††thanks: These two authors contributed equally to this work.††thanks: These
two authors contributed equally to this work.

# Predicting quantum learnability from landscape fluctuation

Hao-Kai Zhang<EMAIL_ADDRESS>Institute for Advanced Study,
Tsinghua University, Beijing 100084, China  Chenghong Zhu  Thrust of
Artificial Intelligence, Information Hub, The Hong Kong University of Science
and Technology (Guangzhou), Guangdong 511453, China  Xin Wang  Thrust of
Artificial Intelligence, Information Hub, The Hong Kong University of Science
and Technology (Guangzhou), Guangdong 511453, China

###### Abstract

The tradeoff between trainability and expressibility is a central challenge
faced by today’s variational quantum computing. Recent studies indicate that
resolving this dilemma necessitates designing specific parametrized quantum
circuits (PQC) tailored for specific problems, which urgently needs a general
and efficient method to assess the learnability of PQCs regarding a given
target. In this Letter, we demonstrate a simple and efficient metric for
learnability by comparing the fluctuations of the given training landscape
with standard learnable landscapes. This metric shows surprising effectiveness
in predicting learnability as it unifies the effects of insufficient
expressibility, barren plateaus, bad local minima, and overparametrization.
Importantly, it does not require actual training and can be estimated
efficiently on classical computers via Clifford sampling. We conduct extensive
numerical experiments to validate its effectiveness regarding both physical
and random Hamiltonians. We also prove a compact lower bound for the metric in
locally scrambled circuits as analytical guidance. Our findings enable
efficient predictions of learnability, allowing fast selection of suitable
PQCs for a given problem without training, which can improve the efficiency of
variational quantum computing especially when access to quantum devices is
limited.

Introduction.— Learning unknown quantum states is a central topic in modern
quantum physics. Many practical problems can be abstracted into state learning
such as ground state preparation of many-body systems in condensed matter
physics and chemistry [1, 2, 3, 4], quantum tomography [5, 6, 7], and
combinatorial optimization with quantum-classical mappings [8, 9]. With rapid
developments of quantum technology [10, 11, 12, 13], parametrized quantum
circuits (PQC) have emerged as a natural and promising approach for such
problems, also known as quantum neural networks in quantum machine learning
(QML), akin to neural networks learning classical distributions [14]. Training
PQCs using classical optimizers to minimize certain cost functions has been a
paradigm as one of the most promising paths to practical quantum advantages on
near-term quantum devices [15, 16].

However, this hybrid quantum-classical paradigm still faces great challenges
because of its trainability and the tradeoff with expressibility. The most
studied trainability issue is the so-called barren plateau phenomenon [17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28], which means that the cost gradient
vanishes exponentially with the system size for deep PQCs under certain
conditions. Another notorious issue is the bad local minimum problem [29, 30,
31], which exists extensively in shallow circuits like quantum convolutional
neural networks [32] and brickwall circuits of moderate constant depth. These
trainability issues impose severe constraints on the expressive power of
trainable PQCs. On the other hand, if the expressive power is too weak to
contain the target, the training will also fail. The learnability, defined by
the overall capability of a PQC to learn a given target and quantified by the
converged error to the ground truth, is adequate only when expressibility and
trainability are both good enough to contain the target and navigate smoothly
toward it respectively. To resolve this dilemma, one of the most promising
directions is to leverage prior information to devise specific PQCs tailored
for specific problems. For example, based on the entanglement area law in
certain locally interacting systems, one can utilize finite local-depth
circuits (FLDC) to avoid barren plateaus [28]. As another example, one can
construct PQCs using special gates inspired by the given problem instead of
universal parametrized gates, e.g., Hamiltonian variational ansatzes (HVA)
[33, 34] like in the quantum approximate optimizing algorithm (QAOA) [8, 9].
To devise problem-specific PQCs systematically necessitates a method to
diagnose which PQCs are suitable for which problems, i.e., to assess the
learnability of a PQC regarding a given target. The direct way is to run an
actual training procedure and see the error, but this is inefficient and
depends on the details of classical optimizers, and sometimes it is hard to
get the exact ground truth for benchmarks. This motivates us to seek a general
and efficient metric to assess learnability.

Figure 1: (a) Schematic workflow for variational quantum algorithms with
learnability prediction. Prediction phase: For each parametrized quantum
circuit (PQC) candidate, the relative fluctuation (RF) $\tilde{\sigma}$ of the
cost function can be efficiently estimated on classical computers by Clifford
sampling typically. High RF heralds better learnability and smaller training
errors. Training phase: Select the PQCs with high RFs and conduct training on
real quantum devices. (b) Cartoons for typical training landscapes. The
density of states in cost value is sketched on the right side. The spread
indicates the landscape fluctuation $\sigma$. The parameter space is very
high-dimensional in practice but here is only 2D for convenience of
visualization.

In this Letter, we demonstrate a simple and efficient metric for the
learnability of PQCs termed “relative fluctuation” (RF). This metric emerges
from comparing the fluctuations of the given training landscape and standard
learnable landscapes, which is a set of decoupled convex functions. We conduct
extensive numerical experiments to validate its effectiveness in predicting
learnability and find this metric can detect all well-known features of PQCs,
such as expressibility (reachability), barren plateaus, bad local minima, and
overparametrization. Hence RF offers a unified picture for us to understand
various crucial concepts in variational quantum algorithms. We also prove a
lower bound for RF in circuits composed of local $2$-designs as analytical
guidance. Notably, RF is highly efficient as it does not require actual
training and can be estimated classically via Clifford sampling. Therefore, RF
enables fast selections of suitable problem-specific PQCs, as in Fig. 1(a),
which can further serve as an indicator or even an objective function for
automatic circuit design and improve the efficiency of variational quantum
computing in the future.

Basic setup.—For brevity, we will focus on the ground state problem and
variational quantum eigensolver (VQE) 111See Supplemental Material for more
preliminaries and background, basic setup, proof of theorems, and details and
explanations of experiments with additional numerical results. Other problems
with different backgrounds can be described under a similar framework. VQE
aims to prepare the ground state of an $N$-qubit system with the Hamiltonian
$H$ given by $H=\sum_{j}\lambda_{j}h_{j}$ where $h_{j}$ is a Pauli string and
$\lambda_{j}$ is a real coefficient. VQE uses a PQC
$\mathbf{U}(\bm{\theta})=\prod_{\mu}U_{\mu}(\theta_{\mu})$ to construct the
ansatz $\ket{\psi(\bm{\theta})}=\mathbf{U}(\bm{\theta})\ket{\bm{0}}$ involving
$M$ tunable parameters with $\ket{\bm{0}}=\ket{0}^{\otimes N}$.
$U_{\mu}(\theta_{\mu})$ represents an elementary quantum gate depending on
parameter $\theta_{\mu}$. A standard workflow of VQE involves running the PQC
on a quantum device, measuring the cost function
$C(\bm{\theta})=\langle\psi(\bm{\theta})|H|\psi(\bm{\theta})\rangle$, and
updating the tunable parameters iteratively to minimize the cost. We clarify
that here expressibility refers to not simply the expressive space size [22],
but the closeness of the lowest energy state within the expressive space to
the true ground state as quantified by the reachability deficit [15]. Bad
local minima mean the cost values of most local minima concentrate far from
the global minimum [29], while good local minima mean they are close. Good
local minima may correspond to physical metastable states [1, 2, 3], while bad
local minima should be seen as a trainability issue that needs to be fixed.

Relative fluctuation.— The basic information used by our metric is the
landscape fluctuation, i.e., the standard deviation of the (approximately)
normalized cost function
$\sigma=\sqrt{\operatorname{Var}_{\Theta}[C]}/\|\bm{\lambda}\|_{1}$, where
$\Theta$ represents the uniform ensemble of parameter points.
$\|\bm{\lambda}\|_{1}$ is the $1$-norm of the coefficient vector
$\bm{\lambda}$. Physically, $\sigma$ can be seen as the energy fluctuation in
the state ensemble induced by $\Theta$. As shown in Fig. 1(b), characterizing
learnability by $\sigma$ is natural and intuitive as landscapes with either
barren plateaus, insufficient expressibility, or bad local minima all tend to
have their cost values concentrated around the averages resulting in small
$\sigma$, albeit to varying degrees. $\sigma$ might seem a bit cliche because
the exponentially small variance of cost derivatives
$\operatorname{Var}_{\Theta}[\partial_{\mu}C]$ has been used to quantify
barren plateaus [17] which is equivalent to the exponential cost concentration
[21, 36, 37]. However, as we will see below, besides exponential scaling
implying barren plateaus, the specific values of $\sigma$ also encode more
useful information on expressibility, bad local minima, and
overparametrization.

The landscape fluctuation $\sigma$ alone is not enough to characterize
learnability, not simply as in Fig. 1(b). Consider an extreme case: for a
fully parametrized unitary, the learnability should be strong while $\sigma$
is exponentially small due to the global information scrambling. This
contradiction arises because $\sigma$ does not correctly account for the
contribution from the dimensionality of the expressive space, resulting in an
unfair comparison of ansatzes with different amounts of parameters. Therefore,
we define the relative fluctuation (RF)

$\tilde{\sigma}=\frac{\sigma}{\sigma_{0}},$ (1)

where $\sigma_{0}$ is the landscape fluctuation of some “standard learnable”
function $C_{0}(\bm{\theta})$. The form of $\sigma_{0}$ can be determined
heuristically through several fundamental assumptions. (i) To ensure good
trainability, $C_{0}(\bm{\theta})$ should be convex. (ii) To ensure sufficient
expressibility, there should exist a point $\bm{\theta}^{*}$ such that
$C_{0}(\bm{\theta}^{*})=-1$ (after normalization). (iii) $C_{0}(\bm{\theta})$
should have a similar form in arbitrary dimensions. A simple method is to
define them recursively. Combined with (i), this recursive operation can be
chosen as the addition since it preserves the convexity for functions of
independent variables. This leads to “decoupled convex functions”
$C_{0}(\bm{\theta})=\sum_{\mu}f_{\mu}(\theta_{\mu})$ where $f_{\mu}$ is a
single-variable convex function. According to the central limit theorem, the
distribution of the cost value of $C_{0}(\bm{\theta})$ will converge to
Gaussian asymptotically with $\sigma_{0}\sim 1/\sqrt{M}$ given similar forms
of $f_{\mu}$. (iv) To conform with the parameter-shift rule [16], $f_{\mu}$
should be sine-shaped which is not strictly convex but is equally trainable.
Combined with (ii), the scaling coefficient can be further determined to
$1/\sqrt{2}$. To incorporate overparametrization, the parameter count that
should go into $\sigma_{0}$ is the effective dimension $M_{\text{eff}}$ of the
expressive space (manifold) instead of the plain dimension $M$, i.e.,

$\sigma_{0}=\frac{1}{\sqrt{2M_{\text{eff}}}},$ (2)

where $M_{\text{eff}}$ is defined by the rank of the quantum Fisher
information (QFI) matrix [38, 23, 34]

$\mathcal{F}_{\mu\nu}=4\operatorname{Re}[\braket{\partial_{\mu}\psi}{\partial_{\nu}\psi}-\braket{\partial_{\mu}\psi}{\psi}\braket{\psi}{\partial_{\nu}\psi}].$
(3)

It holds that $M_{\text{eff}}\leq M$ and $M_{\text{eff}}$ can be replaced by
$M$ especially for underparametrization cases. One is referred to Supplemental
Material for more elaboration [35].

Thus, it turns out that RF is simply the landscape fluctuation $\sigma$
multiplied by a correction factor $\sqrt{2M_{\text{eff}}}$ from the
dimensionality of the expressive space. The numerical experiments below will
show RF is indeed an effective metric with $\tilde{\sigma}=1$ signifying the
computational phase transition or crossover between strong and weak
learnability. Before that, we first give a lower bound as analytical guidance.

###### Theorem 1

For a PQC composed of blocks forming independent local $2$-designs and an
$r$-local Hamiltonian, the landscape fluctuation is lower bounded by

$\sigma\geq 2^{-r\chi\beta}\frac{\|\bm{\lambda}\|_{2}}{\|\bm{\lambda}\|_{1}},$
(4)

where $\chi$ is the maximum local depth and $\beta$ is the maximum block size
of the circuit.

The proof is left to Supplemental Material [35] where a tighter bound is
proven with the picture of information scrambling through paths across the
circuit. Here the local depth refers to the number of blocks acting on a qubit
and $\chi$ refers to the maximum value of the local depths of all qubits [28].
$\chi$ will reduce to the conventional depth $D$ for common circuits with
repeated layers. $\beta$ is the maximum support size of all blocks. For
constant local-depth circuits and extensive Hamiltonians, we have
$\sigma\in\Omega(1/\sqrt{N})$ which is consistent with the $1/\sqrt{M}$
scaling of $\sigma_{0}$ as $M$ scales linearly with $N$ in these cases.

Figure 2: The landscape fluctuation $\sigma$ vs the number of tunable
parameters $M$ in 1D brickwall circuits of depth $D$ and $N=16$ for the 1D
cluster model, where $D=0$ stands for a layer of single-qubit rotation gates.
Each block is a two-qubit random Clifford gate. The dashed line represents a
reference boundary partitioning learnable and unlearnable phases. The upper
inset shows the same data but in terms of the relative fluctuation
$\tilde{\sigma}$. The lower inset compares the data for $N=8,12,16$.

RF is classically simulable via Clifford random circuits for standard PQCs and
stabilizer initial states since RF only involves second-order moments. That is
to say, instead of sampling the Pauli rotation angles from $[0,2\pi)$,
sampling them from the discrete set $\\{0,\pi/2,\pi,3\pi/2\\}$ is enough to
get RF, which is efficient by the stabilizer formalism. We verify the
consistency of the two sampling methods in Supplemental Material [35].

Unified description of learnability.—A typical behavior of $\sigma$ evolving
with circuit depth $D$ is shown in Fig. 2. We use 1D brickwall circuits like
the first two circuits in Fig. 1(a) representing $D=1,3$. The Hamiltonian is
chosen as the 1D cluster model $H_{ZXZ}=-\sum_{j}Z_{j-1}X_{j}Z_{j+1}$ whose
ground state is a 1D symmetry-protected topological state [39] that can be
prepared with a layer of Hadamard gates followed by a layer of nearest-
neighbor CZ gates. $\sigma$ first decreases exponentially with $D$, consistent
with Theorem 1, until the circuit forms a global $2$-design at the linear
scale resulting in the convergence to the value
$\|\bm{\lambda}\|_{2}/\sqrt{2^{N}+1}\|\bm{\lambda}\|_{1}$. Besides, new
information can be decoded from the comparison with $\sigma_{0}$, as depicted
in the inset. At first, a layer of single-qubit rotations is too weak to
include the ground state in line with $\tilde{\sigma}<1$. Then we have
$\tilde{\sigma}\geq 1$ since the expressive power becomes sufficient for
$D=1,2$ without causing severe bad local minima at least for $N\leq 16$. Next,
RF decreases rapidly to $\tilde{\sigma}\ll 1$ with increasing $D$ since the
landscapes start to have extensive bad local minima and even become almost
flat with barren plateaus at $D\sim N$. Finally, RF turns to increase and
again enters the learnable phase with $\tilde{\sigma}\geq 1$ because the
circuit is overparametrized by exponential parameters, resulting in that there
always exists a finite-gradient direction corresponding to the unitary
rotations to the ground state. This instance shows that RF can provide a
unified description of the typical learnability evolution with circuit depth.

Figure 3: Detecting insufficient expressibility. (a) The relative fluctuation
$\tilde{\sigma}$ vs parameter count $M$ for 1D brickwall circuits with
different gate choices and depths $D$ regarding the 1D cluster Hamiltonian.
The colors with increasing intensity correspond to
$N=4,6,8,10,12,16,32,64,128$. (b) The actual training performance of these
circuits with $\epsilon$ representing the training error after convergence.
Figure 4: Detecting bad local minima. (a) The relative fluctuation
$\tilde{\sigma}$ vs parameter count $M$ for random Hamiltonians
$H_{\text{rand}}$ with different brickwall circuit depth $D$ at $N=16$. (b)
The same setting as in (a) but varying the number of qubits. Different points
with the same markers correspond to different samples of $H_{\text{rand}}$.
The results for $H_{Z}$ are also depicted in (a) for comparison.

Gates and expressibility.—Besides circuit depth, RF can also reflect which
kinds of gates are suitable for the given Hamiltonian. As shown in Fig. 3, we
choose different parametrizations for the two-qubit blocks in brickwall
circuits, compute their RFs, and test their real training performances
regarding $H_{ZXZ}$. $\operatorname{SU}(4)$ represents the full
parametrization. $R_{y}$-$\operatorname{CZ}$ represents two $R_{y}$ rotations
followed by a $\operatorname{CZ}$ and others are defined similarly. As
expected, it turns out that higher RFs herald smaller training errors.
$\operatorname{SU}(4)$ and $R_{y}$-$\operatorname{CZ}$ with
$\tilde{\sigma}\geq 1$ indeed contain the exact solution and lead to almost
zero errors, while $R_{y}$-$\operatorname{CX}$ and $R_{x}$-$\operatorname{CX}$
with $\tilde{\sigma}<1$ have insufficient expressibility and relatively large
errors. Moreover, if an external field term is introduced, we have
$\tilde{\sigma}<1$ for $R_{y}$-$\operatorname{CZ}$ as it no longer contains
the exact ground state necessarily [35]. It is worth noticing that RF of
$\operatorname{SU}(4)$ converges below $1$ as $N$ increases while RF of
$R_{y}$-$\operatorname{CZ}$ stays above in Fig. 3. This is attributed to the
bad local minima in locally scrambled circuits with local underparametrization
for large $N$. We discuss this further below.

Figure 5: Detecting overparametrization with polynomial parameters. (a) The
$\sigma$-$M$ diagram for the 1D Hamiltonian variational ansatz with only Pauli
$X$ and nearest-neighbor $ZZ$ rotations and the 1D transverse-field Ising
model at $h=0.5$. The colors with increasing intensity correspond to depth
$D=1,2,4,6,8,12,16,30,60$. (b) The same data as in (a) but in terms of the
effective dimension $M_{\text{eff}}$.

Transition to bad local minima.—The existence of bad local minima in locally
scrambled circuits is determined by the local overparametrization ratio
$\gamma=M_{\text{cone}}/2d_{\text{cone}}$ [29], where $M_{\text{cone}}$ is the
number of tunable parameters within the backward causal cone of an observable
and $d_{\text{cone}}$ is the Hilbert space dimension of the qubits within the
cone. $\gamma$ decays exponentially in $D$ to $\gamma\ll 1$ signifying local
underparametrization and extensive bad local minima for large-size systems.
This suggests a typical depth $D_{c}$ separating phases of good and bad local
minima. This can be witnessed by RFs. We choose the same setting as in Ref.
[29] based on Hamiltonians from random back evolution
$H_{\text{rand}}=\mathbf{V}^{\dagger}H_{Z}\mathbf{V}$ where
$H_{Z}=-\sum_{j}Z_{j}$ and $\mathbf{V}$ is a random 1D (reversed) brickwall
circuit of depth $D$. Such a setting ensures sufficient expressibility for
brickwall circuits of the same depth as ansatzes. As shown in Fig. 4, RF stays
above $1$ even in large-size systems such as $N=64$ for $D=1$ ($\gamma\gtrsim
1$), while for deeper circuits such as $D=4$ ($\gamma\ll 1$) RF falls below
$1$ as $N$ increases, in agreement with the theoretical prediction and
numerical training results in Ref. [29]. On the other hand, RF is insensitive
to good local minima such as the symmetry-breaking states compared to the GHZ-
type true ground states in the ferromagnetic phase of the transverse-field
Ising (TFI) model, which we left to Supplemental Material [35].

Polynomial overparametrization.—It is known that certain PQCs can be
overparametrized (globally) with only a polynomial number of parameters in the
sense that $M_{\text{eff}}$ saturates when increasing $M$, e.g., those with
dynamical Lie algebras (DLA) of polynomial dimensions [34]. This can be also
witnessed by RFs. We use the 1D TFI model
$H_{\text{TFI}}=-\sum_{j}Z_{j}Z_{j+1}-h\sum_{j}X_{j}$ at $h=0.5$ with open
boundary conditions and the 1D HVA circuit like in QAOA where each layer
consists of a layer of $R_{x}$ and a layer of nearest-neighbor $R_{zz}$ gates
with the initial state $\ket{+}^{\otimes N}$ [35]. Fig. 5 shows that as $D$
increases, overparametrization ($M_{\text{eff}}$ saturation) occurs exactly
once the point $(M,\sigma)$ touches the phase boundary of
$\sigma=1/\sqrt{2M}$, which hence offers a natural picture for the conjecture
in Ref. [23] by identifying the cost variance $\sigma^{2}$ with the inverse of
the effective dimension $M_{\text{eff}}$ at overparametrization.

Discussion.—In this Letter, we introduce the relative fluctuation as a metric
of the problem-specific learnability of PQCs based on the intuition of cost
concentration and the universal scaling of standard learnable landscapes. Our
results demonstrate that RF is indeed effective in detecting both insufficient
expressibility and poor trainability. In particular, the ability to detect bad
local minima was rarely addressed before. Conceptually, RF serves as a bridge
connecting various crucial concepts in variational quantum algorithms.
Practically, RF is highly efficient because it does not require actual
training, hence independent of the optimizer details such as learning rate,
and can be classically computed via Clifford sampling for common cases. This
enables fast selection of suitable problem-specific PQCs and possible advanced
automatic search assisted by methods from quantum architecture search [40, 41,
42, 43]. Here we mainly focus on scenarios like VQE where the learning target
is encoded in Hamiltonians. Alternatively, if the target is encoded in initial
states like in QML, learnability should also incorporate overfitting and
generalization issues [44].

###### Acknowledgements.

Acknowledgments.— We acknowledge the stimulating discussions with Shuo Liu and
Shi-Xin Zhang. The numerical simulations in this work are based on the
TensorCircuit [45] and PyClifford [46] packages. This work was partially
supported by the National Key R&D Program of China (Grant No. 2024YFE0102500),
the Guangdong Provincial Quantum Science Strategic Initiative (Grant No.
GDZX2303007), the Guangdong Provincial Key Lab of Integrated Communication,
Sensing and Computation for Ubiquitous Internet of Things (Grant No.
2023B1212010007), the Start-up Fund (Grant No. G0101000151) from HKUST
(Guangzhou), and the Education Bureau of Guangzhou Municipality.

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Supplemental Material for
“Predicting quantum learnability from landscape fluctuation”

In this supplemental material, we first introduce some preliminaries and
elaborate on the basic setup of this work. Next, we provide the rigorous
formulation and proof of the theorem mentioned in the main text, and more
details on the definition of relative fluctuation. Finally, we offer
additional numerical results to showcase our conclusions further.

##  I Preliminaries and basic setup

In this section, we elaborate on the basic setup of our work. For clarity, we
first introduce some preliminaries and background. Throughout this paper, we
consider a $N$-qubit system $\mathcal{H}^{\otimes N}$ with local Hilbert space
being the two-dimensional complex linear space
$\mathcal{H}=\mathbb{C}^{2}=\operatorname{span}\\{\ket{0},\ket{1}\\}$. The
results in this paper can be easily generalized to qudit systems with local
Hilbert space dimension $d$ by replacing Pauli strings with other operator
strings as an orthogonal basis. Here a Pauli string refers to a tensor product
of single-qubit Pauli operators

$X=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\quad\quad Y=\begin{pmatrix}0&-i\\\
i&0\\\ \end{pmatrix},\quad\quad Z=\begin{pmatrix}1&0\\\ 0&-1\\\
\end{pmatrix}.$ (S1)

We register the $N$ qubits as $\\{q_{1},q_{2},\cdots,q_{N}\\}$ and use $s(O)$
to denote the support of a linear operator $O$ on $\mathcal{H}^{\otimes N}$.
Here the support refers to the subset of qubits that are acted by $O$ non-
trivially. For example, $s(I\otimes X\otimes Y\otimes
Z)=s(X_{2}Y_{3}Z_{4})=\\{q_{2},q_{3},q_{4}\\}$. We use $|s|$ to denote the
number of qubits in a subset of qubits $s$.

Ground state problem.—For brevity, we will illustrate the setup using
variational quantum eigensolver (VQE) and quantum physics problems as
examples. Other variational quantum algorithms (VQA) can be described in a
similar manner with some differences depending on the background of related
tasks, such as quantum state learning, combinatorial optimization, quantum
machine learning (QML), etc. VQE aims to prepare the ground state $\ket{G}$ of
a given Hamiltonian $H$. The Hamiltonian is a Hermitian operator on
$\mathcal{H}^{\otimes N}$ representing the total energy of the system. The
Hamiltonian is usually given in the form of its Pauli string decomposition

$H=\sum_{j}\lambda_{j}h_{j},$ (S2)

where $h_{j}$ denotes a Pauli string and $\lambda_{j}$ denotes the
corresponding real coefficient. The ground state $\ket{G}$ is the eigenstate
of $H$ associated with the lowest eigenvalue (ground state energy). Of course,
there may exist multiple degenerate ground states for certain Hamiltonians.
Here we just focus on the non-degenerate case and the degenerate case is
similar. An important feature of a Hamiltonian is its locality. We say a
Hamiltonian is $r$-local if the maximum size of the supports of its subterms
is $r$, i.e.,

$r=\max_{j}|s(h_{j})|.$ (S3)

Most realistic physical models only contain local interactions. That is to
say, $r$ does not scale with the system size $N$. Note that here the term
“local” actually means “few-body” by convention, distinguished from the
spatial or geometrical locality which is more strict by requiring the qubits
in the support of each subterm to be adjacent on the given lattice. Another
common assumption on realistic physical models is that the Hamiltonian is
extensive because energy is supposed to be an extensive quantity of physical
systems. Here an extensive Hamiltonian refers to the case that the support of
the local Hamiltonian covers mainly all the qubits, e.g., $s(H)=N$, which will
result in the number of subterms in $H$ growing at least linearly in the
system size.

As the entire Hilbert space dimension is exponentially large with the system
size $N$, it will take exponential time and memory space to get the ground
state if one chooses to diagonalize the Hamiltonian directly on classical
computers (known as exact diagonalization, ED). For example, on a common
modern classical computer, the ground state of a small-size system can be
easily solved by ED but it will be hard when the system sizes become
moderately large, such as around $20$ qubits. However, this does not mean
these are all classical algorithms can do. Many successful classical
algorithms have been developed in past decades based on insights into specific
quantum systems [47, 48, 49, 50, 51]. A large class of them belongs to the so-
called variational methods, e.g., variational Monte Carlo (VMC), the density
matrix renormalization group (DMRG) also known as the variational matrix
product state (MPS) algorithm and other tensor network (TN) algorithms. These
methods limit themselves to a polynomially large sub-manifold of the entire
Hilbert space and minimize the energy variationally as low as possible to
approximate the ground state. Here the sub-manifold actually refers to a
certain form of parametrized quantum states $\ket{\psi(\bm{\theta})}$ (also
known as ansatz) that depends on polynomially many tunable parameters. If the
ansatz can be efficiently manipulated by classical computers and the ansatz
can contain the true ground state approximately by leveraging some prior
information about the given system, these variational classical methods would
have a fairly high probability of success in finding the ground state.

For example, most systems in quantum many-body physics admit low-entangled
ground states, i.e., obeying the entanglement area law with at most some
logarithmic corrections. Specifically, the area law, also known as the
boundary law, refers to that the entanglement entropy $S_{A}$ of a subsystem
$A$ only scales with the size of the boundary of the subsystem $|\partial A|$,
i.e.,

$S_{A}\propto|\partial A|,$ (S4)

in stark contrast to typical Haar-random states whose entanglement entropy
scales with the volume of the subsystem, i.e., obeying the volume law
$S_{A}\propto|A|$. It is proved that the ground states of local gapped systems
indeed satisfy the area law in one spatial dimension [52] and it is
empirically believed that the same also holds true in two and higher
dimensions based on extensive instances. Therefore, leveraging this feature,
one can design the so-called tensor network states such as projected entangled
paired states (PEPS) and multiscale entanglement renormalization ansatz
(MERA), which follows the entanglement area law with at most logarithmic
corrections and hence can provide good approximations for the ground states of
quantum many-body systems. For one-dimensional systems, the corresponding TN
states, such as MPS (also known as 1D PEPS), will lead to small treewidth and
can be contracted efficiently on classical computers [53]. Hence, it is known
that 1D quantum systems are classically tractable. Nevertheless, for two and
higher-dimensional systems, the corresponding TNs are in general hard to
contract though there exist many heuristic contraction strategies without
controllable errors [54, 55]. As a consequence, many two-dimensional systems
of interest remain difficult to draw definitive conclusions, such as the 2D
Fermi-Hubbard model and $t$-$J$ model that are relevant to the research of
high-temperature superconductivity.

Of course, there also exist many successful non-variational classical methods
for specific quantum systems, but these methods also face their respective
difficulties when dealing with general systems of interest. For example, the
quantum Monte Carlo (QMC) algorithms, which are designed to directly compute
the expectation values of observables of interest without explicitly
representing the wavefunction via certain mappings to classical statistical
models, are very efficient for many systems such as the square-lattice
Heisenberg model. However, QMC will encounter the so-called sign (phase)
problem and become inefficient when dealing with frustrated quantum systems.

Variational quantum eigensolver.—Recent rapid developments of noisy
intermediate-scale quantum (NISQ) devices offer new hope for this ground-state
preparation problem of quantum nature. Based on the above discussion of
classical variational methods, a natural extension is to use quantum circuits
as variational ansatzes. This is exactly the core idea of VQE. On quantum
devices, one can run quantum circuits and read out the measurement results
efficiently, which is believed to be exponentially hard on classical
computers. Because of the limitations of current NISQ devices, textbook
quantum algorithms that require very large-scale quantum computers with very
high precision cannot be implemented in the near term. Thus, VQE with its
variants in VQAs is expected to be one of the most promising routes toward
practical quantum advantage in the NISQ era.

To be specific, VQE uses a parametrized quantum circuit (PQC)

$\mathbf{U}(\bm{\theta})=\prod_{\mu}^{\leftarrow}U_{\mu}(\theta_{\mu}),$ (S5)

to construct the variational ansatz
$\ket{\psi(\bm{\theta})}=\mathbf{U}(\bm{\theta})\ket{\bm{0}}$ by acting on the
initial state $\ket{\bm{0}}=\ket{0}^{\otimes N}$. The left arrow means that
the index $\mu$ follows the increasing order from right to left in the
product. $U_{\mu}(\theta_{\mu})$ represents a single elementary quantum gate
depending on the parameter $\theta_{\mu}$. Note that the quantum gates without
parameter dependence such as CNOT are also included by considering them as
gates with fixed parameters. The common form of parametrized gates is
$U_{\mu}(\theta_{\mu})=e^{-i\Omega_{\mu}\theta_{\mu}/2}$ where $\Omega_{\mu}$
is a Hermitian generator such as a Pauli string. A typical PQC template is the
so-called 1-dimensional brickwall circuit, e.g.,

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(S6)

The above circuit diagram shows a brickwall circuit of $3$ layers on $6$
qubits, where a single “block” represents a grouped continuous series of
elementary gates, denoted by $B_{k}$, which can be seen as the elementary unit
when constructing a PQC. Two typical templates of two-qubit blocks are
depicted as follows,

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{}{{}}{}{{}}{}\pgfsys@moveto{37.59943pt}{-71.9989pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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which is named “$\mathrm{SU}(4)$ Cartan decomposition” and
“$R_{y}$-$\operatorname{CX}$”, respectively. The former is a fully
parametrized two-qubit block that can express any two-qubit unitary. Here
$R_{xx}$, $R_{yy}$ and $R_{zz}$ represent the two-qubit Pauli rotation gates
with generators $X\otimes X$, $Y\otimes Y$ and $Z\otimes Z$, respectively.
$R_{3}$ represents a universal single-qubit gate, e.g., $R_{z}R_{y}R_{z}$
where $R_{y}$ and $R_{z}$ represent the single-qubit Pauli rotation gates with
generators $Y$ and $Z$. Other block templates can be defined and named
similarly. Here we clarify that the spatial dimension of a quantum circuit
actually refers to the dimension of the underlying qubit connectivity lattice,
where the existence of gate locality is assumed.

A standard workflow of VQE involves running the PQC on a quantum device,
measuring the cost, and updating the tunable parameters iteratively to
minimize the cost function using classical optimizers such as gradient descent
algorithms. The cost function in VQE is just the energy expectation value with
respect to the output state

$C(\bm{\theta})=\langle\psi(\bm{\theta})|H|\psi(\bm{\theta})\rangle.$ (S8)

The core idea of VQE is very clear and natural because a quantum circuit is a
quite natural way to characterize and implement a quantum state. However, VQE
and other VQAs still face great challenges on the way to practical quantum
advantages because of their trainability, and the tradeoff between their
trainability and expressibility. To be specific, it is known that a
variational algorithm can learn the true ground state successfully if two
conditions are met: the expressibility of the ansatz is strong enough to
contain the ground state, and the trainability is good enough to navigate
smoothly toward the ground state. However, in the practice of VQAs, these two
conditions seem not easy to be met simultaneously when the system size is
large as compared to classical neural networks.

###  I.0.1 Barren plateaus

To date, the most extensively studied trainability issue in VQAs is the so-
called barren plateau (BP) phenomenon [17, 18], which means that the cost
gradient would vanish exponentially with the system size for deep PQCs under
certain conditions, i.e.,

$\mathbb{E}_{\Theta}[\partial_{\mu}C]=0,\quad\operatorname{Var}_{\Theta}[\partial_{\mu}C]\in
O(b^{-N}),$ (S9)

where $\partial_{\mu}C=\frac{\partial C}{\partial\theta_{\mu}}$ and $b>1$.
$\mathbb{E}_{\Theta}[\cdot]$ and $\operatorname{Var}_{\Theta}[\cdot]$
represent the average and variance taken over the ensemble $\Theta$ that is
defined in the tunable parameter space. This is akin to the vanishing gradient
issue in classical neural networks. The mechanism for the BP phenomenon can be
explained as follows. At the beginning of the VQE workflow, the tunable
parameters are often randomly initialized so that the whole parameter space
can be explored uniformly in the probabilistic sense to search for the
optimum. This random initialization procedure gives rise to a probability
measure that makes the PQC $\mathbf{U}(\bm{\theta})$ become a random quantum
circuit (RQC). An RQC in a generic shape like brickwall will form an
approximate unitary $2$-design if the RQC is sufficiently deep to achieve
linear depth such as $2\times N$ [56]. Here a $t$-design of a certain group,
such as the unitary group $\mathrm{U}(d)$, refers to an ensemble of group
elements that can lead to the same $t$-degree moment as that of the Haar
(uniform) distribution over the group. Thus one can obtain the results for
$2$-design integrals by using the Weingarten formulas for Haar integrals, such
as

$\int d\mu(U)~{}U^{\dagger}(\cdot)U=\frac{1}{d}\operatorname{tr}(\cdot)I,$
(S10) $\displaystyle\int d\mu(U)~{}U^{\dagger\otimes 2}(\cdot)U^{\otimes 2}=$
$\displaystyle\frac{1}{d^{2}-1}\left[\operatorname{tr}(\cdot)I+\operatorname{tr}(S\cdot)S\right]-\frac{1}{d(d^{2}-1)}\left[\operatorname{tr}(\cdot)S+\operatorname{tr}(S\cdot)I\right],$
(S11)

where $\mu(U)$ is the Haar measure and “$\cdot$” is the placeholder for an
arbitrary linear operator. $I$ is the identity, and $S$ is the swap operator
on the two replicas. If the RQC forms a global $2$-design, i.e., a $2$-design
over the entire Hilbert space of the whole system, the factor $d^{2}$ in the
denominator will become $4^{N}$ and result in an exponential vanishing
variance of cost derivatives. This is the reason for the original BP
phenomenon, i.e., a general linear depth RQC is enough to explore the
exponential large Hilbert space in the sense of the second moment, resulting
in the exponential concentration of the expectation value of an observable.
This indicates that a general linear depth circuit (GLDC) is too powerful to
be used as a variational ansatz running on quantum devices. Fortunately, based
on the above discussion regarding the entanglement area law, GLDC seems
overkill for the ground-state preparation task as GLDC typically generates
volume-law states. Utilizing this area-law feature, one can design a new
circuit class called finite local-depth circuit (FLDC), which is proved to be
free from BP for local Hamiltonians [28], and at the same time is generally
hard to simulate classically by tensor network methods in two and higher
dimensions. This creates new possibilities for practical quantum advantages in
VQE.

###  I.0.2 Bad local minima

However, BP is not the only trainability issue in VQAs. Another notorious
issue is the local minimum problem, which is thought to exist extensively in
moderately shallow circuits [29]. It is worth noticing that the scenario that
one really needs to worry about is the landscape with so-called bad local
minima, where the cost values of local minima concentrate far from that of the
global minimum. By contrast, the landscape with good local minima refers to
the case where the cost values of local minima are close to that of the global
minimum. The existence of good local minima does not need to be overly
concerning, as in physical systems, these good local minima might correspond
to some physically meaningful metastable low-energy states compared to the
true ground state, e.g., the competing ordered states in strongly correlated
systems such as high-temperature superconductors [1, 2, 3]. On the other hand,
the existence of bad local minima is a severe trainability issue that needs to
be fixed. Under certain conditions, it is proved [29] that there is a super-
polynomially small fraction of local minima within any constant additive error
of the ground state energy when the so-called local overparameterization ratio
$\gamma$ is much smaller than $1$. Here the local overparameterization ratio
is defined by

$\gamma=\frac{M_{\text{cone}}}{2d_{\text{cone}}},$ (S12)

where $M_{\text{cone}}$ is the number of tunable parameters in the circuit
within the backward causal cone of a certain observable and $d_{\text{cone}}$
is the Hilbert space dimension of the set of qubits within the backward causal
cone. The backward causal cone refers to the set of gates in a circuit that
have an influence on the expectation value of the observable, e.g.,

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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{} {}{} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{tikz@color}{rgb}{0,0,0}\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@stroke@opacity{0.16}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.96078431372549,0.650980392156863,0.137254901960784}\pgfsys@color@rgb@fill{0.96078431372549}{0.650980392156863}{0.137254901960784}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.2}\pgfsys@invoke{
}\pgfsys@setdash{4.5pt,4.5pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{24.56761pt}{-34.34348pt}\pgfsys@lineto{67.59497pt}{-87.70665pt}\pgfsys@lineto{67.53897pt}{-100.62245pt}\pgfsys@lineto{24.04762pt}{-153.60565pt}\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@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad,$ (S13)

where the orange square represents the location of the observable and the
shaded area represents the causal cone. For a 1-dimensional brickwall circuit
of depth $D$ and a single-qubit observable, the ratio can be roughly estimated
as

$\gamma=\frac{2D(2D+1)/2\cdot(d^{2\beta}-1)}{2d^{2D\beta}}\approx
D(D+\frac{1}{2})d^{2\beta(1-D)},$ (S14)

where $\beta$ denotes the block size, i.e., the number of qubits acted by each
block, and $d$ denotes the local Hilbert space dimension with $d=2$ for qubit
systems. For very shallow circuits such as a 1-dimensional brickwall circuit
of depth $D=1$, it holds that $\gamma\geq 1$, which means that the circuit is
overparametrized locally and hence bad local minima are rare. On the other
hand, for constant depth circuits of moderately larger depth $D$, the ratio
decays rapidly to $\gamma\ll 1$, resulting in local underparametrization, and
hence bad local minima become extensive for large-size systems. In other
words, there exists a critical depth $D_{c}$ such that when increasing the
depth of constant depth circuits to $D>D_{c}$, the training landscape
undergoes a computational phase transition (or crossover) from those with good
local minima to those with bad local minima, which would continue to be more
severe for logarithmic depth circuits such as quantum convolutional neural
network (QCNN) and MERA. Nevertheless, we point out that if the support of
each subterm in the Hamiltonian is just one qubit, there is even no need to
consider a 1D brickwall circuit of depth $D=1$, i.e., this is a trivial
polarization Hamiltonian so that one layer of single-qubit rotations is enough
to solve the ground state. In other words, as long as we consider a non-
trivial Hamiltonian including extensive subterms with support larger than two
qubits, even a 1D brickwall circuit of depth $D=1$ can be locally
underparametrized which will lead to bad local minima for large-size systems.

Here, we briefly explain the reason why overparametrization can eliminate
local minima. An intuitive explanation is that when the number of tunable
parameters is equal to or larger than the entire Hilbert space dimension, then
for a certain parameter point away from the global minimum, there will always
exist a direction that allows the cost value to decrease further, which
corresponds to the rotations from the current state to the ground state.
Consequently, the local minima in the underparametrization case will become
saddle points as the parameter space enlarges, leading to the elimination of
local minima. Rigorously, for PQCs consisting of repeated layers,
overparametrization can be defined by the saturate rank of the quantum Fisher
information (QFI) matrix [34], though the overparametrization there is defined
with respect to the maximally reachable sub-manifold instead of the entire
Hilbert space. In general, an exponential number of parameters is required to
achieve overparametrization. For some PQCs with special gates forming a
dynamical Lie algebra (DLA) of polynomially large dimensions [23], it can be
shown that only a polynomial number of parameters is required to achieve
overparametrization [34], such as the PQC consisting of only
$e^{-iX_{j}\theta/2}$ and $e^{-iZ_{j}Z_{j+1}\theta/2}$ gates.

###  I.0.3 Insufficient expressibility

Based on the above discussion, one can see that BP and bad local minima impose
strong constraints on the expressive power of trainable PQCs. On the other
hand, if the expressive power of the PQC is too weak to include the true
ground state of the given Hamiltonian, the PQC will also fail to learn the
ground state successfully. It is worth emphasizing that the expressibility
mentioned here is not simply the size of the expressive space itself, as
measured by metrics like frame potential [22]. Instead, it depends on specific
Hamiltonians, referring to the closeness of the lowest energy state within its
expressive space to the true ground state as quantified by the so-called
reachability deficit [15]

$\Delta_{\text{r}}=\min_{\bm{\theta}}C(\bm{\theta})-E_{\text{G}},$ (S15)

where $E_{\text{G}}$ denotes the true ground state energy of the given
Hamiltonian. By definition, it ensures that $\Delta_{\text{r}}\geq 0$ and the
equality holds if and only if the expressive space contains the true ground
state. Notably, $\Delta_{\text{r}}=0$ only indicates that the ground state is
within the expressive space, while it does not convey any information about
trainability, i.e., it does not guarantee an efficient and smooth optimization
path from the initial state to the ground state. For example, suppose the
expressive power of a PQC is strong enough such as a general polynomial-depth
circuit. In that case, one can almost always get $\Delta=0$ but the PQC still
cannot learn the ground state successfully due to trainability issues.

However, even with such severe constraints from BP, bad local minima, and
insufficient expressibility, there is still a possibility for using quantum
circuits as variational ansatzes to solve problems. The possibility comes from
utilizing the more detailed prior information on the given task to devise the
so-called Hamiltonian informed ansatzes or called problem-inspired ansatzes or
task-specific ansatzes. For example, based on the prior information that
ground states of locally interacting quantum many-body systems usually satisfy
the entanglement area law with at most logarithmic corrections, one can use
FLDC or logarithmic local-depth circuits to avoid barren plateaus [28], which
also satisfy the entanglement area law under the condition of local circuits
and at the same time cannot be efficiently simulated by classical tensor
network methods in two and higher dimensions. On the other hand, one can
increase the expressibility by introducing ancilla qubits and increasing the
block size (like increasing the bond dimension in MPS) instead of directly
appending repeated layers that may promote local underparametrization with bad
local minima. As another example, one can use special block templates that
depend on the given Hamiltonian instead of universal blocks (such as
$\mathrm{SU}(4)$ Cartan decomposition) to construct the PQC, e.g., Hamiltonian
variational ansatzes (HVA) [33] like in the quantum approximate optimization
algorithm (QAOA). This is because the theoretical results mentioned above on
BP and bad local minima are based on the assumption of local scrambled gates.
More broadly, various heuristic algorithms based on the prior knowledge of a
specific task, e.g., a clever initialization, always have the potential to
bypass the above theoretical limitations.

In summary, the above discussion inspires us to design specific PQCs (e.g.,
gate types, circuit depth, qubit connectivity, circuit architectures, etc.)
tailored for specific problems instead of attempting to use a universal PQC to
solve all problems. This necessitates an efficient method to diagnose which
PQCs are suitable for which problems, i.e., to identify the learnability of a
PQC regarding a given target.

Here we clarify the term “learnability” used throughout the paper. We define
learnability as the ability of a PQC to learn a given target state, which can
be quantified by the training error or test error. For example, in the context
of VQE which is also the focus of this paper, the learnability can be
quantified by the gap between the converged energy $E_{\text{conv}}$ in
training and the true ground state energy $E_{\text{G}}$, i.e.,

$\Delta E=E_{\text{conv}}-E_{\text{G}},$ (S16)

or the fidelity between the PQC output state after convergence and the true
ground state $\left|\braket{\psi_{\text{conv}}}{G}\right|$. Note that the
difference between Eq. (S16) and Eq. (S15) is that Eq. (S15) does not care
about whether the minimization process can be efficiently achieved by
training. In these contexts, learnability can be simply regarded as the
combination of trainability and expressibility. That is to say, the
learnability of a given PQC and a Hamiltonian is high if and only if the
trainability and expressibility are both high. However, in other contexts such
as quantum machine learning (QML) where the information of the learning target
is encoded in the initial state instead of the Hamiltonian like in VQE,
learnability also depends on another factor besides trainability and
expressibility, i.e., the number of samples. That is to say, even with
trainable and expressive PQCs, if the sample size of the given dataset is so
small that overfitting occurs and results in a large generalization error
[44], the learnability is still low. Also, here we only consider ideal quantum
devices where the influences of gate noise and measurement error on
learnability are not considered, which we leave to future research.

A direct method for identifying learnability is to run an actual training
procedure and use the training error to assess learnability. Nevertheless,
this method is inefficient, heavily depends on the choice of classical
optimizers and sometimes it is hard to get the exact ground state for
benchmarks. This motivates us to seek an efficient criterion to assess the
learnability of a PQC regarding a given target, leading to the study of the
relative fluctuation (RF) in this work.

##  II Theorem and proof

In this section, we provide a detailed statement and proof of the theorem
shown in the main text. The basic idea is similar to Ref. [28] of one of the
authors, i.e., tracking the back evolution of the local Pauli strings along
different paths across the circuit. One can refer to Ref. [28] for more
pedagogical explanations.

We first clarify some of our notations in advance. We use $M$ to denote the
total number of tunable parameters and $M^{\prime}$ to denote the total number
of blocks. We suppose that the initial state for the PQC is always the zero
state $\rho_{0}=|\bm{0}\rangle\\!\langle\bm{0}|$. Our results below can be
easily generalized to any other product state. We use the Schatten $p$-norm
for linear operators and the vector $l$-norm for vectors. We use $h|_{s}$ to
denote the sub-string of a Pauli string $h$ on the qubit subset $s$, e.g.,
$(X\otimes I\otimes Y)|_{\\{q_{1},q_{2}\\}}=X\otimes I$. We denote the
$t$-degree twirling channel on support $s$ as

$\mathcal{T}_{s}^{(t)}(\cdot)=\int d\mu_{s}(U)~{}U^{\dagger\otimes
t}(\cdot)U^{\otimes t},$ (S17)

where $\mu_{s}$ is the Haar measure over the unitary group
$\mathrm{U}(2^{|s|})$ acting on $s$. We use $\mathcal{P}|_{s}$ to denote the
set of all Pauli strings on $s$, and $\mathcal{P}^{\prime}|_{s}$ to denote the
set of all non-trivial Pauli strings on $s$, i.e.,
$\mathcal{P}^{\prime}|_{s}=\mathcal{P}|_{s}-\\{I|_{s}\\}$.

The main assumption here is that the blocks in the PQC are locally scrambled
independently. That is to say, the ensemble of randomly initialized parameters
within each block of the PQC can induce a unitary ensemble forming a
$2$-design on its local support so that one can calculate the average and
variance by the Weingarten calculus. This can be approximately realized by
uniform-randomly initialized blocks with full parametrization, e.g., the
$\mathrm{SU}(4)$ Cartan decomposition. We denote this circuit ensemble
composed of independent $2$-design blocks as $\mathbb{U}$. Using blocks as the
elementary unit, the PQC can be rewritten as

$\mathbf{U}=\prod_{k}^{\leftarrow}B_{k},$ (S18)

where the dependence on $\bm{\theta}$ is omitted for simplicity. Hence the
average and variance of the cost function can be expressed as

$\displaystyle\mathbb{E}_{\mathbb{U}}\left[C\right]$
$\displaystyle=\operatorname{tr}\left[\rho_{0}\mathcal{T}^{(1)}_{s(B_{1})}\circ\mathcal{T}^{(1)}_{s(B_{2})}\circ\cdots\circ\mathcal{T}^{(1)}_{s(B_{M^{\prime}})}(H)\right],$
(S19) $\displaystyle\operatorname{Var}_{\mathbb{U}}\left[C\right]$
$\displaystyle=\operatorname{tr}\left[\rho_{0}^{\otimes
2}\mathcal{T}^{(2)}_{s(B_{1})}\circ\mathcal{T}^{(2)}_{s(B_{2})}\circ\cdots\circ\mathcal{T}^{(2)}_{s(B_{M^{\prime}})}(H^{\otimes
2})\right]-\left(\mathbb{E}_{\mathbb{U}}\left[C\right]\right)^{2}.$ (S20)

This can be seen as a back-evolution process for the Hamiltonian by a sequence
of twirling channels, like a “stochastic Heisenberg picture”. We assume the
Hamiltonian is always traceless $\operatorname{tr}(H)=0$ since the identity
subterm only contributes to a constant to the energy expectation. Moreover, we
assume $s(\mathbf{U})\supseteq s(H)$ without loss of generality. Under these
assumptions, Eq. (S10) directly leads to
$\mathbb{E}_{\mathbb{U}}\left[C\right]=0$. Our main goal in the following is
to estimate the first term in Eq. (S20). Our main tool is the following lemma
for each single twirling channel and each Pauli string subterm.

###### Lemma S1

Suppose that $h_{1}$ and $h_{2}$ are two Pauli strings that are distinct on
and only on $s^{\prime}$. If $s^{\prime}\cap s=\varnothing$ and $h|_{s}$ is
non-trivial, then it holds that

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