Hamiltonian simulation in the low energy subspace

Our main result is that, for a local Hamiltonian on $N$ spins $H=\sum_{l}H_{l}$ with $H_{l}\geq 0$, the error induced by a $p$-th order product formula is $\mathcal{O}((\Delta^{\prime}s)^{p+1})$, where $s$ is a (short) time parameter and $\Delta^{\prime}$ is an effective low-energy norm of $H$. This norm depends on $\Delta$, which is an energy associated with the initial state, but also depends on $s$ and other parameters that define $H$. The best known error bounds for product formulas that apply to the general case depend on the $\|H_{l}\|$’s CST+19 . (Throughout this paper, $\|.\|$ refers to the spectral norm.) Thus, an improvement in the complexity of product formulas is possible when $\Delta^{\prime}\ll\max_{l}\|H_{l}\|$, which can occur for sufficiently small values of $\Delta$ and $s$. Such values of $s$ appear in low-order product formulas (e.g., first order) or, for larger order, when the overall evolution time $t$ is sufficiently large and/or the desired approximation error $\varepsilon$ is sufficiently small. We summarize some of the complexity improvements in Table I.

To obtain our results, we introduce the notion of effective Hamiltonians that are basically the $H_{l}$’s restricted to act on a low-energy subspace. The relevant norms of these effective operators is bounded by $\Delta^{\prime}$. One could then proceed to simulate the evolution using a product formula that involves effective Hamiltonians and obtain an error bound that matches ours. A challenge is that these effective Hamiltonians are generally non-local and difficult to compute. Methods such as the local Schrieffer-Wolff transformation glazek1993renormalization ; bravyi2011schrieffer work only at the perturbative regime and numerical renormalization group methods for spin systems gu2008tensor ; evenbly2009algorithms have been studied only for a handful of models, while a general analytical treatment does not exist. Thus, efficient methods to simulate time evolution of effective Hamiltonians are lacking. We address this challenge by showing that evolutions under the effective Hamiltonians can be approximated by evolutions under the original $H_{l}$’s with a suitable choice of $\Delta^{\prime}$. This result is key in our construction and may find applications elsewhere.

Our main contributions are based on a number of technical lemmas and corollaries that are given in the Methods section and proven in detail in Supplementary Information.

For a time-independent Hamiltonian $H=\sum_{l=1}^{L}H_{l}$, where each $H_{l}$ is Hermitian, the evolution operator for time $t$ is $U(t)=e^{-itH}$. Product formulas provide a way of approximating $U(t)$ as a product of exponentials, each being a short-time evolution under some $H_{l}$. For $p>0$ integer and $s\in\mathbb{R}$, a $p$-th order product formula is a unitary

$\displaystyle W_{p}(s)=e^{-is_{q}H_{l_{q}}}\cdots e^{-is_{2}H_{l_{2}}}e^{-is_{1}H_{l_{1}}}\;,$

(1)

where each $s_{j}\in\mathbb{R}$ is proportional to $s$ and $1\leq l_{j}\leq L$. The number of terms in the product may depend on $p$ and $L$, and we assume $L=\mathcal{O}(1)$, $q=\mathcal{O}(1)$. (The more general case is analyzed in Supplementary Information.) We define $|{\bf s}|=\sum^{q}_{j=1}|s_{j}|$ and also assume $|{\bf s}|=\mathcal{O}(|s|)$. The $p$-th order product formula satisfies $\|U(s)-W_{p}(s)\|=\mathcal{O}((h|s|)^{p+1})$, where $h=\max_{l}\|H_{l}\|$ BAC07 . One way to construct $W_{p}(s)$ is to apply a recursion in Refs. Suz90 ; Suz91 . These are known as Trotter-Suzuki approximations and satisfy the above assumptions.

By breaking the time interval $t$ into $r$ steps of sufficiently small size $s$, product formulas can approximate $U(t)$ as $U(t)\approx(W_{p}(s))^{r}$. We will refer to $r$ as the Trotter number, and this number will define the complexity of product formulas that simulate $U(t)$ within given accuracy. Note that the total number of terms in the product formula is actually $qMr=\mathcal{O}(Mr)$, where $M$ is the number of terms in the product decomposition of each $e^{-is_{j}H_{l_{j}}}$.

Known error bounds for product formulas that apply to the general case grow with $h$ and can be large. However, error bounds for approximating the evolved state $U(t)\ket{\psi}$ may be better under the additional assumption that $\ket{\psi}$ is supported on a low-energy subspace. We then analyze the case where the initial state satisfies $\Pi_{\leq\Delta}\ket{\psi}=\ket{\psi}$, where $\Pi_{\leq\Lambda}$ is the projector into the subspace spanned by eigenstates of $H$ of energies (eigenvalues) at most $\Lambda\geq 0$. We assume $H_{l}\geq 0$. Our results will be especially useful when $\Delta/h$ vanishes asymptotically, and $\Delta$ will specify the low-energy subspace.

The notion of effective operators will be useful in our analysis. Given a Hermitian operator $X$ and $\Delta^{\prime}\geq\Delta$, the corresponding effective operator is $\bar{X}=\Pi_{\leq\Delta^{\prime}}X\Pi_{\leq\Delta^{\prime}}$, which is also Hermitian. We also define the unitaries $\bar{U}(s)=e^{-is\bar{H}}$ and $\bar{W}_{p}(s)$ by replacing the $H_{l}$’s by $\bar{H}_{l}$’s in $W_{p}(s)$. Note that $\bar{h}=\max_{l}\|\bar{H}_{l}\|\leq\Delta^{\prime}$ and $U(t)\ket{\psi}=\bar{U}(t)\ket{\psi}$. Then, using the known error bound for product formulas, we obtain $\|(U(s)-\bar{W}_{p}(s))\ket{\psi}\|=\mathcal{O}((\Delta^{\prime}s)^{p+1})$. This error bound is a significant improvement over the general case if $\Delta^{\prime}\ll h$, which may occur when $\Delta\ll h$. However, product approximations of $U(t)$ require that each term is an exponential of some $H_{l}$, which is not the case in $\bar{W}_{p}(s)$. We will address this issue and show that the improved error bound is indeed attained by $W_{p}(s)$ for a suitable $\Delta^{\prime}$.

We are interested in simulating the time evolution of a local $N$-spin system on a lattice. Each local interaction term in $H$ is of strength bounded by $J$ and involves, at most, $k$ spins. We do not assume that these interactions are only within neighboring spins but define the degree $d$ as the maximum number of local interaction terms that involve any spin. Next, we write $H=\sum_{l=1}^{L}H_{l}$, where each $H_{l}$ is a sum of $M$ local, commuting terms Bro41 and $LM\leq dN$. Each $e^{-isH_{l}}$ in a product formula can be decomposed as products of $M$ evolutions under the local, commuting terms with no error.

These local Hamiltonians appear as important condensed matter systems, including gapped and critical spin chains, topologically ordered systems, and models with long-range interactions LSM61 ; AKLT04 ; Kit03 ; LMG65 . For example, for a spin chain with nearest neighbour interactions, $L=2$ and each $H_{l}$ may refer to interaction terms associated with even and odd bonds, respectively. In this case, $h=\mathcal{O}(N)$. We will present our results for the case $k=\mathcal{O}(1)$ and $d=\mathcal{O}(1)$ in the main text, which further imply $L=\mathcal{O}(1)$ Bro41 . Nevertheless, explicit dependencies of our results in $k$, $d$, $L$, and other parameters that specify $H$ can be found in Supplementary Note 4.

Table II summarizes the relevant parameters for the simulation of local Hamiltonians with product formulas.

The proof of Thm. 1 is in Supplementary Note 3 and we provide more details about it in the next section, but the basic idea is as follows. There are two contributions to Eq. (2) in our analysis. One comes from approximating the evolution operator with a product formula that involves the effective Hamiltonians and, as long as $\Delta^{\prime}\geq\Delta$, this error is $\mathcal{O}((\Delta^{\prime}|s|)^{p+1})$, as explained. The other comes from replacing such a product formula by the one with the actual Hamiltonians $H_{l}$, i.e., $W_{p}(s)$. However, unlike $\bar{H}_{l}$, the evolution under each $H_{l}$ allows for leakage or transitions from the low-energy subspace to the subspace of energies higher than $\Delta^{\prime}$. In Supplementary Information, we use a result on energy distributions in Ref. AKL16 to show that this leakage can be bounded and decays exponentially with $\Delta^{\prime}$. Thus, this effective norm depends on $\Delta$ and must also depend on $s$, as the support on high-energy states can increase as $s$ increases, resulting in the linear contribution to $\Delta^{\prime}$ in Thm. 1.

The $\log(\beta_{2}/(J|s|))$ factor in $\Delta^{\prime}$ only becomes relevant when $|s|\ll 1$. This term appears in our analysis due to the requirement that both contributions to Eq. (2) discussed above are of the same order. Thus, as $s\rightarrow 0$, we require $\Delta^{\prime}\rightarrow\infty$ to make the error due to leakage zero, which is unnecessary and unrealistic. This term plays a mild role when determining the final complexity of a product formula, as the goal will be to make $s$ as large as possible for a target approximation error. It may be possible that this term disappears in a more refined analysis.

Let $r=t/s$ be the Trotter number, i.e., the number of steps to approximate $U(t)$ as $(W_{p}(s))^{r}$. Since $U(s)\Pi_{\leq\Delta}=\Pi_{\leq\Delta}U(s)\Pi_{\leq\Delta}$ and if $\|(U(s)-W_{p}(s))\Pi_{\leq\Delta}\|\leq\epsilon$, the triangle inequality implies $\|(U(t)-(W_{p}(s))^{r})\Pi_{\leq\Delta}\|\leq 2r\epsilon$. Thus, for overall target error $\varepsilon>0$, it will suffice to satisfy $\|(U(s)-W_{p}(s))\Pi_{\leq\Delta}\|=\mathcal{O}(\varepsilon s/t)$. This condition and Thm. 1 can be used to determine $r$ as follows.

Each term of $\Delta^{\prime}$ in Thm. 1 can be dominant depending on $s$ and $\Delta$. First, we consider the first two terms, and determine a condition in $s$ to satisfy $((\Delta+J)|s|)^{p+1}=\mathcal{O}(\varepsilon s/t)$, by omitting the $\log$ factor. Then, we consider another term and determine a condition in $s$ to satisfy $(J^{2}N|s|^{2})^{p+1}=\mathcal{O}(\varepsilon s/t)$. These two conditions alone can be satisfied with a Trotter number

$\displaystyle r^{\prime}=\mathcal{O}\left(\frac{(t(\Delta+J))^{1+\frac{1}{p}}}{\varepsilon^{\frac{1}{p}}}+\frac{(tJ\sqrt{N})^{1+\frac{1}{2p+1}}}{\varepsilon^{\frac{1}{2p+1}}}\right)\;.$

(3)

Last, we reconsider the second term with $\log$, and we require $(J\log(1/(J|s|))|s|)^{p+1}=\mathcal{O}(\varepsilon s/t)$. As the first two conditions are satisfied with a value for $s$ that is polynomial in $N$ and $tJ/\varepsilon$, this last condition only sets a correction to the first term in $r^{\prime}$ in Eq. (3) that is polylogarithmic in $|t|J/\varepsilon$. Thus, the overall complexity of the product formula for local Hamiltonians is given by Eq. (3), where we need to replace $\mathcal{O}$ by $\tilde{\mathcal{O}}$ to account for the last correction. Note that the number of terms in each $W_{p}(s)$ is constant under the assumptions and $r$ is proportional to the total number of exponentials in $(W_{p}(s))^{r}$.

We give a general result on the complexity of product formulas that provides $r$ as a function of all parameters that specify $H$ in Thm. 2 of Supplementary Note 4.

The error bounds for product formulas used in Thm. 1 depend on the norm of the effective Hamiltonians $\bar{H}_{l}$. The assumption $H_{l}\geq 0$ will then assure that $\|\bar{H}_{l}\|\leq\Delta^{\prime}$, which is sufficient to demonstrate the complexity improvements in Eq. (3).

In general, $H_{l}\geq 0$ can be met after a simple shifting $H_{l}\rightarrow H_{l}+a_{l}$, and the assumption seems irrelevant. However, this shifting could result in a value of $\Delta$ (or $\Delta^{\prime}$) that scales with some parameters such as the system size $N$. In this case, the error bound in Thm. 1 would be comparable to that of the worst case (without the low-energy assumption) and would not provide an advantage.

Nevertheless, for many important spin Hamiltonians, the assumption $H_{l}\geq 0$ is readily satisfied. The Heisenberg model of Fig. 1 is an example, where $H_{l}$ is a sum of terms like ${\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}-X_{i}X_{j}-Y_{i}Y_{j}-Z_{i}Z_{j}\geq 0$. More general (anisotropic) Heisenberg models as well as the so-called frustration-free Hamiltonians that are ubiquitous in many-body physics also satisfy the assumption BT09 ; BOO10 , where our results directly apply. For this class of models, the ground state energy is zero. This class contains interesting low-lying states in the subspace where, e.g., $\Delta=\mathcal{O}(1)$.

We provide more details on potential complexity improvements for the general case ($H_{l}\ngeq 0$) at the end of Supplementary Note 3.

A key ingredient for Thm. 1 is a property of local spin systems, where the leakage to high-energy states due to the evolution under any $H_{l}$ can be bounded. Let $\Pi_{>\Lambda^{\prime}}$ be the projector into the subspace spanned by eigenstates of energies greater than $\Lambda^{\prime}$. Then, for a state $\ket{\phi}$ that satisfies $\Pi_{\leq\Lambda}\ket{\phi}=\ket{\phi}$, we consider a question on the support of $e^{-isH_{l}}\ket{\phi}$ on states with energies greater than $\Lambda^{\prime}$. This question arises naturally in Hamiltonian complexity and beyond, and Lemma 1 below may be of independent interest. A generalization of this lemma will allow one to address the Hamiltonian simulation problem in the low-energy subspace beyond spin systems.

The proof is in Supplementary Note 1. It follows from a result in Ref. AKL16 on the action of a local interaction term on a quantum state of low-energy, in combination with a series expansion of $e^{-isH_{l}}$. While the local interaction term could generate support on arbitrarily high-energy states, that support is suppressed by a factor that decays exponentially in $\Lambda^{\prime}-\Lambda$.

Another key ingredient for proving Thm. 1 is the ability to replace evolutions under the $H_{l}$’s in a product formula by those under their effective low-energy versions (and vice versa) with bounded error. This is addressed by Lemma 2 below, which is a consequence of Lemma 1. The proof is in Supplementary Note 2, where we also provide tighter bounds that depend on $\Delta^{\prime}$.

The consequences of these lemmas for Hamiltonian simulation are many-fold and we only sketch those that are relevant for Thm. 1. Consider any product formula of the form $W=\prod_{j=1}^{q}e^{-is_{j}H_{l_{j}}}$. Then, there exists a sequence of energies $\Lambda_{q}\geq\ldots\geq\Lambda_{0}=\Delta$ such that the action of $W$ on the initial low-energy state $\ket{\psi}$ can be well approximated by that of $W^{\bf\Lambda}=\prod_{j=1}^{q}\Pi_{\leq\Lambda_{j}}e^{-is_{j}H_{l_{j}}}$ on the same state. Furthermore, each $\Pi_{\leq\Lambda_{j}}e^{-is_{j}H_{l_{j}}}$ in $W^{\bf\Lambda}$ can be replaced by $\Pi_{\leq\Lambda_{j}}e^{-is_{j}\bar{H}_{l_{j}}}$ and later by $e^{-is_{j}\bar{H}_{l_{j}}}$ within the same error order, as long as $\Lambda_{q}\leq\Delta^{\prime}$.

In particular, for sufficiently small evolution times $s_{j}$ and $\Delta\ll h$, the resulting effective norm satisfies $\Delta^{\prime}\ll h$ for local Hamiltonians. This is formalized by several corollaries in Supplementary Note 3. Starting from $W$, we can construct the product formula $\bar{W}=\prod_{j=1}^{q}e^{-is_{j}\bar{H}_{l_{j}}}$. Lemmas 1 and 2 imply that both product formulas produce approximately the same state when acting on $\ket{\psi}$, for a suitable choice of $\Delta^{\prime}$ as in Thm. 1. If $\bar{W}$ is a product formula approximation of $\bar{U}(s)=e^{-is\bar{H}}$, it follows that $U(s)\ket{\psi}=\bar{U}(s)\ket{\psi}\approx\bar{W}\ket{\psi}\approx W\ket{\psi}$.

We employ Theorem 2.1 in Ref. [37] that, for an operator $A$, states

$\displaystyle\|\Pi_{>\Lambda^{\prime}}A\Pi_{\leq\Lambda}\|\leq\|A\|\cdot e^{-\lambda(\Lambda^{\prime}-\Lambda-2R)}\;.$

(7)

Here, $\lambda=1/(2gk)$, where $g$ is an upper bound on the sum of the strengths of the interactions associated with any spin. In our case, we take $g=dJ$ and $\lambda=1/(2Jdk)$. If $E_{A}$ is the subset of local interaction terms in $H$ that do not commute with $A$, $R$ is the sum of the strengths of the terms in $E_{A}$. For any $H_{l}$, we note that $(H_{l})^{n}$ is a sum of, at most, $M^{n}$ terms, each of strength bounded by $J^{n}$ and containing, at most, $kn$ spins. For each such term, $R\leq Jdkn$, and we obtain

	$\displaystyle\|\Pi_{>\Lambda^{\prime}}(H_{l})^{n}\Pi_{\leq\Lambda}\|$	$\displaystyle\leq(MJ)^{n}e^{-\frac{1}{2Jdk}(\Lambda^{\prime}-\Lambda-2Jdkn)}$		(8)
		$\displaystyle=(eMJ)^{n}e^{-\frac{1}{2Jdk}(\Lambda^{\prime}-\Lambda)}\;.$		(9)

We now consider the Taylor series expansion of the exponential,

$\displaystyle e^{-isH_{l}}=\sum_{n=0}^{\infty}\frac{(-isH_{l})^{n}}{n!}\;.$

(10)

The triangle inequality and Eq. (9) imply

	$\displaystyle\|\Pi_{>\Lambda^{\prime}}e^{-isH_{l}}\Pi_{\leq\Lambda}\|$	$\displaystyle\leq\sum_{n=1}^{\infty}\frac{|s|^{n}\|\Pi_{>\Lambda^{\prime}}(H_{l})^{n}\Pi_{\leq\Lambda}\|}{n!}$		(11)
		$\displaystyle\leq e^{-\frac{1}{2Jdk}(\Lambda^{\prime}-\Lambda)}\sum_{n=1}^{\infty}\frac{(|s|eMJ)^{n}}{n!}$		(12)
		$\displaystyle=e^{-\frac{1}{2Jdk}(\Lambda^{\prime}-\Lambda)}\left(e^{|s|eMJ}-1\right)$		(13)
		$\displaystyle=e^{-\lambda(\Lambda^{\prime}-\Lambda)}\left(e^{\alpha|s|M}-1\right)\;,$		(14)

where $\alpha=eJ$.

In particular, when $k$ and $d$ are constants, we have $\alpha M\leq eJdN$, and Eq. (14) implies

$\displaystyle\|\Pi_{>\Lambda^{\prime}}e^{-isH_{l}}\Pi_{\leq\Lambda}\|$

$\displaystyle\leq e^{-\alpha_{1}(\Lambda^{\prime}-\Lambda)/J}\left(e^{\alpha_{2}J|s|N}-1\right)\;,$

(15)

where $\alpha_{1}=1/(2dk)$ and $\alpha_{2}=ed$ are also positive constants.

∎

To prove the first result, we will use the identity

$\displaystyle e^{-is\bar{H}_{l}}-e^{-isH_{l}}=-i\int_{0}^{s}ds^{\prime}e^{-i(s-s^{\prime})\bar{H}_{l}}(\bar{H}_{l}-H_{l})e^{-is^{\prime}H_{l}}\;.$

(16)

We note that, from the assumptions, $\Pi_{\leq\Lambda^{\prime}}=\Pi_{\leq\Lambda^{\prime}}\Pi_{\leq\Delta^{\prime}}=\Pi_{\leq\Delta^{\prime}}\Pi_{\leq\Lambda^{\prime}}$ and $[\Pi_{\leq\Delta^{\prime}},e^{-is\bar{H}_{l}}]=0$ for all $s$. Then, we can express $\Pi_{\leq\Lambda^{\prime}}(e^{-is\bar{H}_{l}}-e^{-isH_{l}})\Pi_{\leq\Lambda}$ as

$\displaystyle-i\Pi_{\leq\Lambda^{\prime}}\left(\int_{0}^{s}ds^{\prime}e^{-i(s-s^{\prime})\bar{H}_{l}}\Pi_{\leq\Delta^{\prime}}(\bar{H}_{l}-H_{l})(\Pi_{\leq\Delta^{\prime}}+\Pi_{>\Delta^{\prime}})e^{-is^{\prime}H_{l}}\right)\Pi_{\leq\Lambda}\;.$

We can simplify this expression since $\Pi_{\leq\Delta^{\prime}}(\bar{H}_{l}-H_{l})\Pi_{\leq\Delta^{\prime}}=0$. Observing that $\|\Pi_{\leq\Lambda^{\prime}}\|=1$, $\|e^{-i(s-s^{\prime})\bar{H}_{l}}\|=1$, and using standard properties of the spectral norm, we obtain

	$\displaystyle\|\Pi_{\leq\Lambda^{\prime}}(e^{-is\bar{H}_{l}}-e^{-isH_{l}})\Pi_{\leq\Lambda}\|$	$\displaystyle\leq\int_{0}^{|s|}ds^{\prime}\;\|\Pi_{\leq\Delta^{\prime}}(\bar{H}_{l}-H_{l})\Pi_{>\Delta^{\prime}}\|\|\Pi_{>\Delta^{\prime}}e^{-is^{\prime}H_{l}}\Pi_{\leq\Lambda}\|$		(17)
		$\displaystyle\leq\|\Pi_{\leq\Delta^{\prime}}(\bar{H}_{l}-H_{l})\Pi_{>\Delta^{\prime}}\|e^{-\lambda(\Delta^{\prime}-\Lambda)}\int_{0}^{|s|}ds^{\prime}\;(e^{\alpha s^{\prime}M}-1)$		(18)
		$\displaystyle=\|\Pi_{\leq\Delta^{\prime}}(\bar{H}_{l}-H_{l})\Pi_{>\Delta^{\prime}}\|\frac{e^{-\lambda(\Delta^{\prime}-\Lambda)}(e^{\alpha|s|M}-1-\alpha M|s|)}{\alpha M}\;,$		(19)

where $\lambda=1/(2Jdk)$, $\alpha=eJ$, and Eq. (18) follows from Lemma 1. Note that

	$\displaystyle\|\Pi_{\leq\Delta^{\prime}}(\bar{H}_{l}-H_{l})\Pi_{>\Delta^{\prime}}\|$	$\displaystyle=\|\Pi_{\leq\Delta^{\prime}}H_{l}\Pi_{>\Delta^{\prime}}\|$		(20)
		$\displaystyle\leq\|H_{l}\|$		(21)
		$\displaystyle\leq JM$		(22)
		$\displaystyle\leq\alpha M\;.$		(23)

We obtain

$\displaystyle\|\Pi_{\leq\Lambda^{\prime}}(e^{-is\bar{H}_{l}}-e^{-isH_{l}})\Pi_{\leq\Lambda}\|$

$\displaystyle\leq e^{-\lambda(\Delta^{\prime}-\Lambda)}(e^{\alpha|s|M}-1-\alpha M|s|)\;.$

(24)

To simplify our analysis, we will use a looser upper bound in the statement of the Lemma 2, which follows directly from Eq. (24) and $\Delta^{\prime}\geq\Lambda^{\prime}$:

$\displaystyle\|\Pi_{\leq\Lambda^{\prime}}(e^{-is\bar{H}_{l}}-e^{-isH_{l}})\Pi_{\leq\Lambda}\|$

$\displaystyle\leq e^{-\lambda(\Lambda^{\prime}-\Lambda)}(e^{\alpha|s|M}-1)\;.$

(25)

In particular, when $k$ and $d$ are constants, we have $\alpha M\leq eJdN$, and Eq. (25) implies

$\displaystyle\|\Pi_{\leq\Lambda^{\prime}}(e^{-is\bar{H}_{l}}-e^{-isH_{l}})\Pi_{\leq\Lambda}\|$

$\displaystyle\leq e^{-\alpha_{1}(\Lambda^{\prime}-\Lambda)/J}(e^{\alpha_{2}J|s|N}-1)\;,$

(26)

where $\alpha_{1}=1/(2dk)$ and $\alpha_{2}=ed$ are also constants.

To prove the second result, we use Lemma 1 together with Eq. (25) and standard properties of the spectral norm, and obtain

	$\displaystyle\|\Pi_{>\Lambda^{\prime}}e^{-is\bar{H}_{l}}\Pi_{\leq\Lambda}\|$	$\displaystyle=\|\Pi_{\leq\Delta^{\prime}}\Pi_{>\Lambda^{\prime}}e^{-is\bar{H}_{l}}\Pi_{\leq\Lambda}\|$		(27)
		$\displaystyle=\|\Pi_{\leq\Delta^{\prime}}\Pi_{>\Lambda^{\prime}}(e^{-is\bar{H}_{l}}-e^{-isH_{l}}+e^{-isH_{l}})\Pi_{\leq\Lambda}\|$		(28)
		$\displaystyle\leq\|\Pi_{\leq\Delta^{\prime}}\Pi_{>\Lambda^{\prime}}(e^{-is\bar{H}_{l}}-e^{-isH_{l}})\Pi_{\leq\Lambda}\|+e^{-\lambda(\Lambda^{\prime}-\Lambda)}(e^{\alpha|s|M}-1)$		(29)
		$\displaystyle=\|(\Pi_{\leq\Delta^{\prime}}-\Pi_{\leq\Lambda^{\prime}})(e^{-is\bar{H}_{l}}-e^{-isH_{l}})\Pi_{\leq\Lambda}\|+e^{-\lambda(\Lambda^{\prime}-\Lambda)}(e^{\alpha|s|M}-1)$		(30)
		$\displaystyle\leq\|\Pi_{\leq\Delta^{\prime}}(e^{-is\bar{H}_{l}}-e^{-isH_{l}})\Pi_{\leq\Lambda}\|+\|\Pi_{\leq\Lambda^{\prime}}(e^{-is\bar{H}_{l}}-e^{-isH_{l}})\Pi_{\leq\Lambda}\|$
		$\displaystyle\quad+e^{-\lambda(\Lambda^{\prime}-\Lambda)}(e^{\alpha|s|M}-1)$		(31)
		$\displaystyle\leq(e^{-\lambda(\Delta^{\prime}-\Lambda)}+2e^{-\lambda(\Lambda^{\prime}-\Lambda)})(e^{\alpha|s|M}-1)$		(32)
		$\displaystyle\leq 3e^{-\lambda(\Lambda^{\prime}-\Lambda)}(e^{\alpha|s|N}-1)\;.$		(33)