\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad,$ (S12)

where $R_{xx}$, $R_{yy}$ and $R_{zz}$ are the two-qubit rotation gates with
generators $X\otimes X$, $Y\otimes Y$ and $Z\otimes Z$, respectively. $R_{3}$
is a universal single-qubit gate, e.g., $R_{z}R_{y}R_{z}$ where $R_{y}$ and
$R_{z}$ represent the single-qubit rotation gates with generators $Y$ and $Z$.
This block template from the Cartan decomposition can represent any two-qubit
gate and approximate a $2$-design when the parameters are randomized uniformly
over $[0,2\pi)$ [28].

In the following, we suppose that each block $B_{k}$ in the circuit
$\mathbf{U}$ forms a local $2$-design independently. This will induce an
ensemble of the circuit $\mathbf{U}$ according to Eq. (S11), which we denote
as $\mathbb{U}$. We will estimate the average and variance of the derivatives
of the cost function with respect to this circuit ensemble $\mathbb{U}$. Note
that we do not assume any configuration among these blocks, such as the block
sizes and the relative locations. One of the most commonly used block
configurations is the alternating layered ansatz, also known as the brickwall
ansatz or checkerboard ansatz, which features a periodic layered structure and
alternates the block pattern within one period. A typical 1D example is shown
as follows

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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{
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}{}\pgfsys@moveto{107.09036pt}{-42.00334pt}\pgfsys@curveto{107.09036pt}{-40.89537pt}{107.99034pt}{-39.99539pt}{109.09833pt}{-39.99539pt}\pgfsys@lineto{111.77429pt}{-39.99539pt}\pgfsys@curveto{112.88228pt}{-39.99539pt}{113.77826pt}{-40.89537pt}{113.77826pt}{-42.00334pt}\pgfsys@lineto{113.77826pt}{-58.1111pt}\pgfsys@curveto{113.77826pt}{-59.21909pt}{112.88228pt}{-60.11908pt}{111.77429pt}{-60.11908pt}\pgfsys@lineto{109.09833pt}{-60.11908pt}\pgfsys@curveto{107.99034pt}{-60.11908pt}{107.09036pt}{-59.21909pt}{107.09036pt}{-58.1111pt}\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>(S13)

It is known that when the depth of this circuit reaches $\Omega(N)$, i.e.,
linear with the system size or beyond, the whole circuit approximates a global
$2$-design and hence the training landscape would exhibit barren plateaus
[19], i.e., the variance of the cost derivative vanishing exponentially with
the system size. Otherwise, if the circuit depth is only of order
$\mathcal{O}(\log N)$ or even finite, i.e., does not scale with the qubit
count $N$, the variance of the cost derivative would not vanish exponentially
in the case of local cost functions [27, 28, 29]. However, these circuits of
finite depth have limited expressibility and fail to capture long-range
entanglement, which motivates us to explore other circuit structures, e.g.,
finite local-depth circuits as we will discuss below.

We start with the expression of the cost derivative. The derivative of the
energy expectation with respect to the parameter $\theta_{\mu}$ can be
expressed as

$\displaystyle\partial_{\mu}C=\frac{\partial C}{\partial\theta_{\mu}}$
$\displaystyle=\operatorname{tr}\left(U_{(\mu-1)\leftarrow
1}\rho_{0}U_{(\mu-1)\leftarrow
1}^{\dagger}\left[i\Omega_{\mu},U_{M\leftarrow\mu}^{\dagger}HU_{M\leftarrow\mu}\right]\right)$
(S14) $\displaystyle=\operatorname{tr}\left(U_{\mu\leftarrow
1}\rho_{0}U_{\mu\leftarrow
1}^{\dagger}\left[i\Omega_{\mu},U_{M\leftarrow(\mu+1)}^{\dagger}HU_{M\leftarrow(\mu+1)}\right]\right),$

where we have used the notation
$U_{\nu^{\prime}\leftarrow\nu}=\prod_{\nu^{\prime\prime}=\nu}^{\nu^{\prime}}U_{\nu^{\prime\prime}}=U_{\nu^{\prime}}\cdots
U_{\nu+1}U_{\nu}$ if $\nu^{\prime}\geq\nu$. Otherwise if $\nu^{\prime}<\nu$,
we just set $U_{\nu^{\prime}\leftarrow\nu}$ as the identity. For clarity, we
will use the term “derivative” to refer to a single component of the gradient
and use “gradient” to refer to the entire multi-component vector throughout
the paper. Using twirling channels corresponding to the $2$-design blocks, the
average and variance of the cost derivative can be written as

$\displaystyle\mathbb{E}_{\mathbb{U}}\left[\partial_{\mu}C\right]$
$\displaystyle=\operatorname{tr}\left[\rho_{0}\mathcal{T}^{(1)}_{s(B_{1})}\circ\mathcal{T}^{(1)}_{s(B_{2})}\circ\cdots\circ\mathcal{D}_{\mu}\circ\cdots\circ\mathcal{T}^{(1)}_{s(B_{M^{\prime}})}(H)\right],$
(S15)
$\displaystyle\operatorname{Var}_{\mathbb{U}}\left[\partial_{\mu}C\right]$
$\displaystyle=\operatorname{tr}\left[\rho_{0}^{\otimes
2}\mathcal{T}^{(2)}_{s(B_{1})}\circ\mathcal{T}^{(2)}_{s(B_{2})}\circ\cdots\circ\mathcal{D}_{\mu}^{\otimes
2}\circ\cdots\circ\mathcal{T}^{(2)}_{s(B_{M^{\prime}})}(H^{\otimes
2})\right]-\left(\mathbb{E}_{\mathbb{U}}\left[\partial_{\mu}C\right]\right)^{2},$
(S16)

where $\mathcal{D}_{\mu}(\cdot)=[i\Omega_{\mu},(\cdot)]$ is the quantum
operation induced by the commutator. For convenience, we say that
$\theta_{\mu}$ is the differential parameter, $U_{\mu}$ is the differential
gate and the block $B_{k(\mu)}$ containing $U_{\mu}$ is the differential
block. It is worth emphasizing that there is a detail on the location of
$U_{\mu}$ within $B_{k(\mu)}$. Denote the decomposition of $B_{k(\mu)}$
regarding the gate $U_{\mu}$ as $B_{k(\mu)}=R_{k\mu}U_{\mu}L_{k\mu}$, i.e.,

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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\hbox{\small{\vbox{\halign{#\hfil\cr\cr\vskip
140.0pt\cr\hbox{{$L_{k\mu}$}}\cr}}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{103.19843pt}{-55.99915pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{104.14842pt}{-205.82912pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\small{\vbox{\halign{#\hfil\cr\cr\vskip
140.0pt\cr\hbox{{$R_{k\mu}$}}\cr}}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(S17)

Then the neighboring channels of $\mathcal{D}_{\mu}^{\otimes t}$ in Eqs. (S15)
and (S16) can be written as

$\cdots\circ\mathcal{T}_{s(L_{k\mu})}^{(t)}\circ\mathcal{D}_{\mu}^{\otimes
t}\circ\mathcal{T}_{s(U_{\mu}R_{k\mu})}^{(t)}\circ\cdots\quad\text{or}\quad\cdots\circ\mathcal{T}_{s(L_{k\mu}U_{\mu})}^{(t)}\circ\mathcal{D}_{\mu}^{\otimes
t}\circ\mathcal{T}_{s(R_{k\mu})}^{(t)}\circ\cdots.$ (S18)

depending on the two different expressions in Eq. (S14). Although we assume
that $B_{k(\mu)}$ forms a $2$-design above, the sub-blocks $L_{k\mu}$ and
$R_{k\mu}$ do not necessarily form $2$-designs, which prevents the use of Eqs.
(S5) and (S6). To deal with this detail, we will first focus on the case where
$L_{k\mu}$ and $R_{k\mu}$ indeed form $2$-designs on $s(B_{k(\mu)})$ so that
our calculations could be performed clearly, and then discuss other cases on
top of those calculations. Remember that $s(B)$ represents the support of the
block $B$, i.e., the qubit subset which is acted non-trivially by $B$. We will
call $|s(B)|$ as the block size of the block $B$.

To sum up, based on the assumptions that $\mathbb{U}$ is made up of local
$2$-designs and the differential gate $U_{\mu}$ is sandwiched by two local
$2$-designs, now we are going to estimate the average and variance of the cost
derivative. We first calculate the average of the cost derivative.

###### Lemma S1

If there exist a block or sub-block $B$ within either $U_{\mu\leftarrow 1}$ or
$U_{M\leftarrow\mu}$ forming a local $1$-design with the support $s(B)$
covering $s(U_{\mu})$, then the average of the cost derivative equals to zero,
i.e., $\mathbb{E}_{\mathbb{U}}\left[\partial_{\mu}C\right]=0$.

###### Proof.

According to Eq. (S5), if the $1$-design block $B$ is within $U_{\mu\leftarrow
1}$, the evolved state in Eq. (S14) becomes a maximally mixed state on
$s(U_{\mu})$ after the integration with respect to $B$ so that the trace of a
commutator will give rise to a zero. If $B$ is within $U_{M\leftarrow\mu}$,
the backward evolved Hamiltonian in Eq. (S14) becomes an identity after the
integration with respect to $B$ so that the commutator with an identity will
also give rise to a zero. ∎

Since every $2$-design is also a $1$-design by definition, the condition in
Lemma S1 can be easily satisfied by our assumptions above. Actually, the
vanishing average of the cost derivative can also be obtained by considering
the periodicity of the parameter space. For completeness, we point out that
$1$-designs have the following property.

###### Proposition S2

If each block $B_{k}$ in the circuit $\mathbf{U}$ forms a local $1$-design,
then $\mathbf{U}$ forms a global $1$-design.

###### Proof.

Be definition, the twirling channel of $\mathbf{U}$ with respect to the
induced ensemble $\mathbb{U}$ is composed of twirling channels of each block
$B_{k}$, e.g.,

$\mathcal{T}^{(1)}_{s(\mathbf{U})}=\mathcal{T}^{(1)}_{s(B_{M^{\prime}})}\circ\cdots\circ\mathcal{T}^{(1)}_{s(B_{2})}\circ\mathcal{T}^{(1)}_{s(B_{1})},$
(S19)

where $s(\mathbf{U})=\bigcup_{k}s(B_{k})$. This means that the support of
$\mathbf{U}$ is the union of the supports of all blocks. If $\mathbf{U}$ acts
on all qubits, then $s(\mathbf{U})=N$. We emphasize that the measure in the
definition of $\mathcal{T}^{(1)}_{s(\mathbf{U})}$ is determined by
$\mathbb{U}$ instead of the Haar measure. According to Eq. (S5), we have

$\mathcal{T}^{(1)}_{s(\mathbf{U})}(\cdot)=\operatorname{tr}_{s(B_{M^{\prime}})}\left[\cdots\operatorname{tr}_{s(B_{1})}\left[\cdot\right]\otimes\frac{I|_{s(B_{1})}}{2^{|s(B_{1})|}}\cdots\right]\otimes\frac{I|_{s(B_{M^{\prime}})}}{2^{|s(B_{M^{\prime}})|}}=\operatorname{tr}_{s(\mathbf{U})}\left[\cdot\right]\otimes\frac{I|_{s(\mathbf{U})}}{2^{|s(\mathbf{U})|}}.$
(S20)

This is exactly the definition of global $1$-design. Here “global” means the
ensemble is defined on the support union $s(\mathbf{U})$. $I|_{s}$ is the
identity on support $s$. The last equality means that the composition of a
series of tracing and replacing with maximally mixed states is equivalent to
tracing all the indices first and then replacing with the maximally mixed
state on the support union. ∎

We remark that Proposition S2 holds especially for $1$-designs. That is to
say, for example, the similar statement is not true for $2$-designs, i.e., the
composition of local $2$-designs is not necessarily a global $2$-design. This
is because there are multiple terms in Eq. (S6) which hinders a similar proof
as in Eq. (S20) for $2$-designs. As the integration progresses, the terms
become exponentially numerous with both positive and negative contributions,
making them difficult to estimate. Hence, the estimation regarding the
composition of local $2$-designs, such as the variance of the cost derivative,
is much more complicated than those of $1$-designs. This implies that we need
an alternative method instead of brute-force computation. As we will show
below, our method estimates the variance of the cost derivative in the
Heisenberg picture, i.e., we track the backward evolution of Pauli strings
instead of the forward evolution of the quantum state, which shares the same
start point with that in Ref. [28]. In the following lemma, we will first see
how one Pauli string $h$ evolves by a single twirling channel. We use $h|_{s}$
to denote the sub-string of $h$ on the qubit subset $s$, e.g., $(Z\otimes
I\otimes X)|_{\\{q_{1},q_{2}\\}}=Z\otimes I$. The support of $h$, i.e., the
qubit subset which is acted by $h$ non-trivially, is denoted by $s(h)$, e.g.,
$s(Z\otimes I\otimes X)=\\{q_{1},q_{3}\\}$. Here “non-trivial” means that the
corresponding sub-string is not the identity. We use $\mathcal{P}|_{s}$ to
denote the set of all Pauli strings defined on $s$, and
$\mathcal{P}^{\prime}|_{s}$ to denote the set of all non-trivial Pauli strings
on $s$, i.e., $\mathcal{P}^{\prime}|_{s}=\mathcal{P}|_{s}-\\{I|_{s}\\}$.

###### Lemma S3

Suppose that $h$ is a Pauli string on an $N$-qubit system and
$\mathcal{T}^{(2)}_{s}$ is the $2$-degree twirling channel defined on $s$. If
the sub-string $h|_{s}$ is non-trivial, then the following equality holds

$\mathcal{T}^{(2)}_{s}(h^{\otimes
2})=\frac{1}{2^{|2s|}-1}\sum_{\sigma\in\mathcal{P}^{\prime}|_{s}}\sigma^{\otimes
2}\otimes(h|_{\bar{s}})^{\otimes 2},$ (S21)

where $\bar{s}$ is the complement of $s$ among the $N$ qubits. Otherwise if
$h|_{s}$ is trivial, then $\mathcal{T}^{(2)}_{s}(h\otimes h)=h\otimes h$.

###### Proof.

If $h|_{s}$ is non-trivial, then we have $\operatorname{tr}_{2s}(h\otimes
h)=0$ and $\operatorname{tr}_{2s}[(h\otimes
h)S|_{2s}]=2^{|s|}(h|_{\bar{s}})^{\otimes 2}$ in Eq. (S6) because Pauli
matrices are traceless and the squares of them equal to the identity. Thus,
Eq. (S6) gives rise to

$\mathcal{T}^{(2)}_{s}(h^{\otimes 2})=\frac{2^{|s|}(h|_{\bar{s}})^{\otimes
2}}{2^{|2s|}-1}\otimes\left(S|_{2s}-\frac{I|_{2s}}{2^{|s|}}\right).$ (S22)

Substituting the Pauli decomposition of the swap operator

$S|_{2s}=\frac{1}{2^{|s|}}\sum_{\sigma\in\mathcal{P}|_{s}}\sigma^{\otimes 2},$
(S23)

we arrive at Eq. (S21). If $h|_{s}$ is trivial, Eq. (S6) indicates that
$\mathcal{T}^{(2)}_{s}$ just keeps the input unchanged. ∎

This lemma tells us that, a single step of backward evolution of two copies of
a Pauli string by the twirling channel on the qubit subset $s$ will map the
sub-string inside $s$ to a uniformly weighted sum of two copies of all
possible non-trivial Pauli strings on $s$, while keep the part outside $s$
unchanged. Note that the coefficients are summed to one. We dub this
transformation as the twirling channel “smearing” the sub-strings on $s$. From
this point of view, one can regard a local $2$-design as a “local scrambler”,
which scrambles the information at $s$ uniformly. Besides identical copies of
the form $h^{\otimes 2}$, we also need to pay attention to the evolution of
cross-terms like $h_{1}\otimes h_{2}$ with $h_{1}\neq h_{2}$ from $H^{\otimes
2}$ in Eq. (S16). That is to say, if the Pauli decomposition of the
Hamiltonian is $H=\sum_{j}\lambda_{j}h_{j}$ where $\lambda_{j}\in\mathbb{R}$
and $h_{j}\in\mathcal{P}^{\prime}|_{s(H)}$ represents a non-trivial Pauli
string on the support of $H$, the copied Hamiltonian $H^{\otimes 2}$ can be
expressed by $H^{\otimes 2}=\sum_{ij}\lambda_{i}\lambda_{j}h_{i}\otimes
h_{j}$, which contains the cross-terms $h_{i}\otimes h_{j}$ with $h_{i}\neq
h_{j}$.

###### Lemma S4

Suppose that $h_{1}$ and $h_{2}$ are two distinct Pauli strings and
$s^{\prime}$ is the qubit subset where $h_{1}$ and $h_{2}$ are distinct, i.e.,
$s^{\prime}=\\{q_{i}\mid h_{1}|_{q_{i}}\neq h_{2}|_{q_{i}}\\}$. If
$s^{\prime}\cap s\neq\varnothing$, then $\mathcal{T}^{(2)}_{s}(h_{1}\otimes
h_{2})=0$. Otherwise if $s^{\prime}\cap s=\varnothing$, i.e., $h_{1}$ and
$h_{2}$ are identical on $s$, then Lemma S3 applies.

###### Proof.

Since $h_{1}|_{s}$ and $h_{2}|_{s}$ are distinct, we have
$\operatorname{tr}_{2s}(h_{1}\otimes h_{2})=0$ and
$\operatorname{tr}_{2s}[(h_{1}\otimes h_{2})S|_{2s}]=0$ in Eq. (S6) because
Pauli matrices are traceless and the product of two distinct Pauli matrices is
also traceless. According to Eq. (S6), we have
$\mathcal{T}^{(2)}_{s}(h_{1}\otimes h_{2})=0$. ∎

That is to say, if two Pauli strings are not identical in the scope of the
$2$-degree twirling channel, the output corresponding to the tensor product of
these two Pauli strings is zero. Therefore, for the whole circuit $\mathbf{U}$
composed of local $2$-designs, we have the following corollary.

###### Corollary S5

Suppose that $H$ is a Hamiltonian with the Pauli decomposition
$H=\sum_{j}\lambda_{j}h_{j}$ and $\mathbf{U}$ is a random PQC composed of
local $2$-designs. The differential gate $U_{\mu}$ within the block
$B_{k(\mu)}$ is sandwiched by two local $2$-designs on $s(B_{k(\mu)})$. If the
support of $\mathbf{U}$ covers the support of $H$, i.e.,
$s(\mathbf{U})\supseteq s(H)$, then the following equality holds

$\operatorname{Var}_{\mathbb{U}}[\partial_{\mu}\langle
H\rangle]=\sum_{j}\lambda_{j}^{2}\operatorname{Var}_{\mathbb{U}}[\partial_{\mu}\langle
h_{j}\rangle],$ (S24)

where
$\langle\cdot\rangle=\langle\bm{0}|\mathbf{U}^{\dagger}(\cdot)\mathbf{U}|\bm{0}\rangle$
denotes the expectation value with respect to the output state
$\mathbf{U}|\bm{0}\rangle$.

###### Proof.

Lemma S1 ensures $\mathbb{E}_{\mathbb{U}}\left[\partial_{\mu}C\right]=0$.
Thus, Eq. (S16) reduces to

$\operatorname{Var}_{\mathbb{U}}\left[\partial_{\mu}C\right]=\operatorname{tr}\left[\rho_{0}^{\otimes
2}\mathcal{T}^{(2)}_{s(B_{1})}\circ\mathcal{T}^{(2)}_{s(B_{2})}\circ\cdots\circ\mathcal{D}_{\mu}^{\otimes
2}\circ\cdots\circ\mathcal{T}^{(2)}_{s(B_{M^{\prime}})}(H^{\otimes
2})\right].$ (S25)

Since the maps above are all linear, the Pauli decomposition of $H$ gives

$\operatorname{Var}_{\mathbb{U}}\left[\partial_{\mu}C\right]=\sum_{ij}\lambda_{i}\lambda_{j}\operatorname{tr}\left[\rho_{0}^{\otimes
2}\mathcal{T}^{(2)}_{s(B_{1})}\circ\mathcal{T}^{(2)}_{s(B_{2})}\circ\cdots\circ\mathcal{D}_{\mu}^{\otimes
2}\circ\cdots\circ\mathcal{T}^{(2)}_{s(B_{M^{\prime}})}(h_{i}\otimes
h_{j})\right].$ (S26)

If two Pauli strings $h_{i}$ and $h_{j}$ are distinct on $s(\mathbf{U})$,
there must exist a block $B_{k^{\prime}}$ that first acts on one of the qubits
on which the two Pauli strings are distinct, resulting in
$\mathcal{T}^{(2)}_{s(B_{k^{\prime}})}(h_{i}\otimes h_{j})=0$ by Lemma S4.
Lemma S3 ensures that the distinction is preserved by preceding blocks acting
elsewhere. Therefore, all the cross-terms are eliminated and hence we arrive
at Eq. (S24). Note that the differential channel $\mathcal{D}_{\mu}^{\otimes
2}$ does affect the elimination since $U_{\mu}$ is sandwiched by two local
$2$-designs so that the distinction will be acted by the right one before
meeting $\mathcal{D}_{\mu}^{\otimes 2}$. ∎

In fact, even if the sub-block on the right of the differential gate $U_{\mu}$
does not form a local $2$-design, the cross-terms are still expected to be
very small. This is because $\mathcal{D}_{\mu}$ can not map two distinct non-
trivial Pauli strings to the same one, e.g., $[X,Y]=2iZ$ and $[X,Z]=-2iY$, so
the distinction is preserved by $\mathcal{D}_{\mu}^{\otimes 2}$. The left and
right sub-blocks combine together reforming a $2$-design.

Corollary S5 tells us that in order to give a lower bound on the variance of
the cost derivative, one only needs to bound the variance corresponding to
each Pauli string in the Hamiltonian, because the coefficients in Eq. (S24)
are all non-negative, i.e., $\lambda_{j}^{2}\geq 0$. To see the effect of the
differential channel more clearly, we introduce the following lemma.

###### Lemma S6

Suppose that $h$ and $\Omega$ are two Pauli strings and
$\mathcal{D}(\cdot)=[i\Omega,(\cdot)]$. $s$ is a qubit subset that covers
$s(\Omega)$, i.e., $s(\Omega)\subseteq s$. If $h|_{s}$ is non-trivial, then
the following equality holds

$\mathcal{T}^{(2)}_{s}\circ\mathcal{D}^{\otimes
2}\circ\mathcal{T}^{(2)}_{s}(h^{\otimes
2})=\frac{2^{|2s|+1}}{2^{|2s|}-1}\mathcal{T}^{(2)}_{s}(h^{\otimes 2}).$ (S27)

Otherwise if $h|_{s}$ is trivial,
$\mathcal{T}^{(2)}_{s}\circ\mathcal{D}^{\otimes
2}\circ\mathcal{T}^{(2)}_{s}(h^{\otimes 2})=0$.

###### Proof.

If $h|_{s}=I|_{s}$, Lemma S3 gives $\mathcal{T}^{(2)}_{s}(h^{\otimes
2})=h^{\otimes 2}$ and hence $\mathcal{D}^{\otimes
2}\circ\mathcal{T}^{(2)}_{s}(h^{\otimes 2})=\mathcal{D}^{\otimes 2}(h^{\otimes
2})=0$ because the identity commutes with any operator. If $h|_{s}\neq
I|_{s}$, Lemma S3 gives

$\mathcal{T}^{(2)}_{s}(h^{\otimes
2})=\frac{1}{2^{|2s|}-1}\sum_{\sigma\in\mathcal{P}^{\prime}|_{s}}\sigma^{\otimes
2}\otimes(h|_{\bar{s}})^{\otimes 2}.$ (S28)

Since $\Omega$ is also a Pauli string, hence either $\Omega$ commutes with
$\sigma$ or anti-commutes with $\sigma$. If they commute,
$\mathcal{D}(\sigma)=0$. If they anti-commute, the commutation relation of
Pauli matrices gives $\mathcal{D}(\sigma)=\pm 2\sigma^{\prime}$, where
$\sigma^{\prime}$ is some other non-trivial Pauli string. The number of Pauli
strings in $\mathcal{P}|_{s}$ that anti-commute with $\Omega$ is just one-half
of the total, i.e., $4^{|s|}/2$. This is because one can easily establish a
one-to-one correspondence between the subsets of Pauli strings that commute
and anti-commute with $\Omega$ by multiplying a new Pauli string
$\Omega^{\prime}$ that is only non-trivial on one of the qubits in $s(\Omega)$
and does not equal to $\Omega$. For example, if $\Omega=Z\otimes I$, one can
choose $\Omega^{\prime}=X\otimes I$ and map the commuting subset
$\\{I,Z\\}\otimes\mathcal{P}|_{q_{2}}$ to the anti-commuting subset
$\\{X,Y\\}\otimes\mathcal{P}|_{q_{2}}$ up to some phase factors. As a result,
the final output of the three composed channels is

$\mathcal{T}^{(2)}_{s}\circ\mathcal{D}^{\otimes
2}\circ\mathcal{T}^{(2)}_{s}(h^{\otimes 2})=\frac{(\pm 2)^{2}\times
4^{|s|}/2}{(2^{|2s|}-1)^{2}}\sum_{\sigma\in\mathcal{P}^{\prime}|_{s}}\sigma^{\otimes
2}\otimes(h|_{\bar{s}})^{\otimes 2},$ (S29)

which is exactly Eq. (S27) after simplification. ∎

Lemma S6 reveals that the effect of the differential channel is just to
eliminate the Pauli strings that are trivial on the support of the
differential gate $U_{\mu}$. For those Pauli strings that are non-trivial on
$s(U_{\mu})$, the differential channel just contributes a constant on top of
the two adjacent local $2$-designs. Up to this point, we have introduced all
the lemmas we need to prove the theorems in the main text. Before presenting
the theorems and proofs, we would like to first clarify some concepts that are
used in the formulation of the theorems.

###### Definition S1

The connecting support of two blocks $B_{k}$ and $B_{k^{\prime}}$ with $k\leq
k^{\prime}$ in the circuit $\mathbf{U}$ is defined by
$s_{c}(B_{k},B_{k^{\prime}})=s(B_{k})\cap
s(B_{k^{\prime}})-\bigcup_{k<k^{\prime\prime}<k^{\prime}}s(B_{k^{\prime\prime}})$.
If $k\geq k^{\prime}$, the connecting support is defined by
$s_{c}(B_{k},B_{k^{\prime}})=s_{c}(B_{k^{\prime}},B_{k})$. Non-empty
connecting support means that there is at least a common edge between the two
blocks in the circuit graph. In this case, we call the two blocks are
“connected” within the circuit.

###### Definition S2

The forward residual support $s_{f}(B_{k})$ of a block $B_{k}$ in the circuit
$\mathbf{U}$ is defined by the qubits in $s(B_{k})$ that are not acted by the
blocks in $\mathbf{U}$ that act earlier than $B_{k}$, i.e.,
$s_{f}(B_{k})=s(B_{k})-\bigcup_{k^{\prime}<k}s(B_{k^{\prime}})$.
Correspondingly, the backward residual support $s_{b}(B_{k})$ of $B_{k}$ is
defined by the qubits in $s(B_{k})$ that are not acted by the blocks in
$\mathbf{U}$ that act later than $B_{k}$, i.e.,
$s_{b}(B_{k})=s(B_{k})-\bigcup_{k^{\prime}>k}s(B_{k^{\prime}})$. If the
forward (backward) residual support is not empty, we say the block is a head
(tail) block in the circuit, i.e., the block is the first (last) block in the
entire circuit that acts on some qubit.

###### Definition S3

A path $p$ on the circuit $\mathbf{U}$ is defined as a time-ordered sequence
of blocks $p=\\{B_{k_{1}},B_{k_{2}},B_{k_{3}},\cdots\\}$ with
$k_{1}<k_{2}<k_{3}<\cdots$ where adjacent blocks are connected within the
circuit. The head (tail) of a path $p$ refers to the first (last) block in the
path, denoted as $\mathrm{Head}(p)$ and $\mathrm{Tail}(p)$. The forward
(backward) residual support of a path $p$ is defined by the union of the
forward (backward) residual supports of all the blocks in the path $p$, i.e.,
$s_{f}(p)=\bigcup_{B_{k}\in p}s_{f}(B_{k})$ and $s_{b}(p)=\bigcup_{B_{k}\in
p}s_{b}(B_{k})$. Every path is demanded to traverse the entire circuit, i.e.,
$s_{f}(\mathrm{Head}(p))\neq\varnothing$ and
$s_{b}(\mathrm{Tail}(p))\neq\varnothing$. In other words, the head (tail) of
the path is also a head (tail) block in the circuit.

The motivation for this path definition will be self-evident in the proof
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(S30)

Some measures can be defined to quantify the properties of paths, such as the
“length” below.

###### Definition S4

An edge in a given path $p$ refers to a pair of adjacent blocks
$(B_{k},B_{k^{\prime}})$ in the path with $k<k^{\prime}$. The “length” of the
edge is defined by

$l(B_{k},B_{k^{\prime}})=\log_{4}\left[\frac{4^{|s(B_{k^{\prime}})|}-1}{4^{|s_{c}(B_{k},B_{k^{\prime}})|}-1}\right].$
(S31)

Specifically, we regard $(\rho_{0},B_{k_{1}})$ also as an edge in the path.
Remember that $\rho_{0}$ is the initial state. The length of this edge is
defined by replacing the connecting support with the forward residual support,
i.e.,

$l(\rho_{0},B_{k_{1}})=\log_{4}\left[\frac{4^{|s(B_{k_{1}})|}-1}{4^{|s_{f}(B_{k_{1}})|}-1}\right],$
(S32)

In fact, the forward residual support of $B_{k_{1}}$ can be naturally seen as
the connecting support of $\rho_{0}$ and $B_{k_{1}}$, namely
$s_{f}(B_{k_{1}})=s_{c}(\rho_{0},B_{k_{1}})$, if $\rho_{0}$ is identified as
the “zeroth” block $B_{0}$. The edge set corresponding to the path $p$ is
defined by all edges in the path
$\mathrm{Edge}(p)=\\{(\rho_{0},B_{k_{1}}),(B_{k_{1}},B_{k_{2}}),(B_{k_{2}},B_{k_{3}}),\cdots\\}$.
The “length” of the path is defined by the sum of the lengths of all edges,
i.e.,

$l(p)=\sum_{(B_{k},B_{k^{\prime}})\in\mathrm{Edge}(p)}l(B_{k},B_{k^{\prime}}).$
(S33)

One can see that if the edge length is a constant for every edge in the path,
like in the example above, then the length of the path naturally reduces to
the geometrical length $|p|$ (the number of elements in $p$) times the
constant. In the limit of large $|s(B_{k^{\prime}})|$ and
$|s_{c}(B_{k},B_{k^{\prime}})|$, the edge length can be approximated as
$l(B_{k},B_{k^{\prime}})\approx|s(B_{k^{\prime}})-s_{c}(B_{k},B_{k^{\prime}})|$,
which only depends on the part of $B_{k^{\prime}}$ that is not connected with
$B_{k}$. Actually, even in the minimal case of staggered two-qubit gates with
$|s(B_{k^{\prime}})|=2$ and $|s_{c}(B_{k},B_{k^{\prime}})|=1$, this
approximation is good enough, i.e., $l(B_{k},B_{k^{\prime}})=\log_{4}5\approx
1.16$ which is close to $|s(B_{k^{\prime}})-s_{c}(B_{k},B_{k^{\prime}})|=1$.
As we will see, the edge length defined here signifies the information leakage
by the staggered adjacent blocks in the process of backward evolution. Note
that $l(B_{k},B_{k^{\prime}})=0$ just means that the two adjacent blocks are
connected on the whole support and hence they can be combined into a single
$2$-design.

We will use $s(p)$ to denote the qubit support union of all blocks in $p$,
i.e., $s(p)=\bigcup_{B_{k}\in p}s(B_{k})$. We will use the notation
$p(B_{k(\mu)},h_{j})$ to indicate that the path $p$ goes through the
differential block $B_{k(\mu)}$ and connects to $h_{j}$, i.e., $B_{k(\mu)}\in
p$ and $s_{b}(p)\cap s(h_{j})\neq\varnothing$. Note that the existence of
$p(B_{k(\mu)},h_{j})$ is equivalent to the condition that $B_{k(\mu)}$ is
within the causal cone of $h_{j}$, which means that $B_{k(\mu)}$ would not be
eliminated directly by unitary conjugation when calculating the expectation
value of $h_{j}$. Moreover, $p(B_{k(\mu)},h_{j})$ may also be non-unique,
i.e., there might exist multiple possible paths through $B_{k(\mu)}$ to
$h_{j}$ on the PQC. If so, we just choose one of the paths and denote it as
$p(B_{k(\mu)},h_{j})$.

The concepts defined above will be useful when concerning the factor in the
lower bound on the variance that is related to the deepness of the circuit. On
the other hand, when we are also concerned about the factor in the lower bound
that is related to the locality of the Hamiltonian, the following concepts
will be helpful.

###### Definition S5

A path set $P$ on a given PQC is defined by a collection of paths
$P=\\{p_{1},p_{2},\cdots\\}$ on the PQC. The node set of the path set $P$ is
defined by the union of the paths in $P$, i.e.,
$\mathrm{Node}(P)=\bigcup_{p_{i}\in P}p_{i}$. The edge set of the path set $P$
is defined by the union of the edge sets corresponding to the paths in $P$,
i.e., $\mathrm{Edge}(P)=\bigcup_{p_{i}\in P}\mathrm{Edge}(p_{i})$. The length
of the path set $P$ is defined by the sum of all edges in $P$, i.e.,

$l(P)=\sum_{(B_{k},B_{k^{\prime}})\in\mathrm{Edge}(P)}l(B_{k},B_{k^{\prime}}).$
(S34)

###### Definition S6

The head (tail) of a path in a path set $P$ is also called a head (tail) block
of the path set. The head (tail) set of the path set refers to the set of all
the head (tail) blocks in the path set, i.e.,
$\mathrm{Head}(P)=\\{\mathrm{Head}(p_{i})\mid p_{i}\in P\\}$ and
$\mathrm{Tail}(P)=\\{\mathrm{Tail}(p_{i})\mid p_{i}\in P\\}$. The forward
(backward) residual support of a path set $P$ is defined by the union of the
forward (backward) residual supports of all the paths in $P$, i.e.,
$s_{f}(P)=\bigcup_{p_{i}\in P}s_{f}(p_{i})$ and $s_{b}(P)=\bigcup_{p_{i}\in
P}s_{b}(p_{i})$.

###### Definition S7

The forward width of a block $B_{k}$ is defined by

$w(B_{k})=\log_{2}\left[\frac{4^{|s_{f}(B_{k})|}-1}{2^{|s_{f}(B_{k})|}-1}\right].$
(S35)

The head width of a path $p$ is defined by the forward width of its head
$w(p)=w(\mathrm{Head}(p))$. The head width of a path set $P$ is defined by the
sum of the forward widths of all the head blocks in the path set

$w(P)=\sum_{B_{k}\in\mathrm{Head}(P)}w(B_{k}).$ (S36)

Similar to the case of length, one can see that if the head width is a
constant for every head block in the path set, then the head width is just the
geometrical width (the number of elements in the head set) times the constant.
In the limit of large $|s_{f}(B_{k})|$, the head width of the block $B_{k}$
can be approximated as $w(B_{k})\approx|s_{f}(B_{k})|$, which is just the size
of the forward residual support, i.e., the part of $s(B_{k})$ that is
connected to $\rho_{0}$.

We will use the notation $P(B_{k(\mu)},h_{j})$ to indicate that there exists a
path $p_{i}$ in $P$ that goes through the block $B_{k(\mu)}$ and the backward
residual support of $P$ covers the support of $h_{j}$, i.e.,
$\exists~{}p_{i}\in P,~{}\text{s.t.}~{}B_{k(\mu)}\in p_{i}$ and
$s_{b}(P)\supseteq s(h_{j})$. We will call $P(B_{k(\mu)},h_{j})$ a path set
that goes through $B_{k(\mu)}$ and covers $h_{j}$ for short. An example of the
path set $P(B_{k(\mu)},h_{j})$ is depicted as the following

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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{122.94814pt}{-380.37334pt}\pgfsys@invoke{
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}\hbox{\small{\definecolor{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\cr\vskip
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad,\text{ where
}P=\\{p_{1},p_{2}\\}.$ (S37)

We emphasize that the examples shown above which use the alternating layered
ansatz, are merely to illustrate these newly defined concepts. The
applicability of the concepts together with the theorems below, are not
limited to the scenarios in the examples. So far, we have introduced all the
prerequisites related to Theorem 1 in the main text. Now we are prepared to
prove Theorem 1.

###### Theorem 1

Suppose that $H$ is a Hamiltonian on an $N$-qubit system with the Pauli
decomposition $H=\sum_{j}\lambda_{j}h_{j}$. $\mathbf{U}$ is a PQC composed of
blocks forming independent local $2$-designs with $s(\mathbf{U})\supseteq
s(H)$. The energy expectation with respect to the output state is taken as the
cost function
$C(\bm{\theta})=\operatorname{tr}(\rho_{0}\mathbf{U}^{\dagger}H\mathbf{U})$,
where $\rho_{0}=|\bm{0}\rangle\\!\langle\bm{0}|$. Suppose that the
differential gate $U_{\mu}$ within the block $B_{k(\mu)}$ is sandwiched by two
local $2$-designs on $s(B_{k(\mu)})$. The generator $\Omega_{\mu}$ of
$U_{\mu}$ is a Pauli string. For each $h_{j}$ whose causal cone contains
$B_{k(\mu)}$, there exists at least a path set that goes through $B_{k(\mu)}$
and covers $h_{j}$, denoted as $P_{j}$. Then, the variance of the cost
derivative is lower bounded by

$\operatorname{Var}_{\mathbb{U}}[\partial_{\mu}C]\geq\sum_{j}\lambda_{j}^{2}\cdot
2^{1-2l(P_{j})-w(P_{j})},$ (S38)

where $j$ runs over the indices of $h_{j}$ whose causal cone contains
$B_{k(\mu)}$. $l(P_{j})$ and $w(P_{j})$ denote the length and the head width
of the path set $P_{j}$, respectively. Note that this lower bound holds for
arbitrary choice of $P_{j}$. The choice with the minimum path length and head
width gives rise to the tightest bound.

###### Proof.

According to Lemma S1, the average of the cost derivative equals to zero
$\mathbb{E}_{\mathbb{U}}[\partial_{\mu}C]=0$. Thus we only need to estimate
the first term in Eq. (S16), which is exactly Eq. (S25). According to
Corollary S5, we only need to estimate the variances with respect to
individual Pauli strings in the Hamiltonian

$\operatorname{Var}_{\mathbb{U}}\left[\partial_{\mu}\langle
h_{j}\rangle\right]=\operatorname{tr}\left[\rho_{0}^{\otimes
2}\mathcal{T}^{(2)}_{s(B_{1})}\circ\mathcal{T}^{(2)}_{s(B_{2})}\circ\cdots\circ\mathcal{D}_{\mu}^{\otimes
2}\circ\cdots\circ\mathcal{T}^{(2)}_{s(B_{M^{\prime}})}(h_{j}^{\otimes
2})\right].$ (S39)

and finally add them up via Eq. (S24). An important observation from Lemma S3
is that a twirling channel just maps a doubled Pauli string into a sum of
doubled Pauli strings, with the coefficients summed to one. Namely, the weight
vector is normalized. Then, the subsequent twirling channels just map one sum
of doubled Pauli strings to another sum of doubled Pauli strings, including
the differential block. In the end, among the final sum of doubled Pauli
strings, only those that contain only $Z$ and $I$ operators contribute to the
expectation value with respect to the initial state $|\bm{0}\rangle$ because
$\langle 0|X|0\rangle=\langle 0|Y|0\rangle=0$ while $\langle
0|Z|0\rangle=\langle 0|I|0\rangle=1$. So the variance equals the sum of the
coefficients corresponding to all surviving doubled Pauli strings containing
only $Z$ and $I$ in the output of the final channel. Our goal is to give a
lower bound on the sum of these coefficients. We will omit the term “doubled”
below without confusion.

For clarity, before introducing our method of bounding, we first analyze the
action of the first two channels $\mathcal{T}^{(2)}_{s(B_{M^{\prime}})}$ and
$\mathcal{T}^{(2)}_{s(B_{M^{\prime}-1})}$ in detail, which will be helpful for
introducing our method of bounding later. We start from the action of
$\mathcal{T}^{(2)}_{s(B_{M^{\prime}})}$. Lemma S3 tells us that

  1. (i)

If $s(B_{M^{\prime}})\cap s(h_{j})=\varnothing$, i.e., the block
$B_{M^{\prime}}$ does not act on the support of the Pauli string $h_{j}$ at
all, then the output of the corresponding twirling channel keeps the same as
the input $\mathcal{T}^{(2)}_{s(B_{M^{\prime}})}(h_{j}^{\otimes
2})=h_{j}^{\otimes 2}$.

  2. (ii)

If $s(B_{M^{\prime}})\cap s(h_{j})\neq\varnothing$, then
$\mathcal{T}^{(2)}_{s(B_{M^{\prime}})}$ keeps the sub-string
$(h|_{\bar{s}(B_{M^{\prime}})})^{\otimes 2}$ unchanged, but replaces the sub-
string $(h|_{s(B_{M^{\prime}})})^{\otimes 2}$ with a uniformly weighted sum of
two copies of all non-trivial Pauli strings on $s(B_{M^{\prime}})$, i.e., the
elements in the set $\mathcal{P}^{\prime}|_{s(B_{M^{\prime}})}$. This kind of
replacement is dubbed as “smearing” below. One can see that the qubit support
union of the summed Pauli strings either grows or keeps the same after the
action of $\mathcal{T}^{(2)}_{s(B_{M^{\prime}})}$.

Next we consider the action of $\mathcal{T}^{(2)}_{s(B_{M^{\prime}-1})}$ under
the condition $s(B_{M^{\prime}})\cap s(h_{j})\neq\varnothing$.

  1. (i)

If $s(B_{M^{\prime}-1})\cap s(h_{j})=\varnothing$ and $s(B_{M^{\prime}-1})\cap
s(B_{M^{\prime}})=\varnothing$, the output is the same as the input.

  2. (ii)

If $s(B_{M^{\prime}-1})\supseteq s(B_{M^{\prime}})$, the sub-strings on
$s(B_{M^{\prime}-1})$ are smeared, while the remaining part keeps unchanged.
In this case, the effect of the block is $B_{M^{\prime}}$ is invisible since
it is fully covered by a $2$-design $B_{M^{\prime}}$ with larger support.

  3. (iii)

If $s(B_{M^{\prime}-1})\cap[s(h_{j})-s(B_{M^{\prime}})]=\varnothing$ and
$s(B_{M^{\prime}-1})\cap s(B_{M^{\prime}})\neq\varnothing$ and
$s(B_{M^{\prime}-1})\not\supseteq s(B_{M^{\prime}})$, the output might be not
a uniformly weighted sum anymore. This is because
$\mathcal{T}^{(2)}_{s(B_{M^{\prime}-1})}$ only smears the Pauli strings that
are non-trivial on $s(B_{M^{\prime}-1})$, while not all summed Pauli strings
in $\mathcal{T}^{(2)}_{s(B_{M^{\prime}})}(h_{j}^{\otimes 2})$ satisfy this
condition due to $s(B_{M^{\prime}-1})\not\supseteq s(B_{M^{\prime}})$. For
example, in the set of $2$-qubit non-trivial Pauli strings, the subset
$\\{X\otimes I,Y\otimes I,Z\otimes I\\}$ would not be smeared by the twirling
channel only covering the second qubit. The part of Pauli strings that are
trivial on $s(B_{M^{\prime}-1})$ keep their weights unchanged as
$1/(4^{|s(B_{M^{\prime}})|}-1)$, while the remaining part of strings that are
non-trivial on $s(B_{M^{\prime}-1})$ will be involved in the twirling channel.
The latter forms the subset
$\mathcal{P}|_{s(B_{M^{\prime}})-s(B_{M^{\prime}-1})}\otimes\mathcal{P}^{\prime}|_{s(B_{M^{\prime}})\cap
s(B_{M^{\prime}-1})}$, in which the number of elements is

$\left(4^{|s(B_{M^{\prime}})\cap s(B_{M^{\prime}-1})|}-1\right)\times
4^{|s(B_{M^{\prime}})-s(B_{M^{\prime}-1})|}.$ (S40)

Each string will be smeared into
$\mathcal{P}^{\prime}|_{s(B_{M^{\prime}-1})}$. After collecting the repeated
terms that have the same sub-string on
$s(B_{M^{\prime}})-s(B_{M^{\prime}-1})$, we obtain a summation over the set
$\mathcal{P}|_{s(B_{M^{\prime}})-s(B_{M^{\prime}-1})}\otimes\mathcal{P}^{\prime}|_{s(B_{M^{\prime}-1})}$
with each term having a new uniform weight

$\frac{1}{4^{|s(B_{M^{\prime}})|}-1}\times\frac{1}{4^{|s(B_{M^{\prime}-1})|}-1}\times\left(4^{|s(B_{M^{\prime}-1})\cap
s(B_{M^{\prime}})|}-1\right),$ (S41)

where the last factor comes from collecting repeated terms. Plus those
unchanged strings in
$\mathcal{P}^{\prime}|_{s(B_{M^{\prime}})-s(B_{M^{\prime}-1})}\otimes
I|_{s(B_{M^{\prime}-1})\cap s(B_{M^{\prime}})}$ with the weight
$1/(4^{|s(B_{M^{\prime}})|}-1)$ for each term, we obtain the final output from
$\mathcal{T}^{(2)}_{s(B_{M^{\prime}-1})}$. One can check that these weights
are still summed to one. Therefore, if $s(B_{M^{\prime}-1})\subseteq
s(B_{M^{\prime}})$, then the weights are still uniform since
$s(B_{M^{\prime}-1})\cap s(B_{M^{\prime}})=s(B_{M^{\prime}-1})$. But if
$s(B_{M^{\prime}-1})\not\subseteq s(B_{M^{\prime}})$, namely the two blocks
have an overlap and also have their own exclusive supports, then the weights
are indeed no more uniform.

  4. (iv)

If $s(B_{M^{\prime}-1})\cap[s(h_{j})-s(B_{M^{\prime}})]\neq\varnothing$ and
$s(B_{M^{\prime}-1})\cap s(B_{M^{\prime}})=\varnothing$, the sub-strings are
smeared uniformly with each summed Pauli string non-trivial on
$s(B_{M^{\prime}-1})$ and $s(B_{M^{\prime}})$ simultaneously, i.e., the set
$\mathcal{P}^{\prime}|_{s(B_{M^{\prime}-1})}\otimes\mathcal{P}^{\prime}|_{s(B_{M^{\prime}})}$.

  5. (v)

If $s(B_{M^{\prime}-1})\cap[s(h_{j})-s(B_{M^{\prime}})]\neq\varnothing$ and
$s(B_{M^{\prime}-1})\cap s(B_{M^{\prime}})\neq\varnothing$ and
$s(B_{M^{\prime}-1})\not\supseteq s(B_{M^{\prime}})$, the weights will also
deviate from a uniform distribution. Although all summed strings in
$\mathcal{T}^{(2)}_{s(B_{M^{\prime}})}(h_{j}^{\otimes 2})$ will be involved in
$\mathcal{T}^{(2)}_{s(B_{M^{\prime}-1})}$ due to
$s(B_{M^{\prime}-1})\cap[s(h_{j})-s(B_{M^{\prime}})]\neq\varnothing$, the non-
uniformity arises when collecting repeated terms. In the subset
$\mathcal{P}^{\prime}|_{s(B_{M^{\prime}})-s(B_{M^{\prime}-1})}\otimes\mathcal{P}|_{s(B_{M^{\prime}})\cap
s(B_{M^{\prime}-1})}$, the strings that are the same on
$s(B_{M^{\prime}})-s(B_{M^{\prime}-1})$ contribute the same, with a total
number $4^{|s(B_{M^{\prime}})\cap s(B_{M^{\prime}-1})|}$, while in the subset
$I|_{s(B_{M^{\prime}})-s(B_{M^{\prime}-1})}\otimes\mathcal{P}^{\prime}|_{s(B_{M^{\prime}})\cap
s(B_{M^{\prime}-1})}$, the strings that are the same on
$s(B_{M^{\prime}})-s(B_{M^{\prime}-1})$ contribute the same, with a total
number $\left(4^{|s(B_{M^{\prime}})\cap s(B_{M^{\prime}-1})|}-1\right)$. Thus
the final output is a summation over
$\mathcal{P}^{\prime}|_{s(B_{M^{\prime}})-s(B_{M^{\prime}-1})}\otimes\mathcal{P}^{\prime}|_{s(B_{M^{\prime}-1})}$
with each term having a new weight

$\frac{1}{4^{|s(B_{M^{\prime}})|}-1}\times\frac{1}{4^{|s(B_{M^{\prime}-1})|}-1}\times
4^{|s(B_{M^{\prime}-1})\cap s(B_{M^{\prime}})|},$ (S42)

plus a summation over
$I|_{s(B_{M^{\prime}})-s(B_{M^{\prime}-1})}\otimes\mathcal{P}^{\prime}|_{s(B_{M^{\prime}-1})}$
with each term having a new weight as in Eq. (S41).

One can expect that the classified cases would proliferate exponentially with
the number of twirling channels, and the corresponding weights are in general
nonuniform. Thus it is hard to track the exact backward evolution of the Pauli
string summation at each step in Eq. (S39). Instead, noticing the fact that
the weights are always non-negative, it is possible to track a part of these
Pauli strings to give a lower bound in the end.

According to Lemma S6, only those $h_{j}$ whose causal cone contains the
differential block $B_{k(\mu)}$ can have a non-zero contribution to the
variance $\operatorname{Var}_{\mathbb{U}}[\partial_{\mu}C]$. So we only need
to care about these Pauli strings in the Hamiltonian. For every such $h_{j}$,
as supposed in the theorem, we have chosen a path set $P_{j}$ that goes
through $B_{k(\mu)}$ and covers $h_{j}$. We first focus on the simplest
scenario where $P_{j}$ only contains a single path $p_{1}$. By definition, we
have $B_{k(\mu)}\in p_{1}$ and $s_{b}(p_{1})\supseteq s(h_{j})$. We first
consider the case where $s_{b}(\mathrm{Tail}(p_{1}))\supseteq s(h_{j})$. In
this case, we aim to track the evolution of the Pauli strings that are non-
trivial only along this path. That is to say, we will manually insert a
selection channel $\mathcal{S}_{s(B_{k})}^{(2)}$ just after the twirling
channel $\mathcal{T}_{s(B_{k})}^{(2)}$ of each block $B_{k}$ in the path

$\cdots\circ\mathcal{T}_{s(B_{k})}^{(2)}\circ\cdots\quad\rightarrow\quad\cdots\circ\mathcal{S}_{s(B_{k})}^{(2)}\circ\mathcal{T}_{s(B_{k})}^{(2)}\circ\cdots.$
(S43)

A selection channel $\mathcal{S}_{s(B_{k})}^{(2)}$ on $s(B_{k})$ is defined as
a linear projection channel with the following action. For two adjacent blocks
$(B_{k^{\prime}},B_{k})$ in the path $p_{1}$, after acting
$\mathcal{T}_{s(B_{k})}^{(2)}$, the Pauli strings whose sub-strings on
$s(B_{k})$ are non-trivial only on $s_{c}(B_{k^{\prime}},B_{k})$ will be
mapped to themselves by the selection channel while other strings are directly
mapped to zero, i.e.,

$\mathcal{S}_{s(B_{k})}^{(2)}(\sigma^{\otimes 2})=\begin{cases}\sigma^{\otimes
2}&~{}\text{if}~{}s(\sigma|_{s(B_{k})})\subseteq
s_{c}(B_{k^{\prime}},B_{k})\\\ 0&~{}\text{else}{}\end{cases}.$ (S44)

where $\sigma$ is a Pauli string on the $N$ qubits. By doing so, the
calculation result after inserting selection channels is always smaller than
the original result since the contribution of the strings that are projected
out is always non-negative. This can be utilized to establish a lower bound.
Moreover, the key benefit of such a selection rule is that the tracked strings
would not be influenced by the blocks outside the path since they are trivial
outside, i.e., they are non-trivial only on

$s_{c}(p_{1};B_{k}).$ (S45)

where we have used the notation
$s_{c}(p_{1};B_{k})=s_{c}(B_{k^{\prime}},B_{k})$ to represent the connecting
support of $B_{k}$ and its predecessor $B_{k^{\prime}}$ within the path
$p_{1}$. This will be convenient below for the sake that we do not need to
create a new symbol to represent the predecessor of $B_{k}$. As a consequence,
the twirling channel corresponding to any block outside the path just reduces
to the identity channel. Hence, the variance can be lower bounded by

$\operatorname{Var}_{\mathbb{U}}\left[\partial_{\mu}\langle
h_{j}\rangle\right]\geq\operatorname{tr}\left[\rho_{0}^{\otimes
2}\prod_{B_{k}\in
p_{1}}\left(\mathcal{S}_{s(B_{k})}^{(2)}\circ\mathcal{T}_{s(B_{k})}^{(2)}\right)(h_{j}^{\otimes
2})\right],$ (S46)

where the product of channels follows the increasing order of indices from
left to right, and the differential channel in between is omitted for
simplicity. At the first step of the backward evolution, $h_{j}^{\otimes 2}$
is transformed into a uniform weighted sum of two copies of Pauli strings that
are non-trivial only on $s(\mathrm{Tail}(p_{1}))$. Then, for any two adjacent
blocks $(B_{k},B_{k^{\prime}})$ in the path $p_{1}$, only a certain proportion
of the summed strings in the output of $\mathcal{T}_{s(B_{k^{\prime}})}^{(2)}$
satisfies our selection rule. This proportion is just the number of elements
in $\mathcal{P}^{\prime}|_{s_{c}(B_{k},B_{k^{\prime}})}$ over that in
$\mathcal{P}^{\prime}|_{s(B_{k^{\prime}})}$, i.e.,

$\frac{4^{|s_{c}(B_{k},B_{k^{\prime}})|}-1}{4^{|s(B_{k^{\prime}})|}-1}.$ (S47)

Note that with our tracked strings as the input, the output of each twirling
channel is still a uniform weighted summation because the contribution from
other parts has been neglected. Therefore, this process can be carried out
recursively. Normalizing the weights of the surviving Pauli strings will give
rise to a factor not larger than $1$, which is just Eq. (S47). The normalized
summation will enter the subsequent channels, while the normalization factor
will contribute to our final lower bound. The normalization factor will be
accumulated along the path until meeting the initial state, which contributes
a total factor of

$\prod_{(B_{k},B_{k^{\prime}})\in
p_{1}}\frac{4^{|s_{c}(B_{k},B_{k^{\prime}})|}-1}{4^{|s(B_{k^{\prime}})|}-1}=4^{-l(p_{1})}.$
(S48)

According to Lemma S6, the differential block just contributes a constant
factor

$\frac{2\cdot 4^{|s(B_{k(\mu)})|}}{4^{|s(B_{k(\mu)})|}-1}.$ (S49)

Finally, the tracked Pauli strings are non-trivial only on
$s_{f}(\mathrm{Head}(p_{1}))$. Among these strings, only those that contain
only the Pauli $Z$ operator and the identity operator $I$ will contribute to
the variance by a base number of $1$, whose proportion is

$\frac{2^{|s_{f}(\mathrm{Head}(p_{1}))|}-1}{4^{|s_{f}(\mathrm{Head}(p_{1}))|}-1}=2^{-w(p_{1})}.$
(S50)

To sum up, the final contribution of these selected Pauli strings to the
variance is just

$\frac{2\cdot 4^{|s(B_{k(\mu)})|}}{4^{|s(B_{k(\mu)})|}-1}\times
4^{-l(p_{1})}\times 2^{-w(p_{1})}>2\times 4^{-l(p_{1})}\times 2^{-w(p_{1})},$
(S51)

where we have relaxed the first factor to $2$ for simplicity.

Next, we consider the case where $s_{b}(\mathrm{Tail}(p_{1}))\not\supseteq
s(h_{j})$ but it still holds that $s_{b}(p_{1})\supseteq s(h_{j})$. In this
case, the tracked strings are no longer only non-trivial along the path as in
Eq. (S45), but also on the residual support of $h_{j}$, i.e.,

$s_{c}(p_{1};B_{k})\cup\left[s(h_{j})-\bigcup_{k^{\prime}\geq
k}s_{b}(B_{k^{\prime}})\right],$ (S52)

because $h_{j}$ is not fully covered by the last block of $p_{1}$. However,
this does not make a difference since we still have $s_{b}(p_{1})\supseteq
s(h_{j})$, which means that sooner or later, the residual support of $h_{j}$
will be covered by subsequent blocks in $p_{1}$ before meeting the blocks
outside the path. Thus, the inequality in Eq. (S46) still holds. The
normalization factor of the tracked strings will remain the same as in Eq.
(S47) and hence the lower bound is the same as in Eq. (S51).

Then, we consider the case where $s(h_{j})$ is not fully covered by
$s_{b}(p_{1})$ so that more paths are needed to cover $s(h_{j})$ jointly. We
focus on the case of two paths first, i.e., $s_{b}(P_{j})\supseteq s(h_{j})$
with $P_{j}=\\{p_{1},p_{2}\\}$. We first consider the following three
irreducible configurations of the two paths. The front and back segment of a
path $p$ regarding a block $B_{k}$ is denoted as
$\mathrm{Seg}_{f}(p,B_{k})=\\{B_{k^{\prime}}\in p\mid k^{\prime}\leq k\\}$ and
$\mathrm{Seg}_{b}(p,B_{k})=\\{B_{k^{\prime}}\in p\mid k^{\prime}\geq k\\}$,
respectively.

  1. (i)

If the two paths are separate from each other, i.e., $p_{1}\cap
p_{2}=\varnothing$, we can generalize the selection channel in Eq. (S44) to

$\mathcal{S}_{s(B_{k})}^{(2)}(\sigma^{\otimes 2})=\begin{cases}\sigma^{\otimes
2}&~{}\text{if}~{}s(\sigma|_{s(B_{k})})\subseteq s_{c}(p_{i};B_{k})\\\
0&~{}\text{else}{}\end{cases},$ (S53)

where $p_{i}$ is the path that contains $B_{k}$, and hence the inequality in
Eq. (S46) can be generalized to

$\operatorname{Var}_{\mathbb{U}}\left[\partial_{\mu}\langle
h_{j}\rangle\right]\geq\operatorname{tr}\left[\rho_{0}^{\otimes
2}\prod_{B_{k}\in\mathrm{Node}(P_{j})}\left(\mathcal{S}_{s(B_{k})}^{(2)}\circ\mathcal{T}_{s(B_{k})}^{(2)}\right)(h_{j}^{\otimes
2})\right].$ (S54)

Based on Eq. (S46), this inequality still holds because the selection
regarding the two paths can proceed separately. That is to say, the selected
strings for the block $B_{k}\in p_{1}$ by Eq. (S44) are non-trivial only on

$s_{c}(p_{1};B_{k})\cup
s_{c}(p_{2};B_{k^{\prime}})\cup\left[s(h_{j})-\bigcup_{k^{\prime\prime}\geq
k}s_{b}(B_{k^{\prime\prime}})\right].$ (S55)

where $B_{k^{\prime}}=\mathrm{Head}(\mathrm{Seg}_{b}(p_{2},B_{k}))$ represents
the head block of the back segment of the path $p_{2}$ regarding the block
$B_{k}$. The support of the selected strings for the block in $p_{2}$ can be
written in a symmetric manner. Thus, the factor in Eq. (S47) is accumulated
along the two paths independently because the tracked Pauli string summation
always keeps a tensor product structure. Therefore, the lower bound can be
obtained just by counting the lengths and head widths of the two paths
separately, i.e., generalizing $l(p_{1})$ and $w(p_{1})$ in Eq. (S51) to
$l(P_{j})$ and $w(P_{j})$.

  2. (ii)

If the two paths merge together along the backward direction at some block
$B_{k}$, i.e., $p_{1}\cap
p_{2}=\mathrm{Seg}_{f}(p_{1},B_{k})=\mathrm{Seg}_{f}(p_{2},B_{k})$ and
$B_{k}\in p_{1}\cap p_{2}$, then at the merging node where
$(B_{k},B_{k^{\prime}})\in\mathrm{Edge}(p_{1})$ and
$(B_{k},B_{k^{\prime\prime}})\in\mathrm{Edge}(p_{2})$ with $k^{\prime}\neq
k^{\prime\prime}$, the input of $\mathcal{T}_{s(B_{k})}^{(2)}$ in Eq. (S46) is
just the tensor product of two normalized uniform weighted summation of
doubled Pauli strings over
$\mathcal{P}^{\prime}|_{s_{c}(B_{k},B_{k^{\prime}})}$ and
$\mathcal{P}^{\prime}|_{s_{c}(B_{k},B_{k^{\prime\prime}})}$. The normalization
factor is

$\frac{4^{|s_{c}(B_{k},B_{k^{\prime}})|}-1}{4^{|s(B_{k^{\prime}})|}-1}\times\frac{4^{|s_{c}(B_{k},B_{k^{\prime\prime}})|}-1}{4^{|s(B_{k^{\prime\prime}})|}-1}=4^{-l(B_{k},B_{k^{\prime}})-l(B_{k},B_{k^{\prime\prime}})},$
(S56)

which is just the exponentiated sum of the lengths of the two edges. Moreover,
the accumulated factor does not need to be counted twice for the edges in
$p_{1}\cap p_{2}$. This double-counting issue has already been considered in
the definition of the length and head width of the path set in Eqs. (S34) and
(S36). Therefore, the lower bound can be obtained again by generalizing
$l(p_{1})$ and $w(p_{1})$ in Eq. (S51) to $l(P_{j})$ and $w(P_{j})$.

  3. (iii)

If the two paths first coincide and then split along the backward direction at
some block $B_{k}$, i.e., $p_{1}\cap
p_{2}=\mathrm{Seg}_{b}(p_{1},B_{k})=\mathrm{Seg}_{b}(p_{2},B_{k})$ and
$B_{k}\in p_{1}\cap p_{2}$, then at the splitting node where
$(B_{k^{\prime}},B_{k})\in\mathrm{Edge}(p_{1})$ and
$(B_{k^{\prime\prime}},B_{k})\in\mathrm{Edge}(p_{2})$ with $k^{\prime}\neq
k^{\prime\prime}$, the selection rule should be slightly modified since the
Pauli strings in $\mathcal{P}^{\prime}|_{s_{c}(B_{k^{\prime}},B_{k})}$ and
$\mathcal{P}^{\prime}|_{s_{c}(B_{k^{\prime\prime}},B_{k})}$ should be selected
simultaneously, i.e.,

$\mathcal{S}_{s(B_{k})}^{(2)}(\sigma^{\otimes 2})=\begin{cases}\sigma^{\otimes
2}&~{}\text{if}~{}s(\sigma|_{s(B_{k})})\subseteq s_{c}(p_{1};B_{k})\cup
s_{c}(p_{2};B_{k})\\\ 0&~{}\text{else}{}\end{cases}.$ (S57)

The selected result is just the tensor product of two normalized uniform
weighted summations of doubled Pauli strings over
$\mathcal{P}^{\prime}|_{s_{c}(B_{k^{\prime}},B_{k})}$ and
$\mathcal{P}^{\prime}|_{s_{c}(B_{k^{\prime\prime}},B_{k})}$, respectively. The
normalization factor is

$\frac{(4^{|s_{c}(B_{k^{\prime}},B_{k})|}-1)\times(4^{|s_{c}(B_{k^{\prime\prime}},B_{k})|}-1)}{4^{|s(B_{k})|}-1}\geq
4^{-l(B_{k^{\prime}},B_{k})-l(B_{k^{\prime\prime}},B_{k})},$ (S58)

which is not equal to but can be lower bounded by the exponentiated sum of the
length of the two edges. Moreover, the accumulated factor again should be
counted only once for the edges in $p_{1}\cap p_{2}$. Thus, the lower bound
again can have the same form as in Eq. (S51) after generalizing $l(p_{1})$ and
$w(p_{1})$ to $l(P_{j})$ and $w(P_{j})$.

Other possible configurations of the two paths are just combinations of these
three cases, i.e., separation, merger, and split. Therefore, the bound
$2\times 4^{-l(P_{j})}\times 2^{-w(P_{j})}$ holds generally for arbitrary
configurations of the two paths. Note that due to the tensor product structure
inherited from $h_{j}$ and our selection rule, the two paths could start their
backward evolution independently without causing additional factors.

If $P_{j}$ contains more than two paths, the lower bound can be obtained in a
similar manner by updating the selection rule in Eq. (S44) for the multi-split
nodes as in Eq. (S57), i.e.,

$\mathcal{S}_{s(B_{k})}^{(2)}(\sigma^{\otimes 2})=\begin{cases}\sigma^{\otimes
2}&~{}\text{if}~{}s(\sigma|_{s(B_{k})})\subseteq\bigcup_{i}s_{c}(p_{i};B_{k})\\\
0&~{}\text{else}{}\end{cases},$ (S59)

where $p_{i}$ runs over the paths that contain the block $B_{k}$. This will