give rise to the same lower bound that counts all the factors from the edges
and head blocks together with the differential block, i.e., $2\times
4^{-l(P_{j})}\times 2^{-w(P_{j})}$. Adding the lower bounds corresponding to
the individual Pauli strings $h_{j}$ in the Hamiltonian via Eq. (S24), we
eventually arrive at the desired lower bound corresponding to the entire
Hamiltonian. ∎

We make several remarks on Theorem 1. Firstly, we emphasize that Theorem 1
holds for any random quantum circuits composed of blocks forming independent
local $2$-designs. No assumption is made regarding the number, sizes, and
arrangement of these blocks. One can check that the lower bound in Theorem 1
is consistent with the previous literature on barren plateaus. For example,
compared to the original work on barren plateaus [19] which considers a global
$2$-design, the path set in Eq. (S38) is just this single global block and
hence the head width grows linearly with the system size, i.e., $w(P_{j})\sim
N$, which leads to an exponential small lower bound, consistent with the
exponential small upper bound proposed by Ref. [19]. For the alternating
layered ansatz and local cost functions [27, 28], the head width could be
constant but the path length Eq. (S38) grows linearly with the system size,
i.e., $l(P_{j})\sim N$, which means that linear or even larger depth (deep)
circuits encounter barren plateaus while finite or logarithmic depth (shallow)
circuits do not, consistent with the lower bound in Ref. [27]. For global cost
functions, even a single layer of blocks would require both the path length
and head width to increase linearly with the system size, which leads to an
exponentially small lower bound, again consistent with the exponentially small
upper bound in Ref. [27]. Furthermore, for the quantum convolutional neural
network (QCNN) [29], the path length increases logarithmically with the system
size, which causes a polynomially vanishing lower bound, consistent with the
lower bound proved in Ref. [29]. In summary, Theorem 1 unifies the known
gradient scaling behaviors for all architectures of random quantum circuits
composed of blocks forming $2$-designs. Two geometrical quantities play an
important role in this lower bound, i.e., the path length and head width of
the path set on the circuit. For a given setup of the variational quantum
eigensolver with a certain circuit architecture, as long as there exists a
term $h_{j}$ in $H$ with $\lambda_{j}\in\Omega(1)$ accompanied by a path set
$P_{j}$ satisfying $l(P_{j}),w(P_{j})\in\mathcal{O}(\log N)$, the variance of
the cost derivative would vanish no faster than polynomially, implying that
the corresponding setup does not exhibit barren plateaus.

In particular, we compare Theorem 1 with the lower bound proposed in Ref.
[28], which also utilizes the backward evolution of Pauli strings in the
proof. In the first place, Ref. [28] focuses on the alternating layered
ansatz, while our theorem can be applied to all ansatzes composed of local
$2$-designs. Moreover, even the form of the causal-cone-width-dependent lower
bound considered by Ref. [28] has its own limitation. One can consider the
case when the differential parameter is located at the last block in the
finite local-depth circuit shown in Fig. 1 in the main text and the
Hamiltonian is a single Pauli $Z$ operator on the support of the last block.
In such a case, the width of the causal cone equals $N$, and hence the causal-
cone-dependent bound like in Ref. [28] would give a trivial exponential small
lower bound, while in Theorem 1 we can just choose a short path with the head
width not scaling with the system size, and provide a meaningful constant
lower bound. The lower bound specific to finite local-depth circuits will be
introduced in detail in Theorem 2 below.

Secondly, Theorem 1 defaults the initial state to $|\bm{0}\rangle$, so that
the variance of the cost derivative equals the sum of weights of backward
evolved doubled Pauli strings only containing $Z$ and $I$. In fact, the
theorem can be generalized to the cases with the initial state being an
arbitrary product state
$|\psi_{0}\rangle=\bigotimes_{i=1}^{N}V_{i}|\bm{0}\rangle$ because $V_{i}$ can
be absorbed into the block whose forward residual support covers $q_{i}$,
based on the fact that a $t$-design times any unitary on the same support is
still a $t$-design. More generally, as long as the initial state can be
reduced back to $|\bm{0}\rangle$ by such an absorption procedure, Theorem 1 is
applicable for the corresponding initial state.

Thirdly, Theorem 1 has assumed that the differential gate $U_{\mu}$ is
sandwiched by two local $2$-designs on the support of the differential block
$B_{k(\mu)}$, which in practice means that the theorem holds for the case
where the differential parameter $\theta_{\mu}$ is located in the middle of
the differential block, as shown in Eq. (S17). For the parameters located at
the borders of blocks, it is indeed possible that the lower bound in Eq. (S38)
fails. For example, if $U_{\mu}$ is a single-qubit rotation gate $R_{z}$ and
is the first gate acting on some qubit initialized as $|0\rangle$, then the
derivative will always equal to zero since $R_{z}$ only contributes a global
phase to $|0\rangle$. Similar events occur also for the case where $U_{\mu}$
is the last gate acting on some qubit and $U_{\mu}$ commutes with $h_{j}$.
However, taking these cases into consideration, the proof can still proceed if
either of the following two conditions holds.

  * •

The left sub-block $L_{k\mu}$ forms a $2$-design and $R_{k\mu}$ is the
identity. $s(U_{\mu})$ intersects with the connecting support of $B_{k(\mu)}$
and its successor, or intersects with $s(h_{j})\cap s_{b}(B_{k(\mu)})$ and
$U_{\mu}$ does not commute with $h_{j}$, so that there must exist some
selected Pauli strings surviving from the differential channel
$\mathcal{D}_{\mu}^{\otimes 2}$ and hence can be recovered to a uniform
weighted summation by the twirling channel of the left sub-block.

  * •

The right sub-block $R_{k\mu}$ forms a $2$-design and $L_{k\mu}$ is the
identity. $s(U_{\mu})$ intersects with the connecting support of $B_{k(\mu)}$
and its predecessor, or intersects with $s_{f}(B_{k(\mu)})$ and $U_{\mu}$ does
not commute with $\rho_{0}$, so that there must exist some surviving Pauli
strings from $\mathcal{D}_{\mu}^{\otimes 2}$ still satisfying the selection
rule.

Thus, we expect that the asymptotic behavior of the lower bound would not
change drastically when the differential parameter is located at the borders
of blocks. Taking a step back, this issue does not really matter in practice
if we consider all trainable parameters simultaneously. That is to say, as
long as some of the gradient components do not vanish, the optimization could
still proceed successfully and no barren plateau really occurs.

In addition, Theorem 1 has assumed that the generator of the differential gate
$\Omega_{\mu}$ is a Pauli string. This condition can also be relaxed to the
case where $\Omega_{\mu}$ is a general Hermitian generator by considering the
Pauli decomposition of $\Omega_{\mu}$. The new lower bound is just the linear
combination of the lower bounds corresponding to the Pauli string components
in the generator, after the cross terms caused by $\mathcal{D}_{\mu}^{\otimes
2}$ are eliminated by Lemma S4. In addition, if the gate is commonly not
parameterized as a rotation gate in the exponential form
$e^{-i\Omega_{\mu}\theta_{\mu}}$, e.g., the controlled $R_{y}$ gate, then it
can be seen as the composition of some basic rotation gates with correlating
parameters, e.g., the controlled $R_{y}$ gate can be seen as the composition
of two CNOT gates and two $R_{y}$ gates with opposite rotation angles, and
hence the corresponding derivative can be obtained by the chain rule. This may
cause the left and right sub-blocks in Eq. (S17) to correlate with each other
and be no longer independent $2$-designs. However, since the differential
block only contributes a constant factor for non-trivial backward evolved
Pauli strings in the proof, we expect the asymptotic behavior contributed from
the path length and head width would not be distorted.

Finally, Theorem 1 requires $s(\mathbf{U})\supseteq s(H)$ so that Corollary S5
holds and one can only consider the support of the head of the path set when
counting the Pauli strings that only contains $Z$ and $I$. As a
counterexample, if $h_{j}$ contains a Pauli $X$ operator that is not covered
by $\mathbf{U}$, then the variance of the cost derivative would directly
vanish due to $\langle 0|X|0\rangle=0$. But this issue could be easily fixed
by multiplying the lower bound corresponding to $h_{j}$ by an extra factor,
i.e., the expectation value of the sub-string of $h_{j}$ on
$s(h_{j})-s(\mathbf{U})$ with respect to the initial state.

The proof method in Theorem 1 can be extended to bounding other quantities
besides the variance of the cost derivative. For example, the variance of the
primitive cost function could be lower bounded in a similar way by just
removing the differential channel $\mathcal{D}_{\mu}^{\otimes 2}$ in Eqs.
(S16) and (S46), which will lead to the same result as in Eq. (S38) except
that the path set does not have to go through the differential block. In
general, this path-set approach is promising to be used for estimating the
average value of an arbitrary squared space-time correlator in random quantum
circuits since the cost derivative in Eq. (S14) itself can be seen as a two-
point correlation function between the differential gate and the measured
observable.

Up to this point, we have introduced all the contents on Theorem 1 concerning
a general lower bound for arbitrary circuits composed of $2$-designs. In the
following, we will first make clear some concepts in order to present Theorem
2, which focuses on finite local-depth circuits.

###### Definition S8

The maximum block size of the PQC $\mathbf{U}$ is defined by the maximum value
of block sizes of all blocks in the PQC, i.e.,
$\beta=\max\\{|s(B_{1})|,|s(B_{2})|,\cdots,|s(B_{M^{\prime}})|\\}$.

###### Definition S9

For a Hamiltonian $H$ with the Pauli decomposition
$H=\sum_{j}\lambda_{j}h_{j}$, the maximum interaction range $r$ of the
Hamiltonian is defined by the maximum value of the support sizes of all Pauli
strings $h_{j}$, i.e., $r=\max\\{|s(h_{1})|,|s(h_{2})|,|s(h_{3})|,\cdots\\}$.
A $r$-local Hamiltonian refers to a Hamiltonian with the maximum interaction
range $r$. Note that here the $r$ interacting qubits do not need to be close
to each other spatially in the sense of the lattice geometry associated with
the Hamiltonian.

###### Definition S10

The supported block set of the qubit $q_{i}$ in the PQC contains the block
$B_{k}$ acting on $q_{i}$, denoted as $b(q_{i})=\\{B_{k}\mid q_{i}\in
s(B_{k})\\}$. The supported block set of a subset of qubits $s$ is the union
of the supported sets of the qubits in $s$, i.e., $b(s)=\bigcup_{q_{i}\in
s}b(q_{i})=\\{B_{k}\mid s\cap s(B_{k})\neq\varnothing\\}$. In particular, the
supported block set of all the $N$ qubits is just the set of all the blocks in
$\mathbf{U}$.

###### Definition S11

The local depth (qubit-wise depth) of the PQC at a qubit $q_{i}$ can be
defined by the number of elements in the supported block set $b(q_{i})$,
denoted as $\chi_{i}=|b(q_{i})|$. The maximum local depth of the PQC is
defined by the maximum value of local depths of all qubits, i.e.,
$\chi=\max\\{\chi_{1},\chi_{2},\cdots,\chi_{N}\\}$. Here we use the symbol
$\chi$ to distinguish from the common circuit depth $D$, which is defined by
the minimum number of layers where all blocks within each layer commute with
each other.

###### Definition S12

A finite local-depth circuit (FLDC) refers to a circuit whose maximum local
depth $\chi$ does not scale with the system size $N$.

The above definition of local depth regards blocks as the elementary units
building the PQC. Alternatively, for general quantum circuits taking
elementary gates as units, the local depth at the qubit $q_{i}$ can be defined
by the number of non-commuting gates acting on $q_{i}$, which might be more
complex to estimate because the detailed commutation relation of gates is
involved. Given a block template like in Eq. (S10), these two definitions are
supposed to differ only by a constant coefficient in general. In addition, we
emphasize that the support sizes of the blocks constituting FLDCs are always
finite, i.e., $\beta$ should not scale with the system size by the definition
of “circuit”. Correspondingly, a finite depth circuit (FDC) refers to a
circuit whose global depth does not scale with the system size, while a
general linear depth circuit (GLDC) refers to a circuit whose global depth
scales linearly with the system size.

Note that as long as there is a finite local depth, regardless of how messy
the gate configuration might be, the circuit can be considered as an FLDC. It
is not necessary to arrange the gates uniformly and regularly as shown in Fig.
1 in the main text. Here we point out that FLDCs defined above are more
general than the sequential quantum circuits (SQC) defined in Ref. [47],
especially the instances listed in Ref. [47] because we have not restricted
the gates in the circuit as local gates, which is assumed by Ref. [47]. In
other words, we do not assume an underlying lattice or connectivity to define
the spatial locality of gates. But when it comes to the content of the
entanglement area law, a spatial lattice must be involved, and at that time,
we will restrict our discussion on FLDCs to the circuits composed of spatially
local gates as shown in Fig. 1 in the main text. With the concepts defined
above, we are prepared to prove Theorem 2.

###### Theorem 2

Based on the assumptions in Theorem 1, suppose that the maximum local depth of
$\mathbf{U}$ is $\chi$ and the maximum block size of $\mathbf{U}$ is $\beta$.
Then for any $r$-local Hamiltonian, the variance of the cost derivative is
lower bounded by

$\operatorname{Var}_{\mathbb{U}}[\partial_{\mu}C]~{}\geq~{}4^{-r\chi\beta}\sum_{j}2\lambda_{j}^{2},$
(S60)

where $j$ runs over the indices of the Pauli string $h_{j}$ that is non-
trivial on $s(B_{k(\mu)})$.

###### Proof.

For each $h_{j}$, if $s(h_{j})\cap s(B_{k(\mu)})=\varnothing$, we just neglect
the contribution of $h_{j}$ in Eq. (S38). Otherwise, we can choose the path
set $P_{j}$ to be the straight wires on $s(h_{j})$, i.e.,
$P_{j}=\\{b(q_{i})\mid q_{i}\in s(h_{j})\\}$. Note that the forward residual
supports of the head blocks of the path set appear simultaneously in
expressions for both length and head width in Eqs. (S32) and (S35), we will
extract them and combine together before relaxation to obtain a tighter bound.
We use $l_{0}$ to denote the length of the edges between $\rho_{0}$ and the
head blocks of $P_{j}$. According to the definition in Eq. (S31), the length
of $P_{j}$ apart from $l_{0}$ is upper bounded by

$l(P_{j})-l_{0}\leq
r\cdot(\chi-1)\cdot\log_{4}\left(\frac{4^{\beta}-1}{4-1}\right)\leq
r(\chi-1)\left(\beta-\frac{1}{2}\log_{2}3\right)\leq r(\chi-1)\beta,$ (S61)

where the sizes of the connecting supports are relaxed to one.
Correspondingly, the head width of $P_{j}$ together with $2l_{0}$ is upper
bounded by

$w(P_{j})+2l_{0}=\sum_{B_{k}\in\mathrm{Head}(P_{j})}\log_{2}\left(\frac{4^{|s_{f}(B_{k})|}-1}{2^{|s_{f}(B_{k})|}-1}\right)+\log_{2}\left(\frac{4^{|s(B_{k})|}-1}{4^{|s_{f}(B_{k})|}-1}\right)\leq
r\times\log_{2}\left(\frac{4^{\beta}-1}{2-1}\right)\leq r\cdot 2\beta.$ (S62)

According to Theorem 1, the variance of the cost derivative can be lower
bounded by

$\operatorname{Var}_{\mathbb{U}}[\partial_{\mu}C]\geq\sum_{j}\lambda_{j}^{2}\cdot
2^{1-2l(P_{j})-w(P_{j})}\geq\sum_{j}\lambda_{j}^{2}2^{1-2r(\chi-1)\beta-2r\beta}=4^{-r\chi\beta}\sum_{j}2\lambda_{j}^{2},$
(S63)

where $j$ runs over the indices of $h_{j}$ that is non-trivial on
$s(B_{k(\mu)})$. ∎

Theorem 2 elegantly integrates the factors related to barren plateaus in an
unexpectedly concise manner, i.e., the block locality $\beta$ [19], the
Hamiltonian locality $r$ [27, 28] and the circuit deepness $\chi$. It is
vitally important to note that the relevant quantity characterizing the
circuit deepness is the local depth $\chi$, instead of the common circuit
depth, or say the global depth $D$. For the alternating layered ansatz, the
local depth basically equals the global depth. But for general circuit
structures, the local depth and the global depth are not equivalent and can
differ significantly in specific scenarios, as illustrated in Fig. 1 in the
main text. This implies that the class of circuit architectures that do not
exhibit barren plateaus is larger than finite (or logarithmic) depth circuits
(FDC) [27, 28], which naturally give rise to the class of finite (or
logarithmic) local-depth circuits. In this work, we mainly focus on finite
local-depth circuits (FLDC), which have a clear physical correspondence to
entanglement area-law states as we will introduce below. Logarithmic local-
depth circuits may involve states beyond the area law such as gapless states
with topological orders, which are also interesting to study in the future.

As a superclass, FLDCs have stronger expressibility than FDCs since they can
generate long-range entanglement. To be specific, FDCs can only generate
short-range entangled (SRE) states such as symmetry-protected topological
(SPT) states like the Affleck-Kennedy-Lieb-Tasaki (AKLT) state. These SRE
states are equivalent to the product states in the sense of entanglement
renormalization. Those long-range entangled (LRE) area-law states such as the
Greenberger–Horne–Zeilinger (GHZ) state and the topologically ordered states
like the ground states of the toric code model, can not be generated from
FDCs, while they can be generated from FLDCs with explicit forms [46, 47, 48].

On the other hand, as a subclass, FLDCs have less expressibility than general
linear depth circuits (GLDC), such as the alternating layered ansatz of linear
depth. Given a spatial lattice of qubits, if we restrict that each block can
act only on spatial adjacent qubits, then the output state of an FLDC should
obey the entanglement area law, while GLDCs generate much more entanglement so
that the output states typically exhibit the volume law [30]. Here we give a
simple demonstration to show why an FLDC composed of spatially local blocks
generates area-law output states. Take $\beta=2$ for an example. The number of
blocks acting on two adjacent qubits hence should be not larger than $\chi$.
Suppose that the lattice is partitioned into two continuous regions $A$ and
$\bar{A}$. The number of lattice edges cut by this bipartition is denoted by
$|\partial A|$, which can also be seen as a measure of the boundary between
$A$ and $\bar{A}$. Thus, the number of blocks acting across the boundary is
not larger than $|\partial A|\cdot\chi$, so the Schmidt rank $r_{A}$ of the
output state with respect to this bipartition is not larger than $2^{|\partial
A|\cdot\chi}$. The entanglement entropy with respect to this bipartition could
be upper bounded by $S_{A}=-\operatorname{tr}(\rho_{A}\log\rho_{A})\leq\log
r_{A}\leq\chi|\partial A|$, which means that for a fixed maximum local depth
$\chi$, the entanglement entropy at most grows linearly with the size of the
boundary, i.e., the output state satisfies the entanglement area law. Here the
term “area”, used by convention, refers to the boundary of a three-dimensional
system.

Considering the two aspects mentioned above, FLDC constitutes an intermediate
class of quantum circuits between FDC and GLDC, in the sense of both
expressibility and trainability. FLDC possesses the favorable trainability of
FDC and at the same time, partially exhibits the expressibility of GLDC in
generating long-range entanglement. Under the condition that the target many-
body Hamiltonian has a ground state with entanglement area law, FLDCs hold
promise to serve as an appropriate ansatz in variation quantum eigensolvers to
prepare ground states of gapped many-body systems on quantum devices. In other
words, FLDCs composed of spatially local blocks can be regarded as the quantum
circuit implementation of tensor network states, such as matrix product states
(MPS) and projected entangled paired states (PEPS). Note that some rigorous
theorem in 1D systems and experience in higher dimensions indicate that the
ground states of gapped local Hamiltonians typically satisfy the entanglement
area law. Previously proposed quantum circuit implementation of tensor network
states [51, 52] can be seen as special cases of finite or logarithmic local-
depth circuits.

The above discussion pertains to Hamiltonians with local interactions, i.e.
$r\in\mathcal{O}(1)$, but in practice, it is possible to encounter
Hamiltonians involving non-local interactions. For example, in condensed
matter or quantum chemistry problems, fermionic statistics are usually encoded
to qubit systems via the Jordan-Wigner transformation or the Bravyi-Kitaev
transformation, where the support sizes of the resulting Pauli strings can
scale with the interaction distance of fermions $R$ as $r\in\mathcal{O}(R)$ or
$r\in\mathcal{O}(\log R)$, respectively. If $R\in\mathcal{O}(N)$, the Jordan-
Wigner encoding method would lead to an exponentially small lower bound in Eq.
(S60) even with FLDCs due to $r\in\mathcal{O}(N)$, while the Bravyi-Kitaev
encoding method may still persist with a polynomially vanishing lower bound
with FLDCs due to $r\in\mathcal{O}(\log N)$.

In the end, we remark that one of the preconditions of the absence of barren
plateaus from the lower bound in Eq. (S60) is that, the sum of the
coefficients $\sum_{j}\lambda_{j}^{2}$ does not vanish. That is to say, there
is at least a term $h_{j}$ in the Hamiltonian that is non-trivial on the
support of the differential block with $\lambda_{j}\sim\mathcal{O}(1)$. This
precondition can be satisfied by common locally interacting quantum many-body
systems, where the support of the Hamiltonian covers the whole system, i.e.,
$s(H)=s(\mathbf{U})$. However, if we do encounter cases where there is no term
acting on $s(B_{k(\mu)})$, trivializing the lower bound in Theorem 2, we need
to revert back to Theorem 1 to consider the contribution from the terms acting
elsewhere. For FLDCs composed of spatially local blocks, if there is a term in
the Hamiltonian acting near $s(B_{k(\mu)})$, it is still possible to find a
short path on the circuit connecting to $B_{k(\mu)}$, thereby obtaining a non-
vanishing lower bound. However, if every term in the Hamiltonian acts far from
$s(B_{k(\mu)})$, e.g., the distance grows linearly with the system size, then
the bound in Theorem 1 would be exponentially small because the path length
also grows linearly. To be specific, we can focus on the following “ladder
ansatz”

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{}{}{{}}{}{{}}{}{}{} {}{}{{}}{}{{}}{}{}{} {}{}{{}}{}{{}}{}{}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.901960784313726,1}\pgfsys@color@rgb@fill{0.8}{0.901960784313726}{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{90.01062pt}{-73.28288pt}\pgfsys@curveto{90.01062pt}{-72.0349pt}{91.0186pt}{-71.02692pt}{92.26659pt}{-71.02692pt}\pgfsys@lineto{95.27853pt}{-71.02692pt}\pgfsys@curveto{96.52252pt}{-71.02692pt}{97.5345pt}{-72.0349pt}{97.5345pt}{-73.28288pt}\pgfsys@lineto{97.5345pt}{-93.47057pt}\pgfsys@curveto{97.5345pt}{-94.71454pt}{96.52252pt}{-95.72653pt}{95.27853pt}{-95.72653pt}\pgfsys@lineto{92.26659pt}{-95.72653pt}\pgfsys@curveto{91.0186pt}{-95.72653pt}{90.01062pt}{-94.71454pt}{90.01062pt}{-93.47057pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(S64)

and derive a tighter lower bound than Theorem 2 by considering the
contribution from the Hamiltonian terms acting elsewhere, which is elaborated
in Corollary S9.

###### Corollary S9

Based on the assumptions in Theorem 1, suppose that $H$ is a $r$-local
$1$-dimensional Hamiltonian and $\mathbf{U}$ is a ladder ansatz as in Eq.
(S64) with block size $\beta$. Denote the block which last acts on $s(h_{j})$
as $B_{k^{\prime}(j)}$. The variance of the cost derivative is lower bounded
by

$\displaystyle\operatorname{Var}_{\mathbb{U}}[\partial_{\mu}C]~{}\geq{}$
$\displaystyle\sum_{j}\lambda_{j}^{2}\cdot 2^{1-2\beta(\Delta_{\mu j}+r)}.$
(S65)

where $j$ runs over the indices of $h_{j}$ satisfying $k^{\prime}(j)\geq
k(\mu)$ and $\Delta_{\mu j}=k^{\prime}(j)-k(\mu)$.

###### Proof.

If $k^{\prime}(j)<k(\mu)$, then the differential block is outside the causal
cone of $h_{j}$ and hence the contribution to the variance equals zero. If
$k^{\prime}(j)\geq k(\mu)$, we can choose the path set containing a single
path from $B_{k(\mu)}$ to $B_{k^{\prime}(j)}$ and other $r-1$ path covering
other qubits acted by $h_{j}$. The path length apart from the length $l_{0}$
of the edges between $\rho_{0}$ and the head blocks is upper bounded by

$l(P_{j})-l_{0}\leq\Delta_{\mu
j}\cdot\log_{4}\left(\frac{4^{\beta}-1}{4-1}\right)\leq\Delta_{\mu j}\beta.$
(S66)

The head width of $P_{j}$ together with $2l_{0}$ is upper bounded by

$w(P_{j})+2l_{0}\leq r\cdot\log_{2}\left(\frac{4^{\beta}-1}{2-1}\right)\leq
r\beta.$ (S67)

According to Theorem 1, the variance of the cost derivative can be lower
bounded by

$\operatorname{Var}_{\mathbb{U}}[\partial_{\mu}C]\geq\sum_{j}\lambda_{j}^{2}\cdot
2^{1-2l(P_{j})-w(P_{j})}\geq\sum_{j}2\lambda_{j}^{2}\cdot
4^{-\beta(\Delta_{\mu j}+r)},$ (S68)

where $j$ runs over the indices of $h_{j}$ that is non-trivial on
$s(B_{k(\mu)})$. ∎

In Corollary S9, one can explicitly see that the contribution in the lower
bound indeed decays exponentially with the “distance” $\Delta_{\mu j}$ between
the differential block $B_{k(\mu)}$ and the observable $h_{j}$. Similar
conclusions can be drawn with respect to the circuit structures depicted in
Fig. 1 in the main text, which can be seen as the circuit implementation of
MPS and PEPS. This coincides with the results found in unitary embedding
tensor network states recently [38]. Again, taking a step back, this issue
does not really matter in practice if all trainable parameters are considered
simultaneously. Since there are always blocks on $s(h_{j})$ due to
$s(\mathbf{U})\supseteq s(H)$, as long as some of the gradient components do
not vanish, the optimization could still proceed successfully and no barren
plateau really occurs.

It is worth mentioning that the information spreading in the ladder ansatz
shown in Eq. (S64) is somewhat one-way because the blocks acting on the qubits
with larger indices than $q_{i}$ are outside the causal cone of $q_{i}$. This
disadvantage could be easily fixed by the following two-way ladder ansatz
which still falls in FLDC class

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}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{130.678pt}{-42.53534pt}\pgfsys@curveto{130.678pt}{-41.29137pt}{131.68999pt}{-40.27937pt}{132.93396pt}{-40.27937pt}\pgfsys@lineto{135.94592pt}{-40.27937pt}\pgfsys@curveto{137.19391pt}{-40.27937pt}{138.20187pt}{-41.29137pt}{138.20187pt}{-42.53534pt}\pgfsys@lineto{138.20187pt}{-62.72304pt}\pgfsys@curveto{138.20187pt}{-63.97101pt}{137.19391pt}{-64.979pt}{135.94592pt}{-64.979pt}\pgfsys@lineto{132.93396pt}{-64.979pt}\pgfsys@curveto{131.68999pt}{-64.979pt}{130.678pt}{-63.97101pt}{130.678pt}{-62.72304pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(S69)

##  III Additional Numerical Results and Technical Details

In this section, we provide additional numerical results and technical details
in the numerical experiments. In the experiment of gradient evaluation shown
in Fig. 3 in the main text, we utilize the ladder ansatz like in Eq. (S64)
with $\beta=2$. Each two-qubit block is parameterized by the Cartan
decomposition as in Eq. (S12). The distance $\Delta k$ is just the difference
between the indices of the differential block and the last block, i.e.,
$\Delta_{\mu j}$ in Eq. (S65). The legend in Fig. 3(b) of $R_{y1}$, $R_{yy}$,
and $R_{y2}$ refers to the following three red gates in the Cartan
decomposition respectively.

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(S70)

In Fig. 3(a), we just fix the location of the differential gate within the
differential block as the red block of $R_{yy}$ above. Recall that $R_{xx}$,
$R_{yy}$ and $R_{zz}$ denote the two-qubit rotation gates with the generators
$X\otimes X$, $Y\otimes Y$ and $Z\otimes Z$ respectively as mentioned around
Eq. (S12). $R_{y}$ and $R_{z}$ denote the single-qubit rotation gates with the
generators $Y$ and $Z$ respectively.

In the experiment of VQE on the 2D toric code model shown in Fig. 4 in the
main text, the model Hamiltonian reads

$H=(1-h)\hat{H}_{0}-\sum_{j=1}^{N}\left(h^{x}X_{j}+h^{y}Y_{j}+h^{z}Z_{j}\right).$
(S71)

The additional factor $(1-h)$ arises for the sake of properly approaching the
infinite large field limit. In the two cases on which we focus below, we set
either $(h^{x},h^{y},h^{z})=(h,0,h)$ or $(h^{x},h^{y},h^{z})=(0,0,h)$. Please
do not confuse $(h^{x},h^{y},h^{z})$ with the notation $h_{j}$ representing
the Hamiltonian sub-terms above. The bare toric code Hamiltonian $H_{0}$ reads

$H_{0}=-\sum_{v}A_{v}-\sum_{p}B_{p},$ (S72)

where $v$ and $p$ runs over all the vertices and plaquettes of the 2D square
lattice. The operators $A_{v}$ and $B_{p}$ are defined as the products of four
Pauli operators depicted below

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{}{{}}{}{{}}{}\pgfsys@moveto{143.19781pt}{-47.42728pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(S73)

Note that under open boundary conditions, the vertex operator $A_{v^{\prime}}$
at the boundary should be slightly modified to the products of two or three
Pauli operators as depicted above. The numerical experiments are conducted on
a $3\times 3$ lattice with $N=12$ qubits defined on the edges. The ground
state of $H_{0}$ can be exactly solved and is topologically ordered, which is
signified by the non-vanishing topological entanglement entropy, or say
tripartite mutual information $S_{\text{topo}}(A:B:C)$ defined by

$S_{\text{topo}}(A:B:C)=S_{A}+S_{B}+S_{C}-S_{AB}-S_{BC}-S_{CA}+S_{ABC}.$ (S74)

where $S_{A}=-\operatorname{tr}(\rho_{A}\ln\rho_{A})$ represents the
entanglement entropy of the subsystem $A$ with $\rho_{A}$ being the reduced
density matrix on $A$. The sub-regions $A$, $B$ and $C$ used in our experiment
are chosen as the following colored qubits respectively

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