tr}\left(\hat{n}_{X}(\tau)^{s}\rho\right)\leq\min_{X^{\prime}:X^{\prime}\supseteq
X}{\rm tr}\left(\hat{n}_{X^{\prime}}(\tau)^{s}\rho\right).$ (271)

However, it is not enough to control a single parameter $X^{\prime}$ in
deriving the desired bound below. In deriving a meaningful bound for higher
order moments, the key idea here is to utilize the multi-parameter
optimization by using the inequality (72):

$\displaystyle{\rm
tr}\left(\hat{n}_{X}(\tau)^{s}\rho\right)\leq\min_{X_{1}:X_{1}\supseteq
X}\min_{X_{2}:X_{2}\supseteq X}\cdots\min_{X_{s}:X_{s}\supseteq X}{\rm
tr}\left[\hat{n}_{X_{1}}(\tau)\hat{n}_{X_{2}}(\tau)\cdots\hat{n}_{X_{s}}(\tau)\rho\right].$
(272)

Also, in Subtheorem 1, we have already obtained a general upper bound for
$M_{X_{1},X_{2},\ldots,X_{s}}(\tau)$.

Then, by solving the multi-parameter optimization problem (272), we can prove
the following proposition:

######  Proposition 18.

We here adopt the same setup as that in Proposition 16. Then, we obtain

$\displaystyle[\hat{n}_{X}(\tau)]^{s}$
$\displaystyle\preceq\left[\left(1+5c_{\tau,1}\lambda_{1/2}e^{-f_{\tau}\ell}\right)\hat{n}_{X[\ell]}+c_{\tau,1}e^{-3\ell/16}\hat{\mathscr{D}}_{X[\ell]}+c_{\tau,2}s\right]^{s},$
(273)

where $\hat{\mathscr{D}}_{X[\ell]}$ and $f_{\tau}$ have been defined in Eqs.
(242) and (243).

####  Supplementary Note 3.5.3 Proof of Proposition 18

Without loss of generality, we can assume that $\rho$ is given by a pure
state, i.e., $\rho=|\psi\rangle\langle\psi|$. We note that $O_{1}\preceq
O_{2}$ is derived from
$\langle\psi|O_{1}|\psi\rangle\preceq\langle\psi|O_{2}|\psi\rangle$ for all
the quantum states. Then, by using Subtheorem 1, the multi-parameter
optimization (272) reduces to

$\displaystyle\langle\psi|\hat{n}_{X}(\tau)^{s}|\psi\rangle$
$\displaystyle\leq\min_{X_{1}:X_{1}\supseteq X}\min_{X_{2}:X_{2}\supseteq
X}\cdots\min_{X_{s}:X_{s}\supseteq
X}\langle\psi|\hat{n}_{X_{1}}(\tau)\hat{n}_{X_{2}}(\tau)\cdots\hat{n}_{X_{s}}(\tau)|\psi\rangle$
$\displaystyle\leq\min_{X_{1}:X_{1}\supseteq X}\min_{X_{2}:X_{2}\supseteq
X}\cdots\min_{X_{s}:X_{s}\supseteq
X}\left\langle\psi\right|\prod_{j=1}^{s}\left(\hat{n}_{X_{j}}+c_{\tau,1}\hat{\mathcal{D}}_{X_{j}}+c_{\tau,2}s\right)\left|\psi\right\rangle$
$\displaystyle=\min_{X_{1}:X_{1}\supseteq X}\min_{X_{2}:X_{2}\supseteq
X}\cdots\min_{X_{s}:X_{s}\supseteq
X}\left\langle\psi\right|\prod_{j=1}^{s}\hat{n}_{\tau,X_{j}}\left|\psi\right\rangle,$
(274)

where in the last equation we denote
$\hat{n}_{X_{j}}+c_{\tau,1}\hat{\mathcal{D}}_{X_{j}}+c_{\tau,2}s$ by
$\hat{n}_{\tau,X_{j}}$. We note that each of $\\{\hat{n}_{\tau,X_{j}}\\}$
commutes with $\hat{n}_{i}$ for $\forall i\in\Lambda$.

We define the following unnormalized quantum states $|\psi_{X_{1:j}}\rangle$
($1\leq j\leq s$):

$\displaystyle|\psi_{X_{1:j-1}}\rangle:=\left(\hat{n}_{\tau,X_{1}}\hat{n}_{\tau,X_{2}}\cdots\hat{n}_{\tau,X_{j-1}}\hat{\mathcal{N}}_{\tau,X[\ell]}^{s-j}\right)^{1/2}|\psi\rangle,$
$\displaystyle\hat{\mathcal{N}}_{\tau,X[\ell]}:=\left(1+5c_{\tau,1}\lambda_{1/2}e^{-f_{\tau}\ell}\right)\hat{n}_{X[\ell]}+c_{\tau,1}e^{-3\ell/16}\hat{\mathscr{D}}_{X[\ell]}+c_{\tau,2}s.$
(275)

We then rewrite the optimization problem in (Supplementary Note 3.5.3) as

$\displaystyle\min_{X_{1}:X_{1}\supseteq X}\min_{X_{2}:X_{2}\supseteq
X}\cdots\min_{X_{s}:X_{s}\supseteq
X}\left\langle\psi\right|\prod_{j=1}^{s}\hat{n}_{\tau,X_{j}}\left|\psi\right\rangle=\min_{X_{1}:X_{1}\supseteq
X}\min_{X_{2}:X_{2}\supseteq X}\cdots\min_{X_{s}:X_{s}\supseteq
X}\langle\psi_{X_{1:s-1}}|\hat{n}_{\tau,X_{s}}|\psi_{X_{1:s-1}}\rangle.$ (276)

We can apply Proposition 16 to
$\langle\psi_{X_{1:s-1}}|\hat{n}_{\tau,X_{s}}|\psi_{X_{1:s-1}}\rangle$, which
yields

$\displaystyle\min_{X_{s}:X_{s}\supseteq
X}\langle\psi_{X_{1:s-1}}|\hat{n}_{\tau,X_{s}}|\psi_{X_{1:s-1}}\rangle\leq\langle\psi_{X_{1:s-1}}|\hat{\mathcal{N}}_{\tau,X[\ell]}|\psi_{X_{1:s-1}}\rangle$
(277)

for arbitrary choices of $\\{X_{j}\\}_{j=1}^{s-1}$ in
$|\psi_{X_{1:s-1}}\rangle$. We then reduce Eq. (276) to

$\displaystyle\min_{X_{1}:X_{1}\supseteq X}\min_{X_{2}:X_{2}\supseteq
X}\cdots\min_{X_{s}:X_{s}\supseteq
X}\left\langle\psi\right|\prod_{j=1}^{s}\hat{n}_{\tau,X_{j}}\left|\psi\right\rangle$
$\displaystyle\leq\min_{X_{1}:X_{1}\supseteq X}\min_{X_{2}:X_{2}\supseteq
X}\cdots\min_{X_{s-1}:X_{s-1}\supseteq
X}\langle\psi_{X_{1:s-1}}|\hat{\mathcal{N}}_{\tau,X[\ell]}|\psi_{X_{1:s-1}}\rangle$
$\displaystyle=\min_{X_{1}:X_{1}\supseteq X}\min_{X_{2}:X_{2}\supseteq
X}\cdots\min_{X_{s-1}:X_{s-1}\supseteq
X}\langle\psi_{X_{1:s-2}}|\hat{n}_{\tau,X_{s-1}}|\psi_{X_{1:s-2}}\rangle.$
(278)

In the same way, we apply Proposition 16 to
$\langle\psi_{X_{1:s-2}}|\hat{n}_{\tau,X_{s-1}}|\psi_{X_{1:s-2}}\rangle$,
which further upper-bounds the RHS of the above inequality as

$\displaystyle\min_{X_{1}:X_{1}\supseteq X}\min_{X_{2}:X_{2}\supseteq
X}\cdots\min_{X_{s}:X_{s}\supseteq
X}\left\langle\psi\right|\prod_{j=1}^{s}\hat{n}_{\tau,X_{j}}\left|\psi\right\rangle$
$\displaystyle\leq\min_{X_{1}:X_{1}\supseteq X}\min_{X_{2}:X_{2}\supseteq
X}\cdots\min_{X_{s-2}:X_{s-2}\supseteq
X}\langle\psi_{X_{1:s-2}}|\hat{\mathcal{N}}_{\tau,X[\ell]}|\psi_{X_{1:s-2}}\rangle$
$\displaystyle=\min_{X_{1}:X_{1}\supseteq X}\min_{X_{2}:X_{2}\supseteq
X}\cdots\min_{X_{s-2}:X_{s-2}\supseteq
X}\langle\psi_{X_{1:s-3}}|\hat{n}_{\tau,X_{s-2}}|\psi_{X_{1:s-3}}\rangle.$
(279)

By repeating the same process, we finally obtain

$\displaystyle\min_{X_{1}:X_{1}\supseteq X}\min_{X_{2}:X_{2}\supseteq
X}\cdots\min_{X_{s}:X_{s}\supseteq
X}\left\langle\psi\right|\prod_{j=1}^{s}\hat{n}_{\tau,X_{j}}\left|\psi\right\rangle\leq\langle\psi_{X_{1:0}}|\hat{\mathcal{N}}_{\tau,X[\ell]}|\psi_{X_{1:0}}\rangle=\langle\psi|\hat{\mathcal{N}}_{\tau,X[\ell]}^{s}|\psi\rangle.$
(280)

By applying the definition (275) to the above inequality, we prove the main
inequality (273). This completes the proof. $\square$

##  Supplementary Note 4 Main theorem on the boson concentration

In this section, we consider a boson density after a time evolution for an
arbitrary time $t$. For this purpose, we connect the short-time evolution by
iteratively applying Proposition 18. Our goal is to prove that the boson
number operator in a region $X$ is upper-bounded by a number operator on an
extended region $X[R]$ after a time evolution $e^{-iHt}$, where the length $R$
is almost linear with the time $t$ (see Fig. 5).

Supplementary Figure 5: Schematic picture of our setup for Theorem 1. We
consider the boson number operator $\hat{n}_{X}$ in a region $X$. After a time
evolution, $\hat{n}_{X}(t)$ is upper-bounded by the boson number
$\hat{n}_{X[R]}$ on an extended subset $X[R]$, where the error is given by
$e^{-\Omega(R/t)}$. The obtained inequality is given in the form of the
operator inequality [see (282)], and hence it can be applied to an arbitrary
initial state.

We here prove the following theorem:

######  Theorem 1.

Let $\tau$ be the maximum number such that $\tau\leq 1/(4\gamma\bar{J})$ and
$t/\tau$ becomes integer. Also, for $\ell=\left\lfloor
R/(t/\tau)\right\rfloor=\Theta(R/t)$, we assume

$\displaystyle
e^{-f_{\tau}\ell}\leq\frac{\tau}{5c_{\tau,1}\lambda_{1/2}t}\longrightarrow\ell\geq\frac{1}{f_{\tau}}\log\left(\frac{5c_{\tau,1}\lambda_{1/2}t}{\tau}\right),$
(281)

where $f_{\tau}$ has been defined in Eq. (243). We then upper-bound
$[\hat{n}_{X}(t)]^{s}$ in the form of

$\displaystyle[\hat{n}_{X}(t)]^{s}$
$\displaystyle\preceq\left[\left(1+\frac{3t}{\tau}\delta_{\ell}\right)\hat{n}_{X[R]}+2c_{\tau,1}e^{-3\ell/16}\hat{\mathscr{D}}_{X[R]}+\left(\frac{c_{\tau,2}t}{\tau}+\tilde{\delta}_{\ell}\right)s\right]^{s}$
(282)

with $\hat{\mathscr{D}}_{X[R]}=\sum_{j\in X[R]^{\rm
c}}e^{-3d_{j,X[R]}/4}\hat{n}_{j}$ as in Eq. (242) and

$\displaystyle\delta_{\ell}:=5c_{\tau,1}\lambda_{1/2}e^{-f_{\tau}\ell},\quad\tilde{\delta}_{\ell}:=\frac{c_{\tau,2}t}{\tau}\left(\frac{3t\delta_{\ell}}{\tau}+2c_{\tau,1}\lambda_{3/4}e^{-3\ell/16}|\partial
X[R]|\right).$ (283)

Remark. Because of the condition (281) and $\ell\propto R/t$, the length $R$
should be as large as

$\displaystyle R\gtrsim t\log(t).$ (284)

Then, because of $\delta_{\ell}=e^{-\Theta(\ell)}$ and
$\tilde{\delta}_{\ell}=e^{-\Theta(\ell)}$, we obtain

$\displaystyle[\hat{n}_{X}(t)]^{s}$
$\displaystyle\preceq\left[\hat{n}_{X[R]}+e^{-\Theta(R/t)}\left(\hat{n}_{X[R]}+\hat{\mathscr{D}}_{X[R]}\right)+\Theta(st)\right]^{s},$
(285)

where we use the $\Theta$ notation in Eq. (9). Therefore, for sufficiently
large $R$, we have
$[\hat{n}_{X}(t)]^{s}\lesssim\left[\hat{n}_{X[R]}+\Theta(st)\right]^{s}$.

###  Supplementary Note 4.1 Proof of Theorem 1

For the proof, we iteratively utilize Proposition 18. We decompose the total
time $t$ into $m_{0}$ pieces:

$\displaystyle m_{0}=\frac{t}{\tau}=\mathcal{O}(t),$ (286)

where we have chosen the maximum $\tau$ such that $\tau\leq
1/(4\gamma\bar{J})$ and $t/\tau$ becomes integer. In the proof, we consider
the case that $\ell=\left\lfloor R/m_{0}\right\rfloor=R/m_{0}$, but the
generalization is straightforward. For simplicity of notation, we denote
$C_{0}$ by

$\displaystyle
C_{0}:=1+5c_{\tau,1}\lambda_{1/2}e^{-f_{\tau}\ell}=1+\delta_{\ell},\quad\delta_{\ell}:=5c_{\tau,1}\lambda_{1/2}e^{-f_{\tau}\ell}$
(287)

We then reduce the inequality (273) to

$\displaystyle[\hat{n}_{X}(\tau)]^{s}$
$\displaystyle\preceq\left[C_{0}\hat{n}_{X[\ell]}+c_{\tau,1}e^{-3\ell/16}\hat{\mathscr{D}}_{X[\ell]}+c_{\tau,2}s\right]^{s}$
(288)

for arbitrary $X\subseteq\Lambda$ and $\tau\leq 1/(4\gamma\bar{J})$.

The condition (281) immediately implies the inequality of $\delta_{\ell}\leq
1/m_{0}$ from the definition
$\delta_{\ell}:=5c_{\tau,1}\lambda_{1/2}e^{-f_{\tau}\ell}$. This yields

$\displaystyle C_{0}^{m}=\left(1+\delta_{\ell}\right)^{m}$ $\displaystyle\leq
1+m(e-1)\delta_{\ell}\leq 1+2m\delta_{\ell}\quad\textrm{for}\quad m\leq
m_{0},$ (289)

where we use $(1+z)^{m}\leq 1+m(e-1)z$ for $z\leq 1/m$. Moreover,
$\delta_{\ell}\leq 1/m_{0}\leq 1$ implies

$\displaystyle
e^{-f_{\tau}\ell}=\left(1+\frac{1}{5c_{\tau,1}\lambda_{1/2}+2}\right)^{-\ell/2}\leq\frac{1}{5c_{\tau,1}\lambda_{1/2}}.$
(290)

Because
$\left[1+1/\left(5c_{\tau,1}\lambda_{1/2}+2\right)\right]^{1/2}\leq(8/7)^{1/2}\leq
e^{1/14}$ from $c_{\tau,1}\geq 1$ and $\lambda_{1/2}\geq 1$, we also obtain

$\displaystyle e^{-\ell/14}\leq\frac{1}{5c_{\tau,1}\lambda_{1/2}}.$ (291)

For the time-evolved operator $[\hat{n}_{X}(m\tau)]^{s}$, we prove the
following proposition:

######  Proposition 19.

For an arbitrary time $m\tau$ ($m\leq m_{0}$), we obtain the operator
inequality as

$\displaystyle[\hat{n}_{X}(m\tau)]^{s}\preceq\left[\left(1+3m\delta_{\ell}\right)\hat{n}_{X[m\ell]}+2c_{\tau,1}e^{-3\ell/16}\hat{\mathscr{D}}_{X[m\ell]}+\left(c_{\tau,2}m+\tilde{\delta}_{\ell}\right)s\right]^{s},$
(292)

where we define $\tilde{\delta}_{\ell}$ as

$\displaystyle\tilde{\delta}_{\ell}:=\frac{c_{\tau,2}t}{\tau}\left(\frac{3t\delta_{\ell}}{\tau}+2c_{\tau,1}\lambda_{3/4}e^{-3\ell/16}|\partial
X[R]|\right).$ (293)

From Proposition 19 with $m=m_{0}$ (i.e., $m\tau=m_{0}\tau=t$), we obtain

$\displaystyle[\hat{n}_{X}(t)]^{s}$
$\displaystyle\preceq\left[\left(1+3m_{0}\delta_{\ell}\right)\hat{n}_{X[m_{0}\ell]}+2c_{\tau,1}e^{-3\ell/16}\hat{\mathscr{D}}_{X[m_{0}\ell]}+\left(c_{\tau,2}m_{0}+\tilde{\delta}_{\ell}\right)s\right]^{s}$
$\displaystyle=\left[\left(1+\frac{3t}{\tau}\delta_{\ell}\right)\hat{n}_{X[R]}+2c_{\tau,1}e^{-3\ell/16}\hat{\mathscr{D}}_{X[R]}+\left(\frac{c_{\tau,2}t}{\tau}+\tilde{\delta}_{\ell}\right)s\right]^{s},$
(294)

where we use $m_{0}\ell=R$ and $m_{0}=t/\tau$. This proves the main inequality
(282). $\square$

###  Supplementary Note 4.2 Proof of Proposition 19

For the proof, we upper-bound $[\hat{n}_{X}(m\tau)]^{s}$ in the form of

$\displaystyle[\hat{n}_{X}(m\tau)]^{s}\preceq\left[C_{m,0}\hat{n}_{X[m\ell]}+C_{m,1}\hat{\mathscr{D}}_{X[m\ell]}+C_{m,2}s\right]^{s}.$
(295)

For $m=0$, we trivially obtain the inequality (295) by choosing

$\displaystyle C_{0,0}=1,\quad C_{0,1}=C_{0,2}=0.$ (296)

For $m=1$, the parameters are given by

$\displaystyle C_{1,0}=C_{0},\quad C_{0,1}=c_{\tau,1}e^{-3\ell/16},\quad
C_{0,2}=c_{\tau,2}.$ (297)

In the following, we upper-bound the parameters
$\\{C_{m+1,0},C_{m+1,1},C_{m+1,2}\\}$ under the inequality of (295) for a
fixed $m$.

We start from the time evolution of

$\displaystyle[\hat{n}_{X}(m\tau+\tau)]^{s}=e^{iH\tau}[\hat{n}_{X}(m\tau)]^{s}e^{-iH\tau}\preceq\left[C_{m,0}\hat{n}_{X[m\ell]}(\tau)+C_{m,1}\hat{\mathscr{D}}_{X[m\ell]}(\tau)+C_{m,2}s\right]^{s},$
(298)

where we use $U^{\dagger}O_{1}U\preceq U^{\dagger}O_{2}U$ for an arbitrary
unitary operator $U$ if $O_{1}\preceq O_{2}$. By using Proposition 18, we
obtain

$\displaystyle\left[C_{m,0}\hat{n}_{X[m\ell]}(\tau)+C_{m,1}\hat{\mathscr{D}}_{X[m\ell]}(\tau)+C_{m,2}s\right]^{s}\preceq\Biggl{\\{}C_{m,0}\left[C_{0}\hat{n}_{X[m\ell+\ell]}+c_{\tau,1}e^{-3\ell/16}\hat{\mathscr{D}}_{X[m\ell+\ell]}+c_{\tau,2}s\right]$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+C_{m,1}\sum_{j\in
X[m\ell]^{\rm
c}}e^{-3d_{j,X[m\ell]}/4}\left(\hat{n}_{j}+c_{\tau,1}\hat{\mathcal{D}}_{j}+c_{\tau,2}s\right)+C_{m,2}s\Biggr{\\}}^{s},$
(299)

where we apply Subtheorem 1 to $\hat{\mathscr{D}}_{X[m\ell]}(\tau)=\sum_{j\in
X[m\ell]^{\rm c}}e^{-3d_{j,X[m\ell]}/4}\hat{n}_{j}(\tau)$. First of all, we
consider the coefficient of $s$ in (299), which is given by

$\displaystyle c_{\tau,2}C_{m,0}+C_{m,2}+c_{\tau,2}C_{m,1}\sum_{j\in
X[m\ell]^{\rm c}}e^{-3d_{j,X[m\ell]}/4}$ $\displaystyle\preceq
c_{\tau,2}C_{m,0}+C_{m,2}+c_{\tau,2}C_{m,1}\lambda_{3/4}|\partial X[m\ell]|,$
(300)

where the inequality is derived from

$\displaystyle\sum_{j\in X[m\ell]^{\rm c}}e^{-3d_{j,X[m\ell]}/4}\leq\sum_{j\in
X[m\ell]^{\rm c}}\sum_{i\in\partial
X[m\ell]}e^{-3d_{i,j}/4}\leq\sum_{i\in\partial
X[m\ell]}\sum_{j\in\Lambda}e^{-3d_{i,j}/4}\leq\lambda_{3/4}|\partial
X[m\ell]|.$ (301)

We next consider the following term in the inequality (299):

$\displaystyle C_{m,1}\sum_{j\in X[m\ell]^{\rm
c}}e^{-3d_{j,X[m\ell]}/4}\left(\hat{n}_{j}+c_{\tau,1}\hat{\mathcal{D}}_{j}\right)=$
$\displaystyle\ C_{m,1}\sum_{j\in X[m\ell]^{\rm
c}}e^{-3d_{j,X[m\ell]}/4}\left(\hat{n}_{j}+c_{\tau,1}\sum_{j^{\prime}\in\Lambda}e^{-d_{j,j^{\prime}}}\hat{n}_{j^{\prime}}\right)$
$\displaystyle\preceq$ $\displaystyle\
C_{m,1}\sum_{j\in\Lambda}e^{-3d_{j,X[m\ell]}/4}\left(1+c_{\tau,1}\lambda_{1/4}\right)\hat{n}_{j},$
(302)

where we use the inequality of $d_{j,X}+d_{j,j^{\prime}}\geq d_{j^{\prime},X}$
for $\forall X\subset\Lambda$, which yields

$\displaystyle\sum_{j\in\Lambda}e^{-3d_{j,X[m\ell]}/4}\sum_{j^{\prime}\in\Lambda}e^{-d_{j,j^{\prime}}}\hat{n}_{j^{\prime}}$
$\displaystyle\leq\sum_{j\in\Lambda}\sum_{j^{\prime}\in\Lambda}e^{-3d_{j^{\prime},X[m\ell]}/4}e^{-d_{j,j^{\prime}}/4}\hat{n}_{j^{\prime}}\leq\lambda_{1/4}\sum_{j^{\prime}\in\Lambda}e^{-3d_{j^{\prime},X[m\ell]}/4}\hat{n}_{j^{\prime}}.$
(303)

We further decompose

$\displaystyle\sum_{j\in\Lambda}e^{-3d_{j,X[m\ell]}/4}\hat{n}_{j}=\sum_{j\in
X[m\ell+\ell]}e^{-3d_{j,X[m\ell]}/4}\hat{n}_{j}+\sum_{j\in X[m\ell+\ell]^{\rm
c}}e^{-3d_{j,X[m\ell]}/4}\hat{n}_{j}.$ (304)

For the first term in Eq. (304), we immediately obtain

$\displaystyle\sum_{j\in
X[m\ell+\ell]}e^{-3d_{j,X[m\ell]}/4}\hat{n}_{j}\leq\sum_{j\in
X[m\ell+\ell]}\hat{n}_{j}=\hat{n}_{X[m\ell+\ell]}.$ (305)

For the second term in Eq. (304), by using
$d_{j,X[m\ell]}=d_{j,X[m\ell+\ell]}+\ell$ for $j\in X[m\ell+\ell]^{\rm c}$, we
have

$\displaystyle\sum_{j\in X[m\ell+\ell]^{\rm
c}}e^{-3d_{j,X[m\ell]}/4}\hat{n}_{j}=e^{-3\ell/4}\sum_{j\in X[m\ell+\ell]^{\rm
c}}e^{-3d_{j,X[m\ell+\ell]}/4}\hat{n}_{j}=e^{-3\ell/4}\hat{\mathscr{D}}_{X[m\ell+\ell]}.$
(306)

Then, by combining Eqs. (304), (305), and (306) with the inequality (302), we
obtain

$\displaystyle C_{m,1}\sum_{j\in X[m\ell]^{\rm
c}}e^{-3d_{j,X[m\ell]}/4}\left(\hat{n}_{j}+c_{\tau,1}\hat{\mathcal{D}}_{j}\right)\preceq$
$\displaystyle\
C_{m,1}\left(1+c_{\tau,1}\lambda_{1/4}\right)\left(\hat{n}_{X[m\ell+\ell]}+e^{-3\ell/4}\hat{\mathscr{D}}_{X[m\ell+\ell]}\right).$
(307)

Finally, by applying the inequalities (300) and (307) to (299), we arrive at
the inequality of

$\displaystyle\left[C_{m,0}\hat{n}_{X[m\ell]}(\tau)+C_{m,1}\hat{\mathscr{D}}_{X[m\ell]}(\tau)+C_{m,2}s\right]^{s}$
$\displaystyle\preceq\Biggl{\\{}\left[C_{0}C_{m,0}+C_{m,1}\left(1+c_{\tau,1}\lambda_{1/4}\right)\right]\hat{n}_{X[m\ell+\ell]}+\left[c_{\tau,1}e^{-3\ell/16}+C_{m,1}\left(1+c_{\tau,1}\lambda_{1/4}\right)e^{-3\ell/4}\right]\hat{\mathscr{D}}_{X[m\ell+\ell]}$
$\displaystyle\quad+s\left(c_{\tau,2}C_{m,0}+C_{m,2}+c_{\tau,2}C_{m,1}\lambda_{3/4}|\partial
X[m\ell]|\right)\Biggr{\\}}^{s}.$ (308)

The above inequality yields

$\displaystyle
C_{m+1,0}=C_{0}C_{m,0}+C_{m,1}\left(1+c_{\tau,1}\lambda_{1/4}\right),$
$\displaystyle
C_{m+1,1}=c_{\tau,1}e^{-3\ell/16}+C_{m,1}\left(1+c_{\tau,1}\lambda_{1/4}\right)e^{-3\ell/4},$
$\displaystyle
C_{m+1,2}=c_{\tau,2}C_{m,0}+C_{m,2}+c_{\tau,2}C_{m,1}\lambda_{3/4}|\partial
X[m\ell]|.$ (309)

For an arbitrary numerical sequence $\\{a_{m}\\}$ such that
$a_{m+1}=b_{1}a_{m}+b_{2}$, we have

$\displaystyle
a_{m}=\frac{b_{2}}{1-b_{1}}+b_{1}^{m}\left(a_{0}-\frac{b_{2}}{1-b_{1}}\right)\quad\textrm{for}\quad\forall
m.$ (310)

We first derive the explicit form of $C_{m,1}$. For the second equation in
(309), we use Eq. (310) with $C_{0,1}=0$ as in Eq. (296) to derive

$\displaystyle
C_{m,1}=\left\\{1-\left[\left(1+c_{\tau,1}\lambda_{1/4}\right)e^{-3\ell/4}\right]^{m}\right\\}\frac{c_{\tau,1}e^{-3\ell/16}}{1-\left(1+c_{\tau,1}\lambda_{1/4}\right)e^{-3\ell/4}}\leq
2c_{\tau,1}e^{-3\ell/16}\leq 2c_{\tau,1}e^{-2f_{\tau}\ell},$ (311)

where we use the inequality (291) to derive
$1-\left(1+c_{\tau,1}\lambda_{1/4}\right)e^{-3\ell/4}\geq 1/2$ and use
$e^{-f_{\tau}}>e^{-1/14}$ to derive $e^{-3\ell/16}\leq e^{-2f_{\tau}\ell}$.

We then reduce the first equation in (309) to

$\displaystyle C_{m+1,0}\leq
C_{0}C_{m,0}+2c_{\tau,1}e^{-2f_{\tau}\ell}\left(1+c_{\tau,1}\lambda_{1/4}\right),$
(312)

which yields from $C_{0,0}=1$ in Eq. (296)

$\displaystyle C_{m,0}$ $\displaystyle\leq
C_{0}^{m}+\frac{2c_{\tau,1}e^{-2f_{\tau}\ell}\left(1+c_{\tau,1}\lambda_{1/4}\right)}{1-C_{0}}(1-C_{0}^{m})$
$\displaystyle\leq
1+2m\left[\delta_{\ell}+2c_{\tau,1}e^{-2f_{\tau}\ell}\left(1+c_{\tau,1}\lambda_{1/4}\right)\right]\leq
1+3m\delta_{\ell}.$ (313)

In the second inequality above, we use the inequality (289) and the following
upper bound:

$\displaystyle\frac{C_{0}^{m}-1}{C_{0}-1}=\sum_{m^{\prime}=0}^{m-1}C_{0}^{m^{\prime}}\leq
m+2\delta_{\ell}\sum_{m^{\prime}=0}^{m-1}m^{\prime}=m+\delta_{\ell}m(m-1)\leq
2m,$ (314)

Note that we have $\delta_{\ell}\leq 1/m_{0}\leq 1/m$. In the third inequality
of (313), we use the definition of
$\delta_{\ell}=5c_{\tau,1}\lambda_{1/2}e^{-f_{\tau}\ell}$ and the inequality
(290). Finally, by using the inequalities (311) and (313), we reduce the third
equation in (309) to

$\displaystyle C_{m+1,2}$ $\displaystyle\leq
c_{\tau,2}\left(1+3m\delta_{\ell}\right)+C_{m,2}+2c_{\tau,1}c_{\tau,2}e^{-3\ell/16}\lambda_{3/4}|\partial
X[m\ell]|$ $\displaystyle\leq
C_{m,2}+c_{\tau,2}\left[1+3m\delta_{\ell}+2c_{\tau,1}\lambda_{3/4}e^{-3\ell/16}|\partial
X[R]|\right],$ (315)

where we use $m\leq m_{0}$ and $m\ell\leq R$. The above inequality immediately
gives

$\displaystyle C_{m,2}\leq
c_{\tau,2}m+c_{\tau,2}m_{0}\left(3m_{0}\delta_{\ell}+2c_{\tau,1}\lambda_{3/4}e^{-3\ell/16}|\partial
X[R]|\right)=c_{\tau,2}m+\tilde{\delta}_{\ell},$ (316)

where we use $C_{0,2}=0$.

By using the inequalities (311), (313) and (316), we reduce the upper bound
(295) to

$\displaystyle[\hat{n}_{X}(m\tau)]^{s}\preceq\left[\left(1+3m\delta_{\ell}\right)\hat{n}_{X[m\ell]}+2c_{\tau,1}e^{-3\ell/16}\hat{\mathscr{D}}_{X[m\ell]}+\left(c_{\tau,2}m+\tilde{\delta}_{\ell}\right)s\right]^{s}.$
(317)

We thus prove Proposition 19. $\square$

###  Supplementary Note 4.3 Boson density after time evolution for an initial
state with low-boson density

We apply Theorem 1 to a boson density after a time evolution for an initial
state $\rho_{0}$, which satisfies the following conditions:

$\displaystyle{\rm
tr}\left(\rho_{0}\hat{n}_{X_{0}[r]}^{s}\right)\leq\frac{1}{e}\left[\frac{b_{0}}{e}\left(q_{0}+s^{\kappa}q_{1}\max(r^{D},1)|X_{0}|\right)\right]^{s},$
(318) $\displaystyle{\rm
tr}\left(\rho_{0}\hat{n}_{i[\ell_{0}]}^{s}\right)\leq\frac{1}{e}\left(\frac{b_{0}}{e}s^{\kappa}\ell_{0}^{D}\right)^{s}\quad\textrm{for}\quad
i[\ell_{0}]\in X_{0}^{\rm c},\quad\forall\ell_{0}\geq\bar{\ell}_{0},$ (319)

where $b_{0}$ and $\kappa$ ($\geq 1$) are constants of $\mathcal{O}(1)$. In
this setup, unlike the condition in Definition 3, the boson density can be
large only in the region $X_{0}$. Our purpose is to prove that the boson
density around the site $i$ is still small after a time evolution if $i$ is
sufficiently separated from the region $X$ (see Fig. 6).

Supplementary Figure 6: Schematic picture of our setup for Proposition 20. We
here consider an initial state where the low-boson density condition in
Definition 3 is satisfied outside the region $X_{0}$. After a time evolution,
the concentration of the boson density may occur. Around the region $X_{0}$
(or in the extended region $X_{0}[\bar{\ell}]$), we cannot ensure that the
boson density should be suppressed. On the other hand, in the region
$X_{0}[\bar{\ell}]^{\rm c}$, we can prove that the number of bosons in a ball
region $i[\ell_{0}]$ obeys the (sub)exponentially decaying probability
distribution as long as $\ell_{0}$ satisfies (320). In our theorem, we prove
that the length of $\bar{\ell}$ is roughly given by $t\log(t)$ [see (323)].

We here prove the following corollary:

######  Proposition 20.

Let us consider an initial state $\rho_{0}$ that satisfies the assumptions
(318) and (319). Then, under the condition of

$\displaystyle\ell_{0}\geq\max(\bar{\ell}_{0},2D\bar{\ell}),$ (320)

we obtain

$\displaystyle{\rm
tr}\left[\rho_{0}(t)\hat{n}_{i[\ell_{0}]}^{s}\right]\leq\frac{1}{e}\left(\frac{2b_{0}}{e}s^{\kappa}\ell_{0}^{D}\right)^{s}\quad\textrm{for}\quad
i[\ell_{0}]\in X_{0}[\bar{\ell}]^{\rm c},$ (321)

and

$\displaystyle{\rm
tr}\left[\rho_{0}(t)\hat{n}_{X_{0}[\ell_{0}]}^{s}\right]\leq\frac{1}{e}\left[\frac{2b_{0}}{e}\left(q_{0}+s^{\kappa}q_{1}\ell_{0}^{D}|X_{0}|\right)\right]^{s},$
(322)

where $\bar{\ell}$ is given by

$\displaystyle\bar{\ell}=t\log[\Theta(tq_{0}q_{1}|X_{0}|)].$ (323)

Remark. From the inequality (321) with $s=1$, roughly speaking, the average
boson number in a region $i[\bar{\ell}]$ (i.e., $s=1$) is proportional to
$\bar{\ell}^{D}$, which gives

$\frac{1}{|i[\ell_{0}]|}{\rm
tr}\left[\rho_{0}(t)\hat{n}_{i[\ell_{0}]}^{s}\right]=\mathcal{O}(1).$

In particular, if we have no constraints in the region $X_{0}$, we can
simplify Proposition 20. Here, we obtain the inequality (321) for arbitrary
$i\in\Lambda$ with

$\displaystyle\bar{\ell}=t\log[\Theta(t)].$ (324)

####  Supplementary Note 4.3.1 Proof of Proposition 20

Let us consider an arbitrary operator $Q$ such that

$\displaystyle Q=\nu_{X}\hat{n}_{X}+\sum_{i\in X^{\rm c}}\nu_{i}\hat{n}_{i},$
(325)

where $X$ is arbitrarily chosen and we assume $\nu_{X}\geq 0$ and $\nu_{i}\geq
0$ ($i\in X^{\rm c}$). Then, for arbitrary quantum state $\rho$, we have

$\displaystyle{\rm tr}\left(\rho Q^{s}\right)={\rm
tr}\left[\rho\left(\nu_{X}\hat{n}_{X}+\sum_{i\in X^{\rm
c}}\nu_{i}\hat{n}_{i}\right)^{s}\right]\leq\left(\nu_{X}\left\langle\hat{n}_{X}^{s}\right\rangle_{\rho}^{1/s}+\sum_{i\in\Lambda}\nu_{i}\left\langle\hat{n}_{i}^{s}\right\rangle_{\rho}^{1/s}\right)^{s},$
(326)

where we use the generalized Hölder inequality [3] to obtain

$\displaystyle{\rm
tr}\left(\rho\hat{n}_{X_{1}}\hat{n}_{X_{2}}\cdots\hat{n}_{X_{s}}\right)\leq\prod_{j=1}^{s}\left\langle\hat{n}_{X_{j}}^{s}\right\rangle_{\rho}^{1/s}.$
(327)

Note that $\left\langle\hat{n}_{i}^{s}\right\rangle_{\rho}$ means
$\left\langle\hat{n}_{i}^{s}\right\rangle_{\rho}={\rm
tr}\left(\rho\hat{n}_{i}^{s}\right)$.

We here apply the inequality (326) to ${\rm
tr}\left[\rho_{0}(t)\hat{n}_{X}^{s}\right]={\rm
tr}\left\\{\rho_{0}[\hat{n}_{X}(t)]^{s}\right\\}$. We first use Proposition 18
to derive

$\displaystyle{\rm tr}\left[\rho_{0}(t)\hat{n}_{X}^{s}\right]\leq{\rm
tr}\left\\{\rho_{0}\left[\left(1+\frac{3t}{\tau}\delta_{\ell}\right)\hat{n}_{X[R]}+2c_{\tau,1}e^{-3\ell/16}\hat{\mathscr{D}}_{X[R]}+\left(\frac{c_{\tau,2}t}{\tau}+\tilde{\delta}_{\ell}\right)s\right]^{s}\right\\}$
(328)

with $\ell=\Theta(R/t)$, where we choose $R$ such that the condition (281) is
satisfied, i.e., $R\gtrsim t\log(t)$. The inequality (326) reduces the above
inequality

$\displaystyle\left\\{{\rm
tr}\left[\rho_{0}(t)\hat{n}_{X}^{s}\right]\right\\}^{1/s}\leq\left(1+\frac{3t}{\tau}\delta_{\ell}\right)\left\langle\hat{n}_{X[R]}^{s}\right\rangle_{\rho_{0}}^{1/s}+2c_{\tau,1}e^{-3\ell/16}\left\langle\hat{\mathscr{D}}_{X[R]}^{s}\right\rangle_{\rho_{0}}^{1/s}+\left(\frac{c_{\tau,2}t}{\tau}+\tilde{\delta}_{\ell}\right)s.$
(329)

We now consider the case of $X=i[\ell_{0}]$ such that $i[\ell_{0}+R]\cap
X_{0}=\emptyset$ and $\ell_{0}\geq 2DR$. From the conditions (318) and (319),
we obtain

$\displaystyle\left\langle\hat{n}_{i[\ell_{0}+R]}^{s}\right\rangle_{\rho_{0}}^{1/s}\leq\frac{1}{e^{1/s}}\frac{b_{0}}{e}s^{\kappa}(\ell_{0}+R)^{D},$
(330)

and

$\displaystyle\left\langle\hat{\mathscr{D}}_{i[\ell_{0}+R]}^{s}\right\rangle_{\rho_{0}}^{1/s}$
$\displaystyle\leq\left\langle\hat{n}_{X_{0}}^{s}\right\rangle_{\rho_{0}}^{1/s}+\left\langle\left(\sum_{j\in
i[\ell_{0}+R]^{\rm c}\setminus
X_{0}}e^{-3d_{j,i[\ell_{0}+R]}/4}\hat{n}_{j}\right)^{s}\right\rangle_{\rho_{0}}^{1/s}$
$\displaystyle\leq\frac{1}{e^{1/s}}\frac{b_{0}}{e}(q_{0}+s^{\kappa}q_{1}|X_{0}|)+\frac{1}{e^{1/s}}\left(\frac{b_{0}}{e}s^{\kappa}\ell_{0}^{D}\right)e^{3/2}\left[1+2^{D}D!(4/3)^{D+1}\right],$
(331)

where we use $R\leq\ell_{0}$ from $R\leq\ell_{0}/(2D)$ and

$\displaystyle\left\langle\left(\sum_{j\in i[\ell_{0}+R]^{\rm c}\setminus
X_{0}}e^{-3d_{j,i[\ell_{0}+R]}/4}\hat{n}_{j}\right)^{s}\right\rangle_{\rho_{0}}^{1/s}\leq\sum_{m=1}^{\infty}e^{-3(m-1)\ell_{0}/4}\left\langle\hat{n}_{i[(m+1)\ell_{0}+R]\setminus
i[m\ell_{0}+R]}^{s}\right\rangle_{\rho_{0}}^{1/s}$
$\displaystyle\leq\sum_{m=1}^{\infty}e^{-3(m-1)/4}\left\langle\hat{n}_{i[(m+2)\ell_{0}]}^{s}\right\rangle_{\rho_{0}}^{1/s}\leq\frac{1}{e^{1/s}}\left(\frac{b_{0}}{e}s^{\kappa}\ell_{0}^{D}\right)\sum_{m=1}^{\infty}e^{-3(m-1)/4}(m+2)^{D}$
$\displaystyle\leq\frac{1}{e^{1/s}}\left(\frac{b_{0}}{e}s^{\kappa}\ell_{0}^{D}\right)\sum_{m=1}^{\infty}e^{-3(m+2)/4+9/4}(m+2)^{D}\leq\frac{1}{e^{1/s}}\left(\frac{b_{0}}{e}s^{\kappa}\ell_{0}^{D}\right)e^{3/2}\left[1+2^{D}D!(4/3)^{D+1}\right].$
(332)

By using the above inequalities, we have

$\displaystyle\left\\{{\rm
tr}\left[\rho_{0}(t)\hat{n}_{i[\ell_{0}]}^{s}\right]\right\\}^{1/s}$
$\displaystyle\leq\frac{1}{e^{1/s}}\left(\frac{b_{0}}{e}s^{\kappa}\ell_{0}^{D}\right)\Biggl{[}\left(1+\frac{3t}{\tau}\delta_{\ell}\right)(1+R/\ell_{0})^{D}+2c_{\tau,1}e^{-3\ell/16}\left(\frac{q_{0}s^{-\kappa}+q_{1}|X_{0}|}{\ell_{0}^{D}}+e^{3/2}\left[1+2^{D}D!(4/3)^{D+1}\right]\right)$
$\displaystyle\quad+\frac{e^{1+1/s}}{b_{0}\ell_{0}^{D}}\left(\frac{c_{\tau,2}t}{\tau}+\tilde{\delta}_{\ell}\right)s^{1-\kappa}\Biggr{]}.$
(333)

By using the $\Theta$ notation in Eq. (9), we can obtain

$\displaystyle\left(1+\frac{3t}{\tau}\delta_{\ell}\right)(1+R/\ell_{0})^{D}+2c_{\tau,1}e^{-3\ell/16}\left(\frac{q_{0}s^{-\kappa}+q_{1}|X_{0}|}{\ell_{0}^{D}}+e^{3/2}\left[1+2^{D}D!(4/3)^{D+1}\right]\right)+\frac{e^{1+1/s}}{b_{0}\ell_{0}^{D}}\left(\frac{c_{\tau,2}t}{\tau}+\tilde{\delta}_{\ell}\right)s^{1-\kappa}$
$\displaystyle\leq\left(1+\Theta(t)e^{-\Theta(R/t)}\right)(1+R/\ell_{0})^{D}+e^{-\Theta(R/t)}\Theta\left(q_{0},q_{1}|X_{0}|\right).$
(334)

Thus, under the assumption of $R/\ell_{0}\leq 1/(2D)$ (or $\ell_{0}\geq 2DR$),
we can find a length $R$ such that

$\displaystyle\left(1+\Theta(t)e^{-\Theta(R/t)}\right)(1+R/\ell_{0})^{D}+e^{-\Theta(R/t)}\Theta\left(q_{0},q_{1}|X_{0}|\right)\leq
2,$ (335)

which is achieved by choosing $R=\bar{\ell}$ with

$\displaystyle\bar{\ell}=t\log[\Theta(tq_{0}q_{1}|X_{0}|)].$ (336)

We thus prove the first inequality (321).

We then consider the case where $X=X_{0}[\ell_{0}]$. From the condition (318),
we obtain

$\displaystyle\left\langle\hat{n}_{X_{0}[\ell_{0}+R]}^{s}\right\rangle_{\rho_{0}}^{1/s}\leq\frac{1}{e}\left[\frac{b_{0}}{e}\left(q_{0}+s^{\kappa}q_{1}(\ell_{0}+R)^{D}|X_{0}|\right)\right],$
(337)

and

$\displaystyle\left\langle\hat{\mathscr{D}}_{X_{0}[\ell_{0}+R]}^{s}\right\rangle_{\rho_{0}}^{1/s}$
$\displaystyle\leq\left\langle\left(\sum_{j\in X_{0}[\ell_{0}+R]^{\rm
c}}e^{-3d_{j,X_{0}[\ell_{0}+R]}/4}\hat{n}_{j}\right)^{s}\right\rangle_{\rho_{0}}^{1/s}$
$\displaystyle\leq\frac{1}{e^{1/s}}\left[\frac{b_{0}}{e}\left(q_{0}+s^{\kappa}q_{1}\ell_{0}^{D}|X_{0}|\right)\right]e^{3/2}\left[1+2^{D}D!(4/3)^{D+1}\right],$
(338)

where we use

$\displaystyle\left\langle\left(\sum_{j\in X_{0}[\ell_{0}+R]^{\rm
c}}e^{-3d_{j,X_{0}[\ell_{0}+R]}/4}\hat{n}_{j}\right)^{s}\right\rangle_{\rho_{0}}^{1/s}\leq\sum_{m=1}^{\infty}e^{-3(m-1)/4}\left\langle\hat{n}_{X_{0}[(m+1)\ell_{0}+R]\setminus
X_{0}[m\ell_{0}+R]}^{s}\right\rangle_{\rho_{0}}^{1/s}$
$\displaystyle\leq\sum_{m=1}^{\infty}e^{-3(m-1)/4}\left\langle\hat{n}_{X_{0}[(m+2)\ell_{0}]}^{s}\right\rangle_{\rho_{0}}^{1/s}\leq\sum_{m=1}^{\infty}e^{-3(m-1)/4}\frac{1}{e^{1/s}}\left[\frac{b_{0}}{e}\left(q_{0}+s^{\kappa}q_{1}[(m+2)\ell_{0}]^{D}|X_{0}|\right)\right]$
$\displaystyle\leq\frac{1}{e^{1/s}}\left[\frac{b_{0}}{e}\left(q_{0}+s^{\kappa}q_{1}\ell_{0}^{D}|X_{0}|\right)\right]\sum_{m=1}^{\infty}e^{-3(m+2)/4+9/4}(m+2)^{D}\leq\frac{1}{e^{1/s}}\left[\frac{b_{0}}{e}\left(q_{0}+s^{\kappa}q_{1}\ell_{0}^{D}|X_{0}|\right)\right]e^{3/2}\left[1+2^{D}D!(4/3)^{D+1}\right].$

By applying the above inequalities to (329), we can derive

$\displaystyle\left\\{{\rm
tr}\left[\rho_{0}(t)\hat{n}_{X_{0}[\ell_{0}]}^{s}\right]\right\\}^{1/s}$
$\displaystyle\leq\frac{1}{e}\left[\frac{b_{0}}{e}\left(q_{0}+s^{\kappa}q_{1}\ell_{0}^{D}|X_{0}|\right)\right]\Biggl{[}\left(1+\frac{3t}{\tau}\delta_{\ell}\right)(1+R/\ell_{0})^{D}+2c_{\tau,1}e^{-3\ell/16}\cdot
e^{3/2}\left[1+2^{D}D!(4/3)^{D+1}\right]$
$\displaystyle\quad+\frac{e^{1+1/s}}{b_{0}q_{1}\ell_{0}^{D}|X_{0}|}\left(\frac{c_{\tau,2}t}{\tau}+\tilde{\delta}_{\ell}\right)s^{1-\kappa}\Biggr{]}.$
(339)

Therefore, under the assumption of $\ell_{0}\geq 2DR$, we can obtain

$\displaystyle\left(1+\frac{3t}{\tau}\delta_{\ell}\right)(1+R/\ell_{0})^{D}+2c_{\tau,1}e^{-3\ell/16}e^{3/2}\left[1+2^{D}D!(4/3)^{D+1}\right]+\frac{e^{1+1/s}}{b_{0}q_{1}\ell_{0}^{D}|X_{0}|}\left(\frac{c_{\tau,2}t}{\tau}+\tilde{\delta}_{\ell}\right)s^{1-\kappa}\leq
2$ (340)

by choosing $R=\bar{\ell}$ as in Eq. (336). We thus prove the second
inequality (322). This completes the proof. $\square$

##  Supplementary Note 5 Restriction on the outflow of bosons

In this section, we aim to bound a boson number in a particular region from
below. As in Fig. 7, we consider a setup that many bosons originally
concentrated in a region $X$:

$\displaystyle\Pi_{X,\geq N}\rho=\rho,$ (341)

where $N$ is a fixed number. Unlike the setup in Ref. [2], there exist bosons
outside the region $X$. Then, after a short-time evolution, the bosons are
expected to concentrate around the region $X$, i.e.,

$\displaystyle{\rm tr}\left[\Pi_{X[\ell],\leq N-\delta
N}\rho(\tau)\right]\approx g(\delta N)$ (342)

for sufficiently large $\ell$, where $g(x)$ is a rapidly decaying function. We
need to consider such an upper bound in proving the optimal Lieb-Robinson
bound in one-dimensional systems (see Sec. Supplementary Note 10.5.2).

Supplementary Figure 7: Schematic picture of the setup in Sec. Supplementary
Note 5. We consider an initial state where at least $N$ bosons exist inside
the region $X\subset\Lambda$. After a time evolution, these bosons may escape
outside the region $X$. Even though, if the time is short, we expect that the
bosons still remain in the region of around $X$. Mathematically, as the width
$\ell$ increases, the probability of observing $\mathcal{O}(N)$ bosons in
$X[\ell]$ is exponentially close to $1$ (see Proposition 22).

###  Supplementary Note 5.1 Moment lower bound

First of all, we can prove that in the level of the expectation value. We can
obtain

$\displaystyle\langle\psi_{0}(\tau)|\hat{n}_{X[\ell]}|\psi_{0}(\tau)\rangle\gtrsim\langle\psi_{0}|\hat{n}_{X}|\psi_{0}\rangle-e^{-\Omega(\ell)}\geq
N-e^{-\Omega(\ell)}.$ (343)

Thus, we ensure that the number of bosons is lower-bounded as long as we
consider an extended region $X[\ell]$ with a sufficiently large $\ell$. For
higher-order moments, we can also derive

$\displaystyle\langle\psi_{0}(\tau)|\hat{n}_{X[\ell]}^{s}|\psi_{0}(\tau)\rangle\gtrsim\left(N-e^{-\Omega(\ell)}\right)^{s}.$
(344)

Mathematically, we prove the following corollary by using the same analyses as
in the proof for Proposition 16:

######  Lemma 21.

Let us consider the same setup as that in Proposition 16. For an arbitrary
subset $X$, the moment function $M^{(1)}_{X}(\tau)$ is lower-bounded by

$\displaystyle\hat{n}_{X}(\tau)\succeq\hat{n}_{X[-\ell]}-c_{\tau,1}\lambda_{1/4}e^{-3\ell/16}\hat{\mathscr{D}}_{X}-5c_{\tau,1}\lambda_{1/2}e^{-f_{\tau}\ell}\hat{n}_{X}-c_{\tau,2},$
(345)

where $\hat{\mathscr{D}}_{X}$ and $f_{\tau}$ have been defined in Eqs. (242)
and (243), respectively. Also, for moment functions $M^{(s)}_{X}(\tau)$ with
arbitrary orders, the inequality (186) yields

$\displaystyle
M^{(s)}_{X}(\tau)\geq\left(\hat{n}_{X[-\ell]}-c_{\tau,1}\lambda_{1/4}e^{-3\ell/16}\hat{\mathscr{D}}_{X}-5c_{\tau,1}\lambda_{1/2}e^{-f_{\tau}\ell}\hat{n}_{X}-c_{\tau,2}\right)^{s}.$
(346)

Remark. To derive the inequality (344), we consider the moment function of
$M^{(s)}_{X[\ell]}(\tau)$ and apply the inequality (346), which yields

$\displaystyle M^{(s)}_{X[\ell]}(\tau)$
$\displaystyle\geq\left(\left\langle\hat{n}_{X}\right\rangle_{\rho}-c_{\tau,1}\lambda_{1/4}e^{-3\ell/16}\left\langle\hat{\mathscr{D}}_{X}\right\rangle_{\rho}-5c_{\tau,1}\lambda_{1/2}e^{-f_{\tau}\ell}\left\langle\hat{n}_{X}\right\rangle_{\rho}-c_{\tau,2}\right)^{s}$
$\displaystyle\geq\left(\left\langle\hat{n}_{X}\right\rangle_{\rho}-e^{-\Omega(\ell)}-c_{\tau,2}\right)^{s}\geq\left(N-e^{-\Omega(\ell)}-c_{\tau,2}\right)^{s},$
(347)

where we assume $\left\langle\hat{n}_{X}\right\rangle_{\rho}\geq N$ in the
last inequality.

####  Supplementary Note 5.1.1 Proof of Lemma 21

As shown in Corollary 12, we have ${\rm
tr}(\rho(\tau)\hat{n}_{X}^{s})\geq[{\rm tr}(\rho(\tau)\hat{n}_{X})]^{s}$ for
an arbitrary quantum state $\rho$, and hence it is enough to consider the 1st
order moment in proving the inequality (346) for the $s$th order moment. In
the following, we aim to derive a lower bound for the moment function
$M_{X}^{(1)}(\tau)={\rm tr}[\rho(\tau)\hat{n}_{X}]$ for an arbitrary $\rho$.

We first define the subsets $\\{X_{p}\\}_{p=0}^{{\tilde{p}}+1}$ as follows
(see Fig. 8):

$\displaystyle X_{0}\coloneqq X[-\ell],\quad X_{1}\coloneqq
X_{0}[\ell/4]\setminus X_{0},$ $\displaystyle X_{p}\coloneqq
X_{0}[\ell/4+p-1]\setminus X_{0}[\ell/4+p-2]\quad(2\leq p\leq\tilde{p}),$
$\displaystyle X_{{\tilde{p}}+1}\coloneqq X_{0}[\ell]\setminus
X_{0}[3\ell/4],$ (348)

where $\tilde{p}=\ell/2+1$. We then obtain

$\displaystyle M_{X}^{(1)}(\tau)\geq M_{X_{0:p}}^{(1)}(\tau)$ (349)

for arbitrary $p\geq 0$, where we use $\hat{n}_{X}\succeq\hat{n}_{X_{0:p}}$
from $X\supseteq X_{0:p}$.

Supplementary Figure 8: Choice of the subsets
$\\{X_{p}\\}_{p=0}^{\tilde{p}+1}$. Unlike Fig. 4, we define the subsets inside
the region $X$. We can lower-bound the moment function $M^{(1)}_{X}(\tau)$ by
using $M^{(1)}_{X_{0:p}}(\tau)$ because of $X_{0:p}\subseteq X$. Our task is
to find the optimal $p\in[1,\tilde{p}]$ which gives the lower bound (345).

We start from a similar inequality to (Supplementary Note 3.4):

$\displaystyle M_{X_{0:p}}^{(1)}(\tau)$ $\displaystyle\geq
N^{(\rho)}_{X_{0:p}}-\tilde{N}^{(\rho)}_{X_{0:p}}-c_{\tau,2},\quad\tilde{N}^{(\rho)}_{X_{0:p}}:=\sum_{i\in\partial(X_{0:p})}\sum_{j\in\Lambda}e^{-d_{i,j}}N^{(\rho)}_{j},$
(350)

where we have denoted $\left\langle\hat{n}_{i}\right\rangle_{\rho}$ by
$N^{(\rho)}_{i}$ for $\forall i\in\Lambda$. Then, we aim to lower-bound

$\displaystyle\max_{p\in[1,{\tilde{p}}]}\left(N^{(\rho)}_{X_{0:p}}-c_{\tau,1}\tilde{N}^{(\rho)}_{X_{0:p}}\right),$
(351)

which yields the lower bound for $M_{X}^{(1)}(\tau)$ because of the inequality
(349).

By using Lemma 17, we reduce the maximization problem (351) to the form of

$\displaystyle\max_{p\in[1,{\tilde{p}}]}\left(N^{(\rho)}_{X_{0:p}}-c_{\tau,1}\tilde{N}^{(\rho)}_{X_{0:p}}\right)\geq
N^{(\rho)}_{X_{0}}-c_{\tau,1}\delta
N_{X_{0},\ell}^{(\rho)}+\max_{p\in[1,{\tilde{p}}]}\left(M_{1:p}-c_{\tau,1}\lambda_{1/2}\sum_{j=1}^{{\tilde{p}}}e^{-|p-j|/2}M_{j}\right),$
$\displaystyle M_{j}=\begin{cases}N^{(\rho)}_{X_{j}}&\quad\textrm{for}\quad
1\leq j\leq{\tilde{p}}-1,\\\
N^{(\rho)}_{X_{\tilde{p}}}+N^{(\rho)}_{X_{{\tilde{p}}+1}}&\quad\textrm{for}\quad
j={\tilde{p}},\end{cases}$ (352)

where we use $N^{(\rho)}_{X_{0:p}}:=N^{(\rho)}_{X_{0}}+M_{1:p}$ and $\delta
N_{X,\ell}^{(\rho)}$ has been defined in Eq. (250), i.e.,

$\displaystyle\delta N^{(\rho)}_{X_{0},\ell}=\lambda_{1/4}e^{-3\ell/16}{\rm
tr}\left[\rho\left(\hat{n}_{X_{0}}+\hat{\mathscr{D}}_{X_{0}[\ell]}\right)\right]$
(353)

with $\hat{\mathscr{D}}_{X_{0}[\ell]}$ given by Eq. (242). Then, by using
Lemma 14 with $\bar{\alpha}$ and $\chi_{\alpha}$ as in Eqs. (252) and (253),
we obtain

$\displaystyle\max_{p\in[1,{\tilde{p}}]}\left(M_{1:p}-c_{\tau,1}\lambda_{1/2}\sum_{j=1}^{{\tilde{p}}}e^{-|p-j|/2}M_{j}\right)$
$\displaystyle\geq-5c_{\tau,1}\lambda_{1/2}\left(1+\frac{1}{5c_{\tau,1}\lambda_{1/2}+2}\right)^{-{\tilde{p}}+1}M_{1:{\tilde{p}}}$
$\displaystyle=-5c_{\tau,1}\lambda_{1/2}\left(1+\frac{1}{5c_{\tau,1}\lambda_{1/2}+2}\right)^{-\ell/2}\left(N^{(\rho)}_{X}-N^{(\rho)}_{X_{0}}\right),$
(354)

where we have used ${\tilde{p}}=\ell/2+1$ and
$M_{1:{\tilde{p}}}=N^{(\rho)}_{X}-N^{(\rho)}_{X_{0}}$. By combining the
inequalities (Supplementary Note 5.1.1) and (Supplementary Note 5.1.1), we
obtain

$\displaystyle\max_{p\in[1,{\tilde{p}}]}\left(N^{(\rho)}_{X_{0:p}}-c_{\tau,1}\tilde{N}^{(\rho)}_{X_{0:p}}\right)$
$\displaystyle\geq
N^{(\rho)}_{X_{0}}-c_{\tau,1}\lambda_{1/4}e^{-3\ell/16}\left[N^{(\rho)}_{X_{0}}+{\rm
tr}\left(\rho\hat{\mathscr{D}}_{X_{0}[\ell]}\right)\right]-5c_{\tau,1}\lambda_{1/2}\left(1+\frac{1}{5c_{\tau,1}\lambda_{1/2}+2}\right)^{-\ell/2}\left(N^{(\rho)}_{X}-N^{(\rho)}_{X_{0}}\right),$
$\displaystyle\geq N^{(\rho)}_{X_{0}}-c_{\tau,1}\lambda_{1/4}e^{-3\ell/16}{\rm
tr}\left(\rho\hat{\mathscr{D}}_{X_{0}[\ell]}\right)-5c_{\tau,1}\lambda_{1/2}\left(1+\frac{1}{5c_{\tau,1}\lambda_{1/2}+2}\right)^{-\ell/2}N^{(\rho)}_{X},$
(355)

which yields the main inequality (345) form $X_{0}=X[-\ell]$. This completes
the proof. $\square$

###  Supplementary Note 5.2 Bound on the probability distribution

In the previous section, we have proved that

$\displaystyle\langle\psi_{0}(\tau)|\hat{n}_{X[\ell]}^{s}|\psi_{0}(\tau)\rangle\gtrsim\left(N-e^{-\Omega(\ell)}\right)^{s}.$
(356)

Unfortunately, we cannot utilize it to restrict the boson number distribution
as in Eq. (342). For example, if a time-evolved state $|\psi_{0}(\tau)\rangle$
satisfies

$\displaystyle\left\|\Pi_{X[\ell],2N}|\psi_{0}(\tau)\rangle\right\|^{2}=1/2,\quad\left\|\Pi_{X[\ell],0}|\psi_{0}(\tau)\rangle\right\|^{2}=1/2,$
(357)

the inequality (356) holds, but we still have a probability of
$\mathcal{O}(1)$ that all the bosons flow outside the region $X[\ell]$. In
this sense, it is not enough to consider only the lower bound on the moment
function. By refining the analyses, we can prove the following statement:

######  Proposition 22.

Let $\rho$ be an arbitrary quantum state $\rho$ such that

$\displaystyle\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\rho=\rho,\quad\Pi_{X,\geq
N}\rho=\rho,$ (358)

where the set $\mathcal{S}(L,\xi,N)$ has been defined in Eq. (37)*2*2*2Roughly
speaking, the projection $\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}$ implies
that there does not exist too many bosons around $X[\ell]$.. We then obtain

$\displaystyle{\rm tr}\left[\Pi_{X[\ell],\leq N-\delta
N}\rho(\tau)\right]\leq\frac{2}{1-e^{-1/(8ec_{\tau,2})}}\exp\left[\frac{-\delta
N}{8ec_{\tau,2}}+3c_{\ell}^{\prime}\left(1+c_{\ell}^{\prime}D\right)\log^{2}\left(q_{\ast}\gamma|X|\right)\right],$
(359)

where the length $\ell$ is given by

$\displaystyle\ell=\left\lceil\frac{2}{\tilde{f}_{\tau}}\log\left(\frac{5c_{\tau,1}\lambda_{1/2}e^{2\tilde{f}_{\tau}}\gamma(D/\tilde{f}_{\tau})^{D}q_{\ast}|X|}{2c_{\tau,2}}\right)\right\rceil=c^{\prime}_{\ell}\log\left(q_{\ast}\gamma|X|\right),$
(360)

with

$\displaystyle\tilde{f}_{\tau}\coloneqq\frac{1}{2}\log\left(1+\frac{1}{10ec_{\tau,1}\lambda_{1/2}+2}\right).$
(361)

Here, $c_{\ell}^{\prime}$ is a constant of $\mathcal{O}(1)$.

Remark. From the proposition, as long as $\delta
N\gtrsim\log^{2}\left(q_{\ast}\right)$, we obtain the desired restriction in
Eq. (342). Here, the probability distribution obeys the large deviation bound,
i.e.,

$\displaystyle{\rm tr}\left[\Pi_{X[\ell],\leq N-\delta
N}\rho(\tau)\right]=e^{-\Theta(N)}\quad\textrm{for}\quad\delta N=\Theta(N).$
(362)

We will utilize the inequality (359) in deriving the upper bound (899)
hereafter.

Supplementary Figure 9: We adopt the same notation as in Fig. 4. We decompose
the extended subset $X[\ell]\setminus X$ to $(\tilde{p}+1)$ pieces. In the
spectral decomposition (Supplementary Note 5.3), we assign the boson numbers
$\\{m_{p}\\}_{p=0}^{\tilde{p}+1}$ in the subsets
$\\{X_{p}\\}_{p=0}^{\tilde{p}+1}$, respectively. To prove the main inequality
(359), we derive the upper bound of (Supplementary Note 5.3) and aim to
identify the optimal $p$ which minimizes the RHS of (Supplementary Note 5.3).

###  Supplementary Note 5.3 Proof of Proposition 22

By using the spectral decomposition of $\rho$ as
$\rho=\sum_{j}p_{j}|\psi_{j}\rangle\langle\psi_{j}|$, we have

$\displaystyle{\rm tr}\left(\Pi_{X[\ell],\leq N-\delta
N}\rho(\tau)\right)=\sum_{j}p_{j}\langle\psi_{j}(\tau)|\Pi_{X[\ell],\leq
N-\delta N}|\psi_{j}(\tau)\rangle=\sum_{j}p_{j}\left\|\Pi_{X[\ell],\leq
N-\delta N}|\psi_{j}(\tau)\rangle\right\|^{2},$
$\displaystyle\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}|\psi_{j}\rangle=|\psi_{j}\rangle\quad\textrm{for}\quad\forall
j.$ (363)

Hence, we consider the case that $\rho$ is given by a pure state
$|\psi\rangle$ without loss of generality. In the following, we choose the
length $\ell$ such that

$\displaystyle\delta_{X,\ell,q_{\ast}}:=e^{-\ell/8}q_{\ast}|X|\left(\lambda_{1/2}+\gamma\lambda_{1/4}\ell^{D-1}e^{-\ell/16}\right)\leq\frac{c_{\tau,2}}{c_{\tau,1}}s,$
(364)

where $\delta_{X,\ell,q_{\ast}}$ is going to be defined in Eq. (373).

We adopt the same notation for the subsets $\\{X_{p}\\}_{p=1}^{{\tilde{p}}+1}$
as in Eq. (Supplementary Note 3.4):

$\displaystyle X_{0}\coloneqq X,\quad X_{1}\coloneqq X[\ell/4]\setminus X,$
$\displaystyle X_{p}\coloneqq X[\ell/4+p-1]\setminus X[\ell/4+p-2]\quad(2\leq
p\leq{\tilde{p}}),$ $\displaystyle X_{{\tilde{p}}+1}\coloneqq X[\ell]\setminus
X[3\ell/4],$ (365)

where $\tilde{p}=\ell/2+1$. We here introduce the projection
$\Pi_{m_{0:{\tilde{p}}+1}}$ as

$\displaystyle\Pi_{m_{0:{\tilde{p}}+1}}:=\Pi_{X,m_{0}}\prod_{j=1}^{{\tilde{p}}+1}\Pi_{X_{j},m_{j}}.$
(366)

Note that $\Pi_{X,N}$ has been defined in Eq. (25), i.e.,
$\hat{n}_{X}\Pi_{X,N}=N\Pi_{X,N}$. We then obtain

$\displaystyle\left\|\Pi_{X[\ell],\leq N-\delta N}|\psi(\tau)\rangle\right\|$
$\displaystyle=\left\|\sum_{m_{0},m_{1},\ldots,m_{\tilde{p}+1}}\Pi_{X[\ell],\leq
N-\delta N}e^{-iHt}\Pi_{m_{0:{\tilde{p}}+1}}|\psi\rangle\right\|$
$\displaystyle\leq\sum_{m_{0},m_{1},\ldots,m_{\tilde{p}+1}}\left\|\Pi_{X[\ell],\leq
N-\delta N}e^{-iHt}|\psi_{m_{0:{\tilde{p}}+1}}\rangle\right\|,$ (367)

where we define the unnormalized state
$|\psi_{m_{0:{\tilde{p}}+1}}\rangle:=\Pi_{m_{0:{\tilde{p}}+1}}|\psi\rangle$.
From the condition (358), we have $\Pi_{X,m_{0}}|\psi\rangle=0$ for $m_{0}<N$,
and hence in the summation (Supplementary Note 5.3) we only have to consider
the range of $m_{0}\geq N$.

For the estimation of $\left\|\Pi_{X[\ell],\leq N-\delta
N}e^{-iHt}|\psi_{m_{0:{\tilde{p}}+1}}\rangle\right\|$, we consider

$\displaystyle\left\|\Pi_{X[\ell],\leq N-\delta
N}e^{-iHt}|\psi_{m_{0:{\tilde{p}}+1}}\rangle\right\|$
$\displaystyle\leq\left\|\Pi_{X[\ell],\leq N-\delta
N}|\hat{n}_{X_{0:p}}-N_{p}|^{-s/2}\right\|\cdot\left\||\hat{n}_{X_{0:p}}-N_{p}|^{s/2}e^{-iHt}|\psi_{m_{0:{\tilde{p}}+1}}\rangle\right\|$
(368)

for an arbitrary choice of $p\in[0,\tilde{p}+1]$, where we choose $N_{p}$
appropriately such that $N_{p}$ is larger than $N$. Because of
$X[\ell]\supseteq X_{0:p}$ for $\forall p$, we have $\Pi_{X[\ell],\leq
N-\delta N}=\Pi_{X[\ell],\leq N-\delta N}\Pi_{X_{0:p},\leq N-\delta N}$, which
yields

$\displaystyle\left\|\Pi_{X[\ell],\leq N-\delta
N}|\hat{n}_{X_{0:p}}-N_{p}|^{-s/2}\right\|\leq\frac{1}{|N_{p}-N+\delta
N|^{s/2}},$ (369)

where we use $N_{p}\geq N$ to obtain the inequality. Also, by using
Proposition 10, we obtain an upper bound of

$\displaystyle\left\||\hat{n}_{X_{0:p}}-N_{p}|^{s/2}e^{-iHt}|\psi_{m_{0:{\tilde{p}}+1}}\rangle\right\|^{2}$
$\displaystyle=\langle\psi_{m_{0:{\tilde{p}}+1}}|e^{iHt}|\hat{n}_{X_{0:p}}-N_{p}|^{s}e^{-iHt}|\psi_{m_{0:{\tilde{p}}+1}}\rangle$
$\displaystyle\leq\langle\psi_{m_{0:{\tilde{p}}+1}}|\left(|\hat{n}_{X_{0:p}}-N_{p}|+c_{\tau,1}\hat{\mathcal{D}}_{X_{0:p}}+c_{\tau,2}s\right)^{s}|\psi_{m_{0:{\tilde{p}}+1}}\rangle$
$\displaystyle=\langle\psi_{m_{0:{\tilde{p}}+1}}|\left(|m_{0:p}-N_{p}|+c_{\tau,1}\hat{\mathcal{D}}_{X_{0:p}}+c_{\tau,2}s\right)^{s}|\psi_{m_{0:{\tilde{p}}+1}}\rangle$
$\displaystyle=\langle\psi_{m_{0:{\tilde{p}}+1}}|\left(c_{\tau,1}\hat{\mathcal{D}}_{X_{0:p}}+c_{\tau,2}s\right)^{s}|\psi_{m_{0:{\tilde{p}}+1}}\rangle$
(370)

with $m_{0:p}=\sum_{j=0}^{p}m_{j}$, where we choose $N_{p}$ as $N_{p}=m_{0:p}$
in the last equation. Note that the choice ensures $N_{p}\geq N$ because of
$m_{0}\geq N$.

From
$\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\Pi_{m_{0:{\tilde{p}}+1}}|\psi_{m_{0:{\tilde{p}}+1}}\rangle=|\psi_{m_{0:{\tilde{p}}+1}}\rangle$,
we then consider

$\displaystyle\langle\psi_{m_{0:{\tilde{p}}+1}}|\left(c_{\tau,1}\hat{\mathcal{D}}_{X_{0:p}}+c_{\tau,2}s\right)^{s}|\psi_{m_{0:{\tilde{p}}+1}}\rangle$
$\displaystyle=\langle\psi_{m_{0:{\tilde{p}}+1}}|\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\Pi_{m_{0:{\tilde{p}}+1}}\left(c_{\tau,1}\hat{\mathcal{D}}_{X_{0:p}}+c_{\tau,2}s\right)^{s}\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\Pi_{m_{0:{\tilde{p}}+1}}|\psi_{m_{0:{\tilde{p}}+1}}\rangle$
$\displaystyle\leq\left\|\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\Pi_{m_{0:{\tilde{p}}+1}}\left(c_{\tau,1}\hat{\mathcal{D}}_{X_{0:p}}+c_{\tau,2}s\right)^{s}\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\Pi_{m_{0:{\tilde{p}}+1}}\right\|$
$\displaystyle=\left(c_{\tau,1}\left\|\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\Pi_{m_{0:{\tilde{p}}+1}}\hat{\mathcal{D}}_{X_{0:p}}\Pi_{m_{0:{\tilde{p}}+1}}\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\right\|+c_{\tau,2}s\right)^{s},$
(371)

where we use the fact that $\hat{\mathcal{D}}_{X_{0:p}}$ commutes with
$\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}$ and $\Pi_{m_{0:{\tilde{p}}+1}}$. By
applying the inequalities (369), (Supplementary Note 5.3), (Supplementary Note
5.3) to (368), we obtain

$\displaystyle\left\|\Pi_{X[\ell],\leq N-\delta
N}e^{-iHt}|\psi_{m_{0:{\tilde{p}}+1}}\rangle\right\|$
$\displaystyle\leq\frac{\left(c_{\tau,1}\left\|\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\Pi_{m_{0:{\tilde{p}}+1}}\hat{\mathcal{D}}_{X_{0:p}}\Pi_{m_{0:{\tilde{p}}+1}}\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\right\|+c_{\tau,2}s\right)^{s/2}}{|N_{p}-N+\delta
N|^{s/2}}.$ (372)

By using the same analyses as the proof of Lemma 17, we prove the following
lemma:

######  Lemma 23.

Let us define

$\displaystyle\delta_{X,\ell,q_{\ast}}:=e^{-\ell/8}q_{\ast}|X|\left(\lambda_{1/2}+\gamma\lambda_{1/4}\ell^{D-1}e^{-\ell/16}\right),$
(373)

We then obtain the upper bound of

$\displaystyle\left\|\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\Pi_{m_{0:{\tilde{p}}+1}}\hat{\mathcal{D}}_{X_{0:p}}\Pi_{m_{0:{\tilde{p}}+1}}\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\right\|\leq\lambda_{1/2}\sum_{j=1}^{{\tilde{p}}}e^{-|p-j|/2}M_{j}+\delta_{X,\ell,q_{\ast}}\quad(1\leq
p\leq{\tilde{p}}),$ (374)

with

$\displaystyle M_{j}=\begin{cases}m_{j}&\quad\textrm{for}\quad 1\leq
j\leq{\tilde{p}}-1,\\\ m_{\tilde{p}}+m_{\tilde{p}+1}&\quad\textrm{for}\quad
j={\tilde{p}}.\end{cases}$ (375)

Proof of Lemma 23. From the inequality (Supplementary Note 3.4.1), we can
start from

$\displaystyle\left\|\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\Pi_{m_{0:{\tilde{p}}+1}}\hat{\mathcal{D}}_{X_{0:p}}\Pi_{m_{0:{\tilde{p}}+1}}\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\right\|$
$\displaystyle\leq\lambda_{1/2}\sum_{j=1}^{{\tilde{p}}}e^{-|p-j|/2}M_{j}+\sum_{i\in\partial(X_{0:p})}\sum_{j\in
X_{1:{\tilde{p}}+1}^{\rm
c}}e^{-d_{i,j}}\left\|\hat{n}_{j}\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\right\|$
$\displaystyle\leq\lambda_{1/2}\sum_{j=1}^{{\tilde{p}}}e^{-|p-j|/2}M_{j}+q_{\ast}\sum_{i\in\partial(X_{0:p})}\sum_{j\in
X_{1:{\tilde{p}}+1}^{\rm c}}e^{-d_{i,j}+d_{j,X[\ell]}/2}\quad(1\leq
p\leq{\tilde{p}}),$ (376)

where we use the definition (37) in the last inequality. We then decompose

$\displaystyle\sum_{i\in\partial(X_{0:p})}\sum_{j\in X_{1:{\tilde{p}}+1}^{\rm
c}}e^{-d_{i,j}+d_{j,X[\ell]}/2}=\sum_{i\in\partial(X_{0:p})}\sum_{j\in
X}e^{-d_{i,j}}+\sum_{i\in\partial(X_{0:p})}\sum_{j\in X[\ell]^{\rm
c}}e^{-d_{i,j}+d_{j,X[\ell]}/2},$ (377)

where we use $d_{j,X[\ell]}=0$ for $j\in X$. For the first term, we obtain

$\displaystyle\sum_{i\in\partial(X_{0:p})}\sum_{j\in
X}e^{-d_{i,j}}\leq\sum_{j\in X}\sum_{i:d_{i,j}\geq\ell/4}e^{-d_{i,j}}\leq
e^{-\ell/8}|X|\sum_{i\in\Lambda}e^{-d_{i,j}/2}\leq\lambda_{1/2}e^{-\ell/8}|X|,$
(378)

where, in the first inequality, we use $d_{i,j}\geq\ell/4$ for
$i\in\partial(X_{0:p})$ and $j\in X$ (see Fig. 9), and in the third
inequality, we apply the inequality (12). Also, for the second term in (377),
we use Eq. (37) to obtain

$\displaystyle\sum_{i\in\partial(X_{0:p})}\sum_{j\in X[\ell]^{\rm
c}}e^{-d_{i,j}+d_{j,X[\ell]}/2}$ $\displaystyle\leq
e^{-\ell/8}\sum_{i\in\partial(X_{0:p})}\sum_{j\in X[\ell]^{\rm
c}}e^{-d_{i,j}/2}$ $\displaystyle\leq
e^{-\ell/8}\sum_{i\in\partial(X_{0:p})}\sum_{j:d_{i,j}\geq\ell/4}e^{-d_{i,j}/2}$
$\displaystyle\leq\lambda_{1/4}e^{-3\ell/16}|\partial(X_{0:p})|\leq\gamma\lambda_{1/4}|X|\ell^{D-1}e^{-3\ell/16},$
(379)

where we use $d_{j,X[\ell]}\leq d_{i,j}-\ell/4$ for $i\in\partial(X_{0:p})$
and $j\in X[\ell]^{\rm c}$ in the first inequality, and in the last
inequality, we use $|\partial(X_{0:p})|\leq\left|\partial
X[\ell]\right|\leq\gamma\ell^{D-1}|X|$ from (7). By applying the inequalities
(378) and (Supplementary Note 5.3) to (Supplementary Note 5.3) and (377), we
have

$\displaystyle q_{\ast}\sum_{i\in\partial(X_{0:p})}\sum_{j\in
X_{1:{\tilde{p}}+1}^{\rm c}}e^{-d_{i,j}+d_{j,X[\ell]}/2}$ $\displaystyle\leq
e^{-\ell/8}q_{\ast}|X|\left(\lambda_{1/2}+\gamma\lambda_{1/4}\ell^{D-1}e^{-\ell/16}\right)=\delta_{X,\ell,q_{\ast}},$
(380)

where the last equation is the definition (373) for
$\delta_{X,\ell,q_{\ast}}$. By combining the inequalities (Supplementary Note
5.3) and (380), we prove the main inequality (374). This completes the proof.
$\square$

[ End of Proof of Lemma 23]

By combining the inequalities (372) and (374), we have

$\displaystyle\left\|\Pi_{X[\ell],\leq N-\delta
N}e^{-iHt}|\psi_{m_{0:{\tilde{p}}+1}}\rangle\right\|$
$\displaystyle\leq\frac{1}{|m_{0:p}-N+\delta
N|^{s/2}}\left(c_{\tau,1}\lambda_{1/2}\sum_{j=1}^{{\tilde{p}}}e^{-|p-j|/2}M_{j}+c_{\tau,1}\delta_{X,\ell,q_{\ast}}+c_{\tau,2}s\right)^{s/2}$
$\displaystyle=\left(\frac{\tilde{M}_{p}+2c_{\tau,2}s}{M_{1:p}+m_{0}-N+\delta
N}\right)^{s/2}\quad\textrm{for}\quad\forall p\in[1,\tilde{p}-1],$ (381)

where we define $M_{1:p}=\sum_{j=1}^{p}M_{p}$ and
$\tilde{M}_{p}:=c_{\tau,1}\lambda_{1/2}\sum_{j=1}^{{\tilde{p}}}e^{-|p-j|/2}M_{j}$.
Note that $c_{\tau,1}\delta_{X,\ell,q_{\ast}}\leq c_{\tau,2}s$ from the
condition (364).

Then, we aim to find the optimal $p$ which minimizes the RHS of the inequality
(Supplementary Note 5.3). We here prove the following lemma:

######  Lemma 24.

If the length $\ell$ satisfies

$\displaystyle\ell\geq\frac{2}{\tilde{f}_{\tau}}\log\left(\frac{5c_{\tau,1}\lambda_{1/2}e^{2\tilde{f}_{\tau}}\gamma(D/\tilde{f}_{\tau})^{D}q_{\ast}|X|}{2c_{\tau,2}}\right)$
(382)

with

$\displaystyle\tilde{f}_{\tau}\coloneqq\frac{1}{2}\log\left(1+\frac{1}{10ec_{\tau,1}\lambda_{1/2}+2}\right),$
(383)

we reduce the inequality (Supplementary Note 5.3) to the form of

$\displaystyle\left\|\Pi_{X[\ell],\leq N-\delta
N}e^{-iHt}|\psi_{m_{0:{\tilde{p}}+1}}\rangle\right\|$ $\displaystyle\leq
2\exp\left(\frac{-m_{0}+N-\delta N}{8ec_{\tau,2}}\right)$ (384)

by choosing $p\in[1,\tilde{p}-1]$ appropriately.

####  Supplementary Note 5.3.1 Proof of Lemma 24

We here consider the following two cases:

$\displaystyle\min_{p\in[1:{\tilde{p}}-1]}\left(\frac{\tilde{M}_{p}}{M_{1:p}}\right)\begin{cases}\leq\frac{1}{2e}&\quad\textrm{for}\quad\quad\textrm{Case
1},\\\ \geq\frac{1}{2e}&\quad\textrm{for}\quad\quad\textrm{Case
2}.\end{cases}$ (385)

[Case 1]
In that case, we can find $p_{\ast}$ such that

$\displaystyle\frac{\tilde{M}_{p_{\ast}}}{M_{1:p_{\ast}}}\leq\frac{1}{2e}.$
(386)

By using the inequality of

$\displaystyle\frac{x^{\prime}+y^{\prime}}{x+y}=\frac{x^{\prime}}{x+y}+\frac{y^{\prime}}{x+y}\leq\frac{x^{\prime}}{x}+\frac{y^{\prime}}{y}$
(387)

for arbitrary positive $x,y,x^{\prime},y^{\prime}$, we have

$\displaystyle\frac{\tilde{M}_{p_{\ast}}+2c_{\tau,2}s}{M_{1:p_{\ast}}+m_{0}-N+\delta
N}\leq\frac{\tilde{M}_{p_{\ast}}}{M_{1:p_{\ast}}}+\frac{2c_{\tau,2}s}{m_{0}-N+\delta
N}\leq\frac{1}{2e}+\frac{2c_{\tau,2}s}{m_{0}-N+\delta N}.$ (388)

Therefore, we choose $s$ as

$\displaystyle s=\left\lfloor\frac{m_{0}-N+\delta
N}{4ec_{\tau,2}}\right\rfloor,$ (389)

which reduces the inequality (388) to

$\displaystyle\frac{\tilde{M}_{p_{\ast}}+2c_{\tau,2}s}{M_{1:p_{\ast}}+m_{0}-N+\delta
N}\leq\frac{1}{e}.$ (390)

By applying the above inequality to (Supplementary Note 5.3), we obtain the
desired inequality of

$\displaystyle\left\|\Pi_{X[\ell],\leq N-\delta
N}e^{-iHt}|\psi_{m_{0:{\tilde{p}}+1}}\rangle\right\|$ $\displaystyle\leq
2\exp\left(\frac{-m_{0}+N-\delta N}{8ec_{\tau,2}}\right),$ (391)

where we use $e^{1/2}<2$.

[Case 2]
In this case, we can utilize Lemma 15 with $\Phi=2e$ to obtain

$\displaystyle\tilde{M}_{1}$
$\displaystyle\leq\left(1+\frac{1}{2e\bar{\alpha}_{\infty}+\chi_{\alpha}}\right)^{-{\tilde{p}}+2}\bar{\alpha}_{\infty}M_{1:{\tilde{p}}-1}$
$\displaystyle\leq\left(1+\frac{1}{10ec_{\tau,1}\lambda_{1/2}+2}\right)^{-{\tilde{p}}+2}5c_{\tau,1}\lambda_{1/2}M_{1:{\tilde{p}}-1}=5c_{\tau,1}\lambda_{1/2}e^{-\tilde{f}_{\tau}(2\tilde{p}-4)}M_{1:{\tilde{p}}-1}$
(392)

with $\tilde{f}_{\tau}$ defined in Eq. (383), where the parameters
$\bar{\alpha}_{\infty}$ and $\chi_{\alpha}$ have been estimated as in Eqs.
(252) and (253): $\bar{\alpha}_{\infty}=(4.0829\cdots)c_{\tau,1}\lambda_{1/2}$
and $\chi_{\alpha}=1.5414\cdots$. From ${\tilde{p}}=\ell/2+1$ and the upper
bound of

$\displaystyle M_{1:{\tilde{p}}-1}\leq
M_{0:{\tilde{p}}}\leq\left\|\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\hat{n}_{X[\ell]}\right\|\leq
q_{\ast}|X[\ell]|\leq\gamma\ell^{D}q_{\ast}|X|,$ (393)

we reduce the inequality (Supplementary Note 5.3.1) to

$\displaystyle\tilde{M}_{1}$ $\displaystyle\leq
5c_{\tau,1}\lambda_{1/2}e^{2\tilde{f}_{\tau}}\gamma\ell^{D}q_{\ast}|X|e^{-\tilde{f}_{\tau}\ell}\leq
5c_{\tau,1}\lambda_{1/2}e^{2\tilde{f}_{\tau}}\gamma(D/\tilde{f}_{\tau})^{D}q_{\ast}|X|e^{-\tilde{f}_{\tau}\ell/2},$
(394)

where we use the inequality of $x^{z}e^{-x}\leq z^{z}e^{-x/2}$ for $z>0$.
Therefore, by choosing

$\displaystyle
e^{\tilde{f}_{\tau}\ell/2}\geq\frac{5c_{\tau,1}\lambda_{1/2}e^{2\tilde{f}_{\tau}}\gamma(D/\tilde{f}_{\tau})^{D}q_{\ast}|X|}{2c_{\tau,2}}\longrightarrow\ell\geq\frac{2}{\tilde{f}_{\tau}}\log\left(\frac{5c_{\tau,1}\lambda_{1/2}e^{2\tilde{f}_{\tau}}\gamma(D/\tilde{f}_{\tau})^{D}q_{\ast}|X|}{2c_{\tau,2}}\right),$
(395)

we have $\tilde{M}_{1}\leq 2c_{\tau,2}\leq 2c_{\tau,2}s$ from the inequality
(394).

Then, for $p=1$, the inequality (Supplementary Note 5.3) reduces to

$\displaystyle\left\|\Pi_{X[\ell],\leq N-\delta
N}e^{-iHt}|\psi_{m_{0:{\tilde{p}}+1}}\rangle\right\|$
$\displaystyle\leq\left(\frac{\tilde{M}_{1}+2c_{\tau,2}s}{M_{1}+m_{0}-N+\delta
N}\right)^{s/2}\leq\left(\frac{4c_{\tau,2}s}{m_{0}-N+\delta N}\right)^{s/2}.$
(396)

By choosing

$\displaystyle s=\left\lfloor\frac{m_{0}-N+\delta
N}{4ec_{\tau,2}}\right\rfloor,$ (397)

we reduce the inequality (396) to

$\displaystyle\left\|\Pi_{X[\ell],\leq N-\delta
N}e^{-iHt}|\psi_{m_{0:{\tilde{p}}+1}}\rangle\right\|$ $\displaystyle\leq
2\exp\left(\frac{-m_{0}+N-\delta N}{8ec_{\tau,2}}\right).$ (398)

From the inequalities (391) and (398), we prove the main inequality (384).
This completes the proof. $\square$

[ End of Proof of Lemma 24]

By combining the inequalities (Supplementary Note 5.3) and (384), we obtain

$\displaystyle\left\|\Pi_{X[\ell],\leq N-\delta N}|\psi(\tau)\rangle\right\|$
$\displaystyle\leq\sum_{m_{0},m_{1},\ldots,m_{\tilde{p}+1}}2\exp\left(\frac{-m_{0}+N-\delta
N}{8ec_{\tau,2}}\right).$ (399)

Because of $\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}|\psi\rangle=|\psi\rangle$,
the summation for $m_{p}$ is taken from $m_{p}=0$ to

$\displaystyle
m_{p}=\left\|\Pi_{\mathcal{S}(X[\ell],1/2,q_{\ast})}\hat{n}_{X_{p}}\right\|\leq
q_{\ast}|X_{p}|\leq q_{\ast}\gamma\ell^{D}|X|,$ (400)

where we use $|X_{p}|\leq|X[\ell]|\leq\gamma\ell^{D}|X|$. We recall that
$m_{p}$ is defined as the number of bosons in the region $X_{p}$ as in Eq.
(366). Also, the summation with respect to $m_{0}$ is taken from $m_{0}=N$,
and hence, we have

$\displaystyle\sum_{m_{0}=N}^{\infty}\exp\left(\frac{-m_{0}+N-\delta
N}{8ec_{\tau,2}}\right)\leq\frac{1}{1-e^{-1/(8ec_{\tau,2})}}\exp\left(\frac{-\delta
N}{8ec_{\tau,2}}\right).$ (401)

Therefore, the summation in (399) is upper-bounded by

$\displaystyle\left\|\Pi_{X[\ell],\leq N-\delta N}|\psi(\tau)\rangle\right\|$
$\displaystyle\leq\frac{2\left(q_{\ast}\gamma\ell^{D}|X|\right)^{{\tilde{p}}+1}}{1-e^{-1/(8ec_{\tau,2})}}\exp\left(\frac{-\delta
N}{8ec_{\tau,2}}\right)$
$\displaystyle=\frac{2}{1-e^{-1/(8ec_{\tau,2})}}\exp\left[\frac{-\delta
N}{8ec_{\tau,2}}+\frac{\ell+4}{2}\log\left(q_{\ast}\gamma\ell^{D}|X|\right)\right],$
(402)

where we use $\tilde{p}=\ell/2+1$. Under condition (382), we can choose $\ell$
such that

$\displaystyle\ell=\left\lceil\frac{2}{\tilde{f}_{\tau}}\log\left(\frac{5c_{\tau,1}\lambda_{1/2}e^{2\tilde{f}_{\tau}}\gamma(D/\tilde{f}_{\tau})^{D}q_{\ast}|X|}{2c_{\tau,2}}\right)\right\rceil=c^{\prime}_{\ell}\log\left(q_{\ast}\gamma|X|\right),$
(403)

where $c^{\prime}_{\ell}$ is a constant of $\mathcal{O}(1)$. Then, we obtain
an upper bound for

$\displaystyle\frac{\ell+4}{2}\log\left(q_{\ast}\gamma\ell^{D}|X|\right)$
$\displaystyle=\frac{\ell+4}{2}\left[\log\left(q_{\ast}\gamma|X|\right)+D\log\left(\ell\right)\right]=\frac{\ell+4}{2}\left(\frac{\ell}{c_{\ell}^{\prime}}+D\log\left(\ell\right)\right)$
$\displaystyle\leq
3\left(\frac{1}{c_{\ell}^{\prime}}+D\right)\ell^{2}=3c_{\ell}^{\prime}\left(1+c_{\ell}^{\prime}D\right)\log^{2}\left(q_{\ast}\gamma|X|\right),$
(404)

where in the inequality, we use $\ell\geq 1$, $\ell/2+2\leq 5\ell/2\leq
3\ell$, and $\log(\ell)\leq\ell$. By combining the inequalities (Supplementary
Note 5.3.1) and (Supplementary Note 5.3.1), we obtain the main inequality
(359). This completes the proof. $\square$

##  Supplementary Note 6 Effective Hamiltonian theory

In this section, we consider an effective Hamiltonian in the form of

$\displaystyle\bar{\Pi}H\bar{\Pi},$ (405)

where $\bar{\Pi}$ is an arbitrary projection such that
$[\bar{\Pi},\hat{n}_{i}]=0$ for $\forall i\in\Lambda$. Then, we often
encounter a problem to estimate an error between the time evolutions
$e^{-i\bar{\Pi}H\bar{\Pi}t}$ and $e^{-iHt}$. Usually, we cannot ensure

$\displaystyle\left\|e^{-i\bar{\Pi}H\bar{\Pi}t}-e^{-iHt}\right\|$ (406)

in the level of the operator norm. If we consider specific quantum state
$|\psi\rangle$ and operator $O$, we may be able to prove

$\displaystyle\left\|O(H,\tau)|\psi\rangle-O(\bar{\Pi}H\bar{\Pi},\tau)|\psi\rangle\right\|\approx
0.$ (407)

In general, we can prove the following lemma which upper-bounds the error
using the effective Hamiltonian.

######  Lemma 25 (Generalization of Lemma 13 in Ref. [1]).

Let $|\psi\rangle$ and $O$ be a quantum state and an operator of interest,
respectively. Then, for arbitrary projection $\bar{\Pi}$ and $\tau$, we obtain

$\displaystyle\left\|O(H,\tau)|\psi\rangle-O(\bar{\Pi}H\bar{\Pi},\tau)|\psi\rangle\right\|$
$\displaystyle\leq\left\|\bar{\Pi}^{\rm
c}e^{iH\tau}Oe^{-iH\tau}|\psi\rangle\right\|+\left\|\bar{\Pi}^{\rm
c}Oe^{-iH\tau}|\psi\rangle\right\|+\left\|\bar{\Pi}^{\rm
c}|\psi\rangle\right\|+\left\|\bar{\Pi}^{\rm
c}e^{-iH\tau}|\psi\rangle\right\|$
$\displaystyle+\int_{0}^{\tau}\left\|\bar{\Pi}H_{0}\bar{\Pi}^{\rm
c}e^{iH(\tau-\tau_{1})}Oe^{-iH\tau}|\psi\rangle\right\|d\tau_{1}+\int_{0}^{\tau}\left\|\bar{\Pi}H_{0}\bar{\Pi}^{\rm
c}e^{-iH\tau_{1}}|\psi\rangle\right\|d\tau_{1}$ (408)

with $\bar{\Pi}^{\rm c}:=1-\bar{\Pi}$, where we set $\left\|O\right\|=1$.

We can easily extend the statement to a mixed state $\rho$:

######  Corollary 26.

Let $\rho$ be an arbitrary quantum state. Then, we generalize the inequality
(25) to

$\displaystyle\left\|O(H,\tau)\rho-O(\bar{\Pi}H\bar{\Pi},\tau)\rho\right\|_{1}$
$\displaystyle\leq\left\|\bar{\Pi}^{\rm
c}O(\tau)\sqrt{\rho}\right\|_{F}+\left\|\bar{\Pi}^{\rm
c}O\sqrt{\rho(\tau)}\right\|_{F}+\left\|\bar{\Pi}^{\rm
c}\sqrt{\rho}\right\|_{F}+\left\|\bar{\Pi}^{\rm
c}\sqrt{\rho(\tau)}\right\|_{F}$
$\displaystyle+\int_{0}^{\tau}\left\|\bar{\Pi}H_{0}\bar{\Pi}^{\rm
c}O(\tau-\tau_{1})\sqrt{\rho(\tau_{1})}\right\|_{F}d\tau_{1}+\int_{0}^{\tau}\left\|\bar{\Pi}H_{0}\bar{\Pi}^{\rm
c}\sqrt{\rho(\tau_{1})}\right\|_{F}d\tau_{1},$ (409)

where $\left\|\cdots\right\|_{F}$ is the Frobenius norm, i.e.,
$\left\|O\right\|_{F}=\sqrt{{\rm tr}(OO^{\dagger})}$.

Proof of Corollary 26. Let us decompose the state $\rho$ as

$\displaystyle\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|.$ (410)

Then, for an arbitrary operator $\tilde{O}$, we have

$\left\|\tilde{O}|\psi_{i}\rangle\langle\psi_{i}|\right\|_{1}={\rm
tr}\left(\sqrt{\tilde{O}|\psi_{i}\rangle\langle\psi_{i}|\tilde{O}^{\dagger}}\right)=\sqrt{\langle\psi_{i}|\tilde{O}^{\dagger}\tilde{O}|\psi_{i}\rangle}=\left\|\tilde{O}|\psi_{i}\rangle\right\|,$

and hence

$\displaystyle\left\|O(H,\tau)\rho-O(\bar{\Pi}H\bar{\Pi},\tau)\rho\right\|_{1}$
$\displaystyle\leq\sum_{i}p_{i}\biggl{(}\left\|\bar{\Pi}^{\rm
c}e^{iH\tau}Oe^{-iH\tau}|\psi_{i}\rangle\right\|+\left\|\bar{\Pi}^{\rm
c}Oe^{-iH\tau}|\psi_{i}\rangle\right\|+\left\|\bar{\Pi}^{\rm
c}|\psi_{i}\rangle\right\|+\left\|\bar{\Pi}^{\rm
c}e^{-iH\tau}|\psi_{i}\rangle\right\|$
$\displaystyle+\int_{0}^{\tau}\left\|\bar{\Pi}H_{0}\bar{\Pi}^{\rm
c}e^{iH(\tau-\tau_{1})}Oe^{-iH\tau}|\psi_{i}\rangle\right\|d\tau_{1}+\int_{0}^{\tau}\left\|\bar{\Pi}H_{0}\bar{\Pi}^{\rm
c}e^{-iH\tau_{1}}|\psi_{i}\rangle\right\|d\tau_{1}\biggr{)}.$ (411)

Moreover, for an arbitrary operator $\tilde{O}$, we prove

$\displaystyle\sum_{i}p_{i}\left\|\tilde{O}|\psi_{i}\rangle\right\|=\sum_{i}p_{i}\sqrt{\langle\psi_{i}|\tilde{O}^{\dagger}\tilde{O}|\psi_{i}\rangle}\leq\sqrt{\sum_{i}p_{i}\langle\psi_{i}|\tilde{O}^{\dagger}\tilde{O}|\psi_{i}\rangle}=\sqrt{{\rm
tr}\left(\tilde{O}\rho\tilde{O}^{\dagger}\right)}=\left\|\tilde{O}\sqrt{\rho}\right\|_{F}.$
(412)

By combining the above two inequalities, we derive the desired inequality
(26). This completes the proof. $\square$

As another extension, we can also generalize the results to the time-dependent
Hamiltonian $H_{t}$, which is utilized in proving Lemma 46 below. Here, we
utilize the notations in Eq. (Supplementary Note 1.4), i.e.,
$U_{A_{t},\tau_{1}\to\tau_{2}}=\mathcal{T}e^{-i\int_{\tau_{1}}^{\tau_{2}}A_{t}dt}$
and
$O(A_{t},\tau_{1}\to\tau_{2})=U_{A_{t},\tau_{1}\to\tau_{2}}^{\dagger}OU_{A_{t},\tau_{1}\to\tau_{2}}$.

######  Corollary 27.

Let us assume that $H_{t}$ satisfies $\left\|dH_{t}/dt\right\|<\infty$. For
simplicity, let us denote

$\displaystyle U_{\tau^{\prime}}=U_{H_{t},0\to\tau^{\prime}}$ (413)

for an arbitrary $\tau^{\prime}$. We then obtain

$\displaystyle\left\|O(H_{t},0\to\tau)\rho-O(\bar{\Pi}H_{t}\bar{\Pi},0\to\tau)\rho\right\|_{1}\leq\left\|\bar{\Pi}^{\rm
c}U_{\tau}^{\dagger}OU_{\tau}\sqrt{\rho}\right\|_{F}+\left\|\bar{\Pi}^{\rm
c}OU_{\tau}\sqrt{\rho}\right\|_{F}+\left\|\bar{\Pi}^{\rm
c}\sqrt{\rho}\right\|_{F}+\left\|\bar{\Pi}^{\rm
c}U_{\tau}\sqrt{\rho}\right\|_{F}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\int_{0}^{\tau}\left\|\bar{\Pi}H_{\tau-\tau_{1}}\bar{\Pi}^{\rm
c}U_{\tau_{1}}U_{\tau}^{\dagger}OU_{\tau}\sqrt{\rho}\right\|_{F}d\tau_{1}+\int_{0}^{\tau}\left\|\bar{\Pi}H_{\tau-\tau_{1}}\bar{\Pi}^{\rm
c}U_{\tau_{1}}\sqrt{\rho}\right\|_{F}d\tau_{1}.$ (414)

The proof is exactly the same as for Lemma 25.

####  Supplementary Note 6.0.1 Proof of Lemma 25

We take the same approach as Ref. [1, Supplementary Lemma 13]. For the
convenience of the readers, we show the full proof of this lemma. Throughout
the proof, we denote

$\displaystyle\bar{\Pi}^{\rm
c}=1-\bar{\Pi},\quad\tilde{H}=\bar{\Pi}H\bar{\Pi}$ (415)

for simplicity. We start from the inequality as

$\displaystyle\left\|O(H,\tau)|\psi\rangle-O(\tilde{H},\tau)|\psi\rangle\right\|$
$\displaystyle\leq\left\|e^{iH\tau}Oe^{-iH\tau}|\psi\rangle-e^{i\tilde{H}\tau}Oe^{-iH\tau}|\psi\rangle\right\|+\left\|e^{i\tilde{H}\tau}O(e^{-iH\tau}-e^{-i\tilde{H}\tau})|\psi\rangle\right\|$
$\displaystyle\leq\left\|e^{iH\tau}Oe^{-iH\tau}|\psi\rangle-e^{i\tilde{H}\tau}Oe^{-iH\tau}|\psi\rangle\right\|+\left\||\psi\rangle-e^{i\tilde{H}\tau}e^{-iH\tau}|\psi\rangle\right\|,$
(416)

where the first inequality is derived from the triangle inequality, and we use
$\|O\|=1$ in the second inequality. In order to estimate the RHS in the
inequality (Supplementary Note 6.0.1), we need to upper-bound

$\displaystyle\left\|e^{iH\tau}Oe^{-iH\tau}|\psi\rangle-e^{i\tilde{H}\tau}Oe^{-iH\tau}|\psi\rangle\right\|$
(417)

for an arbitrary $O$. We note that by choosing $O=\hat{1}$ the second term in
the RHS of (Supplementary Note 6.0.1) is also given by the form of (417). For
the estimation of (417), we begin with the inequality of

$\displaystyle\left\|e^{iH\tau}Oe^{-iH\tau}|\psi\rangle-e^{i\tilde{H}\tau}Oe^{-iH\tau}|\psi\rangle\right\|$
$\displaystyle\leq$ $\displaystyle\left\|\bar{\Pi}^{\rm
c}\left(e^{iH\tau}Oe^{-iH\tau}|\psi\rangle-e^{i\tilde{H}\tau}Oe^{-iH\tau}|\psi\rangle\right)\right\|+\left\|\bar{\Pi}\left(e^{iH\tau}Oe^{-iH\tau}|\psi\rangle-e^{i\tilde{H}\tau}Oe^{-iH\tau}|\psi\rangle\right)\right\|$
$\displaystyle=$ $\displaystyle\left\|\bar{\Pi}^{\rm
c}e^{iH\tau}Oe^{-iH\tau}|\psi\rangle\right\|+\left\|\bar{\Pi}^{\rm
c}Oe^{-iH\tau}|\psi\rangle\right\|+\left\|\bar{\Pi}\left(e^{iH\tau}O-e^{i\tilde{H}\tau}O\right)e^{-iH\tau}|\psi\rangle\right\|,$
(418)

where, in the equation, we use

$\displaystyle\left\|\bar{\Pi}^{\rm
c}e^{i\tilde{H}\tau}Oe^{-iH\tau}|\psi\rangle\right\|=\left\|\bar{\Pi}^{\rm
c}Oe^{-iH\tau}|\psi\rangle\right\|$ (419)

because of $\bar{\Pi}^{\rm c}\tilde{H}=\bar{\Pi}^{\rm
c}\bar{\Pi}H\bar{\Pi}=0$.

In order to upper-bound the third term in the RHS of (Supplementary Note
6.0.1), let us decompose the time to infinitely many pieces (i.e.,
$\tau=m_{0}d\tau$ and $m_{0}\to\infty$). We then estimate the norm
$\left\|\bar{\Pi}\left(e^{iH\tau}O-e^{i\tilde{H}\tau}O\right)e^{-iH\tau}|\psi\rangle\right\|$
by using the following equation:

$\displaystyle e^{iH\tau}O-e^{i\tilde{H}\tau}O$
$\displaystyle=\sum_{m=0}^{m_{0}-1}\left(e^{i\tilde{H}md\tau}e^{iH(m_{0}-m)d\tau}O-e^{i\tilde{H}(m+1)d\tau}e^{iH(m_{0}-1-m)d\tau}O\right)$
$\displaystyle=\sum_{m=0}^{m_{0}-1}\left(e^{i\tilde{H}md\tau}e^{iH(\tau-
md\tau)}O-e^{i\tilde{H}(md\tau+d\tau)}e^{iH(\tau-md\tau-d\tau)}O\right)$
$\displaystyle=-i\int_{0}^{\tau}e^{i\tilde{H}\tau_{1}}\left(\tilde{H}-H\right)e^{iH(\tau-\tau_{1})}Od\tau_{1}$
(420)

in the limit of $m_{0}\to\infty$, where, in the last equation, we use

$\displaystyle e^{i\tilde{H}md\tau}e^{iH(\tau-
md\tau)}O-e^{i\tilde{H}(md\tau+d\tau)}e^{iH(\tau-md\tau-d\tau)}O$
$\displaystyle=$ $\displaystyle-
ie^{i\tilde{H}md\tau}\left(\tilde{H}-H\right)e^{iH(\tau-
md\tau)}Od\tau+\mathcal{O}(d\tau^{2}).$ (421)

We consider the decomposition of

$\displaystyle\bar{\Pi}\left(\tilde{H}-H\right)=\bar{\Pi}H\bar{\Pi}-\bar{\Pi}H\left(\bar{\Pi}+1-\bar{\Pi}\right)=-\bar{\Pi}H\bar{\Pi}^{\rm
c}=-\bar{\Pi}H_{0}\bar{\Pi}^{\rm c},$ (422)

where we use $[\bar{\Pi},\tilde{H}]=0$ and $[V,\bar{\Pi}]=0$ which is derived
from $[\hat{n}_{i},\bar{\Pi}]=0$ for $\forall i\in\Lambda$. We recall that
$\bar{\Pi}^{\rm c}$ has been defined as $\bar{\Pi}^{\rm c}:=1-\bar{\Pi}$.
Therefore, from Eq. (Supplementary Note 6.0.1), we obtain

$\displaystyle\left\|\bar{\Pi}\left(e^{iH\tau}O-e^{i\tilde{H}\tau}O\right)e^{-iH\tau}|\psi\rangle\right\|=$
$\displaystyle\left\|i\int_{0}^{\tau}e^{i\tilde{H}\tau_{1}}\bar{\Pi}H_{0}\bar{\Pi}^{\rm
c}e^{iH(\tau-\tau_{1})}d\tau_{1}\cdot Oe^{-iH\tau}|\psi\rangle\right\|$
$\displaystyle\leq$
$\displaystyle\int_{0}^{\tau}\left\|\bar{\Pi}H_{0}\bar{\Pi}^{\rm
c}e^{iH(\tau-\tau_{1})}Oe^{-iH\tau}|\psi\rangle\right\|d\tau_{1},$ (423)

which reduces the inequality (Supplementary Note 6.0.1) to

$\displaystyle\left\|e^{iH\tau}Oe^{-iH\tau}|\psi\rangle-e^{i\tilde{H}\tau}Oe^{-iH\tau}|\psi\rangle\right\|$
$\displaystyle\leq$ $\displaystyle\left\|\bar{\Pi}^{\rm
c}e^{iH\tau}Oe^{-iH\tau}|\psi\rangle\right\|+\left\|\bar{\Pi}^{\rm
c}Oe^{-iH\tau}|\psi\rangle\right\|+\int_{0}^{\tau}\left\|\bar{\Pi}H_{0}\bar{\Pi}^{\rm
c}e^{iH(\tau-\tau_{1})}Oe^{-iH\tau}|\psi\rangle\right\|d\tau_{1}.$ (424)

In the same way, we can derive

$\displaystyle\left\||\psi\rangle-e^{i\tilde{H}\tau}e^{-iH\tau}|\psi\rangle\right\|\leq\left\|\bar{\Pi}^{\rm
c}|\psi\rangle\right\|+\left\|\bar{\Pi}^{\rm
c}e^{-iH\tau}|\psi\rangle\right\|+\int_{0}^{\tau}\left\|\bar{\Pi}H_{0}\bar{\Pi}^{\rm
c}e^{-iH\tau_{1}}|\psi\rangle\right\|d\tau_{1}$ (425)

by choosing $O=\hat{1}$ in the inequality (Supplementary Note 6.0.1).
Therefore, by applying the inequalities (Supplementary Note 6.0.1) and (425)
to (Supplementary Note 6.0.1), we obtain the desired inequality (25). This
completes the proof. $\square$

###  Supplementary Note 6.1 Probability distribution for boson number after
time evolution

In estimating the errors in (25) for the effective Hamiltonian, we have to
estimate the norms like

$\displaystyle\left\|\bar{\Pi}^{\rm
c}e^{iH\tau_{2}}Oe^{-iH\tau_{1}}|\psi\rangle\right\|\quad{\rm
and}\quad\left\|\bar{\Pi}H_{0}\bar{\Pi}^{\rm
c}e^{iH\tau_{2}}Oe^{-iH\tau_{1}}|\psi\rangle\right\|.$ (426)

For this purpose, we let

$\displaystyle O\to O_{X_{0}}\quad(X_{0}\subseteq
i_{0}[r_{0}]),\quad|\psi\rangle\to\rho_{0},$ (427)

where $\rho_{0}$ satisfies the low-boson density condition in Assumption 3
with $b_{\rho}=b_{0}/\gamma$. We also assume that the operator $O_{X_{0}}$
satisfies the following condition, which is characterized by a parameter
$q_{0}$:

$\displaystyle\Pi_{X_{0},>m+q_{0}}O_{X_{0}}\Pi_{X_{0},m}=0,\quad\Pi_{X_{0},<m-q_{0}}O_{X_{0}}\Pi_{X_{0},m}=0\quad\textrm{for}\quad\forall
m>0.$ (428)

Here, the constant $q_{0}$ is the maximum number of boson creation (or
annihilation) by the operator $O_{X_{0}}$. The parameter $q_{0}$ appears in
the Lieb-Robinson bound (see Theorems 2, 3 and 4). For example, if $O_{X_{0}}$
conserve the boson number in the region $X_{0}$, we have $q_{0}=0$. We also
set the norm of $O_{X_{0}}$ as

$\displaystyle\left\|O_{X_{0}}\right\|=\zeta_{0}$ (429)

for simplicity.

In the above setup, we consider a quantum state of*3*3*3In the definition
(431), we consider a time evolution by a time-independent Hamiltonian. But,
all the discussion can be straightforwardly generalized to
$\displaystyle\tilde{\rho}_{\tau_{1},\tau_{2}}:=U_{H^{\prime}_{x},0\to\tau_{2}}^{\dagger}O_{X_{0}}U_{H_{x},0\to\tau_{1}}\rho_{0}U_{H_{x},0\to\tau_{1}}^{\dagger}O_{X_{0}}^{\dagger}U_{H^{\prime}_{x},0\to\tau_{2}},$
(430) where $H_{x}$ and $H_{x}^{\prime}$ are time-dependent Hamiltonians and
we use the notation (Supplementary Note 1.4).

$\displaystyle\tilde{\rho}_{\tau_{1},\tau_{2}}:=e^{iH\tau_{2}}O_{X_{0}}e^{-iH\tau_{1}}\rho_{0}e^{iH\tau_{1}}O_{X_{0}}^{\dagger}e^{-iH\tau_{2}}\quad{\rm
with}\quad|\tau_{1}|,|\tau_{2}|\leq t.$ (431)

Note that $\tilde{\rho}_{\tau_{1},\tau_{2}}$ is simply equal to $\rho_{0}(t)$
by choosing $\tau_{1}=t$, $\tau_{2}=0$ and $O_{X_{0}}=\hat{1}$. Our purpose in
this section is to estimate the upper bound of the probability distribution

$\displaystyle{\rm
tr}\left[\Pi_{i,>q_{i}}\tilde{\rho}_{\tau_{1},\tau_{2}}\right]\quad\textrm{for}\quad
i\in\Lambda.$ (432)

As has been proved in Proposition 20, the distribution decays
(sub)exponentially for $q_{i}\gtrsim[t\log(t)]^{D}$ if the site $i$ is
sufficiently separated from the region $X_{0}$. We here prove the following
lemma:

######  Lemma 28.

We define $\bar{\ell}_{t,R}$ as

$\displaystyle\bar{\ell}_{t,R}:=\bar{c}_{0}t\log(Rq_{0}),$ (433)

where $\bar{c}_{0}$ is an appropriate constant and $R$ is a length scale such
that $R\geq t$ and $R\geq r_{0}$ [see Eq. (427) for the definition of
$r_{0}$]. Then, for a quantum state $\tilde{\rho}_{\tau_{1},\tau_{2}}$ in Eq.
(431), we obtain

$\displaystyle{\rm
tr}\left(\hat{n}_{i[\ell_{0}]}^{p}\Pi_{i[\ell_{0}],>q_{i}}\tilde{\rho}_{\tau_{1},\tau_{2}}\right)\leq\zeta_{0}^{2}q_{i}^{p}e^{-\left[q_{i}/(4b_{0}\ell_{0}^{D})\right]^{1/\kappa}}\quad\textrm{for}\quad
i[\ell_{0}]\subset X_{0}[\bar{\ell}_{t,R}]^{\rm c},$ (434)

and

$\displaystyle{\rm
tr}\left(\hat{n}_{X_{0}[\bar{\ell}_{t,R}]}^{p}\Pi_{X_{0}[\bar{\ell}_{t,R}],>q}\tilde{\rho}_{\tau_{1},\tau_{2}}\right)\leq\zeta_{0}^{2}q^{p}e^{-\left[(q-32e^{2}q_{0}^{3})/\xi_{t,R}\right]^{1/\kappa}},\quad\xi_{t,R}:=16b_{0}\bar{\ell}_{t,R}^{2D}|X_{0}|,$
(435)

where $k\in\mathbb{N}$ and $\ell_{0}$ is assumed to be larger than
$\bar{\ell}_{t,R}$ ($\ell_{0}\geq\bar{\ell}_{t,R}$).

Supplementary Figure 10: For $\forall i[\ell_{0}]\in X[\bar{\ell}_{t,R}]^{\rm
c}$ with $\ell_{0}<\bar{\ell}_{t,R}$, we can find another site $i^{\prime}\in
X^{\rm c}$ such that $i^{\prime}[\bar{\ell}_{t,R}]\subset
X[\bar{\ell}_{t,R}]^{\rm c}$ and $i^{\prime}[\bar{\ell}_{t,R}]\supset
i[\ell_{0}]$.

Remark. In the inequality (434), the condition $\ell_{0}\geq\bar{\ell}_{t,R}$
is assumed. In the case of $\ell_{0}<\bar{\ell}_{t,R}$, we can still prove the
following inequality:

$\displaystyle{\rm
tr}\left(\hat{n}_{i[\ell_{0}]}^{p}\Pi_{i[\ell_{0}],>q_{i}}\tilde{\rho}_{\tau_{1},\tau_{2}}\right)\leq\zeta_{0}^{2}q_{i}^{p}e^{-\left[q_{i}/(4b_{0}\bar{\ell}_{t,R}^{D})\right]^{1/\kappa}}\quad\textrm{for}\quad
i[\ell_{0}]\subset X_{0}[\bar{\ell}_{t,R}]^{\rm c}.$ (436)

The proof is given as follows. First, for $i[\ell_{0}]\subset
X_{0}[\bar{\ell}_{t,R}]^{\rm c}$, there exists a site $i^{\prime}$ such that
$i^{\prime}[\bar{\ell}_{t,R}]\supset i[\ell_{0}]$ and
$i^{\prime}[\bar{\ell}_{t,R}]\subset X_{0}[\bar{\ell}_{t,R}]^{\rm c}$ (see
Fig. 10). Hence, we obtain

$\displaystyle{\rm
tr}\left(\hat{n}_{i[\ell_{0}]}^{p}\Pi_{i[\ell_{0}],>q_{i}}\tilde{\rho}_{\tau_{1},\tau_{2}}\right)$
$\displaystyle\leq{\rm
tr}\left(\hat{n}_{i^{\prime}[\bar{\ell}_{t,R}]}^{p}\Pi_{i^{\prime}[\bar{\ell}_{t,R}],>q_{i}}\tilde{\rho}_{\tau_{1},\tau_{2}}\right),$
(437)

which yields the inequality (436) from (434) by choosing
$\ell_{0}=\bar{\ell}_{t,R}$.

####  Supplementary Note 6.1.1 Proof of Lemma 28

The proof relies on the iterative uses of Proposition 20. In the first step,
we consider the moment function of

$\displaystyle{\rm
tr}\left(\rho_{0}(\tau_{1})\hat{n}_{i[\ell_{0}]}^{s}\right)\quad\textrm{for}\quad|\tau_{1}|\leq
t$ (438)

for $\forall i\in\Lambda$. From Assumption 3 with $b_{\rho}=b_{0}/\gamma$ in
(34), the initial state $\rho_{0}$ satisfies

$\displaystyle{\rm
tr}\left(\rho_{0}\hat{n}_{i[\ell_{0}]}^{s}\right)\leq\frac{1}{e}\left(\frac{b_{0}}{e}s^{\kappa}\ell_{0}^{D}\right)^{s}.$
(439)

Hence, we obtain from Proposition 20

$\displaystyle{\rm
tr}\left(\rho_{0}(\tau_{1})\hat{n}_{i[\ell_{0}]}^{s}\right)\leq\frac{1}{e}\left(\frac{2b_{0}}{e}s^{\kappa}\ell_{0}^{D}\right)^{s}\quad\textrm{for}\quad\forall
i\in\Lambda,\quad\ell_{0}\geq\bar{\ell}_{1},$ (440)

where $\bar{\ell}_{1}$ is determined by the condition (324), i.e.,
$\bar{\ell}_{1}\propto t\log t$.

In the second step, we first consider the moment function of

$\displaystyle{\rm
tr}\left(O_{X_{0}}\rho_{0}(\tau_{1})O_{X_{0}}^{\dagger}\hat{n}_{i[\ell_{0}]}^{s}\right)=:{\rm
tr}\left(\tilde{\rho}_{\tau_{1}}\hat{n}_{i[\ell_{0}]}^{s}\right)\quad\textrm{for}\quad
i[\ell_{0}]\subset X_{0}^{\rm c}.$ (441)

From the inequality (440) and
$\left[O_{X_{0}},\hat{n}_{i[\ell_{0}]}^{s}\right]=0$, we trivially obtain

$\displaystyle{\rm
tr}\left(\tilde{\rho}_{\tau_{1}}\hat{n}_{i[\ell_{0}]}^{s}\right)\leq\frac{\zeta_{0}^{2}}{e}\left(\frac{2b_{0}}{e}s^{\kappa}\ell_{0}^{D}\right)^{s}\quad\textrm{for}\quad
i[\ell_{0}]\in X_{0}^{\rm c},\quad\ell_{0}\geq\bar{\ell}_{1},$ (442)

where we use $\left\|O_{X_{0}}\right\|=\zeta_{0}$. On the other hand, if we
consider a boson number operator $\hat{n}_{Z}$ such that $Z\cap
X_{0}\neq\emptyset$, we cannot obtain the same inequality as (442) due to the
boson creation by $O_{X_{0}}$. We then consider a subset $X[r]$ for generic
$r\geq\bar{\ell}_{1}$ and $X\subset\Lambda$. From the condition (428), the
operator $O_{X_{0}}$ increases the boson number up to $q_{0}$ in the region
$X_{0}$, and hence we utilize Lemma 3 with

$O_{q}\to O_{X_{0}},\quad{\rm and}\quad\left(\nu_{i}=1\quad\textrm{for}\quad
i\in X[r],\quad\And\quad\nu_{i}=0\quad\textrm{for}\quad i\in X[r]^{\rm
c}\right),$

which yields

$\displaystyle{\rm
tr}\left(\tilde{\rho}_{\tau_{1}}\hat{n}_{X[r]}^{s}\right)={\rm
tr}\left(O_{X_{0}}\rho_{0}(\tau_{1})O_{X_{0}}^{\dagger}\hat{n}_{X[r]}^{s}\right)$
$\displaystyle\leq 4^{s}\zeta_{0}^{2}{\rm
tr}\left[\rho_{0}(\tau_{1})(4q_{0}^{3}+\hat{n}_{X[r]})^{s}\right]$
$\displaystyle\leq 4^{s}\zeta_{0}^{2}\left\\{4q_{0}^{3}+\sum_{i\in
X}\left[{\rm
tr}\left(\rho_{0}(\tau_{1})\hat{n}_{i[r]}^{s}\right)\right]^{1/s}\right\\}^{s}.$
(443)

where in the second inequality we use $\hat{n}_{X[r]}\preceq\sum_{i\in
X}\hat{n}_{i[r]}$ and

$\left\langle\left(O_{1}+O_{2}+\cdots+O_{m}\right)^{s}\right\rangle\leq\left(\left\langle
O_{1}^{s}\right\rangle^{1/s}+\left\langle
O_{2}^{s}\right\rangle^{1/s}+\cdots+\left\langle
O_{m}^{s}\right\rangle^{1/s}\right)^{s}$

for arbitrary positive semi-definite Hermitian operators
$\\{O_{1},O_{2},\ldots,O_{m}\\}$ as long as $\\{O_{j}\\}_{j=1}^{m}$ commute
with each other. By applying the inequality (440) to (443), we have

$\displaystyle{\rm tr}(\tilde{\rho}_{\tau_{1}}\hat{n}_{X[r]}^{s})$
$\displaystyle\leq
4^{s}\zeta_{0}^{2}\left(4q_{0}^{3}+\frac{|X|}{e^{1/s}}\cdot\frac{2b_{0}}{e}s^{\kappa}r^{D}\right)^{s}\leq\frac{\zeta_{0}^{2}}{e}\left(16eq_{0}^{3}+|X|\cdot\frac{8b_{0}}{e}s^{\kappa}r^{D}\right)^{s}\quad\textrm{for}\quad\forall
r\geq\bar{\ell}_{1}$
$\displaystyle\leq\frac{\zeta_{0}^{2}}{e}\left(16eq_{0}^{3}+|X|\cdot\frac{8b_{0}\bar{\ell}_{1}^{D}}{e}s^{\kappa}\max(r^{D},1)\right)^{s}\quad\textrm{for}\quad\forall
r\geq 0,$ (444)

where we can choose $\forall X\subset\Lambda$.

In the final step, we consider the moment function for the quantum state

$\displaystyle\tilde{\rho}_{\tau_{1},\tau_{2}}=e^{iH\tau_{2}}O_{X_{0}}e^{-iH\tau_{1}}\rho
e^{iH\tau_{1}}O_{X_{0}}^{\dagger}e^{-iH\tau_{2}}=e^{iH\tau_{2}}\tilde{\rho}_{\tau_{1}}e^{-iH\tau_{2}}=\tilde{\rho}_{\tau_{1}}(-\tau_{2}).$
(445)

Here, the initial state $\tilde{\rho}_{\tau_{1}}(0)$ satisfies the
inequalities (444) and (442), which give the initial conditions (318) and
(319) for Proposition 20, respectively, and hence we can let in Proposition 20

$\displaystyle b_{0}\to 2b_{0},\quad q_{0}\to 16eq_{0}^{3},\quad
q_{1}\to\frac{8b_{0}\bar{\ell}_{1}^{D}}{e},\quad\bar{\ell}_{0}\to\bar{\ell}_{1}\propto
t\log(t).$ (446)

Then, from the inequalities (321) and (322) we have

$\displaystyle{\rm
tr}\left(\tilde{\rho}_{\tau_{1}}(-\tau_{2})\hat{n}_{i[\ell_{0}]}^{s}\right)\leq\frac{\zeta_{0}^{2}}{e}\left(\frac{4b_{0}}{e}s^{\kappa}\ell_{0}^{D}\right)^{s}\quad\textrm{for}\quad
i[\ell_{0}]\subset X_{0}[\bar{\ell}_{2}]^{\rm c},$ (447)

and

$\displaystyle{\rm
tr}\left(\tilde{\rho}_{\tau_{1}}(-\tau_{2})\hat{n}_{X_{0}[\ell_{0}]}^{s}\right)$
$\displaystyle\leq\frac{\zeta_{0}^{2}}{e}\left(32eq_{0}^{3}+\frac{16b_{0}\bar{\ell}_{1}^{D}\ell_{0}^{D}}{e}s^{\kappa}|X_{0}|\right)^{s},$
(448)

where $\ell_{0}\geq\max(2D\bar{\ell_{2}},\bar{\ell}_{1})$ and $\bar{\ell}_{2}$
is given by the condition (323), i.e.,

$\displaystyle\bar{\ell}_{2}=t\log[\Theta(tq_{0}|X_{0}|)],$ (449)

which is also given in the form of Eq. (433) because of
$X_{0}=i_{0}[r_{0}]\subseteq i_{0}[R]$. In the following, we define
$\bar{\ell}_{t,R}$ as $2D\bar{\ell}_{2}$, i.e.,
$\bar{\ell}_{2}:=\bar{\ell}_{t,R}/(2D)$.

By applying Lemma 2 [i.e., the inequality (44)] to the inequality (447), we
obtain

$\displaystyle{\rm
tr}\left(\hat{n}_{i[\ell_{0}]}^{p}\Pi_{i[\ell_{0}],>q_{i}}\tilde{\rho}_{\tau_{1},\tau_{2}}\right)\leq\zeta_{0}^{2}q_{i}^{p}e^{-\left[q_{i}/(4b_{0}\ell_{0}^{D})\right]^{1/\kappa}}\quad\textrm{for}\quad
i[\ell_{0}]\subset X_{0}[\bar{\ell}_{t,R}/(2D)]^{\rm c},$ (450)

where we assume $\ell_{0}\geq\bar{\ell}_{t,R}$. Note that $i[\ell_{0}]\subset
X_{0}[\bar{\ell}_{t,R}/(2D)]^{\rm c}$ is included in the condition
$i[\ell_{0}]\subset X_{0}[\bar{\ell}_{t,R}]^{\rm c}$. Moreover, we apply Lemma
2 to the inequality (448) with $\ell_{0}=\bar{\ell}_{t,R}$, which yields

$\displaystyle{\rm
tr}\left(\hat{n}_{X_{0}[\bar{\ell}_{t,R}]}^{p}\Pi_{X_{0}[\bar{\ell}_{t,R}],>q}\tilde{\rho}_{\tau_{1},\tau_{2}}\right)$
$\displaystyle\leq\zeta_{0}^{2}q^{p}\exp\left[-\left(\frac{q-32e^{2}q_{0}^{3}}{16b_{0}\bar{\ell}_{t,R}^{2D}|X_{0}|}\right)^{1/\kappa}\right],$
(451)

where we use $\bar{\ell}_{1}\leq\bar{\ell}_{t,R}$. We thus prove the main
inequalities (434) and (435). This completes the proof of Lemma 28. $\square$

###  Supplementary Note 6.2 Effective Hamiltonian by the truncation of the
boson number: first result

By combining the obtained statements in Lemma 25 (or Corollary 26) and Lemma
28, we aim to derive an explicit error bound for the effective Hamiltonian. We
here define the projection operator $\bar{\Pi}_{L,\bar{q}}$ as

$\displaystyle\bar{\Pi}_{L,\bar{q}}:=\prod_{i\in L}\Pi_{i,\leq\bar{q}},$ (452)

which uniformly truncates the boson number at each site in the region $L$ by
$\bar{q}$. This type of projection will be considered in proving a non-optimal
Lieb-Robinson bound in Theorem 2 in Sec. Supplementary Note 7.

We here approximate the dynamics $e^{-iHt}$ by using the effective Hamiltonian
$\bar{\Pi}_{L,\bar{q}}H\bar{\Pi}_{L,\bar{q}}$, whose approximation error is
obtained from the following proposition:

######  Proposition 29.

In the same setup as in Sec. Supplementary Note 6.1, we obtain the
approximation error as

$\displaystyle\left\|O_{X_{0}}(H,t)\rho_{0}-O_{X_{0}}(\bar{\Pi}_{L,\bar{q}}H\bar{\Pi}_{L,\bar{q}},t)\rho_{0}\right\|_{1}$
$\displaystyle\leq
4\zeta_{0}\gamma|L|\left(1+\bar{J}t\bar{q}\right)e^{-\left[\bar{q}/(4b_{0}\bar{\ell}_{t,R}^{D})\right]^{1/\kappa}/2},$
(453)

where we assume $d_{X_{0},L}\geq\bar{\ell}_{t,R}+1$ (see Fig. 11).

Supplementary Figure 11: Definition of the projection $\bar{\Pi}_{L,\bar{q}}$.
In the projection, we truncate the boson number at each of the sites in the
region $L$ up to $\bar{q}$, while there are no restrictions on the boson
numbers in $L^{\rm c}$. For the projection, the approximation error for
dynamics by the effective Hamiltonian
$\bar{\Pi}_{L,\bar{q}}H\bar{\Pi}_{L,\bar{q}}$ is estimated in Proposition 29.

####  Supplementary Note 6.2.1 Proof of Proposition 29

In applying Lemma 25 (or Corollary 26) to the estimation of the norm in (453),
we need to estimate the norms of $\left\|\bar{\Pi}_{L,\bar{q}}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$ and
$\left\|\bar{\Pi}_{L,\bar{q}}H_{0}\bar{\Pi}_{L,\bar{q}}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$ for arbitrary
$|\tau_{1}|,|\tau_{2}|\leq t$. For the quantities, we aim to prove the
following upper bounds:

$\displaystyle\left\|\bar{\Pi}_{L,\bar{q}}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}\leq\zeta_{0}|L|^{1/2}e^{-\left[\bar{q}/(4b_{0}\bar{\ell}_{t,R}^{D})\right]^{1/\kappa}/2},$
(454)

and

$\displaystyle\left\|\bar{\Pi}_{L,\bar{q}}H_{0}\bar{\Pi}_{L,\bar{q}}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}\leq
2\gamma\bar{J}\zeta_{0}\bar{q}|L|e^{-\left[\bar{q}/(4b_{0}\bar{\ell}_{t,R}^{D})\right]^{1/\kappa}/2}.$
(455)

By applying the above inequalities with appropriate $\tau_{1}$ and $\tau_{2}$
to the upper bounds in (26), we obtain

$\displaystyle\left\|O_{X_{0}}(H,t)\rho_{0}-O_{X_{0}}(\bar{\Pi}_{L,\bar{q}}H\bar{\Pi}_{L,\bar{q}},t)\rho_{0}\right\|_{1}$
$\displaystyle\leq
4\zeta_{0}|L|^{1/2}e^{-\left[\bar{q}/(4b_{0}\bar{\ell}_{t,R}^{D})\right]^{1/\kappa}/2}+4t\gamma\bar{J}\zeta_{0}\bar{q}|L|e^{-\left[\bar{q}/(4b_{0}\bar{\ell}_{t,R}^{D})\right]^{1/\kappa}/2},$
(456)

which reduces to the main inequality (453) from $\gamma\geq 1$. Therefore, the
remaining tasks are proving the inequalities (454) and (455).

[Proofs of the inequalities (454) and (455)]

By using the inequality (436) with $p=0$ in Lemma 28, we immediately prove the
first inequality (454) as follows:

$\displaystyle\left\|\bar{\Pi}_{L,\bar{q}}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}^{2}={\rm
tr}\left(\bar{\Pi}_{L,\bar{q}}^{\rm
c}O_{X_{0}}(\tau_{2})\rho_{0}(\tau_{1})O_{X_{0}}(\tau_{2})^{\dagger}\bar{\Pi}_{L,\bar{q}}^{\rm
c}\right)$ $\displaystyle\leq\sum_{i\in L}{\rm
tr}\left(\Pi_{i,>\bar{q}}O_{X_{0}}(\tau_{2})\rho_{0}(\tau_{1})O_{X_{0}}(\tau_{2})^{\dagger}\right)$
$\displaystyle\leq\zeta_{0}^{2}|L|e^{-\left[\bar{q}/(4b_{0}\bar{\ell}_{t,R}^{D})\right]^{1/\kappa}},$
(457)

where we use $i\in X_{0}[\bar{\ell}_{t,R}]^{\rm c}$ for $\forall i\in L$ from
$d_{X_{0},L}\geq\bar{\ell}_{t,R}+1>\bar{\ell}_{t,R}$, which is assumed in the
statement of Proposition 29.

Second, to prove the inequality (455), we first decompose

$\displaystyle\bar{\Pi}_{L,\bar{q}}H_{0}\bar{\Pi}_{L,\bar{q}}^{\rm
c}=\bar{\Pi}_{L,\bar{q}}\left(H_{0,L}+\partial
h_{L}\right)\bar{\Pi}_{L,\bar{q}}^{\rm c},$ (458)

where we use the notation (28) for $\partial h_{L}$ and
$\bar{\Pi}_{L,\bar{q}}H_{0,L^{\rm c}}\bar{\Pi}_{L,\bar{q}}^{\rm c}=H_{0,L^{\rm
c}}\bar{\Pi}_{L,\bar{q}}\bar{\Pi}_{L,\bar{q}}^{\rm c}=0$. Then, by using

$\displaystyle|b_{i}b_{j}^{\dagger}|=\sqrt{(b_{i}b_{j}^{\dagger})^{\dagger}b_{i}b_{j}^{\dagger}}=\sqrt{\hat{n}_{i}(\hat{n}_{j}+1)}\preceq\hat{n}_{i}+\hat{n}_{j},$
(459)

and

$\displaystyle\bar{\Pi}_{L,\bar{q}}b_{i}b_{j}^{\dagger}\bar{\Pi}_{L,\bar{q}}^{\rm
c}=\bar{\Pi}_{L,\bar{q}}b_{i}b_{j}^{\dagger}\bar{\Pi}_{i,j,\bar{q}}^{\rm
c},\quad\bar{\Pi}_{i,j,\bar{q}}:=\begin{cases}\Pi_{i,\leq\bar{q}}\Pi_{j,\leq\bar{q}}&\quad\textrm{for}\quad
i,j\in L,\\\ \Pi_{i,\leq\bar{q}}&\quad\textrm{for}\quad i\in L,\quad j\in
L^{\rm c},\end{cases}$ (460)

we obtain

$\displaystyle\left\|\bar{\Pi}_{L,\bar{q}}H_{0}\bar{\Pi}_{L,\bar{q}}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$
$\displaystyle\leq\bar{J}\sum_{i\in
L}\sum_{j:d_{i,j}=1}\left\|\bar{\Pi}_{L,\bar{q}}(b_{i}b_{j}^{\dagger}+{\rm
h.c.})\bar{\Pi}_{i,j,\bar{q}}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$
$\displaystyle\leq\bar{J}\sum_{i\in
L}\sum_{j:d_{i,j}=1}\left\|\left(|b_{i}b_{j}^{\dagger}|+|b_{j}b_{i}^{\dagger}|\right)\bar{\Pi}_{i,j,\bar{q}}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$ $\displaystyle\leq
2\bar{J}\sum_{i\in
L}\sum_{j:d_{i,j}=1}\left\|(\hat{n}_{i}+\hat{n}_{j})\bar{\Pi}_{i,j,\bar{q}}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}.$ (461)

By applying the inequality (436) with $p=2$ to
$\left\|(\hat{n}_{i}+\hat{n}_{j})\bar{\Pi}_{i,j,\bar{q}}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$, we obtain

$\displaystyle\left\|(\hat{n}_{i}+\hat{n}_{j})\bar{\Pi}_{i,j,\bar{q}}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}^{2}$
$\displaystyle={\rm
tr}\left[(\hat{n}_{i}+\hat{n}_{j})^{2}\bar{\Pi}_{i,j,\bar{q}}^{\rm
c}O_{X_{0}}(\tau_{2})\rho_{0}(\tau_{1})O_{X_{0}}(\tau_{2})^{\dagger}\right]$
$\displaystyle\leq{\rm
tr}\left[\hat{n}_{i[1]}^{2}\Pi_{i[1],>\bar{q}}O_{X_{0}}(\tau_{2})\rho_{0}(\tau_{1})O_{X_{0}}(\tau_{2})^{\dagger}\right]$
$\displaystyle\leq\zeta_{0}^{2}\bar{q}^{2}e^{-\left[\bar{q}/(4b_{0}\bar{\ell}_{t,R}^{D})\right]^{1/\kappa}},$
(462)

where we use $\hat{n}_{i}+\hat{n}_{j}\preceq\hat{n}_{i[1]}$ and
$\bar{\Pi}_{i,j,\bar{q}}^{\rm
c}=\Pi_{i[1],>\bar{q}}\bar{\Pi}_{i,j,\bar{q}}^{\rm c}$ from $d_{i,j}=1$ in the
first inequality, and in the second inequality, we ensure $i[1]\subset
X_{0}[\bar{\ell}_{t,R}]^{\rm c}$ for $\forall i\in L$ from the condition
$d_{X_{0},L}\geq\bar{\ell}_{t,R}+1$.

By combining the inequalities (Supplementary Note 6.2.1) and (Supplementary
Note 6.2.1), we arrive at the inequality (455) as follows:

$\displaystyle\left\|\bar{\Pi}_{L,\bar{q}}H_{0}\bar{\Pi}_{L,\bar{q}}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$ $\displaystyle\leq
2\bar{J}\sum_{i\in
L}\sum_{j:d_{i,j}=1}\zeta_{0}\bar{q}e^{-\left[\bar{q}/(4b_{0}\bar{\ell}_{t,R}^{D})\right]^{1/\kappa}/2}$
$\displaystyle\leq
2\gamma\bar{J}\zeta_{0}\bar{q}|L|e^{-\left[\bar{q}/(4b_{0}\bar{\ell}_{t,R}^{D})\right]^{1/\kappa}/2},$
(463)

where we use $\sum_{j:d_{i,j}=1}1\leq|i[1]|\leq\gamma$ for $\forall
i\in\Lambda$. This completes the proof. $\square$

###  Supplementary Note 6.3 Effective Hamiltonian by the truncation of the
boson number: second result

In Eq. (37), we have define the set $\mathcal{S}(L,\xi,N)$ as

$\displaystyle\mathcal{S}(L,\xi,N)\coloneqq\left\\{\mathbb{N}|N_{i}\leq
Ne^{\xi d_{i,L}}\quad\textrm{for}\quad i\in\Lambda\right\\}.$ (464)

We consider the effective Hamiltonian, which is defined by the projection of
(see Fig. 12)

$\displaystyle\Pi_{\mathcal{S}(L,\xi,N)}:=\prod_{i\in L}\Pi_{i,\leq
N}\prod_{i\in L^{\rm c}}\Pi_{i,\leq Ne^{\xi d_{i,L}}}.$ (465)

In our analyses, it is often pretty convenient to restrict the Hilbert space
which is spanned by $\Pi_{\mathcal{S}(L,\xi,N)}$.

Supplementary Figure 12: Definition of the projection
$\Pi_{\mathcal{S}(L,\xi,N)}$. In this projection, we truncate the boson number
at each of the sites in the whole region up to $q_{i}$ ($i\in\Lambda$), where
$q_{i}=N$ for $i\in L$ and $q_{i}=Ne^{\xi d_{i,L}}$ for $i\in L^{\rm c}$. As
the site $i$ is separated from the region $L$, the truncation number rapidly
increases with the distance. The approximation error for dynamics by the
effective Hamiltonian $\Pi_{\mathcal{S}(L,1/2,N)}H\Pi_{\mathcal{S}(L,1/2,N)}$
is given by Proposition 30.

In the following, we choose $\xi=1/2$ and denote the projection

$\displaystyle\Pi_{\mathcal{S}(L,1/2,N)}=\tilde{\Pi}_{(L,N)}$ (466)

for simplicity. We here consider the approximate time evolution by using the
following effective Hamiltonian:

$\displaystyle\tilde{\Pi}_{(L,N)}H\tilde{\Pi}_{(L,N)}=\tilde{H}_{(L,N)}.$
(467)

We aim to prove the following proposition:

######  Proposition 30.

Under the assumption of

$\displaystyle
N\geq\max\left\\{64e^{2}q_{0}^{3},2(8\kappa)^{\kappa}\xi_{t,R}\right\\},$
(468)

the approximation error using the effective Hamiltonian $\tilde{H}_{(L,N)}$ is
upper-bounded by

$\displaystyle\left\|\left[O_{X_{0}}(H,t)-O_{X_{0}}(\tilde{H}_{(L,N)},t)\right]\rho_{0}\right\|_{1}\leq
2^{2D+5}\zeta_{0}\gamma|L|D!\left(1+2c_{\kappa}\gamma\bar{J}t\xi_{t,R}^{2}\right)e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(8\kappa)},$
(469)

where $c_{\kappa}:=2^{4\kappa+3}\kappa\Gamma(2\kappa)$. We also obtain

$\displaystyle\left\|O_{X_{0}}(H,t)\rho_{0}-O_{X_{0}}(\tilde{H}_{(L,N)},t)\tilde{\Pi}_{(L,N)}\rho_{0}\tilde{\Pi}_{(L,N)}\right\|_{1}\leq
2^{2D+6}\zeta_{0}\gamma|L|D!\left(1+c_{\kappa}\gamma\bar{J}t\xi_{t,R}^{2}\right)e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(8\kappa)}.$
(470)

####  Supplementary Note 6.3.1 Proof of Proposition 30

The proof is similar to the one for Proposition 29. To apply Corollary 26 to
the approximation error, we aim to estimate the norms of

$\left\|\tilde{\Pi}_{(L,N)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}\quad{\rm
and}\quad\left\|\tilde{\Pi}_{(L,N)}H_{0}\tilde{\Pi}_{(L,N)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$

for arbitrary $|\tau_{1}|,|\tau_{2}|\leq t$. For the above two quantities, we
prove the following inequalities for $N\geq 2(8\kappa)^{\kappa}\xi_{t,R}$ (see
below for the proofs):

$\displaystyle\left\|\tilde{\Pi}_{(L,N)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$ $\displaystyle\leq
2^{2D+3}\zeta_{0}\gamma|L|D!e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(4\kappa)},$
(471)

and

$\displaystyle\left\|\tilde{\Pi}_{(L,N)}H_{0}\tilde{\Pi}_{(L,N)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}\leq
2^{2D+5}c_{\kappa}\zeta_{0}\bar{J}\xi_{t,R}^{2}\gamma^{2}|L|D!e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(8\kappa)},$
(472)

where $c_{\kappa}:=2^{4\kappa+3}\kappa\Gamma(2\kappa)$.

By applying the inequalities (471) and (472) with appropriate choices of
$\tau_{1}$ and $\tau_{2}$ to the upper bounds in (26), we have

$\displaystyle\left\|O_{X_{0}}(H,t)\rho_{0}-O_{X_{0}}(\tilde{H}_{(L,N)},t)\rho_{0}\right\|_{1}$
$\displaystyle\leq
2^{2D+5}\zeta_{0}\gamma|L|D!e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(4\kappa)}+2^{2D+6}tc_{\kappa}\zeta_{0}\bar{J}\xi_{t,R}^{2}\gamma^{2}|L|D!e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(8\kappa)}$
$\displaystyle\leq
2^{2D+5}\zeta_{0}\gamma|L|D!\left(1+2c_{\kappa}\gamma\bar{J}t\xi_{t,R}^{2}\right)e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(8\kappa)}.$
(473)

We thus prove the first main inequality (469).

In deriving the inequality of (470), because of

$\displaystyle\left\|O_{X_{0}}(H,t)\rho_{0}-O_{X_{0}}(\tilde{H}_{(L,N)},t)\tilde{\Pi}_{(L,N)}\rho_{0}\tilde{\Pi}_{(L,N)}\right\|_{1}$
$\displaystyle=\left\|O_{X_{0}}(H,t)\rho_{0}-O_{X_{0}}(\tilde{H}_{(L,N)},t)\rho_{0}+O_{X_{0}}(\tilde{H}_{(L,N)},t)\rho_{0}-O_{X_{0}}(\tilde{H}_{(L,N)},t)\tilde{\Pi}_{(L,N)}\rho_{0}\tilde{\Pi}_{(L,N)}\right\|_{1}$
$\displaystyle\leq\left\|O_{X_{0}}(H,t)\rho_{0}-O_{X_{0}}(\tilde{H}_{(L,N)},t)\rho_{0}\right\|_{1}+\left\|O_{X_{0}}(\tilde{H}_{(L,N)},t)\left(\rho_{0}-\tilde{\Pi}_{(L,N)}\rho_{0}\tilde{\Pi}_{(L,N)}\right)\right\|_{1},$
(474)

we only have to estimate

$\displaystyle\left\|O_{X_{0}}(\tilde{H}_{(L,N)},t)\left(\rho_{0}-\tilde{\Pi}_{(L,N)}\rho_{0}\tilde{\Pi}_{(L,N)}\right)\right\|_{1}$
$\displaystyle\leq\zeta_{0}\left\|\rho_{0}-\tilde{\Pi}_{(L,N)}\rho_{0}\tilde{\Pi}_{(L,N)}\right\|_{1}$
$\displaystyle\leq 2\zeta_{0}\left\|\rho_{0}\tilde{\Pi}_{(L,N)}^{\rm
c}\right\|_{1}\leq 2\zeta_{0}\left\|\tilde{\Pi}_{(L,N)}^{\rm
c}\sqrt{\rho_{0}}\right\|_{F},$ (475)

where we use

$\displaystyle\left\|\rho_{0}-\tilde{\Pi}_{(L,N)}\rho_{0}\tilde{\Pi}_{(L,N)}\right\|_{1}$
$\displaystyle=\left\|(1-\tilde{\Pi}_{(L,N)})\rho_{0}\tilde{\Pi}_{(L,N)}+\rho_{0}(1-\tilde{\Pi}_{(L,N)})\right\|_{1}\leq
2\left\|\rho_{0}\tilde{\Pi}_{(L,N)}^{\rm c}\right\|_{1}$ (476)

and *4*4*4For a general Schatten $p$ norm, one can prove the generalized
Hölder inequality (see Prop. 2.5 in Ref. [4]):
$\displaystyle\left\|\prod_{j=1}^{s}O_{j}\right\|_{p}\leq\prod_{j=1}^{s}\|O_{j}\|_{p_{j}},$
(477) where $\sum_{j=1}^{s}1/p_{j}=1/p$. The inequality immediately yields
$\displaystyle\left\|O_{1}O_{2}\right\|_{1}\leq\|O_{1}\|_{2}\|O_{2}\|_{2},$
(478) where we set $p=1$, $p_{1}=2$ and $p_{2}=2$ in (477).

$\displaystyle\left\|\rho_{0}\tilde{\Pi}_{(L,N)}^{\rm
c}\right\|_{1}=\left\|\tilde{\Pi}_{(L,N)}^{\rm
c}\rho_{0}\right\|_{1}\leq\left\|\sqrt{\rho_{0}}\right\|_{2}\left\|\tilde{\Pi}_{(L,N)}^{\rm
c}\sqrt{\rho_{0}}\right\|_{2}=\left\|\tilde{\Pi}_{(L,N)}^{\rm
c}\sqrt{\rho_{0}}\right\|_{F}.$ (479)

Note that Schatten $2$ norm $\left\|\cdot\right\|_{2}$ is equivalent to the
Frobenius norm $\left\|\cdot\right\|_{F}$. By using the inequality (471) with
$O_{X_{0}}=\hat{1}$, we immediately obtain

$\displaystyle\left\|\tilde{\Pi}_{(L,N)}^{\rm
c}\sqrt{\rho_{0}}\right\|_{F}\leq
2^{2D+3}\zeta_{0}\gamma|L|D!e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(4\kappa)}.$
(480)

Therefore, by combining the above inequality with (Supplementary Note 6.3.1),

$\displaystyle\left\|O_{X_{0}}(\tilde{H}_{(L,N)},t)\left(\rho_{0}-\tilde{\Pi}_{(L,N)}\rho_{0}\tilde{\Pi}_{(L,N)}\right)\right\|_{1}$
$\displaystyle\leq
2^{2D+5}\zeta_{0}\gamma|L|D!\left(1/2+1+2c_{\kappa}\gamma\bar{J}t\xi_{t,R}^{2}\right)e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(8\kappa)},$

which yields the second main inequality (470). This completes the proof of
Proposition 30. $\square$

[Proof of the inequalities (471) and (472)]

From the inequalities (434) and (435) in Lemma 28, we can obtain the following
looser bound which holds for all $i\in\Lambda$:

$\displaystyle{\rm
tr}\left(\hat{n}_{i[\bar{\ell}_{t,R}]}^{p}\Pi_{i[\bar{\ell}_{t,R}],>q_{i}}\tilde{\rho}_{\tau_{1},\tau_{2}}\right)\leq\zeta_{0}^{2}q_{i}^{p}e^{-\left[(q_{i}-32e^{2}q_{0}^{3})/\xi_{t,R}\right]^{1/\kappa}}\quad\textrm{for}\quad\forall
i\in\Lambda.$ (481)

We utilize the inequality (481) with $p=0$ for the estimation of the norm
$\left\|\tilde{\Pi}_{(L,N)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$. For simplicity,
we here adopt the notation of

$\displaystyle\bar{N}_{i}:=\begin{cases}N&\quad\textrm{for}\quad i\in L,\\\
Ne^{d_{i,L}/2}&\quad\textrm{for}\quad i\in L^{\rm c},\end{cases}$ (482)

and obtain

$\displaystyle\left\|\tilde{\Pi}_{(L,N)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$
$\displaystyle\leq\sum_{i\in\Lambda}\left\|\Pi_{i,>\bar{N}_{i}}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$
$\displaystyle\leq\zeta_{0}\sum_{i\in\Lambda}\exp\left[-\frac{1}{2}\left(\frac{\bar{N}_{i}-32e^{2}q_{0}^{3}}{\xi_{t,R}}\right)^{1/\kappa}\right]$
$\displaystyle\leq\zeta_{0}\sum_{i\in\Lambda}e^{-[\bar{N}_{i}/(2\xi_{t,R})]^{1/\kappa}/2},$
(483)

where in the last inequality we use the condition $N\geq 64e^{2}q_{0}^{3}$ in
(468) to derive $32e^{2}q_{0}^{3}\leq N/2\leq\bar{N}_{i}/2$. For the summation
in the RHS of (Supplementary Note 6.3.1), we have

$\displaystyle\sum_{i\in\Lambda}e^{-\left[\bar{N}_{i}/(2\xi_{t,R})\right]^{1/\kappa}/2}=\sum_{i\in
L}e^{-\left[N/(2^{\kappa+1}\xi_{t,R})\right]^{1/\kappa}}+\sum_{i\in L^{\rm
c}}e^{-[Ne^{d_{i,L}/2}/(2^{\kappa+1}\xi_{t,R})]^{1/\kappa}}$
$\displaystyle\leq|L|e^{-\left[N/(2^{\kappa+1}\xi_{t,R})\right]^{1/\kappa}}+\gamma|L|2^{2D+2}D!e^{-\left[N/(2^{\kappa+1}\xi_{t,R})\right]^{1/\kappa}/(2\kappa)}$
$\displaystyle\leq
2^{2D+3}\gamma|L|D!e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(4\kappa)}\quad\textrm{for}\quad
N\geq 2^{\kappa+1}(2\kappa)^{\kappa}\xi_{t,R}=2(4\kappa)^{\kappa}\xi_{t,R},$
(484)

where we apply the inequality (13) in Lemma 1 to the second term in the first
line by choosing

$\displaystyle\kappa\to\kappa,\quad\Phi\to 2^{\kappa+1}\xi_{t,R},\quad q\to
N,\quad\ell\to 0,\quad\xi\to 2.$ (485)

Note that $N\geq 2(4\kappa)^{\kappa}\xi_{t,R}$ originates from the condition
in (13), i.e., $q\geq(\xi\kappa)^{\kappa}\Phi$. By applying the inequality
(Supplementary Note 6.3.1) to (Supplementary Note 6.3.1), we prove the first
inequality (471).

Second, we estimate the norm of
$\left\|\tilde{\Pi}_{(L,N)}H_{0}\tilde{\Pi}_{(L,N)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$. By using the same
inequality as (Supplementary Note 6.2.1), we obtain

$\displaystyle\left\|\tilde{\Pi}_{(L,N)}H_{0}\tilde{\Pi}_{(L,N)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$
$\displaystyle\leq\bar{J}\sum_{i\in\Lambda}\sum_{j:d_{i,j}=1}\left\|\tilde{\Pi}_{(L,N)}(b_{i}b_{j}^{\dagger}+{\rm
h.c.})\left(\Pi_{i,\leq\bar{N}_{i}}\Pi_{j,\leq\bar{N}_{j}}\right)^{{\rm
c}}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$
$\displaystyle\leq
2\bar{J}\sum_{i\in\Lambda}\sum_{j:d_{i,j}=1}\left\|(\hat{n}_{i}+\hat{n}_{j})\left(\Pi_{i,\leq\bar{N}_{i}}\Pi_{j,\leq\bar{N}_{j}}\right)^{{\rm
c}}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}.$ (486)

Furthermore, from the same inequality as (Supplementary Note 6.2.1), we obtain

$\displaystyle\left\|(\hat{n}_{i}+\hat{n}_{j})\left(\Pi_{i,\leq\bar{N}_{i}}\Pi_{j,\leq\bar{N}_{j}}\right)^{{\rm
c}}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}^{2}$
$\displaystyle={\rm
tr}\left[(\hat{n}_{i}+\hat{n}_{j})^{2}\left(\Pi_{i,\leq\bar{N}_{i}}\Pi_{j,\leq\bar{N}_{j}}\right)^{{\rm
c}}O_{X_{0}}(\tau_{2})\rho_{0}(\tau_{1})O_{X_{0}}(\tau_{2})^{\dagger}\right]$
$\displaystyle\leq{\rm
tr}\left[\hat{n}_{i[1]}^{2}\Pi_{i,>\bar{N}_{i}}O_{X_{0}}(\tau_{2})\rho_{0}(\tau_{1})O_{X_{0}}(\tau_{2})^{\dagger}\right]+{\rm
tr}\left[\hat{n}_{j[1]}^{2}\Pi_{j,>\bar{N}_{j}}O_{X_{0}}(\tau_{2})\rho_{0}(\tau_{1})O_{X_{0}}(\tau_{2})^{\dagger}\right]$
$\displaystyle\leq\zeta_{0}^{2}\left(\bar{N}_{i}^{2}e^{-\left[(\bar{N}_{i}-32e^{2}q_{0}^{3})/\xi_{t,R}\right]^{1/\kappa}}+\bar{N}_{j}^{2}e^{-\left[(\bar{N}_{j}-32e^{2}q_{0}^{3})/\xi_{t,R}\right]^{1/\kappa}}\right),$
(487)

where we use $\hat{n}_{i}+\hat{n}_{j}\preceq\hat{n}_{i[1]}$
($\hat{n}_{i}+\hat{n}_{j}\preceq\hat{n}_{j[1]}$) from $d_{i,j}=1$ in the first
inequality, and in the second inequality, we apply the inequality (481) with
$p=2$. By combining the inequalities (Supplementary Note 6.3.1) and
(Supplementary Note 6.3.1), we have

$\displaystyle\left\|\tilde{\Pi}_{(L,N)}H_{0}\tilde{\Pi}_{(L,N)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$ $\displaystyle\leq
2\zeta_{0}\bar{J}\sum_{i\in\Lambda}\sum_{j:d_{i,j}=1}\left(\bar{N}_{i}e^{-\left[(\bar{N}_{i}-32e^{2}q_{0}^{3})/\xi_{t,R}\right]^{1/\kappa}/2}+\bar{N}_{j}e^{-\left[(\bar{N}_{j}-32e^{2}q_{0}^{3})/\xi_{t,R}\right]^{1/\kappa}/2}\right)$
$\displaystyle\leq
4\zeta_{0}\gamma\bar{J}\sum_{i\in\Lambda}\bar{N}_{i}e^{-\left[(\bar{N}_{i}-32e^{2}q_{0}^{3})/\xi_{t,R}\right]^{1/\kappa}/2}\leq
4\zeta_{0}\gamma\bar{J}\sum_{i\in\Lambda}\bar{N}_{i}e^{-\left[\bar{N}_{i}/(2^{\kappa+1}\xi_{t,R})\right]^{1/\kappa}},$
(488)

where we use $\sqrt{x^{2}+y^{2}}\leq x+y$ in taking the square root of
(Supplementary Note 6.3.1), $\sum_{j:d_{i,j}=1}1\leq|i[1]|\leq\gamma$ for
$\forall i\in\Lambda$, and $32e^{2}q_{0}^{3}\leq N_{i}/2$ from the condition
$N\geq 64e^{2}q_{0}^{3}$. Finally, by employing the similar analyses to
(Supplementary Note 6.3.1), we upper-bound the summation in (Supplementary
Note 6.3.1). For the purpose, we first apply the inequality (15) to
$\bar{N}_{i}e^{-\left[\bar{N}_{i}/(2^{\kappa+1}\xi_{t,R})\right]^{1/\kappa}}$
by letting

$\displaystyle s\to
1,\quad\kappa\to\kappa,\quad\Phi\to(2^{\kappa+1}\xi_{t,R}),$ (489)

which yields

$\displaystyle\bar{N}_{i}e^{-\left[\bar{N}_{i}/(2^{\kappa+1}\xi_{t,R})\right]^{1/\kappa}}$
$\displaystyle\leq 2(2^{\kappa}\cdot
2^{\kappa+1}\xi_{t,R})^{2}\kappa\Gamma(2\kappa)e^{-\left[\bar{N}_{i}/(2^{\kappa+1}\xi_{t,R})\right]^{1/\kappa}/2}$
$\displaystyle=c_{\kappa}\xi_{t,R}^{2}e^{-\left[\bar{N}_{i}/(2^{\kappa+1}\xi_{t,R})\right]^{1/\kappa}/2}\quad\textrm{with}\quad
c_{\kappa}:=2^{4\kappa+3}\kappa\Gamma(2\kappa).$ (490)

By using the above upper bound, we obtain

$\displaystyle\sum_{i\in\Lambda}\bar{N}_{i}e^{-\left[\bar{N}_{i}/(2^{\kappa+1}\xi_{t,R})\right]^{1/\kappa}}\leq
c_{\kappa}\xi_{t,R}^{2}\sum_{i\in
L}e^{-\left[N/(2^{\kappa+1}\xi_{t,R})\right]^{1/\kappa}/2}+c_{\kappa}\xi_{t,R}^{2}\sum_{i\in
L^{\rm
c}}e^{-\left[Ne^{d_{i,L}/2}/(2^{\kappa+1}\xi_{t,R})\right]^{1/\kappa}/2}$
$\displaystyle=c_{\kappa}\xi_{t,R}^{2}|L|e^{-\left[N/(2^{\kappa+1}\xi_{t,R})\right]^{1/\kappa}/2}+c_{\kappa}\xi_{t,R}^{2}\sum_{i\in
L^{\rm
c}}e^{-\left[Ne^{d_{i,L}/2}/(2^{2\kappa+1}\xi_{t,R})\right]^{1/\kappa}}$
$\displaystyle\leq
c_{\kappa}\xi_{t,R}^{2}|L|e^{-\left[N/(2^{\kappa+1}\xi_{t,R})\right]^{1/\kappa}/2}+c_{\kappa}\xi_{t,R}^{2}\gamma|L|2^{2D+2}D!e^{-\left[N/(2^{2\kappa+1}\xi_{t,R})\right]^{1/\kappa}/(2\kappa)}$
$\displaystyle\leq
2^{2D+3}c_{\kappa}\xi_{t,R}^{2}\gamma|L|D!e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(8\kappa)}\quad\textrm{for}\quad
N\geq 2^{2\kappa+1}(2\kappa)^{\kappa}\xi_{t,R}=2(8\kappa)^{\kappa}\xi_{t,R},$
(491)

where in the second inequality, we use the inequality (13) with

$\displaystyle\kappa\to\kappa,\quad\Phi\to 2^{2\kappa+1}\xi_{t,R},\quad q\to
N,\quad\ell\to 0,\quad\xi\to 2.$ (492)

Here, the condition $N\geq 2(8\kappa)^{\kappa}\xi_{t,R}$ in (Supplementary
Note 6.3.1) originates from the condition $q\geq(\xi\kappa)^{\kappa}\Phi$ in
(13). By applying the inequality (Supplementary Note 6.3.1) to (Supplementary
Note 6.3.1), we prove the desired inequality (472).

###  Supplementary Note 6.4 Effective Hamiltonian by the truncation of the
boson number: third result

In this section, we further consider the projection $\Pi_{L^{\prime},\leq Q}$
onto the region $L^{\prime}$ such that (see Fig. 13)

$\displaystyle L^{\prime}\subseteq
i_{1}[\ell_{1}-1]\quad(\ell_{1}\geq\bar{\ell}_{t,R}),\quad
d_{i_{1}[\ell_{1}],X_{0}}\geq\bar{\ell}_{t,R}\to i_{1}[\ell_{1}]\subset
X_{0}[\bar{\ell}_{t,R}]^{\rm c}.$ (493)

We then define the effective Hamiltonian $\tilde{H}_{(L,N,L^{\prime},Q)}$ as

$\displaystyle\tilde{H}_{(L,N,L^{\prime},Q)}=\Pi_{L^{\prime},\leq
Q}\tilde{\Pi}_{(L,N)}H\tilde{\Pi}_{(L,N)}\Pi_{L^{\prime},\leq Q},$ (494)

where the projection $\tilde{\Pi}_{(L,N)}$ has been defined in Eq. (466).

Here, we should note that, in general, the effective Hamiltonian
$\tilde{H}_{(L,N,L^{\prime},Q)}$ is no longer local in the sense that

$\displaystyle[\Pi_{L^{\prime},\leq Q}h_{Z}\Pi_{L^{\prime},\leq
Q},\Pi_{L^{\prime},\leq Q}h_{Z^{\prime}}\Pi_{L^{\prime},\leq Q}]\neq
0\quad\textrm{for}\quad Z\cap Z^{\prime}\neq\emptyset,$ (495)

where $H=\sum_{Z}h_{Z}$ as in Eq. (26). To see the point, let us consider four
interaction terms $h_{Z_{1}}$, $h_{Z_{2}}$, $h_{Z_{3}}$ and $h_{Z_{4}}$ such
that (see Fig. 13)

$Z_{1},Z_{2}\subset L^{\prime}\quad(Z_{1}\cap Z_{2}=\emptyset),\quad
Z_{3}\subset L^{{}^{\prime}{\rm c}},\quad(Z_{4}\cap
L^{\prime})\neq\emptyset\And(Z_{4}\cap L^{{}^{\prime}{\rm c}})\neq\emptyset.$

Here, the effective interaction terms $\Pi_{L^{\prime},\leq
Q}h_{Z_{1}}\Pi_{L^{\prime},\leq Q}$, $\Pi_{L^{\prime},\leq
Q}h_{Z_{2}}\Pi_{L^{\prime},\leq Q}$ and $\Pi_{L^{\prime},\leq
Q}h_{Z_{3}}\Pi_{L^{\prime},\leq Q}$ commute with each other since

$\displaystyle[\Pi_{L^{\prime},\leq Q},h_{Z_{i}}]=0\quad\textrm{for}\quad
i=1,2,3.$ (496)

It is trivially obtained from $[\hat{n}_{i},\Pi_{L^{\prime},\leq Q}]=0$ for
$\forall i\in\Lambda$ and $[b_{i}b_{j}^{\dagger},\Pi_{L^{\prime},\leq Q}]=0$
as long as $i,j\in L^{\prime}$. We thus obtain

$\displaystyle[\Pi_{L^{\prime},\leq Q}h_{Z_{i}}\Pi_{L^{\prime},\leq
Q},\Pi_{L^{\prime},\leq Q}h_{Z_{j}}\Pi_{L^{\prime},\leq
Q}]=\Pi_{L^{\prime},\leq Q}[h_{Z_{i}},h_{Z_{j}}]\Pi_{L^{\prime},\leq Q}=0$
(497)

for $i,j=1,2,3$. On the other hand, we cannot ensure the above equation for
$h_{Z_{4}}$ because of

$\displaystyle[\Pi_{L^{\prime},\leq Q},h_{Z_{4}}]\neq 0.$ (498)

This point leads to the breakdown of the locality of the interactions as in
(495). We also discuss this point in Sec. Supplementary Note 8.6 when we
consider the Lieb-Robinson bound for the effective Hamiltonian.

Supplementary Figure 13: Definition of the projection $\Pi_{L^{\prime},\leq
Q}$. In the projection, we truncate the boson number in the region
$L^{\prime}$ up to $Q$, i.e., $\left\|\hat{n}_{L}\Pi_{L^{\prime},\leq
Q}\right\|=Q$, where the region $i_{1}[\ell_{1}]$ ($\supset L^{\prime}$) is
separated from the region $X_{0}$ at least by the distance of
$\bar{\ell}_{t,R}$. Unlike the other projections $\bar{\Pi}_{L,\bar{q}}$ and
$\tilde{\Pi}_{(L,N)}$ in Figs. 11 and 12, where the boson truncations are
performed for each of the local sites, the boson truncation by
$\Pi_{L^{\prime},\leq Q}$ is non-local in the sense that it cannot be
decomposed into the product form like $\prod_{i\in L^{\prime}}\Pi_{i,\leq
q_{i}}$. This point breaks down the locality of the effective Hamiltonian
around the boundary. In the figure, the interactions $h_{Z_{1}}$, $h_{Z_{2}}$
and $h_{Z_{3}}$ commute with each other by the projection
$\Pi_{L^{\prime},\leq Q}$, while the boundary interaction $h_{Z_{4}}$ becomes
non-local by the projection $\Pi_{L^{\prime},\leq Q}$. In Proposition 32, we
consider the projection $\Pi_{L^{\prime},\leq Q}\tilde{\Pi}_{(L,N)}$ and
upper-bound the approximation error for dynamics by the effective Hamiltonian
$\tilde{H}_{(L,N,L^{\prime},Q)}$ in Eq. (494).

From Eq. (497), we at least prove the following corollary:

######  Corollary 31.

If we consider a subset Hamiltonian $H_{X}$ such that $X\subseteq L^{\prime}$
or $X\subseteq L^{{}^{\prime}{\rm c}}$, the effective Hamiltonian
$\tilde{H}_{X,(L,N,L^{\prime},Q)}$ is still a local Hamiltonian in the sense
that

$\displaystyle[\Pi_{L^{\prime},\leq Q}h_{Z}\Pi_{L^{\prime},\leq
Q},\Pi_{L^{\prime},\leq Q}h_{Z^{\prime}}\Pi_{L^{\prime},\leq
Q}]=0\quad\textrm{for}\quad Z\cap Z^{\prime}=\emptyset,$ (499)

where $h_{Z}$ and $h^{\prime}_{Z}$ are local interaction terms in $H_{X}$,
i.e., $Z,Z^{\prime}\subset X$.

On the approximation error for the effective Hamiltonian
$\tilde{H}_{(L,N,L^{\prime},Q)}$, we prove the following proposition:

######  Proposition 32.

We adopt the same assumption as Eq. (468) for $N$. Let us choose the subset
$L^{\prime}$ such that the condition (493) is satisfied. Then, the
approximation error using the effective Hamiltonian $\tilde{H}_{(L,N)}$ is
upper-bounded by

$\displaystyle\left\|\left[O_{X_{0}}(H,t)-O_{X_{0}}(\tilde{H}_{(L,N,L^{\prime},Q)},t)\right]\rho_{0}\right\|_{1}$
$\displaystyle\leq
2^{2D+5}\zeta_{0}\gamma|L|D!\left(1+2c_{\kappa}\gamma\bar{J}t\xi_{t,R}^{2}\right)e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(8\kappa)}+4\zeta_{0}(1+2t\gamma\bar{J}Q)e^{-\left[Q/(4b_{0}\ell_{1}^{D})\right]^{1/\kappa}/2},$
(500)

where $c_{\kappa}$ has been defined in Proposition 30. Moreover, for
$\tilde{\Pi}_{(L,N,L^{\prime},Q)}\rho_{0}\tilde{\Pi}_{(L,N,L^{\prime},Q)}$, we
obtain the upper bound of

$\displaystyle\left\|\left[O_{X_{0}}(H,t)-O_{X_{0}}(\tilde{H}_{(L,N,L^{\prime},Q)},t)\right]\tilde{\Pi}_{(L,N,L^{\prime},Q)}\rho_{0}\tilde{\Pi}_{(L,N,L^{\prime},Q)}\right\|_{1}$
$\displaystyle\leq
2^{2D+6}\zeta_{0}\gamma|L|D!\left(1+c_{\kappa}\gamma\bar{J}t\xi_{t,R}^{2}\right)e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(8\kappa)}+8\zeta_{0}(1+t\gamma\bar{J}Q)e^{-\left[Q/(4b_{0}\ell_{1}^{D})\right]^{1/\kappa}/2}.$
(501)

####  Supplementary Note 6.4.1 Proof of Proposition 32

In the proof, we denote

$\displaystyle\tilde{\Pi}_{(L,N,L^{\prime},Q)}=\Pi_{L^{\prime},\leq
Q}\tilde{\Pi}_{(L,N)}.$ (502)

In applying Corollary 26, we aim to estimate the norms of

$\displaystyle\left\|\tilde{\Pi}_{(L,N,L^{\prime},Q)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$ (503)

and

$\displaystyle\left\|\tilde{\Pi}_{(L,N,L^{\prime},Q)}H_{0}\tilde{\Pi}_{(L,N,L^{\prime},Q)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$ (504)

for arbitrary $|\tau_{1}|,|\tau_{2}|\leq t$.

First, we have the decomposition of

$\displaystyle\tilde{\Pi}_{(L,N,L^{\prime},Q)}^{\rm c}=\Pi_{L^{\prime},\leq
Q}^{\rm c}\tilde{\Pi}_{(L,N)}+\Pi_{L^{\prime},\leq Q}\tilde{\Pi}_{(L,N)}^{\rm
c}+\Pi_{L^{\prime},\leq Q}^{\rm c}\tilde{\Pi}_{(L,N)}^{\rm
c}=\Pi_{L^{\prime},\leq Q}^{\rm c}\tilde{\Pi}_{(L,N)}+\tilde{\Pi}_{(L,N)}^{\rm
c},$ (505)

where we use $\Pi_{L^{\prime},\leq Q}+\Pi_{L^{\prime},\leq Q}^{\rm c}=1$ in
the second equation. By using the above, we upper-bound Eqs. (503) and (504)
as follows:

$\displaystyle\left\|\tilde{\Pi}_{(L,N,L^{\prime},Q)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}\leq\left\|\tilde{\Pi}_{(L,N)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}+\left\|\Pi_{L^{\prime},\leq
Q}^{\rm c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F},$ (506)

and

$\displaystyle\left\|\tilde{\Pi}_{(L,N,L^{\prime},Q)}H_{0}\tilde{\Pi}_{(L,N,L^{\prime},Q)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$
$\displaystyle\leq\left\|\tilde{\Pi}_{(L,N,L^{\prime},Q)}H_{0}\Pi_{L^{\prime},\leq
Q}^{\rm
c}\tilde{\Pi}_{(L,N)}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}+\left\|\tilde{\Pi}_{(L,N,L^{\prime},Q)}H_{0}\tilde{\Pi}_{(L,N)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$
$\displaystyle\leq\left\|\Pi_{L^{\prime},\leq Q}H_{0}\Pi_{L^{\prime},\leq
Q}^{\rm
c}\tilde{\Pi}_{(L,N)}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}+\left\|\tilde{\Pi}_{(L,N)}H_{0}\tilde{\Pi}_{(L,N)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}.$ (507)

We have already upper-bounded $\left\|\tilde{\Pi}_{(L,N)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$ and
$\left\|\tilde{\Pi}_{(L,N)}H_{0}\tilde{\Pi}_{(L,N)}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$ in the
inequalities (471) and (472), respectively. Therefore, we need to estimate the
norms of $\left\|\Pi_{L^{\prime},\leq Q}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$ and
$\left\|\Pi_{L^{\prime},\leq Q}H_{0}\Pi_{L^{\prime},\leq Q}^{\rm
c}\tilde{\Pi}_{(L,N)}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$,
which are upper-bounded by the following inequalities (see below for the
proofs):

$\displaystyle\left\|\Pi_{L^{\prime},\leq Q}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$
$\displaystyle\leq\zeta_{0}e^{-\left[Q/(4b_{0}\ell_{1}^{D})\right]^{1/\kappa}/2},$
(508)

and

$\displaystyle\left\|\Pi_{L^{\prime},\leq Q}H_{0}\Pi_{L^{\prime},\leq Q}^{\rm
c}\tilde{\Pi}_{(L,N)}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}$
$\displaystyle\leq
4\zeta_{0}\gamma\bar{J}Qe^{-\left[Q/(4b_{0}\ell_{1}^{D})\right]^{1/\kappa}/2}.$
(509)

By applying the upper bounds in the inequalities (471), (472), (508) and (509)
to to Corollary 26, we obtain the upper bound of

$\displaystyle\left\|\left[O_{X_{0}}(H,t)-O_{X_{0}}(\tilde{H}_{(L,N,L^{\prime},Q)},t)\right]\rho_{0}\right\|_{1}$
$\displaystyle\leq
2^{2D+5}\zeta_{0}\gamma|L|D!\left(1+2c_{\kappa}\gamma\bar{J}t\xi_{t,R}^{2}\right)e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(8\kappa)}+\zeta_{0}(4+8t\gamma\bar{J}Q)e^{-\left[Q/(4b_{0}\ell_{1}^{D})\right]^{1/\kappa}/2},$
(510)

where we use the inequality (Supplementary Note 6.3.1) to estimate the
contributions from (471) and (472). We thus prove the main inequality (32).

In deriving the inequality (32), we only have to estimate [see also
(Supplementary Note 6.3.1)]

$\displaystyle\left\|O_{X_{0}}(\tilde{H}_{(L,N,L^{\prime},Q)},t)\left(\rho_{0}-\tilde{\Pi}_{(L,N,L^{\prime},Q)}\rho_{0}\tilde{\Pi}_{(L,N,L^{\prime},Q)}\right)\right\|_{1}$
$\displaystyle\leq 2\zeta_{0}\left\|\tilde{\Pi}_{(L,N,L^{\prime},Q)}^{\rm
c}\sqrt{\rho_{0}}\right\|_{F}$ $\displaystyle\leq
2\zeta_{0}\left(\left\|\tilde{\Pi}_{(L,N)}^{\rm
c}\sqrt{\rho_{0}}\right\|_{F}+\left\|\Pi_{L^{\prime},\leq Q}^{\rm
c}\sqrt{\rho_{0}}\right\|_{F}\right).$ (511)

By applying the upper bounds (471) and (508) to the above inequality, we
obtain the upper bound of

$\displaystyle\left\|O_{X_{0}}(\tilde{H}_{(L,N,L^{\prime},Q)},t)\left(\rho_{0}-\tilde{\Pi}_{(L,N,L^{\prime},Q)}\rho_{0}\tilde{\Pi}_{(L,N,L^{\prime},Q)}\right)\right\|_{1}$
$\displaystyle\leq
2^{2D+4}\zeta_{0}\gamma|L|D!e^{-\left[N/(2\xi_{t,R})\right]^{1/\kappa}/(8\kappa)}+2\zeta_{0}e^{-\left[Q/(4b_{0}\ell_{1}^{D})\right]^{1/\kappa}/2},$
(512)

which yields the inequality (32) by combining the inequality (510). This
completes the proof. $\square$

[Proof of the inequalities (508) and (509)]

We can adopt similar techniques to the proof of Proposition 29. As in the
inequality (Supplementary Note 6.2.1), we utilize the inequality (434) with
$p=0$ in Lemma 28, which yields

$\displaystyle\left\|\Pi_{L^{\prime},\leq Q}^{\rm
c}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}^{2}$
$\displaystyle={\rm
tr}\left[\Pi_{L^{\prime},>Q}O_{X_{0}}(\tau_{2})\rho_{0}(\tau_{1})O_{X_{0}}(\tau_{2})^{\dagger}\right]$
$\displaystyle\leq{\rm
tr}\left[\Pi_{i_{1}[\ell_{1}],>Q}O_{X_{0}}(\tau_{2})\rho_{0}(\tau_{1})O_{X_{0}}(\tau_{2})^{\dagger}\right]\leq\zeta_{0}^{2}e^{-\left[Q/(4b_{0}\ell_{1}^{D})\right]^{1/\kappa}},$
(513)

where in the first inequality we use
$\Pi_{L^{\prime},>Q}=\Pi_{L^{\prime},>Q}\Pi_{i_{1}[\ell_{1}],>Q}$ because of
$L^{\prime}\subseteq i_{1}[\ell_{1}-1]\subset i_{1}[\ell_{1}]$, and in the
second inequality, we use $d_{i_{1}[\ell_{1}],X_{0}}\geq\bar{\ell}_{t,R}$ and
$\ell_{1}\geq\bar{\ell}_{t,R}$. We thus prove the first inequality (508).

Next, to prove the second inequality (509), we first utilize the decomposition
(458) as

$\displaystyle\Pi_{L^{\prime},\leq Q}H_{0}\Pi_{L^{\prime},\leq Q}^{\rm
c}=\Pi_{L^{\prime},\leq Q}\left(H_{0,L^{\prime}}+\partial
h_{L^{\prime}}\right)\Pi_{L^{\prime},\leq Q}^{\rm c}.$ (514)

We then obtain

$\displaystyle\left\|\Pi_{L^{\prime},\leq Q}H_{0}\Pi_{L^{\prime},\leq Q}^{\rm
c}\tilde{\Pi}_{(L,N)}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}^{2}$
$\displaystyle={\rm tr}\left[\left(H_{0,L^{\prime}}+\partial
h_{L^{\prime}}\right)^{\dagger}\Pi_{L^{\prime},\leq
Q}\left(H_{0,L^{\prime}}+\partial h_{L^{\prime}}\right)\Pi_{L^{\prime},\leq
Q}^{\rm
c}\tilde{\Pi}_{(L,N)}O_{X_{0}}(\tau_{2})\rho_{0}(\tau_{1})O_{X_{0}}(\tau_{2})\tilde{\Pi}_{(L,N)}\Pi_{L^{\prime},\leq
Q}^{\rm c}\right].$ (515)

To obtain an operator upper bound for $\left(H_{0,L^{\prime}}+\partial
h_{L^{\prime}}\right)^{\dagger}\Pi_{L^{\prime},\leq
Q}\left(H_{0,L^{\prime}}+\partial h_{L^{\prime}}\right)$, we first derive

$\displaystyle\left|H_{0,L^{\prime}}+\partial
h_{L^{\prime}}\right|\preceq\bar{J}\sum_{i\in
L^{\prime}}\sum_{j:d_{i,j}=1}\left(|b_{i}b_{j}^{\dagger}|+|b_{i}^{\dagger}b_{j}|\right)\preceq
2\bar{J}\sum_{i\in
L^{\prime}}\sum_{j:d_{i,j}=1}\left(\hat{n}_{i}+\hat{n}_{j}\right)\preceq
4\gamma\bar{J}\hat{n}_{L^{\prime}[1]},$ (516)

where we use the operator inequality (459) in the second inequality, and the
last inequality is derived from

$\displaystyle\sum_{i\in
L^{\prime}}\sum_{j:d_{i,j}=1}\hat{n}_{i}\preceq\gamma\sum_{i\in
L^{\prime}}\hat{n}_{i}\preceq\gamma\hat{n}_{L^{\prime}[1]},$
$\displaystyle\sum_{i\in
L^{\prime}}\sum_{j:d_{i,j}=1}\hat{n}_{j}\preceq\sum_{j\in
L^{\prime}[1]}\sum_{i:d_{i,j}=1}\hat{n}_{j}\preceq\gamma\sum_{j\in
L^{\prime}[1]}\hat{n}_{j}\preceq\gamma\hat{n}_{L^{\prime}[1]}.$ (517)

Note that $i\in L^{\prime}$ and $j:d_{i,j}=1$ implies $j\in L^{\prime}[1]$.
Because $[|H_{0,L^{\prime}}+\partial
h_{L^{\prime}}|,\hat{n}_{L^{\prime}[1]}]=0$, we can obtain from (516)

$\displaystyle\left(H_{0,L^{\prime}}+\partial
h_{L^{\prime}}\right)^{\dagger}\Pi_{L^{\prime},\leq
Q}\left(H_{0,L^{\prime}}+\partial h_{L^{\prime}}\right)$
$\displaystyle\preceq\left|H_{0,L^{\prime}}+\partial
h_{L^{\prime}}\right|^{2}\preceq
16\gamma^{2}\bar{J}^{2}\hat{n}_{L^{\prime}[1]}^{2},$ (518)

where we use the fact that for arbitrary Hermitian operators $A$ and $B$ such
that $|A|\preceq|B|$, we have $|A|^{s}\preceq|B|^{s}$ ($s\in\mathbb{N}$) as
long as $[|A|,|B|]=0$*5*5*5In general, we cannot ensure
$|A|^{s}\preceq|B|^{s}$ form $|A|\preceq|B|$ unless $[|A|,|B|]=0$. This point
has made the proof of Subtheorem 1 rather challenging [see the discussion
around Eq. (82)]..

By combining the inequalities (Supplementary Note 6.4.1) and (518), we obtain

$\displaystyle\left\|\Pi_{L^{\prime},\leq Q}H_{0}\Pi_{L^{\prime},\leq Q}^{\rm
c}\tilde{\Pi}_{(L,N)}O_{X_{0}}(\tau_{2})\sqrt{\rho_{0}(\tau_{1})}\right\|_{F}^{2}$
$\displaystyle\leq 16\gamma^{2}\bar{J}^{2}{\rm
tr}\left[\hat{n}_{L^{\prime}[1]}^{2}\Pi_{L^{\prime},\leq Q}^{\rm
c}\tilde{\Pi}_{(L,N)}O_{X_{0}}(\tau_{2})\rho_{0}(\tau_{1})O_{X_{0}}(\tau_{2})\tilde{\Pi}_{(L,N)}\Pi_{L^{\prime},\leq
Q}^{\rm c}\right]$ $\displaystyle\leq 16\gamma^{2}\bar{J}^{2}{\rm
tr}\left[\hat{n}_{L^{\prime}[1]}^{2}\Pi_{L^{\prime}[1],>Q}O_{X_{0}}(\tau_{2})\rho_{0}(\tau_{1})O_{X_{0}}(\tau_{2})\right]$
$\displaystyle\leq 16\gamma^{2}\bar{J}^{2}{\rm
tr}\left[\hat{n}_{i_{1}[\ell_{1}]}^{2}\Pi_{i_{1}[\ell_{1}],>Q}O_{X_{0}}(\tau_{2})\rho_{0}(\tau_{1})O_{X_{0}}(\tau_{2})\right]\leq
16\zeta_{0}^{2}\gamma^{2}\bar{J}^{2}Q^{2}e^{-\left[Q/(4b_{0}\ell_{1}^{D})\right]^{1/\kappa}},$
(519)

where we use $L^{\prime}[1]\subseteq i_{1}[\ell_{1}]$ from
$L^{\prime}\subseteq i_{1}[\ell_{1}-1]$ in the third inequality and apply the
inequality (434) with $p=2$ in the last inequality. Note that we have assumed
$d_{i_{1}[\ell_{1}],X_{0}}\geq\bar{\ell}_{t,R}$ (i.e., $i_{1}[\ell_{1}]\subset
X_{0}[\bar{\ell}_{t,R}]^{\rm c}$).

##  Supplementary Note 7 Bosonic Lieb-Robinson bound: looser results

Based on Proposition 20 (or Proposition 29), we can truncate the boson number
on each of the sites up to
$\mathcal{O}(\bar{\ell}_{t,R}^{D})=\tilde{\mathcal{O}}(t^{D})$ [see Eq. (433)
for the definition of $\bar{\ell}_{t,R}$]. By such a truncation, we can upper-
bound the local energy on each site by $\tilde{\mathcal{O}}(t^{D})$. This
energy upper bound allows us to derive the Lieb-Robinson bound with a velocity
of $\tilde{\mathcal{O}}(t^{D})$. However, as we will prove below, the optimal
Lieb-Robinson velocity is given by $\tilde{\mathcal{O}}(t^{D-1})$ instead of
$\tilde{\mathcal{O}}(t^{D})$. Nevertheless, the following analyses are quite
simple, and we can utilize them to develop a quantum algorithm to simulate the
dynamics by quantum computer (see Result 3 in the main manuscript).

We here prove the following theorem:

######  Theorem 2.

Let us consider an initial state $\rho_{0}$ that satisfies Assumption 3 with
$b_{\rho}=b_{0}/\gamma$ in (34). Also, we consider an arbitrary operator
$O_{X_{0}}$ ($X_{0}\subseteq i_{0}[r_{0}]$*6*6*6Roughly speaking, $r_{0}$ is
the spatial size of the operator $O_{X_{0}}$.,
$\left\|O_{X_{0}}\right\|=\zeta_{0}$) such that the condition (428) holds.
Then, there exists a unitary operator $U_{X[R]}$ such that the following Lieb-
Robinson bound holds:

$\displaystyle\left\|\left[O_{X_{0}}(H,t)-U_{i_{0}[R]}^{\dagger}O_{X_{0}}U_{i_{0}[R]}\right]\rho_{0}\right\|_{1}$
$\displaystyle\leq
4\zeta_{0}\gamma^{2}R^{D}\left(1+R/c_{1}\right)\exp\left[-\frac{1}{2}\left(\frac{R-r_{0}}{8b_{0}c_{1}\bar{J}t\bar{\ell}_{t,R}^{D}}\right)^{1/\kappa}\right]+\zeta_{0}R^{D+1}e^{-(R-r_{0})/(4k)},$
(520)

where $c_{1}:=2^{6-D}e^{2}k\gamma^{3}(2k)^{2D}$ and we assume
$R\geq\max\left[r_{0}+16k,r_{0}+2(\bar{\ell}_{t,R}+1)\right]$.

Remark. By using the $\Theta$ notation in Eq. (9), we can express the
inequality (2) as

$\displaystyle\left\|\left[O_{X_{0}}(H,t)-U_{i_{0}[R]}^{\dagger}O_{X_{0}}U_{i_{0}[R]}\right]\rho_{0}\right\|_{1}$
$\displaystyle\leq\zeta_{0}\Theta(R^{D+1})\left\\{e^{-[\Theta(R-r_{0})/(t\bar{\ell}_{t,R}^{D})]^{1/\kappa}}+e^{-(R-r_{0})/(4k)}\right\\}$
$\displaystyle\leq\zeta_{0}\exp\left[-\Theta\left(\frac{R-r_{0}}{t\bar{\ell}_{t,R}^{D}}\right)^{1/\kappa}+\Theta(\log(R))\right].$
(521)

Because of $\bar{\ell}_{t,R}\propto t\log(Rq_{0})$, for $r_{0}=\mathcal{O}(1)$
and $q_{0}={\rm poly}(R)$, the approximation error becomes sufficiently small
as long as

$\displaystyle R\propto t^{D+1}\log^{D+1}(t),$ (522)

which gives the Lieb-Robinson velocity of $\tilde{\mathcal{O}}(t^{D})$.

On the unitary operator $U_{i_{0}[R]}$, we show the explicit form in the
inequality (533), which is constructed by using the effective Hamiltonian. The
construction only depends on the interaction terms within the region of
$i_{0}[R]$. Hence, when we consider the subset Hamiltonian $H_{i_{0}[R]}$, we
obtain the same inequality as (2) for
$\left\|\left[O_{X_{0}}(H_{i_{0}[R]},t)-U_{i_{0}[R]}^{\dagger}O_{X_{0}}U_{i_{0}[R]}\right]\rho_{0}\right\|_{1}$
by using the same unitary operator $U_{i_{0}[R]}$, which yields

$\displaystyle\left\|\left[O_{X_{0}}(H,t)-O_{X_{0}}(H_{i_{0}[R]},t)\right]\rho_{0}\right\|_{1}$
$\displaystyle\leq\left\|\left[O_{X_{0}}(H,t)-U_{i_{0}[R]}^{\dagger}O_{X_{0}}U_{i_{0}[R]}\right]\rho_{0}\right\|_{1}+\left\|\left[O_{X_{0}}(H_{i_{0}[R]},t)-U_{i_{0}[R]}^{\dagger}O_{X_{0}}U_{i_{0}[R]}\right]\rho_{0}\right\|_{1}$
$\displaystyle\leq
8\zeta_{0}\gamma^{2}R^{D}\left(1+R/c_{1}\right)\exp\left[-\frac{1}{2}\left(\frac{R-r_{0}}{8b_{0}c_{1}\bar{J}t\bar{\ell}_{t,R}^{D}}\right)^{1/\kappa}\right]+2\zeta_{0}R^{D+1}e^{-(R-r_{0})/(4k)}.$
(523)

Therefore, we can obtain the approximation for $O_{X_{0}}(H,t)$ by simply
considering the time evolution by the subset Hamiltonian $H_{i_{0}[R]}$ on the
region $i_{0}[R]$.

###  Supplementary Note 7.1 Proof of Theorem 2: basic strategy

Supplementary Figure 14: Schematic picture of the region $\tilde{X}_{0}$. In
this region, we truncate the boson number up to $\bar{q}$ and define the
projection as $\bar{\Pi}_{\bar{q}}$. We first approximate the original
dynamics $e^{-iHt}$ by using the dynamics
$e^{-i\bar{\Pi}_{\bar{q}}H\bar{\Pi}_{\bar{q}}t}$. Then, because the effective
Hamiltonian $\bar{\Pi}_{\bar{q}}H\bar{\Pi}_{\bar{q}}$ has a finite local
energy in the region $\tilde{X}_{0}$, we can derive the Lieb-Robinson bound
and approximate $e^{-i\bar{\Pi}_{\bar{q}}H\bar{\Pi}_{\bar{q}}t}$ by
$e^{-i\bar{\Pi}_{\bar{q}}H_{X_{0}[2\ell_{0}]}\bar{\Pi}_{\bar{q}}t}$.

For the proof, we consider the projection operator
$\bar{\Pi}_{\tilde{X}_{0},\bar{q}}$ (see Fig. 14), where we use the notation
in Eq. (452), i.e.,

$\displaystyle\bar{\Pi}_{\tilde{X}_{0},\bar{q}}:=\prod_{i\in\tilde{X}_{0}}\Pi_{i,\leq\bar{q}},\quad\tilde{X}_{0}:=X_{0}[2\ell_{0}]\setminus
X_{0}[\ell_{0}]\quad(\ell_{0}\geq 8k),$ (524)

where $X_{0}\subseteq i_{0}[r_{0}]$ and

$\displaystyle R=2\ell_{0}+r_{0},\quad X_{0}[2\ell_{0}]\subseteq
i_{0}[r_{0}+2\ell_{0}]=i_{0}[R].$ (525)

Note that the condition $R\geq r_{0}+16k$ implies $\ell_{0}\geq 8k$. In the
following, we simply denote it as

$\displaystyle\bar{\Pi}_{\tilde{X}_{0},\bar{q}}\to\bar{\Pi}_{\bar{q}}$ (526)

by omitting the region dependence $\tilde{X}_{0}$. We choose $\bar{q}$ ($\geq
2$) such that

$\displaystyle\bar{q}=\frac{\ell_{0}}{c_{1}\bar{J}t},\quad
c_{1}:=2^{6-D}e^{2}k\gamma^{3}(2k)^{2D},$ (527)

where the choice is utilized in ensuring the condition (554) below*7*7*7
Precisely speaking, in Eqs. (525) and $\eqref{choice_of_bar_q_t^d}$ for the
definitions of $\ell_{0}$ and $\bar{q}$, we should define
$\ell_{0}=\left\lfloor(R-r_{0})/2\right\rfloor$ and
$\bar{q}=\left\lfloor\ell_{0}/(c_{1}t)\right\rfloor$, but for simplicity we
assume they become integers without the floor function..

The basic strategy for the proof is similar to the one in Ref. [1], and we
summarize the tasks as follows:

  1. 1.

We approximate the dynamics $e^{-iHt}$ by using the effective Hamiltonian
$\bar{\Pi}_{\bar{q}}H\bar{\Pi}_{\bar{q}}$; that is, our first task is to
estimate the error of

$\displaystyle\left\|\left[O_{X_{0}}(H,t)-O_{X_{0}}(\bar{\Pi}_{\bar{q}}H\bar{\Pi}_{\bar{q}},t)\right]\rho_{0}\right\|_{1}.$
(528)

  2. 2.

For the time evolution of
$O_{X_{0}}(\bar{\Pi}_{\bar{q}}H\bar{\Pi}_{\bar{q}},t)$, we approximate the
dynamics by using the Hamiltonian
$\bar{\Pi}_{\bar{q}}H_{X_{0}[2\ell_{0}]}\bar{\Pi}_{\bar{q}}$ which is
supported on $X_{0}[2\ell_{0}]$. Here, in the Hamiltonian
$\bar{\Pi}_{\bar{q}}H\bar{\Pi}_{\bar{q}}$, the interaction terms are bounded
in the region $\tilde{X}_{0}$ due to the truncation of the boson number.
However, in the region $\tilde{X}_{0}^{\rm c}$, we still encounter the
unboundedness of the norm of the interactions. In Ref. [1], the authors have
developed analyses to derive the Lieb-Robinson bound for
$e^{-i\bar{\Pi}_{\bar{q}}H\bar{\Pi}_{\bar{q}}t}$ only from the finite upper
bound of the Hamiltonian norm in the region $\tilde{X}_{0}$.

We start from the estimation of the norm (528). The error is immediately
obtained by using Proposition 29 with the choice of $L=\tilde{X}_{0}$ in
(453):

$\displaystyle\left\|\left[O_{X_{0}}(H,t)-O_{X_{0}}(\bar{\Pi}_{\bar{q}}H\bar{\Pi}_{\bar{q}},t)\right]\rho_{0}\right\|_{1}\leq
4\zeta_{0}\gamma|\tilde{X}_{0}|\left(1+\bar{J}t\bar{q}\right)e^{-\left[\bar{q}/(4b_{0}\bar{\ell}_{t,R}^{D})\right]^{1/\kappa}/2},$
(529)

where the following condition should be satisfied

$\displaystyle\ell_{0}=d_{X_{0},\tilde{X}_{0}}\geq\bar{\ell}_{t,R}+1.$ (530)

By using $|\tilde{X}_{0}|\leq|X_{0}[2\ell_{0}]|\leq|i_{0}[R]|\leq\gamma
R^{D}$, and $\bar{J}t\bar{q}=\ell_{0}/c_{1}$ ($\leq R/c_{1}$) , we reduce the
inequality (529) to