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# The ground state of electron-doped $t-t^{\prime}-J$ model on cylinders

Yang Shen  Key Laboratory of Artificial Structures and Quantum Control
(Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China  Xiangjian Qian  Key Laboratory of
Artificial Structures and Quantum Control (Ministry of Education), School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
Mingpu Qin  Key Laboratory of Artificial Structures and Quantum Control
(Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China  Hefei National Laboratory, Hefei 230088,
China

###### Abstract

We perform a comprehensive study of the electron-doped $t-t^{\prime}-J$ model
on cylinders with Density Matrix Renormalization Group (DMRG). We adopt both
periodic and anti-periodic boundary conditions along the circumference
direction to explore the finite size effect. We study doping levels of $1/6$,
$1/8$, and $1/12$ which represent the most interesting region in the phase
diagram of electron-doped cuprates. We find that for width-4 and 6 systems,
the ground state for fixed doping switches between anti-ferromagnetic Neel
state and stripe state under different boundary conditions and with system
widths, indicating the presence of large finite size effect in the
$t-t^{\prime}-J$ model. We also have a careful analysis of the $d$-wave
pairing correlations which also changes quantitatively with boundary
conditions and widths of the system. However, the pairing correlations are
enhanced when the system becomes wider for all dopings, suggesting the
existence of possible long-ranged superconducting order in the thermodynamic
limit. The width-8 results are found to be dependent on the starting state in
the DMRG calculation for the kept states we can reach. For width-8 system only
Neel (stripe) state can be stabilized in DMRG calculation for $1/12$ ($1/6$)
doping, while both stripe and Neel states are stable in the DMRG sweep for
$1/8$ doping, regardless of the boundary conditions. These results indicate
that $1/8$ doping is likely to lie in the boundary of a phase transition
between the Neel phase with lower doping and the stripe phase with higher
doping, consistent with the previous study. The sensitivity of ground state on
boundary conditions and size observed in this work is similar to that in the
$t^{\prime}$\- Hubbard model.

##  I Introduction

Hubbard model and its descendant $t-J$ model Hubbard (1963); Zhang and Rice
(1988); Qin et al. (2022); Arovas et al. (2022); LeBlanc et al. (2015) play a
paradigmatic role in the study of strongly correlated quantum many-body
physics Scalapino (2012); Dagotto (1994). Though their forms are simple, many
exotic quantum states are arguably related to these models. It is believed
that they are connected to the high-$T_{c}$ superconductivity in cuprates
Anderson (1987); Dagotto (1994); Zhang and Rice (1988); B. Keimer (2015);
Scalapino (2012). Recent state-of-the-art numerical results show that the
ground state of pure Hubbard model (with only nearest hoppings) is stripe
state Zheng et al. (2017); Xu et al. (2022) without long-ranged
superconductivity Qin et al. (2020), which indicates the necessity to modify
the Hamiltonian if we want to pursue superconducting ground state in the
single-band framework.

It was well known that the next nearest neighboring hopping $t^{\prime}$ term
is necessary to account for the particle-hole asymmetry in the phase diagram
of cuprates Scalapino (2012). Band structure calculations Andersen et al.
(1995); Hirayama et al. (2018) also show the existence of a
$t^{\prime}\approx-0.2$ term in the single band description of cuprates. The
effect of $t^{\prime}$ term on the ground state of Hubbard and $t-J$ models
was extensively explored before White and Scalapino (1999); Jiang et al.
(2020); Dodaro et al. (2017); Chung et al. (2020); Huang et al. (2018); Gong
et al. (2021); Jiang and Kivelson (2021); Jiang et al. (2021); Lu et al.
(2023, 2024); Jiang et al. (2022); Marino et al. (2022); Ponsioen et al.
(2019); Chen et al. (2023). The $t^{\prime}$ term frustrates and weakens the
long-range anti-ferromagnetic Neel order in the parent compound. When the
system is doped, the stripe order should also be suppressed by $t^{\prime}$
term which is believed to benefit superconducting pairing. Calculations on
$t-t^{\prime}-J$ model Jiang and Kivelson (2021); Gong et al. (2021); Jiang et
al. (2021) show the ground state turns to a superconducting one in the
electron doped side. Superconductivity was also found in the $t^{\prime}$\-
Hubbard model Xu et al. (2023) and matches quantitatively with the phase
diagram of cuprates, in which superconductivity is stronger in the hole doped
side than the electron doped side.

One of the important finding in Xu et al. (2023) is that the ground state of
the $t^{\prime}$\- Hubbard model becomes very sensitive to the system width
and boundary conditions, which indicates the presence of large finite size
effect in the model. For this reason, the authors in Xu et al. (2023) utilized
twist averaged boundary conditions and careful finite size scaling to obtain
physical quantity in the thermodynamic limit. For the $t-t^{\prime}-J$ model,
the most recent study Lu et al. (2024); Jiang et al. (2021) on cylinders with
finite width show the superconductivity is stronger in the electron doped side
than the hole doped side if exists, contradictory to the phase diagram of
cuprates and the recent results of $t^{\prime}$-Hubbard model Xu et al.
(2023). Given the presence of very large finite size effect in the
$t^{\prime}$-Hubbard model, it is natural to ask whether $t-t^{\prime}-J$
model also has large finite size effect.

In this work, we have a comprehensive study of the $t-t^{\prime}-J$ model with
Density Matrix Renormalization Group (DMRG) White (1992, 1993); Schollwöck
(2011). We adopt both periodic and anti-periodic boundary conditions along the
circumference direction and study systems with different widths to explore the
finite size effect Xu et al. (2023). We only study the electron doped side in
this work, which was believed to be simper than the hole doped side, without
the competition of different states Scalapino (2012). We study doping levels
of $1/6$, $1/8$, and $1/12$ which represent the most interesting region in the
phase diagram of electron-doped cuprates.

Our results show that the ground state of $t-t^{\prime}-J$ model is also very
sensitive to the boundary conditions and width of the systems, similar to the
$t^{\prime}$-Hubbard model Xu et al. (2023). We find that for width-4 and 6
systems, the ground state for fixed doping switches between anti-ferromagnetic
Neel state and stripe state under different boundary conditions and with
system widths. We find the pairing correlation also changes quantitatively
with boundary conditions and widths. But for all dopings, the pairing
correlations are enhanced when the system becomes wider, suggesting the
existance of possible long-ranged superconducting pairing in the thermodynamic
limit. The width-8 results are found to be dependent on the starting state in
the DMRG calculation for the kept states we can reach. For width-8 system only
Neel (stripe) state can be stabilized in DMRG calculation for $1/12$ ($1/6$)
doping, while both stripe and Neel states are stable in the DMRG sweep for
$1/8$ doping. Although a reliable finite width scaling of the DMRG results is
infeasible at present, these results indicate that $1/8$ doping is likely to
lie in the boundary of the phase transition between the Neel state with lower
doping and stripe state with higher doping, consistent with the previous study
Jiang et al. (2021).

The rest of this paper is organized as follows. In Sec. II we discuss the
model and the method we employ. Our results are presented in Sec. III:
starting with the spin and charge densities, followed by an analysis of the
pairing correlations. In Sec. IV, we have a discussion of the relationship
between superconductivity and stripe. We conclude in Sec. V.

| width-4 | width-6 | width-8
---|---|---|---
PBC | APBC | PBC | APBC |  PBC | APBC
1/12 | spin | Neel, short-ranged | filled stripe | Neel, short-ranged | Neel, (quasi) long-ranged | only Neel stablized
charge | half-filled stripe | filled stripe | 1/3 filled stripe | 1/3 filled stripe
pair | power-law, $K_{sc}\approx 1.1(0.98)$ | exponential, $\xi_{sc}\approx 9.82(9.9)$ | power-law, $K_{sc}\approx 0.49(0.49)$ | power-law, $K_{sc}\approx 0.48(0.41)$ |  |
1/8 | spin | Neel, short-ranged | filled stripe | filled stripe, short-ranged | Neel, (quasi) long-ranged | both filled stripe $\&$ Neel stablized
charge | half-filled stripe | filled stripe | filled stripe, short-ranged | uniform
pair | power-law, $K_{sc}\approx 1.3(1.38)$ | exponential, $\xi_{sc}\approx 13.3(13.4)$ | power-law, $K_{sc}\approx 0.60(0.82)$ | power-law, $K_{sc}\approx 0.59(0.69)$ |  |
1/6 | spin | stripe, short-ranged | filled stripe | filled stripe, short-ranged | Neel,(quasi) long-ranged | only filled stripe stablized
charge | modulated stripe | filled stripe | filled stripe, short-ranged | uniform
pair | power-law, $K_{sc}\approx 1.5(1.8)$ | exponential, $\xi_{sc}\approx 14.4(14.3)$ | power-law, $K_{sc}\approx 1.0(1.2)$ | power-law, $K_{sc}\approx 0.76(0.83)$ |  |

Table 1: Summary of all the results, including the spin and charge densities,
as well as the singlet pair-pair correlations. All systems have $d$-wave
pairing symmetry except the $1/6$ doping case on width-4 cylinder under PBC,
for which the pairing symmetry is plaquette $d$-wave Chung et al. (2020).
$K_{sc}$ and $\xi_{sc}$ are obtained from the fit of pair-pair correlation
with the reference bond set as the 8th (4th for the values in the parentheses)
vertical bond from the left edge.

Figure 1: Staggered spin density and electron density for $1/12$ (top), $1/8$
(middle) and $1/6$ (bottom) dopings on width-4 cylinders (system sizes are
$4\times 48$, $4\times 32$ and $4\times 36$ for $1/12$, $1/8$ and $1/6$
dopings, respectively) under PBC (left two columns) and APBC (right two
columns). Results of finite kept states are plotted to show the convergence of
the DMRG calculations. The insets in (a) and (e) are plotted in semi-
logarithmic scale to show the exponential decay of the staggered spin density
for $1/12$ and $1/8$ dopings under PBC. The dashed horizontal lines represent
zero for the staggered spin density.

Figure 2: Similar as in Fig. 1 but for systems with width 6. Staggered spin
density and electron density for $1/12$ (top), $1/8$ (middle) and $1/6$
(bottom) dopings on width-6 cylinders (system sizes are $6\times 48$, $6\times
32$ and $6\times 36$ for $1/12$, $1/8$ and $1/6$ dopings, respectively) under
PBC (left two columns) and APBC (right two columns). Results of finite kept
states are plotted to show the convergence of the DMRG calculations. The inset
in (a) is plotted in semi-logarithmic scale to show the exponential decay of
the staggered spin density for $1/12$ doping under PBC. The dashed horizontal
lines represent zero for the staggered spin density.

Figure 3: Staggered spin density and electron density for $1/12$ (top)and
$1/6$ (bottom) dopings (system sizes for both dopings are $8\times 24$) on
width-8 cylinders with PBC (left two columns) and APBC (right two columns). We
try both Neel and filled stripe state as the starting states in the DMRG
calculations but only Neel (stripe) state can be stabilized in the DMRG sweeps
for $1/12$ ($1/6$) doping. We only show the numerically stabilized states.

Figure 4: Staggered spin density and electron density for $1/8$ doping levels
on width-8 cylinders ($8\times 16$) with PBC (top) and APBC (bottem). In the
DMRG sweep, both Neel and filled stripe states can be stabilized.

##  II Model and methods

The Hamiltonian for $t-t^{\prime}-J$ model is defined as

$H=-t\sum_{\langle
i,j\rangle,s}c_{i,s}^{\dagger}c_{j,s}-t^{\prime}\sum_{\langle\langle
i,j\rangle\rangle,s}c_{i,s}^{\dagger}c_{j,s}+J\sum_{\langle
i,j\rangle}S_{i}.S_{j}$ (1)

where $\hat{c}_{i,s}^{\dagger}(\hat{c}_{i,s})$ creates (annihilates) an
electron on site $i=(x_{i},y_{i})$ with spin $s$ and
$\hat{S}_{i}=1/2\times\hat{c}^{\dagger}_{i,s}\hat{\sigma}\hat{c}_{i,s^{\prime}}$
($\hat{\sigma}$ is the Pauli matrix). The doubly occupied states are
explicitly excluded in the Hilbert space. We consider electron hopping terms
up to next-nearest neighbors ($t^{\prime}$) and set the nearest neighboring
hopping $t$ as energy unit. We choose $J=1/3$ which is relevant to cuprates.
We intend to study the physics related to the electron doped cuprates in this
work. But in principle, only hole doping is allowed in $t-t^{\prime}-J$ model
because double occupancy is excluded in the local Hilbert space. However, by
connecting the $t-t^{\prime}-J$ model as the large $U$ limit of the
$t^{\prime}$-Hubbard model and through a particle-hole transformation, we can
investigate the electron doped cuprates by studying the hole doped
$t-t^{\prime}-J$ model with the sign of $t^{\prime}$ reversed from $-0.2$ in
cuprates to $0.2$ Andersen et al. (1995); Hirayama et al. (2018). We study
cylinders with size $L_{x}\times L_{y}$ and impose periodic or anti-periodic
boundary conditions (PBC or APBC) along the closed circumference direction. We
apply antiferromagnetic pinning fields with strength $h_{m}=0.5$ on the left
open edges in the calculation, which allows us to calculate spin density
instead of the more demanding correlation function, and make the DMRG
calculation easier to converge. The averaged hole concentration away from
half-filling is defined as $\delta=N_{h}/N$ with $N_{h}=\sum_{i}(1-n_{i})$
($\hat{n}_{i}=\sum_{s}\hat{c}_{i,s}^{\dagger}\hat{c}_{i,s}$ and $N=L_{x}\times
L_{y}$) . We study dopings with $\delta=1/12$, $1/8$, and $1/6$ which
represents the most interesting region in the phase diagram of cuprates.

The method we employ in this work is DMRG. In the calculations, the total
particle number of each spin species and total $S_{z}$ are conserved. For
width-4 and 6 systems, we push the number of the kept states to ensure the
calculation is converged (with truncation error in the order of $10^{-6}$ for
width-4 systems) or in the converged region (with truncation error in the
order of $10^{-5}$ for width-6 systems). For width-8 systems, within the kept
states ($D=30000$) we can reach, the results depend on the starting state in
the DMRG sweep. So we try different states (Neel and stripe) to start the DMRG
calculation and try to determine which one is stable during the sweep.

##  III results

In Table 1, we summarize the results for spin and charge density as well as
pair-pair correlation for the $t-t^{\prime}-J$ model on cylinders with
different boundary conditions and widths for dopings $1/12$, $1/8$, and $1/6$.
For the pair-pair correlation, $K_{sc}$ ($\xi_{sc}$) implies it has a power
(exponential)-law decay. The values of $K_{sc}$ and $\xi_{sc}$ are obtained
from the fit of pair-pair correlation with the reference bond set as the 8th
vertical bond from the left edge, while the values in the parentheses are
results by setting the reference bond as the 4th vertical bond from the left
edge. In Table 1, we can find the ground state switches between stripe and
antiferromagnetic Neel state with different boundary conditions and widths.
For all dopings, the pair-pair correlations are enhanced when the cylinder
becomes wider from $4$ to $6$. The DMRG calculations for width 8 systems are
not well converged for the largest kept states we can reach ($D=30000$). But
we try different starting states in the DMRG calculation and find that only
Neel (stripe) state can be stabilized in DMRG calculation for $1/12$ ($1/6$)
doping, while both stripe and Neel states are stable in the DMRG sweep for
$1/8$ doping, regardless of the boundary conditions. These results indicate
that $1/8$ doping is likely to lie in the boundary of the phase transition
between the Neel and stripe phase, consistent with previous study Jiang et al.
(2021). In the following, we discuss the details of the results.

###  III.1 spin and charge density

The staggered spin density is defined as
$(-1)^{{i_{x}+i_{y}}}\times\langle\hat{S}_{i}^{z}\rangle$ with
$\hat{S}_{i}^{z}=1/2\times(n_{i\uparrow}-n_{i\downarrow})$ and charge density
is defined as the averaged density along the circumference direction
$n_{i}=\sum_{y=1}^{L_{y}}\left\langle{n}(i_{x},i_{y})\right\rangle/L_{y}$.

####  III.1.1 Width-4 results

Fig. 1 shows the staggered spin density and electron density for $1/12$ (top),
$1/8$ (middle) and $1/6$ (bottom) dopings on width-4 cylinders under PBC (left
two columns) and APBC (right two columns). Results for different kept states
in DMRG calculation are plotted and we can find the good convergence of the
results with kept states.

We first discuss results under PBC. For $1/12$ and $1/8$ dopings, the
staggered spin density has the same sign and decays exponentially (see the
insets of Fig. 1 (a) and (e)) with the distance to the antiferromagnetic
pinning fields on the left open edge, indicating the systems have short-ranged
Neel correlation, while the charge density shows a half-filled stripe
structure with periodic modulation with $\lambda=1/(2\delta)$. For $1/6$
doping, the spin density displays short-ranged stripe structure and the charge
density has a small fluctuation without clear pattern.

The ground states are different under APBC. For all the three dopings, we find
that the staggered spin density displays a spatial modulation with a
wavelength ($\lambda=2/\delta$) twice that of the charge density and has a
$\pi$ phase flip at the hole-concentrated sites, indicating the systems all
have a filled stripe ground state.

####  III.1.2 Width-6 results

The sensitivity of ground state to boundary conditions in width-4 system
reflects the finize size effect in the model, so we further study width-6
systems. We show the staggered spin density and electron density for $1/12$
(top), $1/8$ (middle) and $1/6$ (bottom) dopings on width-6 cylinders under
PBC (left two columns) and APBC (right two columns) in Fig. 2. We plot results
with different kept states to show the convergence process.

For $1/12$ doping under both PBC and APBC, we find the spin density is Neel-
type (no sign change in staggered spin density) and charge has $1/3$ filled
stripe structure, though the Neel order in spin is short-ranged under PBC but
(quasi) long-ranged under APBC.

For both $1/8$ and $1/6$ dopings under PBC, spin and charge density results
indicate the systems have short-ranged filled stripe order. But under APBC,
they have quasi-long-ranged Neel order in spin with nearly uniform charge
density.

It seems that the antiferromagnetic Neel correlation in spin density at $1/12$
doping is well established on the width-6 cylinders. But for $1/8$ and $1/6$
dopings, the ground state changes under different boundary conditions. We also
notice that the ground states are different between width-4 and width-6
systems with same doping and boundary conditions, indicating the large finite
size effect in the model.

####  III.1.3 Width-8 results

As expected, the DMRG calculation for width-8 systems is challenging. For the
largest kept states we can reach (D = 30000), the truncation error in DMRG
calculation is still in the order of $10^{-4}$, which indicates the
calculation is not well converged. Nevertheless, we perform DMRG calculations
with different starting states (antiferromagnetic Neel and filled stripe
states) and try to determine whether they are stable in the DMRG sweep. We
find that for width-8 system, only Neel (stripe) state can be stabilized in
DMRG calculation for $1/12$ ($1/6$) doping, while both stripe and Neel states
are stable in the DMRG sweep for $1/8$ doping, regardless of the boundary
condtitions. We show the stable states in Fig. 3 for $1/12$ and $1/6$ dopings
and in Fig. 4 for $1/8$ doping. We notice that the Neel state at $1/12$ doping
persists in the width-8 system from width-6 system and the charge is more
uniform in width-8 system. The fact that both filled stripe and Neel states
are stable in the DMRG sweep for $1/8$ doping indicates that $1/8$ doping is
likely to lie in the boundary of the phase transition between the Neel state
at lower doping and the stripe state at higher doping.

In the electron doped side of the phase diagram of cuprates, the
antiferromagnetic Neel phase in the parent compound stretches to electron
doping larger than $0.1$. In our calculation, even though a reliable
extrapolation with system width is infeasible, it is highly likely that the
ground state for $1/12$ doping has antiferromagnetic Neel order. In this
sense, the $t-t^{\prime}-J$ model is in accordance with cuprates.

Figure 5: Singlet pair-pair correlation for $1/12$ (top), $1/8$ (middle) and
$1/6$ (bottom) dopings on width-4 cylinders under PBC (left) and APBC (right).
The reference bond is placed at the bond consisting of sites (8, 2) and (8,
3). Both results for finite kept states and the extrapolated with truncation
error results are shown. Solid lines are algebraic (left) and exponential
(right) fits. The correlation lengths or exponents are listed in the legend.

Figure 6: Similar as Fig. 5 but for width 6 cylinders. Singlet pair-pair
correlation for $1/12$ (top), $1/8$ (middle) and $1/6$ (bottom) dopings on
width-6 cylinders with PBC (left) and APBC (right). The reference bond is
placed at the bond consisting of sites (8, 2) and (8, 3). Both results for
finite kept states and extrapolated with truncation error results are shown.
Solid lines are algebraic fits and the corresponding exponents are listed in
the legend.

###  III.2 pairing correlation

The singlet pair-pair correlation function between bond $i$ (the reference
bond, consisting of sites ($i_{x}$, $i_{y}$) and ($i_{x}$, $i_{y}+1$)) and
bond $i+r$, defined as
$D(r)=\langle\hat{\Delta}_{i}^{\dagger}\hat{\Delta}_{i+r}\rangle$ with
$\hat{\Delta}_{i}^{\dagger}=1/\sqrt{2}(\hat{c}_{(i_{x},i_{y}),\uparrow}^{\dagger}\hat{c}_{(i_{x},i_{y}+1),\downarrow}^{\dagger}-\hat{c}_{(i_{x},i_{y}),\downarrow}^{\dagger}\hat{c}_{(i_{x},i_{y}+1),\uparrow}^{\dagger})$,
are shown in Fig. 5 (width-4) and Fig. 6 (width-6). The pairing symmetry is
found to be $d$-wave with opposite sign for vertical and horizontal bonds
except the width-4 PBC system at $1/6$ doping, for which the pairing symmetry
is plaquette $d$-wave Chung et al. (2020). As mentioned above, the width-8
calculations are not well converged, we don’t show the pair-pair correlation
for width-8 systems. In Figs. 5 and 6, both results for finite kept states and
the extrapolated with truncation error results are shown.

We only show results for $D(r)$ with reference bond as the 8th vertical bond
(consisting of site(8, 2) and site(8, 3)) from the left open edge. It was
known that the value of $D(r)$ depends on the position of reference bond Shen
et al. (2023). We also calculate $D(r)$ with reference bond as the 4th
vertical bond (consisting of site(4, 2) and site(4, 3)). The results don’t
change qualitatively but there are indeed variations for the correlation
lengths/exponents (see Table. 1).

####  III.2.1 Width-4 results

Fig. 5 shows the pair-pair correlation function $D(r)$ for width-4 systems. We
find that for all dopings, $D(r)$ decays algebraically with $D(r)\propto
r^{-K_{sc}}$ (exponentially with $D(r)\propto e^{-r/\xi_{sc}}$ ) under PBC
(APBC). The difference in ground state between systems under PBC and APBC is
also observed for spin and charge densities in Fig. 1. The fitted exponents
$K_{sc}$ or correlation length $\xi_{sc}$ using the extrapolated with
truncation error data are also shown in Fig. 5. We notice that for system
under PBC, when the spin and charge density show a strong filled stripe state,
$D(r)$ decays exponentially. This correlation indicates strong filled stripe
state is harmful for pairing as seen in the pure Hubbard model Qin et al.
(2020).

####  III.2.2 Width-6 results

Fig. 6 shows the pair-pair correlation function $D(r)$ for width-6 systems.
For all the results in Fig. 6, $D(r)$ has a fast decaying tail which bends
upward with the increase of kept states. We attribute this tail as the
intrinsic consequence of finite number of kept states in DMRG calculations.
From the data we find the extrapolation of $D(r)$ with truncation error at the
tail is not as reliable as at the head. We also notice that the behavior of
$D(r)$ for $r\leq 4$ doesn’t fit in the long-range behavior. Based on these
observations, the small $r$ and very large $r$ data are not included when
performing the fit of $D(r)$ to obtain the exponent $K_{sc}$.

For all doping and boundary conditions, $D(r)$ decays polynomially for width-6
systems as shown in Fig. 6. All the systems display quasi long-range pairing
with $K_{sc}<2$. For fixed doping, the difference of $K_{sc}$ between systems
under PBC and APBC are small. For all the dopings, $K_{sc}$ is reduced
comparing to the width-4 results in Fig. 5, indicating superconducting pairing
is enhanced in wider systems and the existence of possible long-ranged
superconductivity in the thermodynamic limit. We also notice the increase of
$K_{sc}$ when the doping is reduced.

##  IV discussions

It was known that the filled stripe state is stable in the pure Hubbard model
for different widths Qin et al. (2020), where long-ranged pairing is absent.
In the $t-t^{\prime}-J$ model, we also notice that when strong filled stripe
order is found in the ground state for the width-4 system under APBC,
superconducting pairing is also short-ranged. These results indicate the
strong filled stripe order is likely to be harmful for superconductivity. In
other cases, short-ranged stripe states with fractional fillings is
accompanied with quasi-long-ranged superconducting pairing. In general, the
$t^{\prime}$ term makes the electron more mobile, causing the paired electrons
in the stripe state Qin et al. (2020) to gain coherence and display
superconductivity Xu et al. (2023). The sensitivity of the ground state of
$t-t^{\prime}-J$ and $t^{\prime}$-Hubbard models Xu et al. (2023) with
boundary conditions and widths can also be viewed as a sign of the fragility
of stripe state when $t^{\prime}$ term is included. These results underscore
the necessity to check the finite size effect in the future study of systems
on narrow cylinders.

##  V conclusion and perspective

We study the finite size effect in the electron doped $t-t^{\prime}-J$ model
on cylinders. We find the sensitivity of ground state to both boundary
conditions and system width, similar to what was found in the
$t^{\prime}$-Hubbard model Xu et al. (2023). For width-4 and 6 systems, the
ground state switches between Neel and stripe states under different boundary
conditions and with different system widths. For with-8 system and up to the
number of kept states we can reach, only Neel (stripe) state can be stabilized
in DMRG calculation for $1/12$ ($1/6$) doping, while both stripe and Neel
states are stable in the DMRG sweep for $1/8$ doping, regardless of the
boundary conditions. These results indicate that $1/8$ doping is likely to lie
in the boundary of the phase transition between the Neel phase at lower doping
and the stripe phase at higher doping, consistent with previous studies Jiang
et al. (2021). Our results underscore the necessity to check the finite size
effect in the future study of systems on narrow cylinders. It will be
interesting to also analyze the finite size effect in the hole doped region in
the future, which is more complicated with different competing states in the
ground state Scalapino (2012).

###### Acknowledgements.

We thank Shiwei Zhang, Steven R. White, Ulrich Schollwöck, Hao Xu, and Chia-
Min Chung for useful discussions. We thank Kang Wang for useful suggestions on
the manuscript. The calculation in this work is carried out with TensorKit foo
. The computation in this paper were run on the Siyuan-1 cluster supported by
the Center for High Performance Computing at Shanghai Jiao Tong University. Y.
Shen and M. P. Qin thank Weidong Luo for his generosity to provide
computational resources. M. P. Qin acknowledges the support from the National
Key Research and Development Program of MOST of China (2022YFA1405400), the
Innovation Program for Quantum Science and Technology (2021ZD0301902), the
National Natural Science Foundation of China (Grant No. 12274290) and the
sponsorship from Yangyang Development Fund.

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In a similar way, we can obtain the bound on the second term of the right hand side of Eq. (\ref{EqA2:Lemma_symmetry_proof_1}), which results in the above formula under the replacement of $n(l^',l)$ by $n(0,l ⊖l^')$.
Since $n(0,l ⊖l^')$ is always larger than $n(l^',l)$ from their definitions, we obtain the accuracy of the approximate translation symmetry as
\begin{equation}
[\text{l.h.s of Eq. (\ref{EqA2:lemma_translation_symmetry})}] \leq 4 \sum_{n=n(l^\prime, l)}^\infty \frac{(\gamma t)^n}{n!}
\leq 8 \frac{(\gamma t)^{n(l,l^\prime)}}{n(l^\prime,l)!}.
\end{equation}
When we choose the truncation order $l_$ by Eq. (\ref{Eq4B:l_max_choice}), the integer $n(l^',l)$ is always larger than $6 γt$ for indices $l ∈D^4l_$ and $l^'∈D^l_$.
We use Eq. (\ref{Eq4A:sum_Taylor}) for the last inequality. $  □$

This lemma ensures that the approximate translation symmetry in the truncated Floquet-Hilbert space is extremely accurate; the right hand side of Eq. (\ref{EqA2:lemma_translation_symmetry}) is approximately $ε^3$ under the choice of $l_$, Eq. (\ref{Eq4B:l_max_choice}).
As we intuitively discuss in Section \ref{Subsec:amplification_symmetry}, the approximate translation symmetry ensures the amplification by symmetry as Eq. (\ref{Eq5A:approximate_amp1_result}).
We next prove the consequence of Theorem \ref{Thm5:amplification_symmetry}, which gives the exact description of Eq. (\ref{Eq5A:approximate_amp1_result});
\begin{equation}
\norm{\bra{0}\ms{U}_\mr{amp1}^{l_\imax}(t)\ket{0}\ket{\psi(0)}-\frac12 \ket{0}\ket{\psi(t)}} \leq \frac{\varepsilon}{3}.
\end{equation}

\textbf{Proof of Theorem \ref{Thm5:amplification_symmetry}.---} We track Eq. (\ref{Eq5A:amp1_calculation}) in a rigorous way. We begin with the definition,
\begin{eqnarray}
&& \bra{0}\ms{U}_\mr{amp1}^{l_\imax}(t)\ket{0}\ket{\psi(0)} = \nonumber \\
&& \qquad = \frac{1}{4l_\imax}\sum_{l^\prime \in D^{l_\imax}}  \sum_{l \in D^{4l_\imax}} e^{-il\omega t} \bra{l} e^{-i\ms{H}_\eff^{4l_\imax}t} \ket{l^\prime} \ket{\psi(0)}. \nonumber \\
&& \label{EqA2:thm_amp1_proof_1}
\end{eqnarray}
We separate the summation over $l ∈D^4l_$ in the above formula by
\begin{equation} \label{EqA2:thm_amp1_proof_2}
\sum_{l \in D^{4l_\imax}} =  \sum_{l; l-l^\prime \in D^{3l_\imax}} + \sum_{l \in D^{4l_\imax}; l-l^\prime \notin D^{3l_\imax}}.
\end{equation}

Let us focus on the first summation. For each $l^'∈D^l_$, the summation can be approximated as
\begin{eqnarray}
&& \norm{\sum_{l; l-l^\prime \in D^{3l_\imax}} e^{-il\omega t} \bra{l} e^{-i\ms{H}_\eff^{4l_\imax}t} \ket{l^\prime} \ket{\psi(0)} - \ket{\psi(t)} } \nonumber \\
&& \leq \sum_{l; l-l^\prime \in D^{3l_\imax}} 8 \frac{(\gamma t)^{n(l^\prime,l)}}{n(l^\prime,l)!} \nonumber \\
&& \qquad + \norm{ \sum_{l \in D^{3l_\imax}} e^{-il\omega t} \bra{l} e^{-i\ms{H}_\eff^{4l_\imax}t} \ket{0}\ket{\psi(0)} -\ket{\psi(t)}} \nonumber \\
&& \leq \sum_{l \in D^{4l_\imax}} 8 \frac{(\gamma t)^{n(l_\imax,l)}}{n(l_\imax,l)!} + 10 m_\imax \left( \frac{em_\imax \gamma t }{3l_\imax} \right)^{3l_\imax / m_\imax}. \nonumber \\
&&  \label{EqA2:thm_amp1_proof_3}
\end{eqnarray}
In the first inequality, we employ Lemma \ref{LemmaA2:translation_symmetry} with taking $l-l^'= l ⊖l^'$ for $l -l^'∈D^3l_$ into consideration.
The last inequality comes from Theorem \ref{Thm4:Floquet_Hilbert_truncation}.
The first term in Eq. (\ref{EqA2:thm_amp1_proof_3}) is further bounded by
\begin{eqnarray}
&& 16 \sum_{l=3l_\imax-2m_\imax}^\infty \frac{(\gamma t)^{\lceil l/m_\imax \rceil}}{(\lceil l / m_\imax \rceil)!} \nonumber \\
&& \quad \leq  16 m_\imax \left( \frac{e\gamma t}{\lceil 3l_\imax /m_\imax -2 \rceil}\right)^{\lceil 3l_\imax /m_\imax -2 \rceil}, \nonumber
\end{eqnarray}
where we use the relation Eq. (\ref{Eq4A:sum_Taylor_modulo}) and the Stirling's formula Eq. (\ref{Eq4A:Stirling_inequality}).
This accomplishes the evaluation of Eq. (\ref{EqA2:thm_amp1_proof_3}) with the usage of Eq. (\ref{Eq4B:error_condition}) as
\begin{eqnarray}
[\text{Eq. (\ref{EqA2:thm_amp1_proof_3})}] &\leq& 26 m_\imax \left(\frac{  e \gamma t}{3 l_\imax /m_\imax - 3 } \right)^{3 l_\imax / m_\imax - 3} \nonumber \\
&\leq& 26 m_\imax \left[ \left(\frac{  e \gamma t}{l_\imax /m_\imax -1} \right)^{l_\imax / m_\imax -1} \right]^3 \nonumber \\
&\leq& \frac{13}{500 (m_\imax)^2} \varepsilon^3. \label{EqA2:thm_amp1_proof_4}
\end{eqnarray}

We next compute the second summation in Eq. (\ref{EqA2:thm_amp1_proof_2}), which is taken over $l ∈D^4l_$ satisfying $l-l^'∉D^3l_$.
The Lieb-Robinson bound, dictated by Theorem \ref{Thm4:Bound_transition_rate}, immediately concludes its upper bound by
\begin{eqnarray}
&& \norm{\sum_{l \in D^{4l_\imax}; l-l^\prime \notin D^{3l_\imax}} e^{-il\omega t} \bra{l} e^{-i\ms{H}_\eff^{4l_\imax}t} \ket{l^\prime} \ket{\psi(0)}} \nonumber \\
&& \quad \leq \sum_{l \in D^{4l_\imax}; l-l^\prime \notin D^{3l_\imax}} 2 \frac{(\gamma t)^{\lceil |l-l^\prime|/m_\imax \rceil}}{(\lceil |l-l^\prime|/m_\imax \rceil)!} \nonumber \\
&& \quad \leq 4 \sum_{l=3l_\imax}^\infty \frac{(\gamma t)^{\lceil l /m_\imax \rceil}}{(\lceil l / m_\imax \rceil)!} \leq 4 m_\imax \left( \frac{em_\imax \gamma t}{3l_\imax}\right)^{3l_\imax/m_\imax} \nonumber \\
&& \quad \leq \frac{1}{250 (m_\imax)^2} \varepsilon^3. \label{EqA2:thm_amp1_proof_5}
\end{eqnarray}

Finally, we show that the amplification protocol relying on the symmetry reproduces the time-evolved state $|ψ(t)⟩$.
Combining the results of Eqs. (\ref{EqA2:thm_amp1_proof_4}) and (\ref{EqA2:thm_amp1_proof_5}), we arrive at
\begin{eqnarray}
&& \norm{\bra{0}\ms{U}_\mr{amp1}^{l_\imax}(t)\ket{0}\ket{\psi(0)}-\frac12 \ket{0}\ket{\psi(t)}} \nonumber \\
&& \quad \leq \frac{1}{4l_\imax} \sum_{l^\prime \in D^{l_\imax}} \left( \frac{13}{500 (m_\imax)^2} \varepsilon^3 + \frac{1}{250 (m_\imax)^2} \varepsilon^3 \right) \nonumber \\
&& \quad \leq \frac{3}{200 (m_\imax)^2} \varepsilon^3, \label{EqA2:thm_amp1_proof_6}
\end{eqnarray}
which is smaller than $ε/3$ for $m_∈$ and $ε∈[0,1]$. $  □$

% Subsection
\subsection{Validity of refined effective Hamiltonian}\label{Asubsec2:refine_effective_Hamiltonian}
In Section \ref{Subsec:block_encode_H_eff}, we raise an efficient block-encoding protocol for implementing $e^-iH_^l_t$ by the qubitization technique.
The strategy is to substitute the effective Hamiltonian under the periodic boundary conditions, $H_,pbc^4l_$, for the original one $H_^4l_$; The complexity of the oracles reduces from $l_$ to $logl_$.
In order to justify this replacement of $H_^4l_$ with $H_,pbc^4l_$, we provide Theorem \ref{Thm6:refined_H_eff}, which says that its error is bounded from above by
\begin{eqnarray}
\norm{\bra{0} \ms{U}_\mr{amp1,pbc}^{l_\imax} (t) \ket{0} \ket{\psi(0)} - \frac12 \ket{\psi(t)}} &\leq& \frac{\varepsilon}{3}, \label{EqA2:thm_amp1_pbc}\\
\norm{ \ms{U}_\mr{amp2,pbc}^{l_\imax} (t) \ket{0} \ket{\psi(0)} - \ket{0}\ket{\psi(t)}} &\leq& \varepsilon. \label{EqA2:thm_amp2_pbc}
\end{eqnarray}
The amplification protocols under the refined effective Hamiltonian, $U_amp1,pbc^l_ (t)$ and $U_amp2,pbc^l_ (t)$, are designated by Eqs. (\ref{Eq6B:def_amp1_pbc}) and (\ref{Eq6B:def_amp2_pbc}).

\textbf{Proof of Theorem \ref{Thm6:refined_H_eff}.---} The validity of $U_amp2,pbc^l_ (t)$, i.e. Eq. (\ref{EqA2:thm_amp2_pbc}), is derived by Eq. (\ref{EqA2:thm_amp1_pbc}) following the discussion in Section \ref{Subsec:Oblivious_amplification}.
It is sufficient to prove the relation, Eq. (\ref{EqA2:thm_amp1_pbc}).
We first evaluate the upper bound of
\begin{eqnarray}
&& \norm{\bra{0}\ms{U}_\mr{amp1,pbc}^{4l_\imax}(t)\ket{0}\ket{\psi(0)}  - \bra{0}\ms{U}_\mr{amp1}^{4l_\imax}(t)\ket{0}\ket{\psi(0)} } \nonumber \\
&& \leq \frac{1}{4l_\imax} \sum_{l \in D^{4l_\imax}} \sum_{l^\prime \in D^{l_\imax}}\norm{\braket{l|e^{-i \ms{H}_{\eff,\mr{pbc}}^{4l_\imax}  t} - e^{-i \ms{H}_\eff^{4l_\imax}  t}|l^\prime}}. \nonumber  \\
&& \label{Aeq2:thm_refined_proof_1}
\end{eqnarray}
Each transition amplitude in the summation can be computed in a similar way to the proof of Lemma \ref{LemmaA2:translation_symmetry}.
We decompose the refined effective Hamiltonian by $H_,pbc^4l_ = H_^4l_ + H_b$ with the perturbation
\begin{equation}
\ms{H}_b = \sum_{(l,m) \in \partial F^{4l_\imax}} (\ket{l}\bra{l \oplus m} \otimes H_{-m} + \text{h.c.}).
\end{equation}
Following the discussion in the proof of Lemma \ref{LemmaA2:translation_symmetry}, the perturbation $H_b$ which locally acts on the domain $∂F^4l_$ hardly changes the transition amplitude from the original effective Hamiltonian $H_^4l_$.
As a result, the deviation can be bounded by
\begin{eqnarray}
&& \norm{\braket{l|e^{-i \ms{H}_{\eff,\mr{pbc}}^{4l_\imax}  t} - e^{-i \ms{H}_\eff^{4l_\imax}  t}|l^\prime}} \nonumber \\
&& \quad \leq \sum_{n=n(l^\prime,l)}^\infty \frac{t^n}{n!} \left\{ \left(\sup_{t} (\norm{\ms{H}_{\mr{pbc},I}(t)}) \right)^n + \gamma^n \right\}, \nonumber \\
&& \label{Aeq2:thm_refined_proof_2}
\end{eqnarray}
where the Hamiltonian in the interaction picture is defined by
\begin{equation}
\ms{H}_{\mr{pbc},I}(t) = [\ms{U}_0(t)]^\dagger \left( \sum_{m \neq 0} \mr{Add}_{m}^{4l_\imax} \otimes H_m \right) \ms{U}_0(t).
\end{equation}
We again use the property of circulant matrices, and obtain $H_pbc,I(t) ≤γ$, which leads to
\begin{eqnarray}
\norm{\braket{l|e^{-i \ms{H}_{\eff,\mr{pbc}}^{4l_\imax}  t} - e^{-i \ms{H}_\eff^{4l_\imax}  t}|l^\prime}} &\leq& \sum_{n=n(l^\prime,l)}^\infty 2 \frac{(\gamma t)^n}{n!} \nonumber \\
&\leq& 4 \left( \frac{\gamma t}{n(l^\prime,l)} \right)^{n(l^\prime,l)}. \nonumber \\
\end{eqnarray}
As a result, we get the difference between the original and refined effective Hamiltonians as
\begin{eqnarray}
&& \norm{\bra{0}\ms{U}_\mr{amp1,pbc}^{4l_\imax}(t)\ket{0}\ket{\psi(0)}  - \bra{0}\ms{U}_\mr{amp1}^{4l_\imax}(t)\ket{0}\ket{\psi(0)} } \nonumber \\
&& \quad \leq \frac12 \sum_{l \in D^{4l_\imax}} 4 \frac{(\gamma t)^{n(l_\imax,l)}}{n(l_\imax,l)!} \nonumber \\
&& \quad \leq 4 \sum_{l=3l_\imax-2m_\imax}^\infty \frac{(\gamma t)^{\lceil l/m_\imax \rceil}}{(\lceil l/m_\imax \rceil)!} \nonumber \\
&& \quad \leq 4 \left( \frac{3\gamma t }{3l_\imax/m_\imax-2} \right)^{3l_\imax/m_\imax-2} \leq \frac{\varepsilon^3}{250 (m_\imax)^2}. \nonumber \\
&& \label{Aeq2:thm_refined_proof_3}
\end{eqnarray}
As stated by Theorem \ref{Thm5:amplification_symmetry}, the original amplification protocol $U_amp1^4l_(t)$ accurately generates $|ψ(t)⟩$ as Eq. (\ref{EqA2:thm_amp1_proof_6}).
Therefore, the one under the refined effective Hamiltonian is also justified as
\begin{eqnarray}
&&\norm{\bra{0} \ms{U}_\mr{amp1,pbc}^{l_\imax} (t) \ket{0} \ket{\psi(0)} - \frac12 \ket{\psi(t)}} \nonumber \\
&& \quad \leq \norm{\bra{0}\ms{U}_\mr{amp1,pbc}^{4l_\imax}(t)-\ms{U}_\mr{amp1}^{4l_\imax}(t)\ket{0}\ket{\psi(0)} } \nonumber \\
&& \qquad \qquad \qquad + \norm{\bra{0}\ms{U}_\mr{amp1}^{4l_\imax}(t)\ket{0}\ket{\psi(0)} - \frac12 \ket{\psi(t)}} \nonumber \\
&& \quad \leq \frac{19}{1000(m_\imax)^2} \varepsilon^3. \label{Aeq2:thm_refined_proof_4}
\end{eqnarray}
This bound is actually smaller than $ε/3$, and we complete the proof of Theorem \ref{Thm6:refined_H_eff}. $  □$

% Section
\section{Extension to exponentially-decaying Fourier components}\label{Asec:Extension_expo}

In the main text, we focus on time-periodic Hamiltonians which have vanishing Fourier components $H_m=0$ for $|m| > m_$.
Here, we generalize our results to time-periodic Hamiltonians $H(t)$ with exponentially-decaying Fourier components as
\begin{equation}
\norm{H_m} \leq h e^{-|m|/\zeta}, \quad h,\zeta > 0,
\end{equation}
for $|m|>0$.
The norm of $H_0$ is arbitrary as long as it is bounded.
The effective Hamiltonian in the truncated Floquet-Hilbert space, $H_^l_$, is the same as Eq. (\ref{Eq3A:def_effective_H_l_max}), but it is dense in the basis ${|l⟩ }_l ∈D^l_$ compared to the cases where $H_m=0$ is satisfied for $|m| > m_$.

We formulate the protocol in a similar manner to the main text, and show that the computational resources for the time-evolved state have nearly-optimal dependence both in $t$ and $1/ε$.
The difference mainly comes from the form of the Lieb-Robinson bound and the infinite series of ${ H_m }$ needed for designating $H(t)$.
The former one affects the truncation order $l_$ for the Floquet-Hilbert space.
The latter one yields the change in block-encoding so that almost all the information about $H(t)$ can be embedded with keeping the efficiency.

\subsection{Truncation order of Floquet-Hilbert space}

We first determine the proper truncation order $l_$ for the Floquet-Hilbert space, as we did in Section \ref{Sec:truncated_Floquet_Hilbert}.
To this aim, we begin with deriving the Lieb-Robinson bound on the transition rate, corresponding to Theorem \ref{Thm4:Bound_transition_rate}.

\begin{theorem}\label{Athm3:Bound_transition_rate}
\textbf{(Bound on transition rate)}

We assume $\norm{H_m} \leq h e^{-|m|/\zeta}$ with certain positive constants $h$ and $\zeta$.
Then, for $l, l^\prime$ such that $|l|, |l^\prime| \leq l_\imax$, the transition rate is bounded from above by
\begin{equation}
\norm{\bra{l} e^{-i \ms{H}_\eff^{l_\imax} t} \ket{l^\prime}} \leq  \exp \left( - \frac{|l-l^\prime| - 2 \beta \zeta^\prime h t}{\zeta^\prime} + 2/\beta \right).
\end{equation}
Here, $\beta$ and $\zeta^\prime$ are positive constants defined by
\begin{eqnarray}
\beta &=& (1-e^{-1/\zeta})^{-1}, \\
\zeta^\prime &=& (1/\zeta - 1 + e^{-1/\zeta})^{-1}. \label{Aeq3:def_zeta_prime}
\end{eqnarray}
\end{theorem}

\textbf{Proof.---} We prove the theorem in a similar way to Theorem \ref{Thm4:Bound_transition_rate}.
We start from the Dyson series expansion in the interaction picture [See Eq. (\ref{Eq4A:Dyson_transition_path})],
\begin{eqnarray}
&& \norm{\bra{l} e^{-i \ms{H}_\eff^{l_\imax} t} \ket{l^\prime}}  \nonumber \\
&\quad& \leq \sum_{n=0}^\infty \int_0^t \dd t_n \hdots \int_0^{t_2} \dd t_1 \norm{\sum_{\{l_i\}}\prod_{i=1}^n \braket{l_i|\ms{H}_I(t_i)|l_{i-1}}} \nonumber \\
&\quad& \leq \sum_{n=0}^\infty \int_0^t \dd t_n \hdots \int_0^{t_2} \dd t_1 \sum_{\{l_i\}}\prod_{i=1}^n \norm{\braket{l_i|\ms{H}_I(t_i)|l_{i-1}}}, \nonumber \\
&& \label{Aeq3:thm_LRbound_proof_1}
\end{eqnarray}
where we insert the identity $∑_l_i ∈D^l_ |l_i⟩⟨l_i|=I$ for $n-1$ times.
The summation $∑_{l_i}$ is taken over $l_i ∈D^l_$ for $i=1,2,,n-1$ under fixed $l_0=l^'$ and $l_n=l$.
We introduce a new variable $m_i=l_i-l_i-1$ instead of using ${l_i}$, and then the above integrand is bounded by
\begin{eqnarray}
F_n &\equiv& \sum_{\{l_i\}}\prod_{i=1}^n \norm{\braket{l_i|\ms{H}_I(t_i)|l_{i-1}}} \nonumber \\
&\leq& \sum_{\{m_i\}_{i=1}^n \in \bbZ^n} \delta_{l^\prime + m_1+\hdots+m_n, l}  \prod_{i=1}^n \norm{H_{-m_i}} \nonumber \\
&\leq& \sum_{\{m_i\}_{i=1}^n \in \bbZ^n} \delta_{l^\prime + m_1+\hdots+m_n, l} h^n e^{-\sum_{i=1}^n |m_i|/\zeta}. \nonumber \\
\end{eqnarray}
Since $l^'+ m_1++m_n=l$ implies $|m_1|++|m_n| ≥|l-l^'|$, it can be further bounded as follows,
\begin{eqnarray}
F_n &\leq& \sum_{\{m_i\}_{i=1}^n \in \bbZ^n} \theta \left( \sum_{i=1}^n |m_i| -|l-l^\prime| \right) h^n e^{-\sum_{i=1}^n |m_i|/\zeta} \nonumber \\
&\leq& (2h)^n \sum_{m_1,\hdots,m_n=0}^\infty  \theta \left( \sum_{i=1}^n m_i -|l-l^\prime| \right) e^{-\sum_{i=1}^n m_i/\zeta}. \nonumber \\
&\equiv& (2h)^n S_n(|l-l^\prime|).
\end{eqnarray}
Here, $θ(x)$ is a step function, defined by $θ(x) = 1$ for $x ≥0$ and $θ(x)=0$ otherwise.
The summation $S_n (M)$ is defined by
\begin{equation}
S_n (M) = \sum_{m_1,\hdots,m_n=0}^\infty  \theta \left( \sum_{i=1}^n m_i - M \right) e^{-\sum_i m_i/\zeta},
\end{equation}
which satisfies $S_1(M)= ∑_m=M^∞e^-m/ζ = βe^-M/ζ$.

Let us evaluate the upper bound on $S_n(M)$.
We split the summation over $m_n$ into the one over $m_n ≤M-1$ and the one over $m_n ≥M$, which results in
\begin{equation}
S_n (M) = \sum_{m_n=0}^{M-1} e^{-m_n/\zeta} S_{n-1}(M-m_n) + \beta^n e^{-M/\zeta}.
\end{equation}
We use this relation recursively until $S_1$ appears.
After the single use of this equality, we obtain
\begin{eqnarray}
&& S_n(M) \nonumber \\
&& \leq \sum_{m_n=0}^{M-1} \sum_{m_{n-1}=0}^{M-m_n-1} e^{-(m_n+m_{n-1})/\zeta} S_{n-2}(M-m_n-m_{n-1}) \nonumber \\
&& \qquad + \beta^{n-1} e^{-M/\zeta} \sum_{m_n=0}^{M-1} 1 + \beta^n e^{-M/\zeta},
\end{eqnarray}
where we use $e^-m_n/ζ ≤1$ for the second term.
By repeating this calculation $n-1$ times, we arrive at
\begin{eqnarray}
&& S_n(M) \nonumber \\
&& \leq \left( \sum_{m_1=0}^{M-1} \hdots \sum_{m_{n-1}=0}^{M-1-M_{n-2}} \right) e^{-M_{n-1}/\zeta} S_1 (M-M_{n-1}) \nonumber \\
&& \qquad + e^{-M/\zeta} \sum_{k=1}^{n-1} \beta^{n-k+1} \left( \sum_{m_1=0}^{M-1}  \hdots \sum_{m_{k-1}=0}^{M-1-M_{k-2}} \right) 1 \nonumber \\
&& = e^{-M/\zeta} \sum_{k=1}^{n} \beta^{n-k+1} \left( \sum_{m_1=0}^{M-1}  \hdots \sum_{m_{k-1}=0}^{M-1-M_{k-2}} \right) 1,
\end{eqnarray}
where $∑_i=1^k m_k$ is denoted by $M_k$.
In the above formula, the summation of $1$ over ${ m_i }_i=1^k-1$ represents the number of lattice points included in a $(k-1)$-dimensional pyramid $P_k-1^M = { (x_1,,x_k-1) ∈[0,M-1]^k   |   0 ≤∑_i x_i ≤M-1 }$, and hence it is bounded by the volume of $P_k-1^M+2$.
This leads to the relation,
\begin{eqnarray}
S_n(M) &\leq& \beta^n e^{-M/\zeta} \sum_{k=1}^n \frac{\{ \beta^{-1} (M+2) \}^{k-1}}{(k-1)!} \nonumber \\
&\leq& \beta^n e^{-(1/\zeta-1/\beta)M + 2/\beta},
\end{eqnarray}

We define a positive constant $ζ^'$ by $1/ζ^'= 1/ζ- 1/β$ as  Eq. (\ref{Aeq3:def_zeta_prime}).
The above inequality enables to evaluate the integrand $F_n$ as
\begin{equation}
F_n \leq (2h)^n S_n (|l-l^\prime|) \leq  (2\beta h)^n e^{-|l-l^\prime|/\zeta^\prime + 2/\beta}.
\end{equation}
Finally, going back to the inequality Eq. (\ref{Aeq3:thm_LRbound_proof_1}), we arrive at the bound on the transition rate,
\begin{eqnarray}
\norm{\bra{l} e^{-i \ms{H}_\eff^{l_\imax} t} \ket{l^\prime}} &\leq&
\sum_{n=0}^\infty \frac{t^n}{n!}  (2\beta h)^n e^{-|l-l^\prime|/\zeta^\prime + 2/\beta} \nonumber \\
&\leq& e^{2\beta h t - |l-l^\prime|/\zeta^\prime + 2/\beta}.
\end{eqnarray}
This completes the proof of the theorem. $  □$

Theorem \ref{Athm3:Bound_transition_rate} says that the bound on the transition amplitude from $|l^'⟩$ to $|l⟩$ exponentially decays in the distance $|l-l^'|$ for Hamiltonians with $H_m ≲e^-|m|$.
The decay is relatively slow compared to the cases $H_m=0$ ($|m|>m_$) showing $|l-l^'|^-|l-l^'|$.
In fact, this result is reminiscent of the Lieb-Robinson bound for short-ranged interacting systems \cite{Robinson1976-wd,Nachtergaele2006-ok,Nachtergaele2006-il}, which provides exponentially-decaying correlation functions under interactions $U_ij ∼e^-|i-j|$.
Based on Theorem \ref{Athm3:Bound_transition_rate}, we assess how the exact time-evolved state can be approximated by the truncated Floquet-Hilbert space, and determine the proper truncation order $l_$.
We summarize the result by the following theorem, which is a counterpart of Theorem \ref{Thm4:Floquet_Hilbert_truncation}.

\begin{theorem}\label{Athm3:Floquet_Hilbert_truncation}
\textbf{(Floquet-Hilbert space truncation)}

We assume $\norm{H_m} \leq C e^{-|m|/\zeta}$.
The exact time-evolved state $\ket{\psi(t)}$ is approximated by the truncated state $\ket{\psi^{l_\imax}(t)}$ [See Eq. (\ref{Eq3A:def_psi_l_max})] as
\begin{equation}\label{Aeq3:thm_truncation_deviation}
\norm{\ket{\psi(t)}-\ket{\psi^{l_\imax}(t)}} \leq  4 \zeta^\prime e^{2\beta h t - (l_\imax - 1 )/\zeta^\prime + 2/\beta}.
\end{equation}
\end{theorem}

\textbf{Proof.---}
The derivation is similar to the one for Theorem \ref{Thm4:Floquet_Hilbert_truncation}.
According to Eqs. (\ref{Eq4A:thm_truncation_proof_1})-(\ref{Eq4A:thm_truncation_proof_3}), we should evaluate the following two values,
\begin{eqnarray}
\varepsilon_1 &=& \sum_{l \in (D^{l_\imax^\prime} \backslash D^{l_\imax})} \norm{\bra{l} e^{i\ms{H}_\eff^{l_\imax^\prime}t} \ket{0}}, \\
\varepsilon_2 &=& \sum_{l \in D^{l_\imax}} \norm{\bra{l} e^{i\ms{H}_\eff^{l_\imax^\prime}t}-e^{i\ms{H}_\eff^{l_\imax}t} \ket{0}}, \label{Aeq3:thm_truncation_epsilon_2}
\end{eqnarray}
where the left hand side of Eq. (\ref{Aeq3:thm_truncation_deviation}) is bounded by $ε_1 + ε_2$ under $l_^'→∞$.
Theorem \ref{Athm3:Bound_transition_rate} soon concludes the upper bound of $ε_1$;
\begin{eqnarray}
\varepsilon_1 &\leq& \sum_{l \in (D^{l_\imax^\prime} \backslash D^{l_\imax})} e^{2 \beta h t - |l|/\zeta^\prime + 2/\beta} \nonumber \\
&\leq& 2  \int_{l_\imax-1}^\infty \dd x e^{2\beta h t - x /\zeta^\prime + 2/\beta} \nonumber \\
&\leq& 2 \zeta^\prime e^{2\beta h t - (l_\imax-1)/\zeta^\prime + 2/\beta}. \label{Aeq3:thm_truncation_epsilon_1_result}
\end{eqnarray}

The second error $ε_2$ can be evaluated by the Dyson series expansion like Eq. (\ref{Eq4A:Dyson_transition_path}); each term of $ε_2$ in Eq. (\ref{Aeq3:thm_truncation_epsilon_2}) is bounded from above by
\begin{eqnarray}
&& \sum_{n=0}^\infty \int_0^t \dd t_n \hdots \int_0^{t_2} \dd t_1 \nonumber \\
&\quad&  \qquad \norm{\braket{l|\prod_{i=1}^n \ms{H}_I^{l_\imax^\prime}(t_i) - \prod_{i=1}^n \ms{H}_I^{l_\imax}(t_i)|0}}. \label{Aeq3:thm_truncation_proof_1}
\end{eqnarray}
As stated in the proof of Theorem \ref{Thm4:Bound_transition_rate}, each of $⟨l|∏_i=1^n H_I^l_^'(t_i) |0|$⟩ and $\braket{l|\prod_{i=1}^n \ms{H}_I^{l_\imax}(t_i) |0}$ is decomposed into a product of transition amplitudes via a path $|0⟩ →|l_1⟩ →→|l_n-1⟩ →|l⟩$.
Their difference appears only when the path goes across $D^l_^' \D^l_$.
In other words, denoting the summation over the set of ${l_i}$ taking such nontrivial paths by $∑_{ l_i }^'$, we reach
\begin{eqnarray}
&& [\text{Eq. (\ref{Aeq3:thm_truncation_proof_1})}] \leq \nonumber \\
&& \quad \sum_{n=0}^\infty \int_0^t \dd t_n \hdots \int_0^{t_2} \dd t_1 \sum_{\{ l_i \}} \,^\prime \prod_{i=1}^n \norm{\braket{l_i | \ms{H}_I^{l_\imax^\prime}(t_i) | l_{i-1}}}. \nonumber \\
&& \label{Aeq3:thm_truncation_proof_2}
\end{eqnarray}
When we define the hopping distance $m_i$ by $m_i=l_i -l_i-1$, it should satisfy $∑_i=1^n |m_i| ≥(l_- |l|) + (l_- 0)$ for ${ l_i }$ such that a nontrivial path is organized.
This results in the relation described by
\begin{eqnarray}
\varepsilon_2 &\leq& \sum_{l \in D^{l_\imax}}\sum_{n=0}^\infty \frac{t^n}{n!} \sum_{\{m_i\}_{i=1}^n \in \bbZ^n} \left(\prod_{i=1}^n  \norm{H_{-m_i}} \right) \nonumber  \nonumber \\
&\quad& \qquad \qquad  \qquad \quad \times \theta \left( \sum_{i=1}^n |m_i| - 2l_\imax + |l| \right) \nonumber \\
&\leq&  \sum_{l \in D^{l_\imax}} \sum_{n=0}^\infty \frac{(2 h t)^n}{n!} S_n (2l_\imax - |l|) \nonumber \\
&\leq& \sum_{l \in D^{l_\imax}} \sum_{n=0}^\infty \frac{(2 \beta h t)^n}{n!} e^{-(2l_\imax - |l|)/\zeta^\prime + 2/\beta} \nonumber \\
&\leq& 2 \zeta^\prime e^{2\beta h t - (l_\imax - 1 )/\zeta^\prime + 2/\beta}. \label{Aeq3:thm_truncation_proof_3}
\end{eqnarray}
Combining this inequality with $|ψ(t)⟩-|ψ^l_(t)⟩ ≤ε_1 + ε_2$ and Eq. (\ref{Aeq3:thm_truncation_epsilon_1_result}), we obtain Eq. (\ref{Aeq3:thm_truncation_deviation}), which completes the proof. $  □$

Theorem \ref{Athm3:Floquet_Hilbert_truncation} determines the proper truncation order of the Floquet-Hilbert space for the algorithm.
When we aim at the desirable accuracy as $|ψ(t)⟩-|ψ^l_(t)⟩ ≤ε$, it is sufficient to choose the truncation order $l_$ by
\begin{eqnarray}
l_\imax &=& \left\lceil 2 \beta \zeta^\prime ht + \zeta^\prime \log (1/\varepsilon) + \zeta^\prime \log (4\zeta^\prime )+ \frac{2 \zeta^\prime}{\beta}  + 1 \right\rceil  \nonumber \\
&\in&\Theta \left( ht + \log (1/\varepsilon) \right). \label{Aeq3:l_max_choice_exp}
\end{eqnarray}
The result has similar scaling in $t$ and $1/ε$ to the one for Hamiltonians with a finite number of Fourier components, described by Eq. (\ref{Eq4B:l_max_choice}), other than the factor of $loglog(1/ε)$.
While the actual $m_$ is infinite for exponentially-decaying Fourier components as $H_m ≤h e^-|m|/ζ$, the parameters $h$ and $ζ^'$ play a role of $γ$ and $m_$ respectively.

\subsection{Amplitude amplification}

We next verify the validity of the amplitude amplification protocols, discussed in Section \ref{Sec:Amplification}, for time-periodic Hamiltonians with exponentially-decaying Fourier components $H_m ≤h e^-|m|/ζ$.

The first amplification relies on the translation symmetry of the effective Hamiltonian, as discussed in \ref{Subsec:amplification_symmetry}.
To show its extension, we begin with discussing the approximate translation symmetry in the truncated Floquet-Hilbert space, which is a counterpart of Lemma \ref{LemmaA2:translation_symmetry}.

\begin{lemma}\label{LemmaA3:translation_symmetry}
\textbf{(Approximate translation symmetry)}

We assume $\norm{H_m} \leq h e^{-|m|/\zeta}$, and consider the truncated Floquet-Hilbert space $\bbC^{8l_\imax} \otimes \mcl{H}$.
The transition rate has an approximate translation symmetry in that it satisfies
\begin{eqnarray}
&& \norm{\bra{l} e^{-i \ms{H}_\eff^{4 l_\imax}  t} \ket{l^\prime} - e^{il^\prime \omega t} \bra{l \ominus l^\prime} e^{-i \ms{H}_\eff^{4 l_\imax}  t} \ket{0}} \nonumber \\
&\qquad& \qquad \leq 2  e^{2\beta h t - (8l_\imax - |l|-|l^\prime|)/\zeta^\prime + 2/ \beta}. \label{Aeq3:lemma_approximate_symmetry}
\end{eqnarray}
\end{lemma}

\textbf{Proof.---} The proof is done in a similar manner to that of Lemma \ref{LemmaA3:translation_symmetry}.
We consider a perturbation $H̃_b(t)$ designated by
\begin{eqnarray}
\tilde{\ms{H}}_b(t) &=& \sum_{(l,m) \in \partial \tilde{F}^{4l_\imax}} \ket{l}\bra{l \oplus m} \otimes e^{8 i l_\imax \omega t} H_{-m} + \text{h.c.}, \nonumber \\
&&  \label{Aeq3:perturbed_H_b(t)}
\end{eqnarray}
with $ ∂F̃^4l_ = { (l,m)   |   l ∈D^4_,   8l_-l + 1 ≤m ≤8l_-1 }$.
This Hamiltonian indicates hopping terms that go across the boundaries $|4l_⟩$ and $|-4l_+1⟩$.
Let $Ũ_pert(t)$ denote a time evolution operator under $H_^4l_+H̃_b(t)$.
Then, due to the exact translation symmetry in the interaction picture, the transition amplitude $⟨l |Ũ_pert(t)|l^'|$⟩ satisfies

\begin{equation}\label{Aeq3:lemma_exact_symmetry_int}
\braket{l |\tilde{\ms{U}}_\mr{pert}(t)|l^\prime} = e^{i l^\prime \omega t} \braket{l \ominus l^\prime |\tilde{\ms{U}}_\mr{pert}(t)|0}.
\end{equation}

The difference of transition amplitudes between $\ms{H}_\eff^{4l_\imax}+\tilde{\ms{H}}_b(t)$ and $\ms{H}_\eff^{4l_\imax}$ is bounded in a similar way to Eqs. (<ref>) and (<ref>).
The difference survives only when the trajectory $\ket{l^\prime} \to \ket{l_1} \to \ket{l_{n-1}} \to \ket{l}$ pass through the boundaries $\ket{4 l_\imax}$ and $\ket{- 4 l_\imax +1 }$ via $\tilde{\ms{H}}_b(t)$, and then its length $\sum_{i=1}^n |m_i|$ with $m_i = l_i - l_{i-1}$ should be equal to or larger than $(4 l_\imax - |l|)+(4 l_\imax-|l^\prime|)$.
We obtain its upper bound similarly to Eq. (<ref>),

\begin{eqnarray}
&& \norm{\braket{l |\tilde{\ms{U}}_\mr{pert}(t)-e^{-i \ms{H}_\eff^{4 l_\imax}t}|l^\prime}} \qquad \nonumber \\
&& \qquad \leq  \sum_{n=0}^n \frac{(2ht)^n}{n!} S_n(8 l_\imax - |l|-|l^\prime|) \nonumber \\
&& \qquad \leq e^{2\beta h t - (8l_\imax - |l|-|l^\prime|)/\zeta^\prime + 2/ \beta}. \label{Aeq3:transition_difference_H_b(t)}
\end{eqnarray}

Using this relation twice and the symmetry Eq. (<ref>) leads to Eq. (<ref>). $\quad \square$

With the usage of this approximate translation symmetry, we can organize the amplitude amplification by symmetry like Theorem <ref>.
We prepare the truncated Floquet-Hilbert space $\bbC^{8l_\imax} \otimes \mcl{H}$, and make the initial state uniform in $\ket{l}$ with $l_\imax \in \Theta ( h t + \log (1/\varepsilon))$.
The state resulting from the time evolution $\exp (-i \ms{H}_\eff^{4l_\imax} t)$ outputs the target time-evolved state $\ket{\psi(t)}$ with amplitude $1/2$ as follows.

(Amplification by symmetry)

We assume $\norm{H_m} \leq h e^{-|m|/\zeta}$, and choose the truncation order $l_\imax \in \order{ht+\log(1/\varepsilon)}$ by Eq. (<ref>).
Let us summarize the amplification protocol relying on the symmetry by $\ms{U}_\mr{amp1}^{l_\imax}(t)$, whose explicit formula is given by Eq. (<ref>).
Then, it generates the time-evolved state $\ket{\psi(t)}$ with $\order{1}$ amplitude as

\begin{equation}\label{Aeq3:amp1_deviation_result_expo}
\norm{\braket{0|\ms{U}_\mr{amp1}^{l_\imax}(t)|0}\ket{\psi(0)} - \frac{1}{2} \ket{\psi(t)}} \leq \frac{5}{64(\zeta^\prime)^2} \varepsilon^3.
\end{equation}

It is reasonable to assume that the desirable error $\varepsilon$ is sufficiently small, not greater than the constant $8 \zeta^\prime / \sqrt{15}$. Then, the right hand side of the above formula is bounded by $\varepsilon/3$, which reproduces Theorem <ref>.

Proof.— The proof follows that of Theorem <ref>, described in Appendix <ref>.
The left hand side of the above inequality can be bounded by two contributions determined by Eq. (<ref>).
The first contribution, which corresponds to Eq. (<ref>), is given by

\begin{eqnarray}
&& \norm{\sum_{l; l-l^\prime \in D^{3l_\imax}} e^{-il\omega t} \bra{l} e^{-i\ms{H}_\eff^{4l_\imax}t} \ket{l^\prime} \ket{\psi(0)} - \ket{\psi(t)} } \nonumber \\
&& \leq \sum_{l; l-l^\prime \in D^{3l_\imax}} 2 e^{2\beta h t - \frac{7 l_\imax - |l|}{\zeta^\prime} + \frac{2}{\beta}} + \norm{\ket{\psi^{3l_\imax}(t)} - \ket{\psi(t)}} \nonumber \\
&& \leq 8 \zeta^\prime e^{2\beta h t - (3 l_\imax - 1 )/\zeta^\prime + 2/\beta} \nonumber \\
&& \leq \frac{1}{8 (\zeta^\prime)^2} \varepsilon^3.
\end{eqnarray}

Here, we used Lemma <ref> in the second line, Theorem <ref> in the third line, and the choice of $l_\imax$, Eq. (<ref>), in the last line.
The second contribution, which is a counterpart of Eq. (<ref>), is bounded by

\begin{eqnarray}
&& \norm{\sum_{l \in D^{4l_\imax}; l-l^\prime \notin D^{3l_\imax}} e^{-il\omega t} \bra{l} e^{-i\ms{H}_\eff^{4l_\imax}t} \ket{l^\prime} \ket{\psi(0)}} \nonumber \\
&& \quad \leq \sum_{l \in D^{4l_\imax}; l-l^\prime \notin D^{3l_\imax}} e^{2\beta ht - |l-l^\prime|/\zeta^\prime+2/\beta} \nonumber \\
&& \quad \leq 2 \zeta^\prime e^{2\beta ht - (3l_\imax-1)/\zeta^\prime + 2/\beta} \leq \frac{1}{32 (\zeta^\prime)^2} \varepsilon^3.
\end{eqnarray}

By combining these results like Eq. (<ref>), we complete the proof of Theorem <ref>. $\quad \square$

The above theorem ensures the amplification by symmetry also for time-periodic Hamiltonians with $\norm{H_m} \leq h e^{-|m|/\zeta}$; it enhances the amplitude from $\order{l_\imax^{-1}}$ to $1/2$ only with additional $\order{\log l_\imax}$ elementary gates.
In order to bring the amplitude up to $1-\order{\varepsilon}$, we need the oblivious amplitude amplification.
Its validity immediately follows from Theorem <ref>, as we discussed in <ref>.
In other words, when we apply the protocol $\ms{U}_\mr{amp2}^{l_\imax}(t)$, defined by Eq. (<ref>), the time-evolved state $\ket{\psi(t)}$ can be obtained with accuracy $1-\order{\varepsilon}$;

\begin{equation}
\norm{\ms{U}_\mr{amp2}^{l_\imax}(t)\ket{0}\ket{\psi(0)} - \ket{0} \ket{\psi(t)}} \leq \frac{15}{64(\zeta^\prime)^2} \varepsilon^3.
\end{equation}

The right hand side is bounded by $\varepsilon$ for sufficiently small $\varepsilon$.

§.§ Qubitization technique for effective Hamiltonian

Simulating $\ket{\psi(t)}$ via the amplitude amplification $\ms{U}_\mr{amp2}^{l_\imax}(t)$ requires the time evolution operators $\exp (-i \ms{H}_\mr{LP}^{4l_\imax}t)$ and $\exp (-i \ms{H}_\eff^{4l_\imax}t)$.
The former one is the same as Section <ref>.
We hereby present how the latter one is implemented for the cases $\norm{H_m} \leq h e^{-|m|/\zeta}$.
We take a similar strategy to Section <ref>, that is, we first organize a refined effective Hamiltonian $\ms{H}_{\eff,\mr{pbc}}^{4l_\imax}$, which can accurately reproduce the dynamics under $\ms{H}_\eff^{4l_\imax}$.
After that, we compose its block-encoding which can be efficiently achievable.

§.§.§ Refined effective Hamiltonian

We introduce a refined effective Hamiltonian $\ms{H}_{\eff,\mr{pbc}}^{4l_\imax}$, which acts on the truncated Floquet-Hilbert space $\bbC^{8l_\imax} \otimes \mcl{H}$, by

\begin{equation}\label{Aeq3:refined_H_eff}
\ms{H}_{\eff,\mr{pbc}}^{4l_\imax} = \ms{H}_\eff^{4l_\imax} + \tilde{\ms{H}}_b,
\end{equation}

\begin{equation}\label{Aeq3:refined_H_eff_boundary}
\tilde{\ms{H}}_b = \sum_{(l,m) \in \partial \tilde{F}^{4l_\imax}} \ket{l}\bra{l \oplus m} \otimes H_{-m} + \text{h.c.},
\end{equation}

where $\partial \tilde{F}^{4l_\imax}$ shares the definition with the one in Eq. (<ref>).
Owing to the additional term $\ms{H}_b$, the hopping terms induced by $H_{-m}$ become translation symmetric in $\ket{l}$ as

\begin{equation}\label{Aeq3:refined_H_eff_result}
\ms{H}_{\eff,\mr{pbc}}^{4l_\imax} = \sum_{m \in D^{4l_\imax}} \mr{Add}_m^{4l_\imax} \otimes H_m - \ms{H}_\mr{LP}^{4l_\imax}.
\end{equation}

Here, a full quantum adder $\mr{Add}_m^{4l_\imax}$, defined by Eq. (<ref>), appears and it allows efficient implementation of block-encoding as discussed later.
Before going to its block-encoding, we prove the validity of the refined effective Hamiltonian as a counterpart of Theorem <ref>.

(Refined effective Hamiltonian)

We assume $\norm{H_m} \leq h e^{-|m|/\zeta}$, and organize the two amplification protocols $\ms{U}_\mr{amp1,pbc}^{l_\imax}(t)$ and $\ms{U}_\mr{amp2,pbc}^{l_\imax}(t)$ respectively based on Eqs. (<ref>) and (<ref>) with using the refined effective Hamiltonian $\ms{H}_{\eff,\mr{pbc}}^{4l_\imax}$ given by Eqs. (<ref>) and (<ref>).
When the truncation order $l_\imax \in \Theta (ht+\log(1/\varepsilon))$ is chosen by Eq. (<ref>), they also provide the exact time-evolved state $\ket{\psi(t)}$ as

\begin{eqnarray}
\norm{\bra{0} \ms{U}_\mr{amp1,pbc}^{l_\imax} (t) \ket{0} \ket{\psi(0)} - \frac12 \ket{\psi(t)}} &\leq& \frac{11}{128 (\zeta^\prime)^2} \varepsilon^3 , \nonumber \\
&& \label{EqA3:thm_amp1_pbc} \\
\norm{ \ms{U}_\mr{amp2,pbc}^{l_\imax} (t) \ket{0} \ket{\psi(0)} - \ket{0}\ket{\psi(t)}} &\leq& \frac{33}{128 (\zeta^\prime)^2} \varepsilon^3, \nonumber \\
&& \label{EqA3:thm_amp2_pbc}
\end{eqnarray}

for arbitrary initial states $\ket{\psi(0)} \in \mcl{H}$.
For a allowable error $\varepsilon$ sufficiently small compared to the constant $\zeta^\prime$, both of the left hand sides are smaller than $\varepsilon$.

Proof.— We prove the theorem as we do for Theorem <ref> in Appendix <ref>.
First, we evaluate the difference of the transition rates between $\ms{H}_{\eff,\mr{pbc}}^{4l_\imax}$ and $\ms{H}_{\eff}^{4l_\imax}$.
We replace the perturbation $\tilde{H}_b(t)$ by the boundary term $\tilde{H}_b$ in the proof of Lemma <ref> (See Appendix <ref>).
We obtain the same result as Eq. (<ref>),

\begin{eqnarray}
&& \norm{\bra{l} e^{-i \ms{H}_{\mr{eff,pbc}}^{4l_\imax}  t} \ket{l^\prime} - \bra{l} e^{-i \ms{H}_{\mr{eff}}^{4l_\imax}  t} \ket{l^\prime}} \nonumber \\
&\qquad& \qquad \leq  e^{2 \beta h t - (8l_\imax -|l|-|l^\prime|)/\zeta^\prime + 2/\beta}.
\end{eqnarray}

Once we obtain this bound, we can track the proof of Theorem <ref>, composed of Eqs. (<ref>)-(<ref>).
First, we evaluate the deviation from $\ms{U}_\mr{amp1}^{l_\imax}(t)$ as

\begin{eqnarray}
&& \norm{\bra{0}\ms{U}_\mr{amp1,pbc}^{4l_\imax}(t) - \ms{U}_\mr{amp1}^{4l_\imax}(t)\ket{0}\ket{\psi(0)} } \nonumber \\
&& \quad \leq \frac{1}{4l_\imax} \sum_{l \in D^{4l_\imax}} \sum_{l^\prime \in D^{l_\imax}} e^{2 \beta h t - (8l_\imax -|l|-|l^\prime|)/\zeta^\prime + 2/\beta} \nonumber \\
&& \quad \leq \frac{1}{2} \zeta^\prime e^{2\beta h t - (3l_\imax -1 )/\zeta^\prime + 2/\beta} \leq \frac{1}{128 (\zeta^\prime)^2} \varepsilon^3.
\end{eqnarray}

Since $\bra{0} \ms{U}_\mr{amp1}^{4l_\imax}(t)\ket{0}\ket{\psi(0)}$ accurately provides the state $\ket{\psi(t)}/2$ as Eq. (<ref>) according to Theorem <ref>, a triangle inequality concludes Eq. (<ref>).
Since the oblivious amplitude amplification generates the error at-most three times larger than Eq. (<ref>) according to the discussion in Section <ref>, Eq. (<ref>) is immediately derived. $\quad \square$

§.§.§ Block-encoding

We compose block-encoding of the refined effective Hamiltonian $\ms{H}_{\eff,\mr{pbc}}^{4l_\imax}$ to implement its time evolution operator by qubitization.

We begin with describing the assumption on time-periodic Hamiltonians $H(t)$.
If we adopt the one for the main text as Eq. (<ref>), we should have the knowledge about block-encoding for the infinite series $\{ H_m \}_{m \in \bbZ}$.
To avoid this complexity, we instead consider time-periodic Hamiltonians $H(t)$ written by

\begin{equation}
H(t) = \sum_{j=1}^{j_\imax} \alpha_j(t) M_j, \quad \alpha_j(t+T) = \alpha_j (t).
\end{equation}

with coefficients $\{ \alpha_j (t) \}_{j=1}^{j_\imax}$ and operators $\{ M_j \}_{j=1}^{j_\imax}$.
We assume the knowledge of block-encoding for each operator $M_j$ as

\begin{equation}\label{EqA3:block_encode_M_j}
\braket{G_j|O_j|G_j} = M_j, \quad \ket{G_j}= G_j \ket{0}^{\otimes n_a},
\end{equation}

with an oracle state $\ket{G_j} \in \bbC^{2^{n_a}}$ and an oracle unitary gate $O_j$ on $\bbC^{2^{n_a}} \otimes \mcl{H}$.
We can always set the denominator in block-encoding by $1$ since its absolute value and its phase can be absorbed respectively into the coefficient $\alpha_j(t)$ and the oracle $O_j$.
The Fourier components are given by

\begin{equation}\label{Aeq3:block_encoding_H_m_expo}
H_m = \sum_{j=1}^{j_\imax} \alpha_j^m M_j, \quad \alpha_j^m = \frac{1}{T}\int_0^T \dd t
\alpha_j(t) e^{im\omega t},
\end{equation}

and it is always possible to impose $\alpha_j^m \geq 0$ for every $j,m$.
This is because, when a certain coefficient $\alpha_j^m$ is complex, we can divide it into $\alpha_j^m= \Re \alpha_j^m + i \Im \alpha_j^m$ and redefine $M_j$ with the signs of $\Re \alpha_j^m, \Im \alpha_j^m$ and the phase $i$.
In other words, adding $-M_j$, $iM_j$, and $-iM_j$ to the set $\{M_j \}_j$ reproduces the form of Eq. (<ref>) with $\alpha_j^m \geq 0$, where $j_\imax$ is replaced by at-most $4l_\imax$.
In the following discussion, the oracles $O_j$ and $G_j$ are respectively supposed to be composed of at-most $C$ elementary gates.
In addition, we also assume the access to the coefficients $\alpha_j^m$ for $m \in D^{l_\imax}$ by the oracle unitary gate $G_\mr{coef}^{l_\imax}$ as

\begin{eqnarray}
G_\mr{coef}^{l_\imax} \ket{0} \ket{0} &=&  \sum_{m \in D^{l_\imax}}  \sum_{j=1}^{j_\imax}\sqrt{\frac{\alpha_j^m}{\alpha^{l_\imax}}} \ket{m} \ket{j}, \label{Aeq3:G_coeff_expo} \\
\alpha^{l_\imax} &=& \sum_{m \in D^{l_\imax}} \sum_{j=1}^{j_\imax} \alpha_j^m.
\end{eqnarray}

The oracle state $G_\mr{coef}^{l_\imax} \ket{0} \ket{0} \in \bbC^{2l_\imax+j_\imax}$ is a $\order{\log l_\imax + \log j_\imax}$-qubit ancillary state, and its preparation $G_\mr{coef}^{l_\imax}$ is supposed to require $C^{l_\imax}$ elementary gates.
The parameter $\alpha^{l_\imax}$ is bounded by the whole-system energy scale as

\begin{equation}\label{Aeq3:alpha_l_max_bound_expo}
\alpha^{l_\imax} \leq \sum_{m \in \bbZ} \sum_{j=1}^{j_\imax} \alpha_j^m \equiv \alpha.
\end{equation}

Although the definition of $\alpha$ is different from that of the main text, it again provides the bound on the scale of time-dependent terms as

\begin{equation}
\gamma \leq \sum_{m \in \bbZ} \sum_{j=1}^{j_\imax} |\alpha_j^m| \cdot \norm{M_j} \leq \alpha,
\end{equation}

where we use $\norm{M_j} \leq 1$ from Eq. (<ref>).

We move to how we organize an oracle state and oracle unitary gate for the refined $\ms{H}_{\eff,\mr{pbc}}^{4l_\imax}$.
Substituting the Fourier components $H_m$ into Eq. (<ref>) results in

\begin{equation}
\ms{H}_{\eff,\mr{pbc}}^{4l_\imax} = \sum_{m \in D^{4l_\imax}} \sum_{j=1}^{j_\imax} \alpha_j^m \mr{Add}_m^{4l_\imax} \otimes M_j - \ms{H}_\mr{LP}^{4l_\imax}.
\end{equation}

We introduce five kinds of ancillary systems labeled by $a,b,c,d$, and $e$.
The systems $a$, $b$, and $e$ are respectively described by $2^{n_a}$-, $8l_\imax$-, and $2$-dimensional Hilbert spaces, as we consider in Section <ref>.
States of the system $c$ is spanned by $\{ \ket{m} \}_{m \in D^{4l_\imax}}$ due to the absence of $m_\imax$.
The system $d$ is characterized by $\{\ket{j}\}_{j=1}^{j_\imax}$.
The oracle unitary gate for the refined effective Hamiltonian $\ms{H}_{\eff,\mr{pbc}}^{4l_\imax}$ is given by

\begin{eqnarray}
\ms{O}_\eff^{4l_\imax} &=& \ket{0}\bra{0}_e \otimes O_J \otimes \mr{Add}^{4l_\imax} \otimes I_b \nonumber \\
&\qquad& \qquad - \ket{1}\bra{1}_e \otimes I_d \otimes I_c \otimes \ms{O}_\mr{LP}^{4l_\imax} \otimes I_a, \nonumber \\
&& \\
O_J &=& \sum_{j=1}^{j_\imax} \ket{j}\bra{j}_d \otimes O_j, \\
\mr{Add}^{4l_\imax} &=& \sum_{m \in D^{4l_\imax}} \ket{m}\bra{m}_c \otimes \mr{Add}_m^{4l_\imax}.
\end{eqnarray}

Here, the unitary gates $O_J$, $\mr{Add}_m^{4l_\imax}$ and $\ms{O}_\mr{LP}^{4l_\imax}$ are respectively implemented by $\order{j_\imax C}$, $\order{\log l_\imax}$, and $\order{\log l_\imax}$ elementary gates (the second one represents addition of a variable $m \in D^{4l_\imax}$ <cit.>).
Next, we provide the oracle state $\ket{G_\eff^{4l_\imax}}$ on the auxiliary systems $a,b,c,d,$ and $e$ by

\begin{eqnarray}
\ket{G_\eff^{4l_\imax}} &=& G_\eff^{4l_\imax}\ket{w^{4l_\imax}}_e (G_\mr{coef}^{4l_\imax}\ket{0}_d \ket{0}_c) \ket{a^{4l_\imax}}_b \ket{0}_a, \nonumber \\
&& \\
G_\eff^{4l_\imax}  &=& I_e \otimes \sum_{j=1}^{j_\imax} \ket{j}\bra{j}_d \otimes I_c \otimes I_b \otimes (G_j)_a, \\
\ket{w^{4l_\imax}}_e &=& \frac{\sqrt{\alpha^{4l_\imax}}\ket{0}_e + \sqrt{8l_\imax \omega} \ket{1}_e}{\sqrt{\alpha^{4l_\imax} + 8l_\imax \omega }}.
\end{eqnarray}

We can confirm the relation,

\begin{equation}
\braket{G_\eff^{4l_\imax}|\ms{O}_\eff^{4l_\imax}|G_\eff^{4l_\imax}} = \frac{\ms{H}_{\eff,\mr{pbc}}^{4l_\imax}}{\alpha^{4l_\imax}+8l_\imax \omega},
\end{equation}

by directly substituting the above equations.
The cost of implementing the oracle state $\ket{G_\eff^{4l_\imax}}$ from the trivial state $\ket{0}_e \ket{0}_d \ket{0}_c \ket{0}_b \ket{0}_a$ is $\order{j_\imax C + C^{4l_\imax} + \log l_\imax}$ elementary gates.

Therefore, implementing the time evolution operator $\exp(-i \ms{H}_{\eff,\mr{pbc}} t)$ with the qubitization technique requires the following resources for time-periodic Hamiltonians satisfying $\norm{H_m} \leq h e^{-|m|/\zeta}$;

* Number of ancillary qubits; $\order{\log j_\imax + \log l_\imax}$

* Number of overall gates;

\begin{eqnarray}
&& \mathcal{O} \biggl( \left\{ (\alpha + l_\imax \omega) t +  o(\log (1/\varepsilon)) \right\} \biggr. \nonumber \\
&& \qquad   \biggl. \times \left( j_\imax C + C^{4l_\imax} + n_a + \log j_\imax + \log l_\imax  \right) \biggr). \nonumber
\end{eqnarray}

We replace $\alpha^{4l_\imax}$ by $\alpha$ based on the inequality, Eq. (<ref>).

§.§ Algorithm and comparison

The algorithm for computing the time-evolved state for time-periodic Hamiltonian with $\norm{H_m} \leq h e^{-|m|/\zeta}$ is almost the same as those in Section <ref>.
We separately consider adiabatic-like cases and generic long-time cases.
The computational resources are obtained by replacing $\gamma$ in Table <ref> by $h$.
The query complexity is determined by the coefficients of $j_\imax C$ and $C^{4l_\imax}$ in the number of overall elementary gates.
Their scaling is summarized as follows;

Adiabatic-like cases.—

The scaling of the query complexity is optimal in the time $t$ and nearly-optimal in the inverse error $1/\varepsilon$.

* Number of ancillary qubits;

\begin{equation*}
n_a + \order{\log j_\imax + \log (\gamma t) + \log \log (1/\varepsilon)}.
\end{equation*}

* Query complexity; $\order{\alpha t + \log (1/\varepsilon)}$.

* Additional gates per query;

\begin{equation}
\order{ n_a + \log j_\imax + \log (\gamma t)  + \log \log (1  /\varepsilon) }. \nonumber
\end{equation}

Generic long-time cases.—

The scaling of the query complexity is nearly-optimal in the time $t$ and the inverse error $1/\varepsilon$.
For practical problems up to $\poly{N}$-time, it becomes optimal in the time $t$.

* Number of ancillary qubits;

\begin{equation*}
n_a + \order{\log j_\imax + \log (\gamma/\omega) + \log \log (\omega t/\varepsilon)}.
\end{equation*}

* Query complexity; $\order{\alpha t + \omega t \log (\omega t/\varepsilon)}$.

* Additional gates per query;

\begin{equation}
\order{ n_a + \log j_\imax + \log (\gamma / \omega)  + \log \log (\omega t/\varepsilon)}. \nonumber
\end{equation}

We note that $\log \log (1/\varepsilon)$ factor in the query complexity is buried by $l_\imax$.
This difference from the cases $H_m=0$ ($|m|>m_\imax$) essentially originates from the form of the Lieb-Robinson bound dictated by <ref>.
A slightly slow decay in the transition amplitude results in slightly larger cost for the cases $\norm{H_m} \leq h e^{-|m|/\zeta}$, compared to the results in the main text.
This leads to the nearly-optimal dependence of the query complexity in $1/\varepsilon$.
Anyway, since the factor $\log \log (1/\varepsilon)$ is not so large compared to others, the relation to the quantum algorithms for time-independent and generic time-dependent Hamiltonians, discussed in Section <ref>, is maintained.
Importantly, in the query complexity, the whole energy scales $\alpha t$ and $\gamma t$ are separated from the $\order{\log (1/\varepsilon)}$ term; the query complexity is usually much smaller than that of the truncated Dyson-series algorithm, and close to that of the qubitization technique.
The number of ancillary qubits, which scales as $\order{\log \log (1/\varepsilon)}$ for an allowable error $\varepsilon$, also lies between those for these algorithms.

We also mention about the complexity of the oracles for the cases $\norm{H_m} \leq h e^{-|m|/\zeta}$.
In contrast to the cases $H_m=0$ ($|m|>m_\imax$), we require a quantum circuit which embeds the function $\alpha_j^m$ to a quantum state with $\order{j_\imax (\gamma t + \log (1/\varepsilon))}$ or $\order{j_\imax (\gamma/\omega + \log (\omega t/\varepsilon))}$ degrees of freedom, as the oracle $G_\mr{coef}^{4l_\imax}$ defined by Eq. (<ref>).
While integrable functions are efficiently implemented <cit.>, the number of elementary gates for the worst cases amounts to the dimension of the quantum state, i.e. $C^{4l_\imax} \in$ $\order{j_\imax (\gamma t + \log (1/\varepsilon))}$ or $\order{j_\imax (\gamma/\omega + \log (\omega t/\varepsilon))}$.
The preparation of oracles is relatively difficult compared to the cases $H_m=0$ ($|m|>m_\imax$).
However, even in such cases, we emphasize that it is much easier than that of the truncated Dyson-series algorithm.
In the truncated Dyson-series algorithm, we need more or less an oracle that embeds the information of time-dependent Hamiltonians at every discretized time into a quantum state.
Such an oracle is exemplified by Eq. (<ref>) for generic Hamiltonians <cit.>.
For time-dependent LCU Hamiltonians or sparse-access Hamiltonians, it requires a quantum state holding all the coefficients $\alpha_j(t)$ at every discretized time $t$ as an oracle <cit.>.
The number of elementary gates per oracle can be proportional to that of the discretized time, $\order{j_\imax \omega \gamma t/ \alpha \varepsilon}$, at the worst case.
Therefore, our algorithm also improves the complexity of the oracles as well as the query complexity, compared to the truncated Dyson-series algorithm.

§.§ Example

For the cases $\norm{H_m} \leq h e^{-|m|/\zeta}$, we adopt the assumption of block-encoding as Eq. (<ref>), in contrast to the cases $H_m=0$ ($|m|>m_\imax$).
To show that it is reasonable and versatile in condensed matter physics and quantum chemistry, we provide a simple example.

We consider an $N$-site Fermi-Hubbard model $H_\mr{Hub}$, given by Eq. (<ref>).
While we consider an idealized laser light which has a constant amplitude in Section <ref>, we focus on a rather realistic case where Gaussian wave packets of lights are shone to materials <cit.>.
Then, if the time scale of the wave packets is sufficiently larger than the light with the frequency $\Omega$, the external drive $H_\mr{ext}(t)$ is well described by

\begin{equation}
H_\mr{ext}(t) = \left( \sum_{n \in \bbZ} e^{-(t-(n+1/2)T)^2/2 \tau^2} \sin \Omega t \right) \sum_{x,\sigma} V_x \hat{n}_{x\sigma},
\end{equation}

where peaks of the wave packets are located at $t = (n+1/2)T$.
The period $T$ is designated by $T=p(2\pi/\Omega)$ with the number of the waves of the light for each wave packet $p \in \bbN$.

Let us organize the block-encoding required for simulation.
By carefully computing Fourier components of $H(t)=H_\mr{Hub}+H_\mr{ext}(t)$, we arrive at $H_0=H_\mr{Hub}$ and

\begin{eqnarray}
H_m &=& i (-1)^{p+m} \mr{sgn}(m) A_m \sum_{x,\sigma} V_x \hat{n}_{x\sigma}, \\
A_m &=&
\frac{\omega \tau}{\sqrt{2\pi}} e^{-\frac12 (p^2+m^2)(\omega \tau)^2} \sinh \left\{ p |m| (\omega \tau)^2 \right\}, \nonumber \\
\end{eqnarray}

for $m \neq 0$. We define $\mr{sgn}(m)$ by $\mr{sgn}(m)=m/|m|$ for $m \neq 0$ and $\mr{sgn}(0)=0$.
Since $A_m \geq 0$ rapidly decays as $e^{-\order{m^2}}$, this Hamiltonian is suitable for our algorithm, satisfying $\norm{H_m} \leq h e^{-|m|/\zeta}$.
This satisfies the assumption of Eq. (<ref>) with non-negative coefficients $\alpha_j^m$ by choosing $j_\imax = 3$;

\begin{eqnarray}
M_{1} &=& \frac{H_\mr{Hub}}{2\sum_x \epsilon_x +UN}, \quad \alpha_{1}^m = \left( 2\sum_x \epsilon_x +UN \right) \delta_{m0}, \nonumber \\
&& \\
M_{2} &=& + i \frac{\sum_{x,\sigma} V_x \hat{n}_{x\sigma}}{2 \sum_x V_x}, \quad \alpha_{2}^m = 2 \sum_x V_x A_m \delta_{(-1)^{p+m}\mr{sgn}(m), +1} \nonumber \\
&& \\
M_{3} &=& - i \frac{\sum_{x,\sigma} V_x \hat{n}_{x\sigma}}{2 \sum_x V_x}, \quad \alpha_{3}^m = 2 \sum_x V_x A_m \delta_{(-1)^{p+m}\mr{sgn}(m), -1}. \nonumber \\
\end{eqnarray}

The block-encoding for $\{ M_j \}$ can be composed by Eqs. (<ref>)-(<ref>) in Section <ref>.
Therefore, by resorting to our algorithm for the adiabatic-like cases, we can efficiently simulate the response to a realistic wave packet of laser light with the number of ancillary qubits and the query complexity provided in Appendix <ref>.

§ EFFICIENT ALGORITHM FOR
HIGH-FREQUENCY REGIMES WITH $\OMEGA \SIMEQ \ALPHA, \GAMMA$

In Section <ref>, we concentrate on the generic long-time regime with the frequency $\omega \in \order{N^0}$ for the number of sites, while the whole energy scales $\alpha,\gamma$ are given by $\poly{N}$.
Here, we show its reasonableness by showing that quantum simulation of broad time-periodic Hamiltonians under the frequency comparable to the whole energy scale can be almost attributed to that of time-independent Hamiltonians.

As a generic setup for time-periodic Hamiltonians, we assume the locality [= the system involves at-most $\order{1}$-body interactions] and the extensiveness.
The latter one means that the energy scale per site, represented by $\lambda$, is bounded.
In our definitions, the scale $\lambda$ is approximated by $\order{\alpha/N, \gamma/N}$.
When the ratio of the frequency $\omega$ to the local energy scale $\lambda$ is sufficiently large as $\omega/\lambda \gg 1$, we can apply the perturbative expansion in $(\omega/\lambda)^{-1}$, called the high-frequency expansion <cit.>.
Recent developments in generic local and extensive Floquet systems have revealed the accurate upper bound on the error as follows <cit.>;

\begin{eqnarray}
&& \norm{U(nT) - e^{-i H_\mr{FM}^{l} nT}} \nonumber \\
&\qquad& \qquad \leq C_1 \lambda N nT e^{-C_2 l_0} + C_3 n N (\lambda T)^{l+2} (l+1)!. \nonumber \\
&& \label{Aeq4:high_frequency_precision}
\end{eqnarray}

In the above inequality, $C_1$, $C_2$, and $C_3$ represent some positive constants, and the integer $l_0 \in \order{\omega/\lambda}$ provides the optimal truncation order of the perturbation theory, while the actual truncation order $l$ ($\leq l_0$) is optional.
The time-independent Hamiltonian $H_\mr{FM}^l$ is the $l$-th order Floquet-Magnus expansion, given by

\begin{eqnarray}
H_\mr{FM}^{l} &=& \sum_{l^\prime = 0}^l H_\mr{FM}^{(l^\prime)}, \quad H_\mr{FM}^{(0)} = H_0, \\
H_\mr{FM}^{(1)} &=& \frac{1}{2i T} \int_0^T \dd t_1 \int_0^{t_1} \dd t_2 [H(t_1),H(t_2)],
\end{eqnarray}

where each $l^\prime$-th order term in $\omega/\lambda$, represented by $H_\mr{FM}^{(l^\prime)}$, involves $l^\prime$-fold multi-commutators of $H(t)$ (See Ref. <cit.> for the explicit formula).

Based on the above formalism, we see that the long-time dynamics at $t=(n+\delta)T \gg T$ with $n \in \bbN$ and $\delta \in [0,1)$ can be computed in a trivial way when $\omega$ is comparable to $\alpha,\gamma \in \poly{N}$.
In this regime, the integer $l_0$ is $\order{N}$ due to the local energy scale $\lambda \in \order{\alpha/N, \gamma/N}$.
Then, we simulate the dynamics $U(t)=U(\delta T) U(nT)$ with approximating the latter one $U(nT)$ by Eq. (<ref>).
We choose the truncation order $l$ by $\order{N^0}$.
Since $H(t)$ generally has $\poly{N}$ local terms, the Floquet-Magnus expansion $H_\mr{FM}^{l}$ for such $l \in \order{N^0}$ can be identified with $\poly{N}$-time classical computation, and $H_\mr{FM}^{l}$ is also composed of $\poly{N}$ local terms.
Therefore, we can apply the qubitization technique to the implementation of $\exp (-i H_\mr{FM}^l nT)$ with the cost designated by Eq. (<ref>).
Since the right hand side of Eq. (<ref>) is bounded by $\order{n N^{-l-1}}$, the replacement of $U(nT)$ by $\exp (-i H_\mr{FM}^l nT)$ with an allowable error $\varepsilon$ is valid as long as

\begin{equation}
n \leq \mr{Const.} \times \varepsilon N^{l+1}.
\end{equation}

We usually employ a polynomial accuracy $\varepsilon \in \order{N^{-\nu_1}}$ in quantum computation.
The assumption for the frequency, represented by $\omega \in \order{N^{\nu_2}}$, implies the period $T \in \order{N^{-\nu_2}}$.
Therefore, by properly choosing the $\order{N^0}$ truncation order $l \geq \nu_1 + \nu_2 -1$, we can efficiently simulate $U(nT)$ for arbitrary polynomial time $nT \in \order{N^{l+1-\nu_1-\nu_2}}$ with the qubitization technique for time-independent Hamiltonians.
The remaining micromotion $U(\delta T)$ can be implemented with our protocol for the adiabatic-like regime (See Section <ref>), but it does not affects the computational cost due to $\delta T \ll nT$.

In short, the dynamics under time-periodic Hamiltonians $H(t)$ with $\omega \sim \alpha, \gamma$ can be simulated with the time-independent Hamiltonian approaches.
As a result, it is reasonable to concentrate on the cases where $\omega$ is negligible compared to $\alpha$ and $\gamma$, as we do in Section <ref>.