{\bar{s}}\sum_{s_{2}=s_{1}+1}^{\bar{s}}\cdots\sum_{s_{m}=s_{m-1}+1}^{\bar{s}}\Delta
O_{\mathbb{s}_{m}}.$ (851)

Here, the operator $\bar{O}_{X_{\bar{s}}}$ is supported on the subset
$X_{\bar{s}}\subseteq i_{0}[R]$ [see Eq. (832)]. Therefore, we need to upper-
bound the norm of

$\displaystyle\left\|\left[O_{X_{0}}(t)-\bar{O}_{X_{\bar{s}}}\right]\rho_{0}\right\|_{1}\leq\sum_{\mathbb{s}_{\bar{m}}\in
V_{\bar{s}+1}}\|\Delta O_{\mathbb{s}_{\bar{m}}}\rho_{0}\|_{1},$ (852)

where $V_{\bar{s}+1}$ is a set of $\mathbb{s}_{\bar{m}}$ such that at least
one $s_{p}\in\mathbb{s}_{\bar{m}}$ is equal to $\bar{s}+1$.

In general, for the estimation of $\|\Delta O_{\mathbb{s}_{m}}\rho_{0}\|_{1}$
($1\leq m\leq\bar{m}$), we need to estimate the norms like

$\displaystyle\left\|\left(U_{{s_{m}}}^{\dagger}\Delta
O_{\mathbb{s}_{m-1}}U_{{s_{m}}}-U_{{s_{m}-1}}^{\dagger}\Delta
O_{\mathbb{s}_{m-1}}U_{{s_{m}-1}}\right)\rho_{0}\right\|_{1}.$ (853)

However, we here encounter the following obstacle. By using the short-time
Lieb-Robinson bound in Proposition 45, we can only obtain an upper bound in
the form of

$\displaystyle\left\|\left(U_{{s_{m}}}^{\dagger}\Delta
O_{\mathbb{s}_{m-1}}U_{{s_{m}}}-U_{{s_{m}-1}}^{\dagger}\Delta
O_{\mathbb{s}_{m-1}}U_{{s_{m}-1}}\right)\rho_{0}\right\|_{1}\leq\left\|\Delta
O_{\mathbb{s}_{m-1}}\right\|\varepsilon_{s_{m-1},s_{m}}$ (854)

instead of

$\displaystyle\left\|\left(U_{{s_{m}}}^{\dagger}\Delta
O_{\mathbb{s}_{m-1}}U_{{s_{m}}}-U_{{s_{m}-1}}^{\dagger}\Delta
O_{\mathbb{s}_{m-1}}U_{{s_{m}-1}}\right)\rho_{0}\right\|_{1}\leq\left\|\Delta
O_{\mathbb{s}_{m-1}}\rho_{0}\right\|_{1}\varepsilon_{s_{m-1},s_{m}},$ (855)

where $\varepsilon_{s_{m-1},s_{m}}$ is a small constant which depends on
$s_{m-1}$ and $s_{m}$. If we could obtain the bound as (855), we can
iteratively upper-bound $\left\|\Delta O_{\mathbb{s}_{m}}\rho_{0}\right\|_{1}$
for $m\geq 1$, which may yield the desired inequality (830). However, the
inequality (855) does not seem to hold in general.

In the following sections, we aim to refine the decomposition (850)
furthermore.

###  Supplementary Note 10.4 Proof of Proposition 48: inserting projections

Supplementary Figure 26: New decomposed operator
$\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}$ in Eq. (Supplementary Note
10.4). We here adopt the projection operators $P^{(\sigma)}_{s,s^{\prime}}$
with $0\leq s\leq s^{\prime}\leq\bar{s}+1$ and $\sigma=0,1$, which is
supported on $X_{s^{\prime}}\setminus X_{s}$. The explicit definition of the
projection will be given in Eq. (Supplementary Note 10.5). In every step from
the time $(m-1)\tau$ to $m\tau$, we insert the projection
$P^{(0)}_{s_{m-1},s_{m}}$ or $P^{(1)}_{s_{m-1},s_{m}}$. Regarding the
projection indices $\bm{\sigma}_{m}=\\{\sigma_{j}\\}_{j=1}^{m}$, we redefine
the new decomposed operators
$\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}$ as in Eq. (Supplementary
Note 10.4).

To overcome the difficulty, we introduce a projection operator
$P^{(\sigma)}_{s,s^{\prime}}$ ($0\leq s\leq s^{\prime}$, $\sigma=0,1$) such
that $P^{(0)}_{s,s^{\prime}}+P^{(1)}_{s,s^{\prime}}=1$ which is supported on
$X_{s^{\prime}}\setminus X_{s}$ (see also Fig. 26). By using the projection,
we first decompose $\Delta O_{s_{1}}$ as

$\displaystyle\Delta O_{s_{1}}$ $\displaystyle=\Delta
O_{s_{1}}P^{(0)}_{0,s_{1}}+\Delta
O_{s_{1}}P^{(1)}_{0,s_{1}}=:\Delta\tilde{O}_{s_{1},0}+\Delta\tilde{O}_{s_{1},1}.$
(856)

In the same way, we decompose $\Delta O_{s_{1},s_{2}}$ in Eq. (Supplementary
Note 10.3) as

$\displaystyle\Delta O_{s_{1},s_{2}}$
$\displaystyle=\sum_{\sigma_{1}=0,1}\left(U_{{s_{2}}}^{\dagger}\Delta\tilde{O}_{s_{1},\sigma_{1}}U_{{s_{2}}}-U_{{s_{2}-1}}^{\dagger}\Delta\tilde{O}_{s_{1},\sigma_{1}}U_{{s_{2}-1}}\right)\left(P^{(0)}_{s_{1},s_{2}}+P^{(1)}_{s_{1},s_{2}}\right)$
$\displaystyle=:\sum_{\sigma_{1}=0,1}\sum_{\sigma_{2}=0,1}\Delta\tilde{O}_{s_{1},s_{2},\sigma_{1},\sigma_{2}},$
(857)

where we consider the case of $s_{1}+2\leq s_{2}\leq\bar{s}$ in Eq.
(Supplementary Note 10.3) for simplicity. The above decomposition reduces Eq.
(846) to

$\displaystyle O_{X_{0}}(2\tau)$
$\displaystyle=\sum_{s_{1}=1}^{\bar{s}+1}\sum_{s_{2}=\min(\bar{s}+1,s_{1}+1)}^{\bar{s}+1}\sum_{\sigma_{1}=0,1}\sum_{\sigma_{2}=0,1}\Delta\tilde{O}_{s_{1},s_{2},\sigma_{1},\sigma_{2}}.$
(858)

By repeating the same procedure, we decompose
$\Delta\tilde{O}_{\mathbb{s}_{m}}$ as

$\displaystyle\Delta\tilde{O}_{\mathbb{s}_{m}}=\sum_{\bm{\sigma}_{m}\in\\{0,1\\}^{\otimes
m}}\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}},$
$\displaystyle\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}:=\left(U_{{s_{m}}}^{\dagger}\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}U_{{s_{m}}}-U_{{s_{m}-1}}^{\dagger}\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}U_{{s_{m}-1}}\right)P^{(\sigma_{m})}_{s_{m-1},s_{m}}.$
(859)

By using the notation of $\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}$,
we reduce Eqs. (850) and (851) to

$\displaystyle
O_{X_{0}}(t)=\sum_{\bm{\sigma}_{\bar{m}}\in\\{0,1\\}^{\otimes\bar{m}}}\sum_{s_{1}=1}^{\bar{s}+1}\sum_{s_{2}=\min(\bar{s}+1,s_{1}+1)}^{\bar{s}+1}\cdots\sum_{s_{\bar{m}}=\min(\bar{s}+1,s_{\bar{m}-1}+1)}^{\bar{s}+1}\Delta\tilde{O}_{\mathbb{s}_{\bar{m}},\bm{\sigma}_{\bar{m}}},$
$\displaystyle\bar{O}_{X_{\bar{s}}}=\sum_{\bm{\sigma}_{\bar{m}}\in\\{0,1\\}^{\otimes\bar{m}}}\sum_{s_{1}=1}^{\bar{s}}\sum_{s_{2}=s_{1}+1}^{\bar{s}}\cdots\sum_{s_{\bar{m}}=s_{\bar{m}-1}+1}^{\bar{s}}\Delta\tilde{O}_{\mathbb{s}_{\bar{m}},\bm{\sigma}_{\bar{m}}}.$
(860)

By using the above equations, we have the approximation of

$\displaystyle\left\|\left[O_{X_{0}}(t)-\bar{O}_{X_{\bar{s}}}\right]\rho_{0}\right\|_{1}\leq\sum_{\bm{\sigma}_{\bar{m}}\in\\{0,1\\}^{\otimes\bar{m}}}\sum_{\mathbb{s}_{\bar{m}}\in
V_{\bar{s}+1}}\left\|\Delta\tilde{O}_{\mathbb{s}_{\bar{m}},\bm{\sigma}_{\bar{m}}}\rho_{0}\right\|_{1}.$
(861)

Remember that $V_{\bar{s}+1}$ has been defined as the set of
$\mathbb{s}_{\bar{m}}$ such that at least one $s_{p}\in\mathbb{s}_{\bar{m}}$
is equal to $\bar{s}+1$. We aim to upper-bound the norm
$\left\|\left[O_{X_{0}}(t)-\bar{O}_{X_{\bar{s}}}\right]\rho_{0}\right\|_{1}$
based on this decomposition by appropriately defining the projection
$P^{(\sigma)}_{s,s^{\prime}}$.

Supplementary Figure 27: Schematic picture of the definition (Supplementary
Note 10.5) for the projections $P_{s_{m-1},s_{m}}^{(0)}$ and
$P_{s_{m-1},s_{m}}^{(1)}$. Each of the projections characterizes the number of
bosons in the region $L_{m}$, which is the half region between $X_{s_{m}}$ and
$X_{s_{m-1}}$, i.e., $L_{m}=X_{s_{m}}\setminus
X_{s_{m-1}}[(s_{m}-s_{m-1})\Delta r/2]$. Here, the projection
$P_{s_{m-1},s_{m}}^{(0)}$ implies a small boson number in the region $L_{m}$,
while the projection $P_{s_{m-1},s_{m}}^{(1)}$ implies a large boson number.

###  Supplementary Note 10.5 Proof of Proposition 48: estimation of the
decomposed operators $\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}$

In the following, we aim to upper-bound the norm of

$\displaystyle\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|$
(862)

for arbitrary fixed choices of $\mathbb{s}_{m}$ ($\in V_{\bar{s}+1}$) and
$\bm{\sigma}_{m}$. We here choose the projection operator
$P_{s_{m-1},s_{m}}^{(\sigma_{m})}$ as follows (see Fig. 27):

$\displaystyle P_{s_{m-1},s_{m}}^{(0)}=\Pi_{L_{m},\leq 16\check{q}|L_{m}|},$
$\displaystyle P_{s_{m-1},s_{m}}^{(1)}=\Pi_{L_{m},>16\check{q}|L_{m}|},$
$\displaystyle L_{m}:=X_{s_{m}}\setminus X_{s_{m-1}}[(s_{m}-s_{m-1})\Delta
r/2],$ (863)

where the constant $\check{q}$ is an appropriately defined parameter as in Eq.
(829). We recall that $\Pi_{L_{m},\leq 16\check{q}|L_{m}|}$ has been defined
as a projector onto the eigenspace of $\hat{n}_{L_{m}}$ with the eigenvalues
smaller than $16\check{q}|L_{m}|$. Here, $L_{m}$ is a half region from the
point $s_{m-1}$ to $s_{m}$ (see Fig. 27). Roughly speaking, the projection
$P_{s_{m-1},s_{m}}^{(0)}$ restricts the boson number in the region $L_{m}$,
while the projection $P_{s_{m-1},s_{m}}^{(1)}$ permits unlimited number of
bosons in the region $L_{m}$.

Based on the above definition, we take the following strategy (see also Fig.
28):

  1. 1.

The operator $\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|$
includes $m$ projections $\\{P_{s_{j-1},s_{j}}^{(\sigma_{j})}\\}_{j=1}^{m}$.
If the projection is given by $P_{s_{j-1},s_{j}}^{(0)}$, we can apply the
Lieb-Robinson bound for the short-time evolution (i.e., Proposition 44) from
time $(j-1)\tau$ to $j\tau$ in the region $X_{s_{m}}\setminus X_{s_{m-1}}$. On
the other hand, if the projection is given by $P_{s_{j-1},s_{j}}^{(1)}$, we
cannot obtain a meaningful Lieb-Robinson bound.

  2. 2.

If many of the region have the indices $\sigma_{j}=0$ [Fig. 28 (a)], we can
ensure that the norm
$\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|$ is
sufficiently small by connecting the short-time Lieb-Robinson bound [see Lemma
49 and Corollary 50].

  3. 3.

Conversely, if many of the region have the indices $\sigma_{j}=1$ [Fig. 28
(b)], we have to take another approach to upper-bound the norm
$\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|$. We have
already upper-bounded the total number of bosons in the region $\tilde{X}_{0}$
by $Q$ as in Eq. (828). Hence, such a constraint allows us to upper-bound
$\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|$ when many of
the region have large boson numbers [see Lemma 51].

(a) Many low-boson-density projections (b) Many high-boson-density projections

Supplementary Figure 28: The two cases of the projection patterns
$\\{P_{s_{j-1},s_{j}}^{(\sigma_{j})}\\}_{j=1}^{m}$. Now, the parameter
$M_{\mathbb{s}_{m},\bm{\sigma}_{m}}=\sum_{j=1}^{m}\sigma_{j}(s_{j}-s_{j-1})$
in Eq. (875) is used to characterize the ratio of the region with low (or
high)-boson density. Note that $0\leq M_{\mathbb{s}_{m},\bm{\sigma}_{m}}\leq
s_{m}$. In the first case (a), the majority of the projections have the index
$\sigma_{j}=0$; that is, we have $M_{\mathbb{s}_{m},\bm{\sigma}_{m}}\ll
s_{m}$. We then apply the short-time Lieb-Robinson bound in Proposition 44 to
the time evolution with the projections
$\\{P_{s_{j-1},s_{j}}^{(\sigma_{j})}\\}_{j=1}^{m}$ such that $\sigma_{j}=0$,
which yields Lemma 49 and Corollary 50. On the other hand, in the second case
(b), the majority of the projections have the index $\sigma_{j}=1$, i.e.,
$M_{\mathbb{s}_{m},\bm{\sigma}_{m}}\approx s_{m}$. In this case, it means that
so many bosons exist in the region $X_{s_{m}}\setminus
X_{0}[\bar{\ell}_{t,R}+1]$. However, we have truncated the number of bosons in
the region $\tilde{X}_{0}:=i_{0}[R]\setminus X_{0}[\bar{\ell}_{t,R}+1]$ by $Q$
as in Eq. (828). This allows us to efficiently upper-bound the norm
$\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|$ as in Lemma
51.

We first consider the case with many low-boson-density projections [Fig. 28
(a)]. By using Proposition 44, we prove the following lemma:

######  Lemma 49.

Let us choose $\check{q}$ as in (837). Then, we can upper-bound the norm
$\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|$ by using
$\left\|\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}\right\|$ as
follows:

$\displaystyle\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|\leq
2\left\|\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}\right\|\quad\textrm{for}\quad\sigma_{m}=1,$
$\displaystyle\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|\leq\lambda_{Q}\delta_{\Delta
r}^{s_{m}-s_{m-1}-2}\left\|\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}\right\|\quad\textrm{for}\quad\sigma_{m}=0,$
(864)

where $\delta_{\Delta r}$ and $\lambda_{Q}$ are constants such that

$\displaystyle\delta_{\Delta r}=e^{-\Theta(\Delta
r)},\quad\lambda_{Q}=\Theta(Q^{4}R^{D}N_{0}).$ (865)

####  Supplementary Note 10.5.1 Proof of Lemma 49

From the definition (Supplementary Note 10.4), the first inequality in (49) is
trivial, and hence we need to prove the second inequality. By using the
expression (Supplementary Note 10.4) and the triangle inequality, we first
obtain

$\displaystyle\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|\leq$
$\displaystyle\left\|\left[\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H,\tau)-U_{{s_{m}}}^{\dagger}\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}U_{{s_{m}}}\right]P^{(\sigma_{m})}_{s_{m-1},s_{m}}\right\|$
$\displaystyle+\left\|\left[\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H,\tau)-U_{{s_{m}-1}}^{\dagger}\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}U_{{s_{m}-1}}\right]P^{(\sigma_{m})}_{s_{m-1},s_{m}}\right\|$
$\displaystyle=$
$\displaystyle\left\|\left[\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H,\tau)-\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H_{X_{s_{m}}},\tau)\right]P^{(\sigma_{m})}_{s_{m-1},s_{m}}\right\|$
$\displaystyle+\left\|\left[\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H,\tau)-\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H_{X_{s_{m}-1}},\tau)\right]P^{(\sigma_{m})}_{s_{m-1},s_{m}}\right\|,$
(866)

where we use the definition (842) of $U_{s}$. From the abbreviation of the
notation as in (828), $H$, $H_{X_{s_{m}}}$ and $H_{X_{s_{m}-1}}$ are given by
the effective Hamiltonian under the projection
$\tilde{\Pi}_{(i_{0}[R],N_{0},\tilde{X}_{0},Q)}$. For $\sigma_{m}=0$, we need
to estimate

$\displaystyle\left\|\left[\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H,\tau)-\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H_{X_{s_{m}}},\tau)\right]P^{(0)}_{s_{m-1},s_{m}}\right\|$
$\displaystyle=$
$\displaystyle\left\|\left[\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H,\tau)-\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H_{X_{s_{m}}},\tau)\right]\Pi_{L_{m},\leq
16\check{q}|L_{m}|}\right\|,$ (867)

where we use the definition (Supplementary Note 10.5) for
$P^{(0)}_{s_{m-1},s_{m}}$.

Because of the abbreviated notation (828), we here consider the effective
Hamiltonian which is constructed by the projections $\Pi_{\tilde{X}_{0},\leq
Q}$ and $\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}$. We hence apply Proposition 44
to $\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(\tau)$ instead of
Subtheorem 2. The operator
$\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}$ is supported on
$X_{s_{m-1}}\subseteq i_{0}[R]$, and hence the boson creation/annihilation by
the operator is upper-bounded by

$\displaystyle N_{0}|X_{s_{m-1}}|\leq\gamma R^{D}N_{0},\quad{\rm i.e.,}\quad
q_{Z_{0}}\to\gamma R^{D}N_{0},$ (868)

because of the constraint $\mathcal{S}(i_{0}[R],1/2,N_{0})$. In the inequality
(44) of Proposition 44, we set

$\displaystyle
O_{Z_{0}}\to\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}},\quad\zeta_{Z_{0}}\to\left\|\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}\right\|,\quad\ell_{0}\to|L_{m}|,\quad
Q\to 16\check{q}|L_{m}|,\quad\tilde{L}\to L_{m},\quad\bar{L}=i_{0}[R].$ (869)

Here, the condition $\ell_{0}\geq\Theta[\log(q_{Z_{0}}N_{0})]$ in Proposition
44 reduces to $|L_{m}|\geq\Theta[\log(N_{0})]$ because of $q_{Z_{0}}\leq{\rm
poly}(N_{0})$ from Eq. (826). We thus obtain

$\displaystyle\left\|\left[\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H,\tau)-\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H_{X_{s_{m}}},\tau)\right]P^{(0)}_{s_{m-1},s_{m}}\right\|$
$\displaystyle\leq\left\|\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}\right\|\Theta(Q^{4}R^{D}N_{0})\left\\{e^{-\Theta(|L_{m}|)}+e^{\Theta(|L_{m}|)}\left[\Theta(\tau)+\Theta\left(\frac{\check{q}\tau\log(\check{q}|L_{m}|)}{|L_{m}|}\right)\right]^{\Theta(|L_{m}|)}\right\\}.$
(870)

By using the condition (837), i.e., $\check{q}\leq\Theta[\Delta r/\log(R)]$,
and $|L_{m}|\geq\Theta(\Delta r)$, we have

$\displaystyle\Theta(\tau)+\Theta\left(\frac{\check{q}\tau\log(\check{q}|L_{m}|)}{|L_{m}|}\right)=\Theta(\tau),$
(871)

which reduces the inequality (Supplementary Note 10.5.1) to

$\displaystyle\left\|\left[\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H,\tau)-\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H_{X_{s_{m}}},\tau)\right]P^{(0)}_{s_{m-1},s_{m}}\right\|$
$\displaystyle\leq\left\|\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}\right\|\Theta(Q^{4}R^{D}N_{0})e^{-\Theta(|L_{m}|)}$
(872)

under an appropriate choice of $\tau$. In the same way, we obtain

$\displaystyle\left\|\left[\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H,\tau)-\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}(H_{X_{s_{m}-1}},\tau)\right]P^{(0)}_{s_{m-1},s_{m}}\right\|$
$\displaystyle\leq\left\|\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}\right\|\Theta(Q^{4}R^{D}N_{0})e^{-\Theta(|L_{m}|-\Delta
r)},$ (873)

where we use $X_{s_{m}}=X_{s_{m}-1}[\Delta r]$. By applying the inequalities
(Supplementary Note 10.5.1) and (Supplementary Note 10.5.1) to (Supplementary
Note 10.5.1), we finally obtain

$\displaystyle\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|$
$\displaystyle\leq\left\|\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}\right\|\Theta(Q^{4}R^{D}N_{0})e^{-\Theta(|L_{m}|-\Delta
r)}$
$\displaystyle=\Theta(Q^{4}R^{D}N_{0})e^{-\Theta\left[(s_{m}-s_{m-1}-2)\Delta
r/2\right]}\left\|\Delta\tilde{O}_{\mathbb{s}_{m-1},\bm{\sigma}_{m-1}}\right\|,$
(874)

where we use the definition (Supplementary Note 10.5) of $L_{m}$, which
implies $|L_{m}|=(s_{m}-s_{m-1})\Delta r/2$. We thus prove the inequality
(49). $\square$

[ End of Proof of Lemma 49]

By iteratively applying Lemma 49 to
$\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|$, we can
immediately prove the following corollary:

######  Corollary 50.

Let us define $M_{\mathbb{s}_{m},\bm{\sigma}_{m}}$ as

$\displaystyle
M_{\mathbb{s}_{m},\bm{\sigma}_{m}}:=\sum_{j=1}^{m}\sigma_{j}(s_{j}-s_{j-1}),$
(875)

where we set $s_{0}=0$. We then obtain the upper bound of

$\displaystyle\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|\leq\zeta_{0}\left(2\lambda_{Q}\delta_{\Delta
r}^{-2}\right)^{m}\delta_{\Delta
r}^{s_{m}-M_{\mathbb{s}_{m},\bm{\sigma}_{m}}}\left\|O_{X_{0}}\right\|,$ (876)

where $\zeta_{0}$ has been defined as $\zeta_{0}=\left\|O_{X_{0}}\right\|$.

Proof of Corollary 50. We here define $m_{0}$ and $m_{1}$ as the numbers of
$\\{\sigma_{j}\\}_{j=1}^{m}$ such that $\sigma_{j}=0$ and $\sigma_{j}=1$,
respectively. Note that $m_{0}+m_{1}=m$. Then, by recursively applying the
inequality (49) to
$\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|$, we obtain

$\displaystyle\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|\leq
2^{m_{1}}\cdot\left(\lambda_{Q}\delta_{\Delta
r}^{-2}\right)^{m_{0}}\delta_{\Delta r}^{M^{\rm
c}_{\mathbb{s}_{m},\bm{\sigma}_{m}}}\leq\left(2\lambda_{Q}\delta_{\Delta
r}^{-2}\right)^{m}\delta_{\Delta r}^{M^{\rm
c}_{\mathbb{s}_{m},\bm{\sigma}_{m}}}\left\|O_{X_{0}}\right\|,$ (877)

where we define $M^{\rm c}_{\mathbb{s}_{m},\bm{\sigma}_{m}}$ as

$\displaystyle M^{\rm
c}_{\mathbb{s}_{m},\bm{\sigma}_{m}}:=\sum_{j=1}^{m}(1-\sigma_{j})(s_{j}-s_{j-1}).$
(878)

By using $M_{\mathbb{s}_{m},\bm{\sigma}_{m}}+M^{\rm
c}_{\mathbb{s}_{m},\bm{\sigma}_{m}}=\sum_{j=1}^{m}(s_{j}-s_{j-1})=s_{m}-s_{0}=s_{m}$
from $s_{0}=0$, we prove the inequality (876). This completes the proof.
$\square$

[ End of Proof of Corollary 50]

From the above corollary, we can ensure that the norm
$\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|$ decays
exponentially as $M_{\mathbb{s}_{m}}$ decreases (i.e.,
$s_{m}-M_{\mathbb{s}_{m}}$ increases). Then, if
$M_{\mathbb{s}_{\bar{m}},\bm{\sigma}_{\bar{m}}}\ll s_{m}$, the RHS of the
inequality (876) is sufficiently small as long as

$\left(s_{m}-M_{\mathbb{s}_{\bar{m}},\bm{\sigma}_{\bar{m}}}\right)\Delta
r\gtrsim m\log\left(\lambda_{Q}\delta_{\Delta r}^{-2}\right).$

On the other hand, for $M_{\mathbb{s}_{\bar{m}},\bm{\sigma}_{\bar{m}}}\approx
s_{m}$, the above condition cannot be satisfied unless we set $\Delta r$
sufficiently large beyond the condition $\Delta r=\Theta[\log^{2}(N_{0})]$ as
in (839). To obtain meaningful bound without taking $\Delta r$ so large, we
need another analysis to upper-bound
$\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|$. To see the
point, let us expand $\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}$ in Eq.
(Supplementary Note 10.4). Then, the expanded terms, for example, include

$\displaystyle\left(U_{s_{1}}\cdots
U_{s_{m-1}}U_{s_{m}}\right)^{\dagger}O_{X_{0}}\left(U_{s_{1}}P^{(\sigma_{1})}_{0,s_{1}}\cdots
U_{s_{m-1}}P^{(\sigma_{m-1})}_{s_{m-2},s_{m-1}}U_{s_{m}}P^{(\sigma_{m})}_{s_{m-1},s_{m}}\right).$
(879)

In more detail, all the expanded terms are given in the form of

$\displaystyle\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\rho_{0}=\sum_{\zeta_{1},\zeta_{2},\ldots,\zeta_{m}=0,1}\left(U_{s_{1}-\zeta_{1}}\cdots
U_{s_{m-1}-\zeta_{m-1}}U_{s_{m}-\zeta_{m}}\right)^{\dagger}O_{X_{0}}\Phi(\mathbb{s}_{m},\bm{\sigma}_{m},\bm{\zeta}_{m})\rho_{0}$
$\displaystyle\Phi(\mathbb{s}_{m},\bm{\sigma}_{m},\bm{\zeta}_{m})=\prod_{j=1}^{m}(-1)^{\zeta_{j}}U_{{s_{j}-\zeta_{j}}}P^{(\sigma_{j})}_{s_{j-1},s_{j}},\quad\bm{\zeta}_{m}:=\\{\zeta_{1},\zeta_{2},\ldots,\zeta_{m}\\}$
(880)

where the term (879) corresponds to the case of $\zeta_{j}=0$ for
$j=1,2,\ldots,m$. In the case where
$M_{\mathbb{s}_{\bar{m}},\bm{\sigma}_{\bar{m}}}\ll m$, the operator
$\Phi(\mathbb{s}_{m},\bm{\sigma}_{m},\bm{\zeta}_{m})$ roughly plays a role of
projection onto quantum states with large boson number in the region
$(X_{s_{m}}\setminus X_{0}[\bar{\ell_{t,R}+1}])\subseteq\tilde{X}_{0}$ [see
also Fig. 28 (b)]. Therefore, because the number of bosons is truncated up to
$Q$ in the region $\tilde{X}_{0}$, we expect that

$\displaystyle\left\|\Phi(\mathbb{s}_{m},\bm{\sigma}_{m},\bm{\zeta}_{m})\rho_{0}\right\|\ll
1\quad\textrm{for}\quad M_{\mathbb{s}_{\bar{m}},\bm{\sigma}_{\bar{m}}}\approx
s_{m}.$ (881)

The difficulty stems from the unitary operator $U_{{s_{j}-\zeta_{j}}}$ that is
inserted between $P^{(\sigma_{j})}_{s_{j-1},s_{j}}$ and
$P^{(\sigma_{j})}_{s_{j-2},s_{j-1}}$ [see (884) below].

In the following, we aim to derive a complementary statement to Corollary 50,
where the norm
$\left\|\Delta\tilde{O}_{\mathbb{s}_{m},\bm{\sigma}_{m}}\right\|$ decays
exponentially as $M_{\mathbb{s}_{m}}$ increases. Indeed, we can prove the
following lemma:

######  Lemma 51.

The norm of $\Delta O_{\mathbb{s}_{m},\bm{\sigma}_{m}}\rho_{0}$ is bounded
from above by

$\displaystyle\left\|\Delta
O_{\mathbb{s}_{m},\bm{\sigma}_{m}}\rho_{0}\right\|_{1}\leq
2^{2m}\zeta_{0}\exp\left[-\Theta(1)\left(\frac{M_{\mathbb{s}_{m},\bm{\sigma}_{m}}\check{q}\Delta
r}{2}-\frac{Q}{8}\right)\right],$ (882)

where $M_{\mathbb{s}_{m},\bm{\sigma}_{m}}$ has been defined in Eq. (875).

####  Supplementary Note 10.5.2 Proof of Lemma 51

For arbitrary choices of $\bm{\zeta}_{m}$, we aim to prove

$\displaystyle\left\|\Phi(\mathbb{s}_{m},\bm{\sigma}_{m},\bm{\zeta}_{m})\rho_{0}\right\|_{1}\leq
2^{m}\exp\left[-\Theta(1)\left(\frac{M_{\mathbb{s}_{m},\bm{\sigma}_{m}}\check{q}\Delta
r}{2}-\frac{Q}{8}\right)\right],$ (883)

which immediately yields the desired inequality (882) by counting all the
patterns of $\\{\bm{\zeta}_{m}\\}$ in Eq. (Supplementary Note 10.5.1).

We here consider the case of $\zeta_{j}=0$ for $j=1,2,\ldots,m$ for
simplicity, but the same calculations are applied to the other cases. We then
need to consider the norm of

$\displaystyle\left\|U_{s_{1}}P^{(\sigma_{1})}_{0,s_{1}}\cdots
U_{s_{m-1}}P^{(\sigma_{m-1})}_{s_{m-2},s_{m-1}}U_{s_{m}}P^{(\sigma_{m})}_{s_{m-1},s_{m}}\rho_{0}\right\|_{1}.$
(884)

If we simply consider the product of the projections
$\\{P^{(\sigma_{j})}_{s_{j-1},s_{j}}\\}_{j=1}^{m}$, it becomes a projection
onto a region $X_{m}\setminus X_{0}[\bar{\ell}_{t,R}+1]$ with a large boson
number. However, due to the unitary operators $\\{U_{s_{j}}\\}_{j=1}^{m}$,
such a simple discussion cannot be applied. For example, let us consider the
case of $\sigma_{m}=1$ (see also Fig. 29). Then, by the projection
$P^{(1)}_{s_{m-1},s_{m}}$, a quantum state
$P^{(1)}_{s_{m-1},s_{m}}|\psi\rangle$ has a large boson number in the region
$X_{s_{m}}\setminus X_{s_{m-1}}$. However, because of the unitary operator
$U_{s_{m}}$, the bosons in $X_{s_{m}}\setminus X_{s_{m-1}}$ may outflow from
the region, and hence the quantum state
$U_{s_{m}}P^{(1)}_{s_{m-1},s_{m}}|\psi\rangle$ may no longer have many bosons
in $X_{s_{m}}\setminus X_{s_{m-1}}$.

Supplementary Figure 29: Schematic picture of the decomposition of
$U_{s_{j}}P^{(1)}_{s_{j-1},s_{j}}$. We have mentioned that the operator
$\Phi(\mathbb{s}_{m},\bm{\sigma}_{m},\bm{\zeta}_{m})$ in Eq. (Supplementary
Note 10.5.1) roughly plays a role in projection onto states with a large boson
number. However, we cannot apply this intuitive discussion due to the outflow
of bosons by the unitary operators $\\{U_{s_{j}}\\}_{j=1}^{m}$. To identify
the influence of the unitary operators quantitatively, we introduce new
projections $\tilde{P}_{s_{j-1},s_{j}}^{(1,0)}$ and
$\tilde{P}_{s_{j-1},s_{j}}^{(1,1)}$ after $U_{s_{j}}P^{(1)}_{s_{j-1},s_{j}}$.
The projection $\tilde{P}_{s_{j-1},s_{j}}^{(1,0)}$ characterizes the case that
the over half of the bosons outflow from the region $X_{s_{j}}\setminus
X_{s_{j-1}}$, while the projection $\tilde{P}_{s_{j-1},s_{j}}^{(1,1)}$
characterizes the case that many of the bosons still remains in the region
$X_{s_{j}}\setminus X_{s_{j-1}}$.

To avoid the above difficulty, we decompose $U_{s_{j}}P^{(1)}_{s_{j-1},s_{j}}$
as follows:

$\displaystyle
U_{s_{j}}P^{(1)}_{s_{j-1},s_{j}}=\left(\tilde{P}_{s_{j-1},s_{j}}^{(1,0)}+\tilde{P}_{s_{j-1},s_{j}}^{(1,1)}\right)U_{s_{j}}P^{(1)}_{s_{j-1},s_{j}},$
(885)

where $\tilde{P}_{s_{j-1},s_{j}}^{(1,0)}$ and
$\tilde{P}_{s_{j-1},s_{j}}^{(1,1)}$ are defined as follows (see Fig. 29):

$\displaystyle\tilde{P}_{s_{j-1},s_{j}}^{(1,0)}=\Pi_{X_{s_{j}}\setminus
X_{s_{j-1}},\leq 8\check{q}|L_{j}|},$
$\displaystyle\tilde{P}_{s_{j-1},s_{j}}^{(1,1)}=\Pi_{X_{s_{j}}\setminus
X_{s_{j-1}},>8\check{q}|L_{j}|},\quad 8\check{q}|L_{j}|=4\check{q}\Delta
r(s_{j}-s_{j-1}),$ (886)

where $L_{j}$ has been defined in Eq. (Supplementary Note 10.5), i.e.,
$L_{j}:=X_{s_{j}}\setminus X_{s_{j-1}}[(s_{j}-s_{j-1})\Delta r/2]$. Note that
original projection $P^{(1)}_{s_{j-1},s_{j}}$ ensures at least
$16\check{q}|L_{j}|$ bosons in the region $L_{j}$ as in Eq. (Supplementary
Note 10.5). Hence, the projection $\tilde{P}_{s_{j-1},s_{j}}^{(1,0)}$ implies
that over half of the bosons outflows from the region $X_{s_{j}}\setminus
X_{s_{j-1}}$, while the projection $\tilde{P}_{s_{j-1},s_{j}}^{(1,1)}$ means
that many of the bosons still remains in the region $X_{s_{j}}\setminus
X_{s_{j-1}}$.

Because the projection $\tilde{P}_{s_{j-1},s_{j}}^{(1,0)}$ is supported on
$X_{s_{j}}\setminus X_{s_{j-1}}$, it commutes with
$U_{s_{j^{\prime}}}P^{(\sigma_{j^{\prime}})}_{s_{j^{\prime}-1},s_{j^{\prime}}}$
for $\forall j^{\prime}\leq j-1$ (see also Fig. 29):

$\displaystyle\left[\tilde{P}_{s_{j-1},s_{j}}^{(1,\theta_{j})},U_{s_{j^{\prime}}}P^{(\sigma_{j^{\prime}})}_{s_{j^{\prime}-1},s_{j^{\prime}}}\right]=0\quad\textrm{for}\quad\forall
j^{\prime}\leq j-1,\quad\theta_{j}=0,1.$ (887)

For the fixed $\\{\sigma_{j}\\}_{j=1}^{m}$, we define the indices
$\\{j_{1},j_{2},\ldots,j_{m_{1}}\\}$ such that

$\displaystyle\sigma_{j_{1}}=\sigma_{j_{2}}=\cdots=\sigma_{j_{m_{1}}}=1.$
(888)

To see how to obtain an upper bound of
$\left\|U_{s_{1}}P^{(\sigma_{1})}_{0,s_{1}}\cdots
U_{s_{m-1}}P^{(\sigma_{m-1})}_{s_{m-2},s_{m-1}}U_{s_{m}}P^{(\sigma_{m})}_{s_{m-1},s_{m}}\rho_{0}\right\|_{1}$
using Eq. (887), let us consider a simple case of $m=4$ and the norm
$\left\|U_{s_{1}}P^{(0)}_{0,s_{1}}U_{s_{2}}P^{(1)}_{s_{1},s_{2}}U_{s_{3}}P^{(0)}_{s_{2},s_{3}}U_{s_{4}}P^{(1)}_{s_{3},s_{4}}\rho_{0}\right\|_{1}$
($\sigma_{1}=\sigma_{3}=0$ and $\sigma_{2}=\sigma_{4}=1$). We first obtain

$\displaystyle\left\|U_{s_{1}}P^{(0)}_{0,s_{1}}U_{s_{2}}P^{(1)}_{s_{1},s_{2}}U_{s_{3}}P^{(0)}_{s_{2},s_{3}}U_{s_{4}}P^{(1)}_{s_{3},s_{4}}\rho_{0}\right\|_{1}=\left\|\sum_{\theta_{2},\theta_{4}}U_{s_{1}}P^{(0)}_{0,s_{1}}\tilde{P}_{s_{1},s_{2}}^{(1,\theta_{2})}U_{s_{2}}P^{(1)}_{s_{1},s_{2}}U_{s_{3}}P^{(0)}_{s_{2},s_{3}}\tilde{P}_{s_{3},s_{4}}^{(1,\theta_{4})}U_{s_{4}}P^{(1)}_{s_{3},s_{4}}\rho_{0}\right\|_{1}.$
(889)

Then, because the Hilbert space has been restricted by the projections
$\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}$ and $\Pi_{\tilde{X}_{0},\leq Q}$ [see
Eq. (828)], we have the following two inequalities:

$\displaystyle\left\|U_{s_{1}}P^{(0)}_{0,s_{1}}\tilde{P}_{s_{1},s_{2}}^{(1,\theta_{2})}U_{s_{2}}P^{(1)}_{s_{1},s_{2}}U_{s_{3}}P^{(0)}_{s_{2},s_{3}}\tilde{P}_{s_{3},s_{4}}^{(1,\theta_{4})}U_{s_{4}}P^{(1)}_{s_{3},s_{4}}\rho_{0}\right\|_{1}$
$\displaystyle=\left\|\Pi_{\tilde{X}_{0},\leq
Q}\tilde{P}_{s_{1},s_{2}}^{(1,\theta_{2})}\tilde{P}_{s_{3},s_{4}}^{(1,\theta_{4})}U_{s_{1}}P^{(0)}_{0,s_{1}}P^{(1)}_{s_{1},s_{2}}U_{s_{2}}U_{s_{3}}P^{(0)}_{s_{2},s_{3}}U_{s_{4}}P^{(1)}_{s_{3},s_{4}}\rho_{0}\right\|_{1}$
$\displaystyle\leq\left\|\Pi_{\tilde{X}_{0},\leq
Q}\tilde{P}_{s_{1},s_{2}}^{(1,\theta_{2})}\tilde{P}_{s_{3},s_{4}}^{(1,\theta_{4})}\right\|\cdot\left\|U_{s_{1}}P^{(0)}_{0,s_{1}}P^{(1)}_{s_{1},s_{2}}U_{s_{2}}U_{s_{3}}P^{(0)}_{s_{2},s_{3}}U_{s_{4}}P^{(1)}_{s_{3},s_{4}}\rho_{0}\right\|_{1}\leq\left\|\Pi_{\tilde{X}_{0},\leq
Q}\tilde{P}_{s_{1},s_{2}}^{(1,\theta_{2})}\tilde{P}_{s_{3},s_{4}}^{(1,\theta_{4})}\right\|$
(890)

and

$\displaystyle\left\|U_{s_{1}}P^{(0)}_{0,s_{1}}\tilde{P}_{s_{1},s_{2}}^{(1,\theta_{2})}U_{s_{2}}P^{(1)}_{s_{1},s_{2}}U_{s_{3}}P^{(0)}_{s_{2},s_{3}}\tilde{P}_{s_{3},s_{4}}^{(1,\theta_{4})}U_{s_{4}}P^{(1)}_{s_{3},s_{4}}\rho_{0}\right\|_{1}$
$\displaystyle\leq\left\|U_{s_{1}}P^{(0)}_{0,s_{1}}\right\|\cdot\left\|\tilde{P}_{s_{1},s_{2}}^{(1,\theta_{2})}U_{s_{2}}P^{(1)}_{s_{1},s_{2}}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}\right\|\cdot\left\|U_{s_{3}}P^{(0)}_{s_{2},s_{3}}\right\|\cdot\left\|\tilde{P}_{s_{3},s_{4}}^{(1,\theta_{4})}U_{s_{4}}P^{(1)}_{s_{3},s_{4}}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}\right\|$
$\displaystyle\leq\left\|\tilde{P}_{s_{1},s_{2}}^{(1,\theta_{2})}U_{s_{2}}P^{(1)}_{s_{1},s_{2}}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}\right\|\cdot\left\|\tilde{P}_{s_{3},s_{4}}^{(1,\theta_{4})}U_{s_{4}}P^{(1)}_{s_{3},s_{4}}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}\right\|,$
(891)

where we use the commutation relation (887) in the first inequality. We then
obtain

$\displaystyle\left\|U_{s_{1}}P^{(0)}_{0,s_{1}}U_{s_{2}}P^{(1)}_{s_{1},s_{2}}U_{s_{3}}P^{(0)}_{s_{2},s_{3}}U_{s_{4}}P^{(1)}_{s_{3},s_{4}}\rho_{0}\right\|_{1}$
$\displaystyle\leq\sum_{\theta_{2},\theta_{4}}\min\left(\left\|\Pi_{\tilde{X}_{0},\leq
Q}\tilde{P}_{s_{1},s_{2}}^{(1,\theta_{2})}\tilde{P}_{s_{3},s_{4}}^{(1,\theta_{4})}\right\|,\left\|\tilde{P}_{s_{1},s_{2}}^{(1,\theta_{2})}U_{s_{2}}P^{(1)}_{s_{1},s_{2}}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}\right\|\cdot\left\|\tilde{P}_{s_{3},s_{4}}^{(1,\theta_{4})}U_{s_{4}}P^{(1)}_{s_{3},s_{4}}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}\right\|\right).$

By generalizing the above inequality, we can derive

$\displaystyle\left\|U_{s_{1}}P^{(\sigma_{1})}_{0,s_{1}}\cdots
U_{s_{m-1}}P^{(\sigma_{m-1})}_{s_{m-2},s_{m-1}}U_{s_{m}}P^{(\sigma_{m})}_{s_{m-1},s_{m}}\rho_{0}\right\|_{1}$
$\displaystyle\leq\sum_{\theta_{j_{1}},\theta_{j_{2}}\cdots\theta_{j_{m_{1}}}=0,1}\min\left(\left\|\prod_{p=1}^{m_{1}}\tilde{P}_{s_{j_{p}-1},s_{j_{p}}}^{(1,\theta_{j_{p}})}\Pi_{\tilde{X}_{0},\leq
Q}\right\|,\prod_{p=1}^{m_{1}}\left\|\tilde{P}_{s_{j_{p}-1},s_{j_{p}}}^{(1,\theta_{j_{p}})}U_{s_{j_{p}}}P^{(1)}_{s_{j_{p}-1},s_{j_{p}}}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}\right\|\right).$
(892)

In the following, we consider the terms

$\displaystyle\left\|\prod_{p=1}^{m_{1}}\tilde{P}_{s_{j_{p}-1},s_{j_{p}}}^{(1,\theta_{j_{p}})}\Pi_{\tilde{X}_{0},\leq
Q}\right\|\quad{\rm
and}\quad\prod_{p=1}^{m_{1}}\left\|\tilde{P}_{s_{j_{p}-1},s_{j_{p}}}^{(1,\theta_{j_{p}})}U_{s_{j_{p}}}P^{(1)}_{s_{j_{p}-1},s_{j_{p}}}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}\right\|,$
(893)

separately.

We first consider the first term in (893). We here define
$Q(\theta_{1},\ldots,\theta_{m_{1}})$ as

$\displaystyle
Q(\theta_{j_{1}},\ldots,\theta_{j_{m_{1}}})=\sum_{p=1}^{m_{1}}8\theta_{j_{p}}\check{q}|L_{j_{p}}|.$
(894)

Then, from the definition (Supplementary Note 10.5.2), the produce of the
projections
$\prod_{p=1}^{m_{1}}\tilde{P}_{s_{j_{p}-1},s_{j_{p}}}^{(1,\theta_{j_{p}})}$
ensures at least more than $Q(\theta_{1},\ldots,\theta_{m_{1}})$ bosons in the
region $X_{s_{m}}\setminus X_{0}[\bar{\ell}_{t,R}+1]\subseteq\tilde{X}_{0}$.
Therefore, we have

$\displaystyle\prod_{p=1}^{m_{1}}\tilde{P}_{s_{j_{p}-1},s_{j_{p}}}^{(1,\theta_{j_{p}})}\Pi_{\tilde{X}_{0},\leq
Q(\theta_{1},\ldots,\theta_{m_{1}})}=0,$ (895)

which yields the equation of

$\displaystyle\left\|\prod_{p=1}^{m_{1}}\tilde{P}_{s_{j_{p}-1},s_{j_{p}}}^{(1,\theta_{j_{p}})}\Pi_{\tilde{X}_{0},\leq
Q}\right\|=0\quad\textrm{for}\quad Q(\theta_{1},\ldots,\theta_{m_{1}})\geq Q.$
(896)

Supplementary Figure 30: Estimation of the norm (897), i.e.,
$||\tilde{P}_{s_{j-1},s_{j}}^{(1,0)}U_{s_{j}}P^{(1)}_{s_{j-1},s_{j}}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}||$.
The setup is similar to the one in Proposition 22. We consider the setup that
a quantum state initially has more than $16\check{q}|L_{j}|$ bosons in the
region $L_{j}$. Then, the unitary operator $U_{s_{j}}$ induces the outflow of
the bosons from the region $L_{j}$. Proposition 22 provides the probability of
the number of bosons that still remain in the region $X_{s_{j}}\setminus
X_{s_{j-1}}$. From the statement, we can ensure that the probability to
observe less than $8\check{q}|L_{j}|$ bosons is smaller than
$e^{-\Theta(\check{q}|L_{j}|)}$ as in the inequality (899).

We next consider the second term in (893). In the estimation, we can apply
Proposition 22 to the norm of (see Fig. 30)

$\displaystyle\left\|\tilde{P}_{s_{j-1},s_{j}}^{(1,0)}U_{s_{j}}P^{(1)}_{s_{j-1},s_{j}}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}\right\|.$
(897)

For an arbitrary quantum state $|\psi\rangle$, the quantum state
$|\tilde{\psi}\rangle=P^{(1)}_{s_{j-1},s_{j}}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}|\psi\rangle$
satisfies

$\displaystyle\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}|\tilde{\psi}\rangle=|\tilde{\psi}\rangle,\quad\Pi_{L_{j},>16\check{q}|L_{j}|}|\tilde{\psi}\rangle=|\tilde{\psi}\rangle.$
(898)

By using the inequality (359) with

$N\to 16\check{q}|L_{j}|,\quad q_{\ast}\to N_{0},\quad\delta N\to
8\check{q}|L_{j}|,\quad X\to L_{j},\quad\ell\to\frac{(s_{j}-s_{j-1})\Delta
r}{2},$

we obtain for the time evolution $U_{s_{j}}=e^{-iH_{X_{s_{j}}}\tau}$

$\displaystyle\left\|\Pi_{X_{j}\setminus X_{j-1},\leq
8\check{q}|L_{j}|}U_{s_{j}}|\tilde{\psi}\rangle\right\|^{2}\leq\frac{2}{1-e^{-1/(8ec_{\tau,2})}}\exp\left[\frac{-8\check{q}|L_{j}|}{8ec_{\tau,2}}+3c_{\ell}^{\prime}\left(1+c_{\ell}^{\prime}D\right)\log^{2}\left(N_{0}\gamma|L_{j}|\right)\right]$
(899)

as long as the following condition from Eq. (360) is satisfied:

$\displaystyle\frac{(s_{j}-s_{j-1})\Delta
r}{2}\geq\left\lceil\frac{2}{\tilde{f}_{\tau}}\log\left(\frac{5c_{\tau,1}\lambda_{1/2}e^{2\tilde{f}_{\tau}}\gamma(D/\tilde{f}_{\tau})^{D}N_{0}|L_{j}|}{2c_{\tau,2}}\right)\right\rceil=c^{\prime}_{\ell}\log\left(N_{0}\gamma|L_{j}|\right),$
$\displaystyle\tilde{f}_{\tau}\coloneqq\frac{1}{2}\log\left(1+\frac{1}{10ec_{\tau,1}\lambda_{1/2}+2}\right).$
(900)

Because of the condition (839), i.e., $\Delta
r=\Theta[\log^{1+\kappa}(N_{0})]$, the condition (900) is satisfied.

By applying the inequality (899) to the norm (897), we have

$\displaystyle\left\|\tilde{P}_{s_{j-1},s_{j}}^{(1,0)}U_{s_{j}}P^{(1)}_{s_{j-1},s_{j}}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}\right\|=\sup_{\psi}\left\|\Pi_{X_{j}\setminus
X_{j-1},\leq 8\check{q}|L_{j}|}U_{s_{j}}|\tilde{\psi}\rangle\right\|\leq
e^{-\Theta(1)\check{q}|L_{j}|}.$ (901)

On the other hand, for $\theta_{j}=1$, we only obtain the trivial inequality
of

$\displaystyle\left\|\tilde{P}_{s_{j-1},s_{j}}^{(1,\theta_{j})}U_{s_{j}}P^{(1)}_{s_{j-1},s_{j}}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}\right\|\leq
1\quad\textrm{for}\quad\theta_{j}=1.$ (902)

By applying the inequalities (901) and (902) to the second term in (893), we
obtain the following upper bound:

$\displaystyle\prod_{p=1}^{m_{1}}\left\|\tilde{P}_{s_{j_{p}-1},s_{j_{p}}}^{(1,\theta_{j_{p}})}U_{s_{j_{p}}}P^{(1)}_{s_{j_{p}-1},s_{j_{p}}}\Pi_{\mathcal{S}(i_{0}[R],1/2,N_{0})}\right\|\leq\exp\left[-\Theta(1)\sum_{p=1}^{m_{1}}\left(1-\theta_{j_{p}}\right)\check{q}|L_{j_{p}}|\right]$
$\displaystyle\leq\exp\left[-\Theta(1)\left(\frac{M_{\mathbb{s}_{m},\bm{\sigma}_{m}}\check{q}\Delta
r}{2}-\frac{Q(\theta_{1},\ldots,\theta_{m_{1}})}{8}\right)\right],$ (903)

where we use the definition (894) and the inequality of

$\displaystyle\sum_{p=1}^{m_{1}}|L_{j_{p}}|=\frac{\Delta
r}{2}\sum_{p=1}^{m_{1}}(s_{j_{p}}-s_{j_{p}-1})=\frac{\Delta
r}{2}\sum_{j=1}^{m}\sigma_{j}(s_{j}-s_{j-1})=\frac{M_{\mathbb{s}_{m},\bm{\sigma}_{m}}\Delta
r}{2}.$ (904)

Note that we have defined as in Eqs. (888) and (875).

By combining the inequalities (896) and (903), we reduce the inequality
(Supplementary Note 10.5.2) to

$\displaystyle\left\|U_{s_{1}}P^{(\sigma_{1})}_{0,s_{1}}\cdots
U_{s_{m-1}}P^{(\sigma_{m-1})}_{s_{m-2},s_{m-1}}U_{s_{m}}P^{(\sigma_{m})}_{s_{m-1},s_{m}}\rho_{0}\right\|_{1}$
$\displaystyle\leq\sum_{\theta_{j_{1}},\theta_{j_{2}}\cdots\theta_{j_{m_{1}}}=0,1}\exp\left[-\Theta(1)\left(\frac{M_{\mathbb{s}_{m},\bm{\sigma}_{m}}\check{q}\Delta
r}{2}-\frac{Q}{8}\right)\right]\leq
2^{m}\exp\left[-\Theta(1)\left(\frac{M_{\mathbb{s}_{m},\bm{\sigma}_{m}}\check{q}\Delta
r}{2}-\frac{Q}{8}\right)\right],$ (905)

where we use
$\sum_{\theta_{j_{1}},\theta_{j_{2}}\cdots\theta_{j_{m_{1}}}=0,1}1\leq
2^{m_{1}}\leq 2^{m}$. We thus prove the inequality (883). This completes the
proof. $\square$

[ End of Proof of Lemma 51]

###  Supplementary Note 10.6 Proof of Proposition 48: completing the proof

We have finally obtained all the ingredients to prove Proposition 48. By
combining Corollary 50 and Lemma 51, we obtain

$\displaystyle\left\|\Delta
O_{\mathbb{s}_{m},\bm{\sigma}_{m}}\rho_{0}\right\|_{1}$
$\displaystyle\leq\zeta_{0}\min\left(2^{2m}\exp\left[-\Theta(1)\left(\frac{M_{\mathbb{s}_{m},\bm{\sigma}_{m}}\check{q}\Delta
r}{2}-\frac{Q}{8}\right)\right],\left(2\lambda_{Q}\delta_{\Delta
r}^{-2}\right)^{m}\delta_{\Delta
r}^{s_{m}-M_{\mathbb{s}_{m},\bm{\sigma}_{m}}}\right)$
$\displaystyle\leq\zeta_{0}\max\left(2^{2m}\exp\left[-\Theta(1)\left(\frac{\check{q}s_{m}\Delta
r}{4}-\frac{\check{q}R^{\prime\prime}}{8}\right)\right],\left(2\lambda_{Q}\delta_{\Delta
r}^{-2}\right)^{m}\delta_{\Delta r}^{s_{m}/2}\right),$ (906)

where we use $Q=\check{q}R^{\prime\prime}$ as in Eq. (823). Note that the
condition $0\leq M_{\mathbb{s}_{m},\bm{\sigma}_{m}}\leq s_{m}$ from the
definition (875) implies that either $M_{\mathbb{s}_{m},\bm{\sigma}_{m}}$ or
$(s_{m}-M_{\mathbb{s}_{m},\bm{\sigma}_{m}})$ is greater than $s_{m}/2$.

We now consider the case of $m=\bar{m}$ and $\mathbb{s}_{\bar{m}}\in
V_{\bar{s}+1}$ [see (861)]. Then, we can replace $s_{m}$ with $\bar{s}$ in the
inequality (Supplementary Note 10.6). Also, because of the definition of
$\Delta r$ in Eq. (833), we have

$\displaystyle\Delta
r\geq\frac{R^{\prime\prime}}{\bar{s}}-1\longrightarrow\bar{s}\Delta r\geq
R^{\prime\prime}-\bar{s}.$ (907)

Then, the above inequality reduces the inequality (Supplementary Note 10.6) to

$\displaystyle\left\|\Delta
O_{\mathbb{s}_{\bar{m}},\bm{\sigma}_{\bar{m}}}\rho_{0}\right\|_{1}$
$\displaystyle\leq\zeta_{0}\max\left(2^{2\bar{m}}\exp\left[\Theta(1)\left(\frac{-\check{q}R^{\prime\prime}+2\check{q}\bar{s}}{8}\right)\right],\left(2\lambda_{Q}\delta_{\Delta
r}^{-2}\right)^{\bar{m}}\delta_{\Delta r}^{\bar{s}/2}\right)$
$\displaystyle=\zeta_{0}\left(2\lambda_{Q}\delta_{\Delta
r}^{-2}\right)^{\bar{m}}\delta_{\Delta r}^{\bar{s}/2}$ (908)

where $\left(2\lambda_{Q}\delta_{\Delta r}^{-2}\right)^{\bar{m}}\delta_{\Delta
r}^{\bar{s}/2}$ is larger than $e^{-\Theta(\check{q}R^{\prime\prime})}$
because of $\delta_{\Delta r}^{\bar{s}/2}=e^{-\Theta(R^{\prime\prime})}$ from
the definition (865), i.e., $\delta_{\Delta r}=e^{-\Theta(\Delta r)}$. By
applying the above inequality to (861), we obtain

$\displaystyle\left\|\left[O_{X_{0}}(t)-\bar{O}_{X_{\bar{s}}}\right]\rho_{0}\right\|_{1}$
$\displaystyle\leq\sum_{\bm{\sigma}_{\bar{m}}\in\\{0,1\\}^{\otimes\bar{m}}}\sum_{\mathbb{s}_{\bar{m}}\in
V_{\bar{s}+1}}\zeta_{0}\left(2\lambda_{Q}\delta_{\Delta
r}^{-2}\right)^{\bar{m}}\delta_{\Delta r}^{\bar{s}/2}$
$\displaystyle\leq\zeta_{0}\left[4(\bar{s}+1)\lambda_{Q}\delta_{\Delta
r}^{-2}\right]^{\bar{m}}\delta_{\Delta r}^{\bar{s}/2}.$ (909)

Finally, by choosing $\bar{s}=8\bar{m}$ in the inequality (Supplementary Note
10.6), we arrive at the desired inequality of

$\displaystyle\left\|\left[O_{X_{0}}(t)-\bar{O}_{X_{\bar{s}}}\right]\rho_{0}\right\|_{1}$
$\displaystyle\leq\zeta_{0}\exp\left\\{\bar{m}\log\left[4(\bar{s}+1)\lambda_{Q}\right]-2\bar{m}\log(\delta_{\Delta
r})\right\\}=\zeta_{0}e^{\Theta\left[t\log(N_{0})\right]-\Theta(R^{\prime\prime})},$
(910)

where we use $\log(\lambda_{Q})=\Theta[\log(N_{0})]$ from Eq. (865) and
$\bar{m}\log(\delta_{\Delta r})=\bar{m}\Theta(\Delta
r)=\Theta(R^{\prime\prime})$. This completes the proof of Proposition 48.
$\square$

##  Supplementary Note 11 Possible Scenario for Acceleration by Time-
Independent Hamiltonian

Finally, we discuss the possibility of realizing the acceleration of
information propagation through more natural Hamiltonian dynamics. In the
protocol of the main manuscript, we employed a highly artificial Hamiltonian
by fine-tuning local interactions. Therefore, considering the standard Bose-
Hubbard Hamiltonian with a simple initial state (e.g., the Mott state), it is
natural to expect that the linear light cone should be established, and
acceleration no longer occurs. Additionally, even if a 1D information path
with high boson densities is realized, it should not persist for a long time
under Hamiltonian dynamics with repulsive interactions. However, these simple
intuitions are extremely challenging to make mathematically
rigorous*14*14*14Indeed, even for the homogeneous Bose-Hubbard model with the
Mott state as the initial state, there have been no mathematical tools to
treat information propagation in general..

Here, we demonstrate that a special choice of the initial state can induce the
acceleration of information propagation. To illustrate this, we first
highlight the essence of acceleration, summarized by the following two points:
i) concentrating bosons on a particular one-dimensional region, ii) the speed
of information propagation is proportional to local boson numbers (see Ref.
[11]). While the first step is unlikely in natural dynamics, using reverse
time evolution allows us to consider a situation where bosons automatically
concentrate on a one-dimensional region (Fig. 31):

  1. 1.

We first consider a quantum state $|\psi_{0}\rangle$ that has an information
path with high boson densities on a 1D region.

  2. 2.

We then evolve the state by a translation-invariant Bose-Hubbard Hamiltonian,
resulting in a steady-state $e^{-iHt}|\psi_{0}\rangle$ after a long time.

  3. 3.

We choose the state $e^{-iHt}|\psi_{0}\rangle$ as our initial state of
interest $|\psi_{\rm ini}\rangle$, which is expected to satisfy the low-boson
density condition.

  4. 4.

Considering the Hamiltonian $-H$, the time evolution $e^{iHt}|\psi_{\rm
ini}\rangle$ yields boson concentration onto the one-dimensional path.

Importantly, the initial state $|\psi_{\rm ini}\rangle$ is quite rare in the
whole phase space, but we always have to exclude the possibility that the
initial state of interest is not included in this class of states. In the
above situation, it might be possible to observe the acceleration of
information propagation in the above setup.

At this stage, the computational cost for the numerical simulation of the
above dynamics is quite substantial. However, it remains an intriguing open
question to identify whether we can observe the acceleration of information
propagation under the above dynamics or other natural situations.

Supplementary Figure 31: In process (a), the bosons diffuse, and the state
transitions to a steady state with low-boson density. By reversing the time
evolution, we induce boson concentration (b). During this process, the
acceleration of information propagation can occur.

## Supplementary References

  * Kuwahara and Saito [2021a] T. Kuwahara and K. Saito, Lieb-Robinson Bound and Almost-Linear Light Cone in Interacting Boson Systems, Phys. Rev. Lett. 127, 070403 (2021a).
  * Schuch  _et al._ [2011] N. Schuch, S. K. Harrison, T. J. Osborne, and J. Eisert, Information propagation for interacting-particle systems, Phys. Rev. A 84, 032309 (2011).
  * Finner [1992] H. Finner, A Generalization of Holder’s Inequality and Some Probability Inequalities, The Annals of Probability 20, 1893 (1992), full publication date: Oct., 1992.
  * Sutter [2018] D. Sutter, Approximate quantum Markov chains, arXiv preprint arXiv:1802.05477 (2018), arXiv:1802.05477 .
  * Kuwahara [2015] T. Kuwahara, Fundamental inequalities in quantum many-body systems, PhD thesis, University of Tokyo (2015).
  * Hastings and Koma [2006] M. B. Hastings and T. Koma, Spectral Gap and Exponential Decay of Correlations, Communications in Mathematical Physics 265, 781 (2006).
  * Nachtergaele  _et al._ [2006] B. Nachtergaele, Y. Ogata, and R. Sims, Propagation of Correlations in Quantum Lattice Systems, Journal of Statistical Physics 124, 1 (2006).
  * Kuwahara [2016] T. Kuwahara, Exponential bound on information spreading induced by quantum many-body dynamics with long-range interactions, New Journal of Physics 18, 053034 (2016).
  * Kuwahara and Saito [2021b] T. Kuwahara and K. Saito, Absence of Fast Scrambling in Thermodynamically Stable Long-Range Interacting Systems, Phys. Rev. Lett. 126, 030604 (2021b).
  * Yin and Lucas [2022] C. Yin and A. Lucas, Finite Speed of Quantum Information in Models of Interacting Bosons at Finite Density, Phys. Rev. X 12, 021039 (2022).
  * Barmettler  _et al._ [2012] P. Barmettler, D. Poletti, M. Cheneau, and C. Kollath, Propagation front of correlations in an interacting Bose gas, Phys. Rev. A 85, 053625 (2012).