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# Multimodal deep representation learning for quantum cross-platform
verification

Yang Qian  School of Computer Science, Faculty of Engineering, University of
Sydney, NSW 2008, Australia  Yuxuan Du  JD Explore Academy, Beijing 101111,
China  Zhenliang He  JD.COM, Beijing 101111, China  Min-Hsiu Hsieh  Hon Hai
(Foxconn) Research Institute, Taipei 114699, Taiwan  Dacheng Tao  School of
Computer Science, Faculty of Engineering, University of Sydney, NSW 2008,
Australia

###### Abstract

Cross-platform verification, a critical undertaking in the realm of early-
stage quantum computing, endeavors to characterize the similarity of two
imperfect quantum devices executing identical algorithms, utilizing minimal
measurements. While the random measurement approach has been instrumental in
this context, the quasi-exponential computational demand with increasing qubit
count hurdles its feasibility in large-qubit scenarios. To bridge this
knowledge gap, here we introduce an innovative multimodal learning approach,
recognizing that the formalism of data in this task embodies two distinct
modalities: measurement outcomes and classical description of compiled
circuits on explored quantum devices, both enriched with unique information.
Building upon this insight, we devise a multimodal neural network to
independently extract knowledge from these modalities, followed by a fusion
operation to create a comprehensive data representation. The learned
representation can effectively characterize the similarity between the
explored quantum devices when executing new quantum algorithms not present in
the training data. We evaluate our proposal on platforms featuring diverse
noise models, encompassing system sizes up to 50 qubits. The achieved results
demonstrate a three-orders-of-magnitude improvement in prediction accuracy
compared to the random measurements and offer compelling evidence of the
complementary roles played by each modality in cross-platform verification.
These findings pave the way for harnessing the power of multimodal learning to
overcome challenges in wider quantum system learning tasks.

##  I Introduction

The imperfections inherent in contemporary quantum computers [1, 2, 3]
underscore the need for efficient control, calibration, and validation
protocols, a prerequisite to reach their practical utility and advance the
development of next-generation quantum computing [4]. In pursuit of validation
protocols, cross-platform verification emerges as a pivotal endeavor,
primarily concerned with quantifying the similarity between quantum states
generated by two quantum computers while executing nominally identical
arbitrary circuits [5, 6, 7, 8]. A leading strategy in tackling cross-platform
verification is the random measurement approach [9, 10, 11, 12]. Nevertheless,
recent studies have revealed significant challenges of this strategy when
applied to large-qubit quantum systems, as the estimation error inversely and
quasi-exponentially scales with the required number of measurements [9]. In
light of these challenges, a fundamental question surfaces: can a solution be
found to address cross-platform verification beyond quantum devices comprising
tens of qubits?

In this study, we overcome this challenge by harnessing the potential of
multimodal deep representation learning, underpinned by two fundamental
tenets. The first tenet is that the practical interest of cross-platform
verification generally revolves around quantifying the similarity between the
outputs of two quantum devices as they execute a class of nominally identical
circuits drawn from a certain distribution. Recognizing this, we transition
into a learning paradigm, with the objective of training a learner capable of
using a minimal set of measurements to predict the similarity between two
devices when operating on new circuits sampled from the same distribution. The
second tenet is that cross-platform verification inherently entails two
distinct modalities of data: the measurement outcomes and the classical
descriptions of the compiled circuits on each quantum device. These modalities
carry rich, unique, and complementary information. A learner trained on this
multimodal data can discern cross-platform verification with a deeper insight
compared to reliance on a solitary modality.

For concreteness, we devise a novel multimodal deep learning model for quantum
cross-platform verification, dubbed Measurement-Circuit Driven Neural Network
(MC-Net). Conceptually, MC-Net adopts a permutation-invariant neural network
and a graph neural network to separately distill knowledge from the modalities
of measurement outcomes and circuit descriptions, respectively. Then, a fusion
operation is applied to merge the distilled knowledge into a comprehensive and
compact data representation. This representation, after training, reveals how
factors such as varying system noise, qubit connectivity, and differences in
circuit layout influence the similarity of two quantum devices, and the cross-
platform verification can be efficiently completed by calculating the cosine
similarity of two representations. The knowledge distillation on two
modalities enables the superiority of MC-Net in the realm of cross-platform
verification compared to prior learning-based methods, centering on the single
modality data. As an additional benefit, MC-Net can be directly extended to
accomplish purity and entanglement entropy estimation tasks, which may be of
independent interest.

To evaluate the efficiency of MC-Net, we focus on a specific class of quantum
circuits tailored for cross-platform verification on near-term quantum
devices. Envisioned by the fact that most near-term quantum algorithms are
inherently hardware-efficient, optimized for effective operation on these
devices, the explored class of quantum circuits consists of variational
single-qubit and fixed two-qubit gates arranged in diverse configurations
within a predefined circuit depth. This circuit category is versatile and
encompasses a broad range of near-term quantum algorithms, including quantum
neural networks [13, 14, 15], variational quantum Eigen-solvers [16, 17],
quantum simulations (Trotter methods or other high-order methods) [18, 19, 20,
4], and random circuit sampling [21, 2]. We conduct a systematic analysis on
this class of circuits. Numerical results reveal a remarkable improvement in
terms of prediction accuracy achieved by our proposal compared to the random
measurement approach [12]. Additionally, we conduct an extensive ablation
study to demonstrate the benefits of harnessing quantum circuit architecture
information in cross-platform verification.

##  II Main Results

We first introduce the formal definition of cross-platform verification.
Suppose that there are two $N$-qubit devices $\mathcal{Q}_{i}$ and
$\mathcal{Q}_{j}$ whose noise models are described by $\mathcal{E}_{i}$ and
$\mathcal{E}_{j}$, respectively. When these two devices receive an identical
quantum algorithm described by a unitary $U\in\mathcal{SU}(2^{N})$, cross-
platform verification concerns the similarity of their output states, named
cross-platform fidelity [22], i.e.,

$\mathcal{F}(\rho_{i},\rho_{j})=\frac{\text{Tr}(\rho_{i}\rho_{j})}{\sqrt{\text{Tr}(\rho_{i}^{2})\text{Tr}(\rho_{j}^{2})}}\in[0,1],$
(1)

where $\rho_{i}=\mathcal{E}_{i}(U\rho_{0}U^{\dagger})$,
$\rho_{j}=\mathcal{E}_{j}(U\rho_{0}U^{\dagger})$, and
$\rho_{0}=(\ket{0}\bra{0})^{\otimes N}$. Considering that $\mathcal{E}_{i}$
and $\mathcal{E}_{j}$ are generally distinct, caused by the variations in
hardware architectures, different compiling strategies, qubit configurations,
noise levels, and other factors, the quantification of
$\mathcal{F}(\rho_{i},\rho_{j})$ provides us with deeper insights into
hardware-specific errors and the reliability and adaptability of quantum
algorithms. As shown in Fig. 1(a), conventional methods towards this problem
include quantum state tomography and random measurements, where the classical
description of $\rho_{i}$ and $\rho_{j}$ is (approximately) obtained followed
by an explicit calculation of $\mathcal{F}(\rho_{i},\rho_{j})$ [9, 12].
However, the runtime and memory cost of both methods (quasi) exponentially
scale with $N$, which presents a formidable challenge for manipulating the
large-qubit scenario.

Figure 1: The framework of MC-Net for cross-platform fidelity estimation. (a)
Conventional methods such as quantum state tomography and random measurements
in solving cross-platform fidelity estimation. (b) Instead of (approximately)
reconstructing quantum states, MC-Net completes this task by harnessing neural
networks to learn the states’ low-dimensional and comprehensive
representations $\bm{v}_{i}$ and $\bm{v}_{j}$ from two modalities, where the
cross-platform fidelity is predicted by computing the similarity of the
learned representations with $\langle\bm{v}_{i},\bm{v}_{j}\rangle$. (c) The
implementation of MC-Net consists of two parts. First, MC-Net employs a
permutation-invariant neural network and a graph neural network to separately
distill knowledge from the measurement modality and circuit-encoded modality,
generating representations for each (denoted as a piece of puzzle). Second,
these two feature vectors are fused to create a unified representation of the
input state (put the two puzzle pieces together), i.e., $\bm{v}_{i}$ or
$\bm{v}_{j}$, providing complementarily richer information than individual
modalities for downstream tasks.

Besides the infeasibility for the large $N$, there is another deficiency of
prior methods in cross-platform verification from a practical perspective.
That is, the overarching goal of cross-platform verification revolves
estimating the similarity of two devices for a class of quantum circuits
rather than a single circuit. In this respect, it is desired to develop an
efficient algorithm minimizing the risk

$\mathbb{E}_{\mathcal{E}_{i},\mathcal{E}_{j},U\sim\mathbb{U}}[|\mathcal{A}(\bm{x}_{i},\bm{x}_{j};\bm{w})-\mathcal{F}(\rho_{i},\rho_{j})|],$
(2)

where $\mathcal{A}(\bm{x}_{i},\bm{x}_{j};\bm{w})$ refers to the predicted
cross-platform fidelity returned by a classical learner, $\bm{x}_{i}$ (or
$\bm{x}_{j}$) amounts to the classical description of $\rho_{i}$ (or
$\rho_{j}$), $\bm{w}$ denotes trainable parameters, and $\mathbb{U}$ and
$\mathcal{E}_{i}$ (or $\mathcal{E}_{j}$) represents the distribution over
quantum algorithms and the noise model on $\mathcal{Q}_{i}$ (or
$\mathcal{Q}_{j}$) respectively that will be elucidated later. For instance,
when a user deploys a quantum neural network on two platforms for image
classification, the flexibility to alter the network layout and update
trainable parameters necessitates the evaluation of
$\mathcal{F}(\rho_{i},\rho_{j})$ with the varied $U\sim\mathbb{U}$.

To compensate for the deficiency of conventional methods, here we propose a
multimodal learning method, dubbed Measurement-Circuit Driven Neural Network
(MC-Net), to address the cross-platform verification problem in Eq. (2) when
$N$ becomes large. The proposed MC-Net falls into the supervised learning
framework and comprises three fundamental components: multimodal dataset
collection, model implementation, and model optimization. In the following, we
briefly recap the key insight and realization of these three components, and
defer the omitted details to SM B and C.

The multimodal dataset used for training MC-Net, denoted by
$\mathcal{D}_{\text{Tr}}=\\{(\bm{x}_{i}^{(s)},\bm{x}_{j}^{(s)},\mathcal{F}^{(s)}_{ij})\\}_{s=1}^{S}$
consists of $S$ samples randomly drawn from $\mathbb{U}$, where
$\bm{x}^{(s)}_{i}$ (or $\bm{x}^{(s)}_{j}$) represents data features of the
$s$-th sample for $\mathcal{Q}_{i}$ (or $\mathcal{Q}_{j}$), and the label
$\mathcal{F}^{(s)}_{ij}=\mathcal{F}(\rho_{i},\rho_{j})$ corresponds to the
cross-platform fidelity defined in Eq. (1). A central challenge in
constructing $\mathcal{D}_{\text{Tr}}$ is identifying the specific information
to retain from an exponentially large state space, i.e., the data features
must strike a balance, encompassing adequate information for the task while
maintaining memory and computational efficiency. To address this challenge, we
design a multimodal strategy to engineer the data features, where each sample
contains two parts: the measurement modality and circuit modality.
Particularly, the measurement modality is gathered by conducting the classical
shadow of the quantum state $\rho^{(s)}$ from $M$ random Pauli measurements
[11], which warrants a succinct classical representation of a quantum state.
The circuit modality is obtained by encoding the circuit layout and the device
configuration from $U^{(s)}$ and $\mathcal{E}^{(s)}$ as a directed acyclic
graph (DAG). The handcrafted multimodal data features contain a more
comprehensive and accurate description of a quantum system compared to the
single modality features, enabling MC-Net to use fewer training samples and
measurements to reach a satisfied prediction accuracy. See SM C for detailed
construction of $\mathcal{D}_{\text{Tr}}$.

Before moving on to present the implementation details of MC-Net, let us
emphasize the intrinsic hardness in the design of a multimodal deep learning
model for handling the dataset $\mathcal{D}_{\text{Tr}}$. Note that the
original formats for both the measurement and circuit modalities in
$\mathcal{D}_{\text{Tr}}$ are incompatible with neural networks. Accordingly,
a major challenge is to determine how to effectively preprocess these raw data
features, making them conducive to integration within the chosen neural
network architecture. Furthermore, the process of leveraging prior knowledge
to enhance the efficiency of the learning model for extracting knowledge from
multimodal features remains a largely uncharted realm. Additionally, the means
to synergize the extracted knowledge from diverse modalities to bolster cross-
platform verification has yet to be explored.

To conquer the aforementioned challenges, we devise the architecture
illustrated in Fig. 1(c) to implement MC-Net. In particular, the measurement
modality is processed by a permutation-invariant neural network similar to
PointNet [23] to output a feature vector related to the measurements. The
permutation-invariance allows MC-Net to well respect the inherent disorder of
measurement data obtained from multiple random snapshots. Besides, the circuit
modality is processed by a graph neural network to obtain a feature vector
related to the circuit-induced DAG. These two feature vectors are aggregated
by a fusion operation using bilinear pooling [24], ultimately yielding a
unified, comprehensive, and low-dimensional representation $\bm{v}$. Denote
the distilled representation of MC-Net for the $i$-th ($j$-th) quantum device
on the $s$-th sample as $\bm{v}_{i}^{(s)}$ ($\bm{v}_{j}^{(s)}$), the estimated
cross-platform fidelity amounts to the cosine similarity of these two
representations, i.e.,

$\hat{\mathcal{F}}_{ij}^{(s)}=\frac{\left\langle\bm{v}_{i}^{(s)},\bm{v}_{j}^{(s)}\right\rangle}{\left\|\bm{v}_{i}^{(s)}\right\|_{2}\left\|\bm{v}_{j}^{(s)}\right\|_{2}}.$
(3)

Given the assembled multimodal training dataset $\mathcal{D}_{\text{Tr}}$, MC-
Net utilizes it to optimize the trainable parameters $\bm{w}$ to minimize the
objective function

$\epsilon(\bm{w})=\frac{1}{S}\sum_{s=1}^{S}\mathrm{D}\left(\hat{\mathcal{F}}_{ij}^{(s)},\mathcal{F}^{(s)}_{ij}\right),$
(4)

where $\mathrm{D}(\cdot,\cdot)$ is the per-sample loss, e.g., the mean squared
error, and
$\hat{\mathcal{F}}_{ij}=\mathcal{A}(\bm{x}_{i}^{(s)},\bm{x}_{j}^{(s)},\bm{w})$
refers to the predicted cross-platform fidelity returned by MC-Net. The
optimization of $\bm{w}$ is completed by the gradient-descent methods.

During inference, once we have the profile $(\bm{x}_{i},\bm{x}_{j})$ of an
unknown pair of states $(\rho_{i},\rho_{j})$ prepared by the same $U$ sampled
from $\mathbb{U}$, we can efficiently estimate the cross-platform fidelity by
performing a fast feedforward through the optimized MC-Net. MC-Net’s ability
of efficiently generalizing to new states enables the assessment of state
consistency on different platforms after changing the preparation circuit,
e.g., updating the parameters of variational quantum circuits, in a
computationally friendly manner.

Remark. MC-Net breaks down the procedure of cross-platform fidelity estimation
into two subroutines: the multimodal representation learning of a quantum
state and the similarity calculation. This decoupling allows for easy
adaptation of MC-Net to various tasks. Namely, by replacing the task-related
head responsible for cross-platform verification with other tasks, such as
state purity and entanglement, MC-Net can be repurposed to address different
aspects of quantum system analysis [25, 26, 27]. Additionally, this decoupling
design facilitates extending MC-Net to verify multiple devices
$\\{\mathcal{Q}_{k}\\}$ simultaneously, a feature that will be demonstrated in
the subsequent experiments. See SM D for elaborations.

Figure 2: Cross-platform fidelity estimated by classical shadows (abbreviated
as ‘CS’), the single measurement branch MeaNet, the single architecture branch
CircuitNet and the multi-modal model MC-Net for $6$-qubit quantum systems. (a)
The correlation between ground truth cross-platform fidelity and fidelity
predicted by four approaches with $500$ random Pauli measurements and $100$
circuits. $R^{2}$ represents the coefficient of determination. The black solid
line $y=x$ refers to perfect prediction. (b) The mean square error (MSE)
comparison by four approaches. The labels ‘Mean’ and ‘Std’ represent the
arithmetic mean and standard deviation of MSE respectively. (c) The scaling
behavior of classical shadow based cross-platform fidelity estimation as a
function of the shadow size (number of measurements). The top and bottom
panels denote the $R^{2}$ and MSE respectively. The red star represents the
performance of MC-Net. (d) The behaviors of four approaches concerning various
noise levels.

##  III Numerical Simulations

To assess our proposal against prior methods, we compare classical shadow-
based cross-platform verification (CS) [12] with MC-Net under diverse
settings. Moreover, to highlight the benefits of multimodal representation
learning, in some experiments, we conduct the ablation study, completed by
benchmarking the performance of MC-Net when only a single branch is preserved,
dubbed MC-Net’s measurement branch (MeaNet) and MC-Net’s circuit branch
(CircuitNet).

We initially conduct cross-platform verification on platforms simulated with
two-qubit depolarizing noise, which have the worst effect on the state [28].
The explicit form of the explored class of circuits takes a hardware-efficient
characteristic, i.e.,

$\left\\{U=\prod_{l=1}^{L}\prod_{n=1}^{N}e^{-i\theta_{l,n}H_{l,n}}V_{l}\Big{|}\theta_{l,n}\sim\text{Unif}(0,2\pi)\right\\},$
(5)

where $H_{l}$ is a random Pauli operator and $V_{l}\sim\mathbb{V}$ is an
entanglement block consisting of multiple CNOT gates connecting physically
neighboring qubits. Both the number of CNOT gates and the index of their
control qubit are generated randomly (See SM B.1 for details). When creating
the training dataset, we initially sample $100$ circuits from Eq. (5)
randomly. For each circuit, we introduce $10$ levels of depolarizing noise,
uniformly sampled in the range $[0.01,0.1]$. Subsequently, we create pairs of
circuits with two different noise strengths applied to the same circuit,
resulting in a dataset with a size of $S=4500$. The process of constructing
the test dataset follows a similar procedure. The circuits are sampled from
the same distribution as those in the training dataset, but they have
different layouts compared to the training data.

We evaluate the model’s performance using two standard metrics: the
coefficient of determination $R^{2}$ and Mean Square Error (MSE), i.e.,
$R^{2}=1-\sum_{s}(\mathcal{F}^{(s)}-\hat{\mathcal{F}}^{(s)})^{2}/\sum_{s}(\mathcal{F}^{(s)}-\bar{\mathcal{F}})^{2}$,
where $\bar{\mathcal{F}}$ is the average fidelity over all samples;
$\text{MSE}=\frac{1}{S}\sum_{s}(\mathcal{F}^{(s)}-\hat{\mathcal{F}}^{(s)})^{2}$.
Intuitively, $R^{2}$ measures how well the model’s estimations match the
actual values, with a perfect fit having an $R^{2}$ value of 1, and MSE
quantifies the average squared difference between our model’s predictions and
the actual values.

The overall performance of CS, MC-Net, CircuitNet, and MeaNet on $6$-qubit
systems with $M=500$ is illustrated in Fig. 2(a-b). With such a limited number
of measurements, CS exhibits diminished accuracy in estimating cross-platform
fidelity, as evidenced by its low $R^{2}$ score of $-2836$ and high MSE of
$0.6727$. In contrast, MeaNet, CircuitNet, and MC-Net consistently demonstrate
the satisfied fidelity prediction accuracy, boosting $R^{2}$ values exceeding
$0.9$ and MSE values below $0.005$. Among them, MC-Net outperforms the rest,
achieving the highest $R^{2}$ score of $0.995$ and the smallest MSE of
$0.0003$, a three-orders-of-magnitude enhancement in terms of MSE precision
than CS.

We further exploit the resource efficiency of MC-Net in comparison to CS in
terms of the snapshot $M$. As illustrated in Fig. 2(c), when exponentially
increasing shot number $M$ from $500$ to $64000$, the resulting $R^{2}$ of CS
increases from $-2836$ to $-7$, and the resulting MSE of CS experiences
approximate linear reduction from $0.67$ to $0.2$. Such accuracy still remains
impractical for real-world applications. Nevertheless, MC-Net with only $500$
measurements achieves an extremely low MSE of $10^{-4}$ and a high $R^{2}$ of
$0.995$, which would require over $100,000$ measurements for CS to match.

We subsequently dive into the contribution of measurement modality and circuit
modality to the final prediction by zooming in on a single circuit prepared
with various noise strengths. We take the state prepared by the circuit with a
noise level of $0.01$ as the reference state and calculate the fidelity
between the reference state and states prepared by the same circuit with the
noise level growing from $0.02$ to $0.1$. The behaviors of four approaches are
shown in Fig. 2(d). While the fidelity predicted by MeaNet and CircuitNet
exhibits an apparent gap compared to the exact fidelity, the behavior of their
combination, MC-Net, is almost identical to the exact value. Although previous
results suggest that CircuitNet statistically outperforms MeaNet, Fig. 2(d)
reveals that there are cases where MeaNet performs better than CircuitNet,
such as for devices with a two-qubit depolarizing noise level of $0.07$ and
$0.1$. This phenomenon indicates the complementary nature of measurement
modality and circuit modality in enhancing fidelity estimation. Taken
together, they contribute to a more accurate prediction of fidelity in quantum
systems.

Figure 3: The performance of the proposed model concerning the training
dataset size, the number of measurements, and the number of qubits. (a) The
sensitivity of MSE achieved by CS, MeaNet, and MC-Net to the reduction of
measurements. The CircuitNet is ignored in this context because it requires no
measurement. (b) The sensitivity of MSE achieved by MeaNet, CircuitNet, and
MC-Net to the reduction of training data. (c) The performance of MC-Net when
handling larger-scale systems, including a $30$-qubit platform and a
$50$-qubit platform.

We next examine how MC-Net behaves with the varied number of measurements and
the size of the training dataset. As depicted in Fig. 3(a), fixing the number
of circuits in the training data $S=4500$, increasing the number of
measurements $M$ from $5$ to $500$ leads to a substantial reduction of MSE
from $10^{-3}$ to $10^{-4}$ for MC-Net. In contrast, MeaNet shows a slower
decrease in MSE, from $10^{-2}$ to $5\times 10^{-3}$, and CS exhibits a only
marginal change, with the value remaining at the level of $10^{-1}$. Fig. 3(b)
underscores the minimal training data requirements of our proposed methods
with fixed $M=500$ measurements. By exponentially decreasing the training
dataset size $S$ from $4500$ to $450$, MC-Net witnesses a one-order-of-
magnitude increase in MSE but still attains the same low MSE level as MeaNet
trained on $10^{3}$ circuits. In contrast, when trained on just $10$ samples,
MeaNet struggles to predict cross-platform fidelity, resulting in a
significantly higher MSE of $10^{-2}$. MC-Net’s ability to deliver valid
estimations with a small training dataset size and a modest number of
measurements makes it a viable choice for assessing large-scale quantum
platforms.

To show the compatibility of our proposed model as the system size increases,
we then expand our experiments to encompass larger-scale quantum systems,
specifically involving $30$ qubits and $50$ qubits. It is important to note
that the dimensions of both modalities constructed in our model scale
polynomially with the system size, alleviating any undue burden on the
requisite number of trainable parameters within MC-Net. The noise simulation
for these experiments is the same as in the experiments described earlier,
with the depolarizing channel strength sampled from a range of smaller values
$[10^{-4},10^{-2}]$. Considering the high complexity of large-scale systems,
we set the number of measurements $M$ to $10000$ for the $30$-qubit system and
$20000$ for the $50$-qubit system. Fig. 3(c) shows the predictions of MC-Net
on circuits randomly sampled from Eq. (5) after training on nearly $100$
circuits following the same distribution. Although 20,000 shots fall short of
achieving precise fidelity estimation for large-scale systems using CS, it is
evident that the estimated cross-platform fidelity returned by MC-Net still
exhibits a strong correlation with the ground truth, with $R^{2}$ greater than
$0.95$. See SM E.1 for more results about the scaling behavior of MC-Net with
system size.

Figure 4: The performance of MC-Net for platforms with real device noise
models. (a) The cross-platform fidelity between four devices estimated by
quantum state tomography and proposed MC-Net. The devices ‘A’, ‘B’, ‘C’ and
‘G’ represent fake_almaden, fake_boeblingen, fake_cambridge, and
fake_guadalupe respectively. Fidelity values in the heatmap are obtained by
averaging on $100$ random circuits. The lower triangle records the results of
QST which refers to quantum state tomography, and the upper triangle records
the results of MC-Net. (b) The t-SNE visualization of the low-dimensional
representation predicted by MC-Net.

We last assess the performance of MC-Net on real near-term devices. To do so,
we conducted cross-platform verification using MC-Net on four offline
platforms that mimic the behaviors of the IBM Quantum system [29]. These
platforms, namely fake_almaden (A), fake_boeblingen (B), fake_cambridge (C),
and fake_guadalupe (G) [30], exhibit variations in qubit connection maps,
basis gates, and noise configurations. To ensure comprehensive consideration
of the quantum platform’s characteristics, each hardware-efficient circuit is
compiled to match the target platform’s specified basis gates and qubit
connectivity. Subsequently, we transform the compiled circuit into a DAG. In
addition, we incorporated calibration data for each physical qubit, such as
T1, T2, and state-preparation-and-measurement error, into our model, encoding
them as feature vectors for each graph node. We sample $S=500$ and $S=100$
random $4$-qubit circuits from the same distribution as in previous
experiments and execute them on the four platforms mentioned above to
construct the training and test datasets.

The simulation results are illustrated in Fig. 4(a-b). The lower triangle in
subplot (a) represents outcomes obtained via quantum state tomography with
10,000 measurements. In comparison, the upper triangle displays results
generated by our model using 1000 random Pauli measurements for each circuit.
Remarkably, MC-Net adeptly discerns disparities across different physical
devices, achieving nearly equivalent cross-platform fidelity estimations as
quantum state tomography. The average estimation error between MC-Net and QST
remains consistently at the level of $10^{-4}$. Furthermore, Fig. 4(b) employs
the t-distributed stochastic neighbor embedding (t-SNE) technique [31] to
visualize original the measurement features and the low-dimensional
representations of all test states predicted by MC-Net. The original features
based on random measurements, denoted by the hollow markers, are uniformly
distributed, showing no device-specific clues. In contrast, the
representations learned by MC-Net, denoted by solid markers, reveal distinct
clusters for states prepared on different devices, with no overlap among these
clusters. This observation underscores MC-Net’s ability to effectively extract
features from mixed states, enabling the discrimination of various quantum
platforms.

##  IV Discussion and Outlook

In this work, we present a multimodal representation learning framework for
quantum cross-platform verification, emphasizing the complementary roles of
measurement and circuit modalities to enhance prediction performance. The
proposed MC-Net contributes to the fast evaluation of the reliability and
adaptability of quantum algorithms across different near-term quantum
platforms. Numerical results up to 50 qubits validate the efficacy of MC-Net.

Our work stands as a catalyst for further study on multimodal learning for
quantum tasks. There is an opportunity to explore new modalities that provide
more comprehensive information about quantum systems, thus enhancing the
model’s representation accuracy. In addition to the measurement modality and
circuit modality used in our model, the geometrical structure of the
Hamiltonians can be considered as another modality, which proves essential in
predicting the ground state properties of a quantum many-body system [32].
Furthermore, the concrete format of each modality can be updated to improve
the model’s performance for specific tasks. For instance, the random Pauli
measurements employed in our work could potentially be substituted with more
advanced strategies [33, 34, 35, 36]. Besides upgrading the dataset, we can
introduce more powerful neural networks and advanced fusion strategies that
have been proven successful in classical tasks, such as the Transformer [37],
to handle larger quantum datasets more efficiently. These refinements could
open up new avenues for research in quantum system learning.

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###### Contents

  1. I Introduction
  2. II Main Results
  3. III Numerical Simulations
  4. IV Discussion and Outlook
  5. A Related work
    1. A.1 Random measurement based protocols for cross-platform verification
    2. A.2 AI-driven quantum system learning
  6. B The construction of a class of quantum circuits towards near-term quantum machines
    1. B.1 Implementation of a class of circuits
    2. B.2 Two quantum noise models
  7. C Implementation details of MC-Net
    1. C.1 Dataset Construction
    2. C.2 MC-Net
      1. C.2.1 Measurement Branch
      2. C.2.2 Circuit Branch
      3. C.2.3 Fusion module
    3. C.3 Training details
  8. D Extension of MC-Net for cross-platform verification with multiple devices and the prediction of other properties of quantum states
  9. E More numerical results
    1. E.1 Scaling behavior of MC-Net with respect to the system size
    2. E.2 Ablation study on the measurement data aggregation
    3. E.3 Comparison of fusion module

The outline of supplementary materials is structured as follows. In Sec. A, we
provide a review of prior literature related to our work. Sec. B introduces
the construction of a class of quantum circuits and quantum noise models
employed in our experiments. Sec. C elaborates on the details of MC-Net. In
Sec. D, we explore several extensions of MC-Net for new tasks. Finally, Sec. E
presents additional numerical results obtained with MC-Net.

##  Appendix A Related work

In this section, we will review the prior literature related to our work and
elucidate the connections and distinctions between our proposed model and
existing methods. Initially, we will clarify two random measurement based
protocols for cross-platform verification. Following this, a concise survey of
the field of artificial intelligence driven quantum system learning is
provided.

###  A.1 Random measurement based protocols for cross-platform verification

Random measurements offer a practical solution in approximately expressing a
multi-qubit quantum state by an efficient classical representation, which can
be subsequently processed on classical computers to predict many properties of
the state [38, 39, 11, 40, 41, 12, 9]. Refer to Refs. [42, 33] for a
comprehensive survey of fundamental theory of random measurements and the
relevant applications. In this context, two studies particularly pertinent to
our work are protocols grounded in cross correlations and classical shadow
approaches, specifically designed for the assessment of cross-platform
fidelity.

Cross correlations based protocol [9]. The fundamental principle of this
protocol is leveraging statistical correlations between random measurement
results on distinct devices to infer the overlap of two states. Following
notations in the main text, two mixed $N$-qubit states $\rho_{i}$ and
$\rho_{j}$ are prepared on two quantum devices $\mathcal{Q}_{i}$ and
$\mathcal{Q}_{j}$. The cross correlations based protocol first applies the
same random unitary $U$ to both $\rho_{i}$ and $\rho_{j}$ and then performs
projective measurements in the computational basis
$\ket{\bm{b}}=\ket{b_{1},...,b_{N}}$, where $\bm{b}$ represents a bit string
of possible binary measurement outcomes. Then, after multiple measurements,
the estimation of the probabilities
$P_{U}^{i}(\bm{b})=\text{Tr}(U\rho_{i}U^{\dagger}\ket{\bm{b}}\bra{\bm{b}})$
and
$P_{U}^{j}(\bm{b})=\text{Tr}(U\rho_{j}U^{\dagger}\ket{\bm{b}}\bra{\bm{b}})$
can be acquired by using the collected frequencies. Through randomly sampling
different unitaries $U$ and repeating the above procedure, the overlap
$\text{Tr}(\rho_{i}\rho_{j})$ in Eq. (1) is calculated by the second-order
cross-correlations between $\mathcal{Q}_{i}$ and $\mathcal{Q}_{j}$, formulated
as

$\text{Tr}(\rho_{i}\rho_{j})=2^{N}\sum_{\bm{b},\bm{b}^{\prime}}(-2)^{-D[\bm{b},\bm{b}^{\prime}]}\overline{P_{U}^{i}(\bm{b})P_{U}^{j}(\bm{b}^{\prime})},$
(6)

where the symbol $\overline{\cdot\cdot\cdot}$ denotes the average over random
unitaries $U$, and $D(\bm{b},\bm{b}^{\prime})$ defines the Hamming distance
between bit strings $\bm{b}$ and $\bm{b}^{\prime}$. The denominator of Eq.
(1), i.e., $\text{Tr}(\rho_{i}^{2})$ and $\text{Tr}(\rho_{j}^{2})$, can be
inferred by assigning the same subscript in Eq. (6).

The estimation error of cross-platform fidelity in this protocol arises from
two parts: the finite number of random unitaries $N_{U}$ used to infer the
overlap in Eq. (6) and the finite number of measurements performed for each
unitary $N_{M}$ used to infer the probability $P_{U}$. An empirical study
conducted by Ref. [9] discovered that the required number of measurement
budget $N_{U}N_{M}$ needed to reduce the statistical error below a fixed value
exhibits exponential scaling with the system size $N$, i.e.,

$N_{U}N_{M}\sim 2^{bN},~{}\text{with}~{}b\lesssim 1.$ (7)

This scaling behavior highlights the expensive or even unaffordable
computational demands for larger quantum systems when aiming to achieve a
certain level of fidelity estimation accuracy.

Classical shadows based protocol [11]. This protocol targets reconstructing a
quantum state from random measurement results in a memory and computation
efficient manner. For clarity, we first present the mechanism of classical
shadow and then elucidate the implementation of classical shadows based
protocol.

The classical shadow is a classical description of a quantum state derived
from finite random measurements, serving for efficiently estimating the
expectation values of certain observables of the state. The procedure of
obtaining the classical shadow is as follows. For an $N$-qubit state $\rho$
prepared on a quantum device $\mathcal{Q}$, a unitary $U$ randomly sampled
from a fixed ensemble $\mathbb{U}$ is applied on this state and then measured
in the computational basis to obtain the binary measurement outcomes
$\ket{\bm{b}}=\ket{b_{1},...,b_{N}}$. Based on the measurement results, a
classical description of the reverse operation
$U^{\dagger}\ket{\bm{b}}\bra{\bm{b}}U$ is stored in classical memory, which is
called the snapshot of the state. The expectation of the snapshot over the
randomness of the unitary $U$ and measurement outcome $\bm{b}$ can be viewed
as a quantum channel $\mathcal{M}$ mapping $\rho$ to its snapshot:

$\mathbb{E}[U^{\dagger}\ket{\bm{b}}\bra{\bm{b}}U]=\mathcal{M}(\rho).$ (8)

If the ensemble of unitaries is a tomographically complete set of
measurements, we can recover the state $\rho$ by inverting the channel

$\rho=\mathbb{E}[\mathcal{M}^{-1}(U^{\dagger}\ket{\bm{b}}\bra{\bm{b}}U)].$ (9)

Performing the random measurements for $M$ times, the state $\rho$ can be
approximated by taking the statistical average:

$\hat{\rho}=\frac{1}{M}\sum_{m=1}^{M}\mathcal{M}^{-1}(U_{m}^{\dagger}\ket{\bm{b}_{m}}\bra{\bm{b}_{m}}U_{m}).$
(10)

A general choice of the random unitary $U$ is the single-qubit Pauli group
$\\{U=\otimes_{n=1}^{N}U_{n}\big{|}U_{n}\in\\{\sigma_{I},\sigma_{X},\sigma_{Y},\sigma_{Z}\\}\\}$,
which is conveniently implemented and highly efficient as suggested in [11].
With this configuration, the classical shadow of the density matrix $\rho$ is
expressed as

$\hat{\rho}=\frac{1}{M}\sum_{m=1}^{M}\otimes_{n=1}^{N}(3U^{\dagger}_{m,n}\ket{b_{m,n}}\bra{b_{m,n}}U_{m,n}-\mathbbm{I}),$
(11)

where $U_{m,n}$ and $b_{m,n}$ represents the selected unitary and measurement
outcome for the $n$-th qubit in the $m$-the measurement.

We next introduce the classical shadow based protocol for cross-platform
verification proposed by [12]. Once the classical shadow $\hat{\rho}_{i}$ and
$\hat{\rho}_{j}$ are obtained for states $\rho_{i}$ and $\rho_{j}$
respectively, the cross-platform fidelity estimation can be calculated by

$\hat{\mathcal{F}}(\rho_{i},\rho_{j})=\frac{\text{Tr}(\hat{\rho}_{i}\hat{\rho}_{j})}{\sqrt{\text{Tr}(\hat{\rho}_{i}\hat{\rho}_{i})\text{Tr}(\hat{\rho}_{j}\hat{\rho}_{j})}}.$
(12)

With increasing the number of measurements $M$, the error between
$\hat{\mathcal{F}}$ and $\mathcal{F}$ in Eq. (1) gradually diminishes and
tends towards zero. When the reconstructed state $\hat{\rho}$ is too large to
store classically for a large-scale quantum system, the Pauli-based random
measurements allow us to employ a memory-efficient version of Eq. (12) to
estimate the fidelity, i.e.,

$\hat{\mathcal{F}}(\rho_{i},\rho_{j})=\frac{\sum_{m=1,m^{\prime}=1}^{M}\prod_{n=1}^{N}\text{Tr}(\hat{\rho}_{i,m,n}\hat{\rho}_{j,m^{\prime},n})}{\sqrt{\sum_{m=1,m^{\prime}=1}^{M}\prod_{n=1}^{N}\text{Tr}(\hat{\rho}_{i,m,n}\hat{\rho}_{i,m^{\prime},n})*\sum_{m=1,m^{\prime}=1}^{M}\prod_{n=1}^{N}\text{Tr}(\hat{\rho}_{j,m,n}\hat{\rho}_{j,m^{\prime},n})}},$
(13)

where
$\hat{\rho}_{i,m,n}=3U_{m,n}^{\dagger}\ket{b_{m,n}}\bra{b_{m,n}}U_{m,n}-\mathbbm{I}$
denotes a local classical shadow of the whole state on the $n$-th qubit in
system $\mathcal{Q}_{i}$ when conducting the $m$-th measurement. The
computational complexity of this classical shadow-based cross-platform
fidelity estimation also scales exponentially in the system’s size [11, 12].

Remark. Our work complements and extends the existing literature in this
domain, offering broader applications and a more efficient approach. First,
one of the modalities employed in MC-Net is the classical shadow of a quantum
state. This implies that an advanced version of classical shadow can also
enhance our model. Simultaneously, our model can effectively address the
versatile properties supported by classical shadow techniques. Second, MC-Net
incorporates additional modalities, such as circuit information, to complement
the classical shadow. This results in a more efficient understanding of
quantum systems and a reduction of the required number of measurements.

###  A.2 AI-driven quantum system learning

Artificial intelligence (AI), known for its powerful ability of distilling
useful knowledge from complicated data, is playing an increasingly significant
role in expediting traditionally challenging research [43, 44, 45, 46, 47, 48,
49, 50]. Recent endeavors have ventured into utilizing AI techniques to
enhance quantum science. In the following, we introduce several seminal works
in this context and highlight their relations and differences with our work.

Quantum state tomography. AI-based quantum state tomography (AI-QST) [51, 52,
53, 54, 55] can be categorized into two primary groups based on the type of
data processed by the neural networks. In the first category, researchers
utilize a neural network, such as the restricted Boltzmann machine and deep
Boltzmann machine, to represent the wave function of the target state, which
takes as input a certain reference basis and outputs the corresponding density
measurement [51, 53]. In the second category, neural networks, such as multi-
layer perceptrons and convolutional neural networks, are employed as a black
box to infer the target density matrix from a set of measurement results [56,
57, 58].

In contrast to AI-QST methods, which primarily revolve around the design of
neural networks for simulating quantum systems, our work places a distinct
emphasis on the significance of data modalities in characterizing quantum
systems. Furthermore, our approach focuses on learning the classical
representation of a family of quantum states, as opposed to directly
reconstructing the density matrix of a specific state, as done in previous
studies. These distinctions enable a more efficient and practical solution for
cross-platform verification.

Quantum system property prediction. In addition to fully characterize quantum
systems, AI is commonly employed to predict specific properties of these
systems. These AI-driven methodologies harness the strong capacities of neural
networks to derive partial information about quantum systems from a small
amount of data when the quantum systems belong to a family of systems with
similar structures, achieving high accuracy with fewer measurements and
holding the potential to generalize to unknown quantum systems. Recent
research has focused on predicting various properties of quantum systems using
neural networks, including nonlocality detection [59, 60], estimation of
quantum state entanglement [61, 62, 63, 64], entropy estimation [65, 66] and
fidelity prediction [67, 68, 69, 70, 71]. Despite these initial successes,
using AI techniques for quantum cross-platform verification remains an open
challenge.

Remark. There are limitations when directly transferring these AI-based
approaches to cross-platform verification. Although theoretical evidence has
shown that estimating fidelity between an ideal state and a mixed state can be
achieved by polynomial measurements [11], it remains impractical to estimate
the fidelity between two mixed states. When solely utilizing measurement data
as the input of neural networks, their potential remains confined by the
finite information encapsulated within the measurement outcomes.
Alternatively, [68] takes the architectural features of quantum circuits as
quantum system descriptors to predict the probability of successful trials
(PST) [72], underscoring the significance of circuit layout information in
learning quantum states. In contrast to these methods which consider single
modality, our work adopts a multi-modal learning paradigm that simultaneously
incorporates measurement data and circuit data, leading to a reduced required
number of measurements for cross-platform verification.

##  Appendix B The construction of a class of quantum circuits towards near-
term quantum machines

In this section, we will introduce the construction of a class of quantum
circuits for near-term quantum machines. Considering the constraints of
contemporary hardware and the inevitable quantum noise, we focus on a class of
circuits that can be efficiently implemented on the devices. In the following,
we start by discussing the design details of a class of quantum circuits
tailored for cross-platform verification on near-term devices and elucidating
the rationale behind our choice of these circuits. Afterward, we will
introduce the different noise models employed in our experiments.

###  B.1 Implementation of a class of circuits

In the noisy intermediate-scale quantum (NISQ) era [1], it is expected that
the majority of near-term quantum algorithms will be tailored to align with
the hardware constraints of near-term quantum devices. This alignment is
critical to ensure that these algorithms remain practical and efficient for
real-world application. Specifically, these circuits are tailor-made to
leverage the available quantum operations offered by the hardware, such as the
basis gates and the restricted set of effective two-qubit gates dictated by
the qubit connectivity. In this way, this class of circuits mitigates the
negative effects of device noise while making efficient use of the limited
quantum resources. A diverse range of quantum algorithms are rooted in this
circuit category, such as quantum neural networks [13, 73, 74, 75, 76, 15],
variational quantum Eigen-solvers [16, 17], quantum simulations (Trotter
methods or other high-order methods) [18, 19, 20, 4], and random circuit
sampling [21, 2]. Given their efficiency and versatile applications, we choose
them as the quantum circuits for cross-platform verification in this paper.
Following the general structure of circuits used in many near-term quantum
algorithms, our implementation follows an alternating layered configuration,
i.e.,

$U=\prod_{l=1}^{L}U_{l}(\bm{\theta}_{l}),~{}\text{with}~{}U_{l}=\prod_{n=1}^{N}e^{-i\theta_{l,n}H_{l,n}}V_{l},$
(14)

where $\theta_{l,n}\sim\text{Unif}(0,2\pi)$ represents the angle of a rotation
gate, $H_{l}\in\\{\sigma_{X},\sigma_{Y},\sigma_{Z}\\}$, $V_{l}\sim\mathbb{V}$
is an entanglement block consisting of multiple CNOT gates connecting
physically neighboring qubits. An example of the circuit with $L=2$ is shown
in Fig. M1.

Figure M1: An example of HEA with $2$ layers in the ideal scenario. Each layer
consists of two blocks: a single-qubit gate rotation block and a two-qubit
gate entanglement block. The single-qubit gates represent rotation gates that
are efficiently supported by most quantum devices, while the two-qubit gates
in the entanglement block connect only physically adjacent qubits.

The construction of a set of circuits following the form in Eq. (14) is as
follows. Fixing the maximum layer $L$ and the system size $N$, we randomly
generate a quantum circuit by randomly selecting $\theta_{l,n}$, ${H_{l}}$ and
$V_{l}$. To be concrete, for each layer of the circuit, $\theta_{l,n}$ is
determined by sampling from the distribution $\text{Unif}(0,2\pi)$ and
${H_{l}}$ is equally sampled from the set
$\\{\sigma_{X},\sigma_{Y},\sigma_{Z}\\}$. For the entanglement block $V_{l}$,
we randomly determine whether a CNOT gate exists between two neighboring
qubits. Repeating this stochastic process allows us to generate multiple
quantum circuits with diverse architectures, capturing the variability in
circuit designs.

###  B.2 Two quantum noise models

Due to the imperfections of quantum devices, the state $\rho$ prepared by the
quantum circuit undergoes additional alterations. This process can be modeled
by a device-relevant noise model $\mathcal{E}$ applied to the quantum state
after the unitary transformation $U$:
$\rho=\mathcal{E}(U^{\dagger}\rho_{0}U)$, where
$\rho_{0}=(\ket{0}\bra{0})^{\otimes N}$ represents the initial state. In our
experiments, we introduce various noise models to comprehensively validate the
performance of our model. We first incorporate a two-qubit depolarizing
channel after each two-qubit entanglement gate to simulate quantum noise as
shown in Fig. M2, which is the primary source of noise in physical devices.
The mathematical form of a two-qubit depolarizing channel is

$\Phi_{p}(\rho)=(1-p)\rho+p\frac{\mathbb{I}}{4},$ (15)

where $p$ is the strength of the depolarizing channel and ${\mathbb{I}}/{4}$
is the maximally mixed state for a two-qubit system. This approach enables us
to subject circuits with varying numbers of entanglement gates to different
levels of noise. Moreover, by adjusting the noise strength $p$ of the
depolarizing channel, we can simulate various noise scenarios on different
quantum hardware.

Figure M2: An example of the circuit with $2$ layers when exposed to two-qubit
depolarizing noise. The two-qubit depolarizing noise is modeled by applying a
two-qubit depolarizing channel, denoted as $\Phi_{p}$, after each CNOT gate.
The effect of this noise intensifies with an increase in the number of CNOT
gates in the circuit.

Another type of noise explored in our experiments is extracted from the real
device noise. It encompasses a broader range of errors typical in near-term
quantum devices, including factors like gate fidelity, qubit decoherence and
relaxation times, and the specific chip topology of the device. In our
numerical simulations, we acquire such noise models from four offline devices
in IBM Quantum system. The qubit connection maps of a pair of quantum chips
ibmq_guadalupe and ibmq_boeblingen are shown in Fig. M3, which describes where
a subset of qubits is physically connected to neighboring qubits, while others
are not. These two devices display different patterns on the qubit connection.
Fig. M4 records the device parameters that influence the fidelity of the
prepared state. Unlike depolarizing noise, the noise associated with the
ibmq_guadalupe device cannot be characterized by a single parameter. To be
concrete, the values of T1 and T2 for a qubit determine the performance of
CNOT gate, while a single error rate controls the remaining single-qubit
gates. Additionally, we also consider the measurement error of each qubit,
named as ‘prob_meas0_prep1’ and ‘prob_meas1_prep0’. The value of
prob_meas0_prep1 refers to the probability of returning $0$ when measuring on
a prepared $\ket{1}$ state, while prob_meas0_prep1 refers to the probability
of returning $1$ when measuring on a prepared $\ket{0}$ state. This level of
detail provides a comprehensive noise model that more accurately reflects the
complexities of real quantum hardware, which is essential for our cross-
platform fidelity verification experiments.

Figure M3: Comparison of the qubit connection map of two quantum devices
ibmq_guadalupe (left panel) and ibmq_boeblingen (right panel) [29].

Parameters | Min | Median | Max
---|---|---|---
T1(us) | 38.89 | 71.86 | 119.14
T2(us) | 14.55 | 88.30 | 142.49
RZ error | 0 | 0 | 0
SX error | 0.0002 | 0.0003 | 0.0018
$\sigma_{X}$ error | 0.0002 | 0.0003 | 0.0018
CNOT error | 0.0068 | 0.0101 | 0.0199
prob_meas0_prep1 | 0.0186 | 0.025 | 0.09
prob_meas1_prep0 | 0.0016 | 0.0057 | 0.0298

Parameters | Min | Median | Max
---|---|---|---
T1(us) | 24.11 | 73.76 | 98.44
T2(us) | 9.60 | 79.06 | 153.18
u1 error | 0 | 0 | 0
u2 error | 0.0002 | 0.0005 | 0.0014
u3 error | 0.0005 | 0.0010 | 0.0029
CNOT error | 0.0061 | 0.0165 | 0.0277
prob_meas0_prep1 | 0.0202 | 0.0481 | 0.2128
prob_meas1_prep0 | 0.0062 | 0.0227 | 0.2128

Figure M4: Comparison of qubit and gate parameters of two quantum devices
ibmq_guadalupe (left panel) and ibmq_boeblingen (right panel) [29]. These two
devices offer different sets of basis gates and various hardware errors. For
ibmq_guadalupe, the supported basis gates include RZ, SX, PauliX, and CNOT,
while for ibmq_boeblingen, they encompass u1, u2, u3, and CNOT.

##  Appendix C Implementation details of MC-Net

In this section, we elaborate on the implementation details of MC-Net. Recall
the illustration in Fig. 1(c), the overall architecture of MC-Net comprises
two branches, i.e., the measurement branch and the circuit branch, where each
is responsible for handling distinct modality of data features. Namely, the
measurement branch distills knowledge from sets of the measured bit strings;
the circuit branch distills knowledge from the layout information of quantum
circuits and the noise parameters associated with a specific quantum device.
These two branches work in tandem to fuse the extracted features and learn a
comprehensive representation of the quantum mixed state, ultimately
facilitating the estimation of fidelity. In the subsequent sections, we will
provide a detailed description of the dataset construction, and MC-Net’s
measurement branch and circuit branch architecture. Following that, we will
introduce and compare several fusion strategies. Finally, we will present the
training details.

###  C.1 Dataset Construction

We build the dataset as follows. The input $\bm{x}$ for MC-Net encompasses two
distinct modalities: measurement results and circuit layout. For the
measurement modality, we utilize the random measurement technique to obtain an
efficient representation of a quantum state $\rho$. Specifically, we initially
conduct random Pauli measurements on each qubit of $\rho$. Let
$\ket{\bm{b}_{m}^{(s)}}=\ket{b_{m,1}^{(s)},...,b_{m,n}^{(s)},...,b_{m,N}^{(s)}}$
and $U_{m}^{(s)}=\otimes_{n=1}^{N}U_{m,n}^{(s)}$ be the binary outcomes and
the corresponding measurement bases of the $m$-th measurements for the $s$-th
sample. In the following explanation, we will omit the subscripts $m$ and $s$
for clarity. Next, we construct the classical shadow [11] of the state $\rho$
based on the measurement data. For the $n$-th qubit, the shadow is expressed
as $\hat{\rho}_{n}=3U_{n}^{\dagger}\ket{b_{n}}\bra{b_{n}}U_{n}-\mathbbm{I}$,
which simultaneously encodes the information of both measurement basis and
measurement outcomes. Following this, we flatten $\hat{\rho}_{n}$ into an
eight-dimensional vector (each complex value is represented by two real
values) and concatenate the shadows for all qubits, generating a feature
vector $\bm{x}$ of length $8N$. Furthermore, to enhance the sensitivity of
measurement outputs to the noise configuration, we can explicitly incorporate
available noise-specific parameters into the feature vector $\bm{x}$. This
process is repeated for $M$ times, resulting in a set of measurement vectors
$\\{\bm{x}_{m}\\}_{m=1}^{M}$ that serve as the input for the measurement
branch of MC-Net. For clarity, the subscript of $\bm{x}$ only specifies the
$m$-th measured outcomes, omitting the device index.

To represent the device-specific circuit architecture, we encode the topology
of a quantum circuit as a DAG, with device-specific noise information encoded
into the node features. Following the convention in [30, 68], each node in the
DAG corresponds to a quantum gate, and directed edges signify the temporal
order of quantum gate execution. Additionally, two special node types, the
input node, and the output node, represent the circuit’s starting and ending
points on a wire. For noisy devices simulated by depolarizing channels, we set
the noise strength as an element of the feature vector of the CNOT gate node,
while we set $0$ for others. For physical noise models, we consider the value
of $T1$ and $T2$ of a qubit, as well as the gate and readout errors. (See SM
C.2.2 for a typical sample of encoding a circuit as a DAG).

###  C.2 MC-Net

MC-Net employs two branches, the measurement branch and the circuit branch, to
separately distill the valid information that contributes to the verification
of cross-platform from the two modalities. To handle the measurement modality
$\\{\bm{x}_{m}\\}_{m=1}^{M}$, we design a convolutional neural network similar
to PointNet [23] as the measurement branch. Specifically, each measurement
vector $\bm{x}_{m}$ is mapped to a higher-dimensional feature vector through
shared convolutional kernels. Subsequently, we create a unified measurement
feature vector by averaging over all $M$ feature vectors. This approach
ensures that our model retains the permutation invariance of multiple
measurements.

For the circuit modality, we employ a graph neural network to extract the
feature of the circuit-induced DAG. Following several layers of graph
convolution, we perform average aggregation on all node features, resulting in
a unified circuit feature vector. The measurement feature vector and circuit
feature vector are then combined by low-rank bilinear pooling [24] to generate
a unified representation of the quantum state (See SM C for more details).

####  C.2.1 Measurement Branch

In MC-Net, the measurement branch independently processes the measurement
data. As explained in the main text, recognizing that the measurement modality
$\\{\bm{x}_{m}\\}_{m=1}^{M}$ collected from multiple snapshots is unordered,
it is necessary that the measurement branch has the capability of preserving
the permutation invariance of quantum measurements. To address this challenge,
we draw inspiration from the successful application of PointNet [23] in
handling point clouds in computer vision. In particular, we design a novel
neural network termed ‘MeaNet’ as the measurement branch of MC-Net,
specifically designed to effectively capture the measurement features of a
quantum state.

Figure M5: Implementation of measurement branch of MC-Net. (a) The
architecture of MeaNet. The block ‘MLP’ represents a multi-layer perceptron.
The symbols ‘M’, ‘N’, and ‘D’ refer to the number of measurements, the number
of qubits, and the length of the output measurement vector, respectively. (b)
The implementation of 1D convolution block in the MeaNet.

The architecture of MeaNet is illustrated in Fig. M5. For each training
example, the network receives the modality of measurement in data features,
i.e., a set of measurement vectors $\\{\bm{x}_{m}\\}_{m=1}^{M}$ generated from
$M$ random Pauli measurements as introduced in SM. Initially, each original
vector is projected to a higher-dimensional space using multiple 1D
convolution blocks, and each of these blocks comprises several 1D
convolutional kernels, a nonlinear activation function and a batch norm layer
shown in Fig. M5(b). Let $\\{\mathcal{K}_{d}\\}_{d=1}^{D}$ denote the set of
$D$ convolutional kernels. After applying convolution on a single measurement
vector $\bm{x}_{m}$, we obtain a new vector
$[\mathcal{K}_{1}(\bm{w}_{1},\bm{x}_{m}),...,\mathcal{K}_{D}(\bm{w}_{D},\bm{x}_{m})]$
with a length of $D$, where $\bm{w}_{d}$ is the weight of the $d$-th kernel.
This process is repeated for all measurement vectors, yielding a set of
measurement features
$[[\mathcal{K}_{1}(\bm{w}_{1},\bm{x}_{1}),...,\mathcal{K}_{D}(\bm{w}_{D},\bm{x}_{1})],...,[\mathcal{K}_{1}(\bm{w}_{1},\bm{x}_{M}),...,\mathcal{K}_{D}(\bm{w}_{D},\bm{x}_{M})]]$
with the size $M\times D$. It is essential to highlight that each vector,
corresponding to an individual measurement, shares the same convolution
kernels. Subsequently, the features from convolution blocks are aggregated
using average pooling followed by a multi-layer perceptron (MLP), resulting in
a single vector with a length of $D$. This pooling operation ensures the
preservation of measurement permutation invariance. The unified representation
of a quantum state from multiple measurements is used for direct cross-
platform fidelity estimation and can be integrated with additional information
for further predictions.

####  C.2.2 Circuit Branch

The circuit branch of MC-Net is designed to process the circuit data. Unlike
measurement data, quantum circuits can be better represented as structured
graphs rather than vectors. As introduced above, we transform a quantum
circuit into a Directed Acyclic Graph (DAG), where the edge and node
connections precisely correspond to the circuit’s layout and the execution
order of quantum gates. To incorporate device-specific information into this
DAG, we utilize a systematic rule for encoding device parameters as node
features. We employ a Graph Neural Network (GNN), named ‘CircuitNet’ to manage
this circuit-induced DAG, enabling the model to learn the intrinsic impact of
device noise on the final prepared state. In the following sections, we will
first provide a rigorous explanation of the rule used to encode a quantum
circuit as a DAG. Then, we will describe the design of the GNN in the circuit
branch of MC-Net.

Directed Acyclic Graph reformulation. A typical example of encoding an
original quantum circuit as a DAG is depicted in Fig. M6(a). This
representation encompasses six types of nodes: an input node, an output node,
and four types of gate nodes. The input node (colored in grey) and the output
node (colored in white) symbolize the initial state and measurement on a wire,
respectively. The remaining nodes encompass three types of single-qubit
rotation gates $\\{R_{X},R_{Y},R_{Z}\\}$ and a two-qubit CNOT gate. The
directed edges that link neighboring quantum gates indicate the temporal order
of quantum gate execution.

Figure M6: Conversion from a quantum circuit to a DAG. (a) An example of
transforming an original quantum circuit into a DAG. (b) Construction of the
node feature vector. (c) An example of transforming a quantum circuit
transpiled for a platform into a DAG.

As shown in Fig. M6(b), the feature vector of each graph node is composed of
four key components: the type of the node, the noise features of the gate, the
qubit index that the gate acts on and the node index. The type of the node is
represented by a one-hot vector with a length equal to the number of gate
types. In this representation, the $i$-th element is set to $1$, while the
other elements remain $0$ to categorize the node into the appropriate type.
The noise features of the gate depend on the specific device. For instance, in
the case of numerical simulation with two-qubit depolarizing noise models, the
noise feature of two-qubit gates is expressed as the strength parameter $p$ of
the depolarizing channel, while the noise feature of other gates remains $0$.
For real device noise models, the noise feature comprises additional
information about each node, such as the values of $T1$ and $T2$ for the
qubit, as well as gate and readout error rates. The qubit index is encoded as
a one-hot vector, while the node index is encoded as an integer.

The construction rule of the DAG can be easily adjusted to accommodate the
various basis gates supported by different quantum devices. When running a
quantum circuit on a real quantum device that does not support the exact
quantum gates used in the original circuit, it is necessary to transpile the
circuit [77, 78, 79] to suppress the hardware noise. This ensures that the
circuit is implemented using gates that are compatible with the capabilities
of the target device. As shown in Fig. M6(c), the $R_{X}$ and $R_{Y}$ gates of
the original circuit in Fig. M6(a) are implemented with $R_{Z}$ and
$\sqrt{X}$. While the original circuit and the transpiled circuit are
theoretically equivalent in an ideal scenario, they can differ in practice due
to variations in noise levels for different gate types in noisy conditions.
This discrepancy in noise characteristics and basis gates results in distinct
DAGs for the two versions of the same circuit. This encoding scheme captures
the device-specific variations of quantum circuits, allowing for meaningful
distinctions between different quantum platforms.

Figure M7: The architecture of the circuit branch of MC-Net. The block ‘MLP’
represents a multi-layer perceptron.

Implementation of circuit branch. The architecture of the circuit branch of
MC-Net is depicted in Fig. M7. Specifically, this branch takes the circuit-
induced DAG as input and processes it using a graph convolution network. The
graph convolution operation enables message passing among neighboring gates,
effectively modeling the impact of gate errors in the quantum state evolution
process. After several graph convolutions, we can obtain the circuit’s feature
vector by averaging the features of all nodes and projecting them into a
higher-dimensional space using a multi-layer perceptron (MLP). This unified
representation serves as a classical proxy for quantum circuits, ready for
downstream tasks.

####  C.2.3 Fusion module

Fusion plays a crucial role in the framework of multi-modal learning as it
combines information from different modalities into a unified representation.
In this context, consider two vectors: $\bm{x}\in\mathbb{R}^{D_{1}}$ and
$\bm{y}\in\mathbb{R}^{D_{2}}$, representing feature of distinct modalities
indexed by $1$ and $2$ respectively. The objective is to design a mapping
$f:(\bm{x},\bm{y})\rightarrow\bm{v}\in\mathbb{R}^{D}$ such that the fused
feature $\bm{v}$ outperforms any individual unimodal data $\bm{x}$ or $\bm{y}$
in terms of performance of downstream tasks. Note that the dimensions of each
feature vector ($D_{1}$, $D_{2}$, and $D$) need not be the same. Various
fusion strategies for the selection of $f$ exist, including simple operation-
based fusion, bilinear pooling-based fusion, and attention-based fusion, as
discussed in [80].

Figure M8: Overview of various fusion strategies. (a) Summation-based fusion.
(b) Concatenation-based fusion. (c) Low-rank bilinear pooling-based fusion.
(d) Attention-based fusion.

Simple operation-based fusion [81, 82, 83]. In classical multi-modal learning,
simple operations can be employed as the mapping function $f$, such as
concatenation and summation. These operations have few or no trainable
parameters and limited interactions of elements of different modality
features.

  * •

Summation: As shown in Fig. M8(a),
$\bm{v}=\bm{W}_{1}\bm{x}+\bm{W}_{2}\bm{y}\in\mathbb{R}^{D}$, where
$\bm{W}_{1}\in\mathbb{R}^{D\times D_{1}}$ and
$\bm{W}_{2}\in\mathbb{R}^{D\times D_{2}}$ are the weights that project
$\bm{x}$ and $\bm{y}$ to the same dimension to satisfy the condition of
element-wise addition.

  * •

Concatenation: As shown in Fig. M8(b),
$\bm{v}=[\bm{x},\bm{y}]\in\mathbb{R}^{D_{1}+D_{2}}$.

Bilinear pooling-based fusion. Bilinear pooling involves calculating the outer
product of two vectors, which can capture the complete set of interactions
between every element of the feature vectors. Mathematically, the $i$-th
element of the fused vector $\bm{v}$ after applying bilinear pooling on
$\bm{x}$ and $\bm{y}$ can be expressed as

$\bm{v}_{i}=\bm{x}^{T}\bm{W}_{i}\bm{y},$ (16)

where $\bm{W}_{i}\in\mathbb{R}^{D_{1}\times D_{2}}$ is the $i$-th block of
weight tensor $\bm{W}=[\bm{W}_{1},...,\bm{W}_{D}]\in\mathbb{R}^{D_{1}\times
D_{2}\times D}$. It is worth noting that for large values of $D_{1}$, $D_{2}$,
and $D$, the computation cost of $\bm{W}$ can be quite high, which limits its
practical feasibility.

Various works have been proposed to reduce the rank of the weight tensor
$\bm{W}$ to restrict the number of parameters of $\bm{W}$. One typical
approach [84], called low-rank bilinear pooling (LRBP), is to decompose
$\bm{W}_{i}$ into the product of two low-rank matrices:
$\bm{W}_{i}=\bm{A}_{i}\bm{B}_{i}^{T}$, where
$\bm{A}_{i}\in\mathbb{R}^{D_{1}\times D_{3}}$,
$\bm{B}_{i}\in\mathbb{R}^{D_{2}\times D_{3}}$ and
$D_{3}\leq\min(D_{1},D_{2})$. This allows us to rewrite $\bm{v}_{i}$ as:

$\bm{v}[i]=\bm{x}^{T}\bm{A}_{i}\bm{B}_{i}^{T}\bm{y}=\mathbbm{1}^{T}(\bm{A}_{i}^{T}\bm{x}\odot\bm{B}_{i}^{T}\bm{y}),$
(17)

where $\mathbbm{1}\in\mathbb{R}^{D_{3}}$ is a column vector of ones and
$\odot$ represents Hadamard product. To further simplify the weight tensor
$\bm{W}$, we use the same weight matrix $\bm{W}_{i}$ when calculating each
element of the vector $\bm{v}$ and replace $\mathbbm{1}$ with a projection
matrix $\bm{P}\in\mathbb{R}^{D_{3}\times D}$, as formulated by

$\bm{v}=\bm{P}^{T}(\bm{A}^{T}\bm{x}\odot\bm{B}^{T}\bm{y}),$ (18)

where $\bm{A}\in\mathbb{R}^{D_{1}\times D_{3}}$,
$\bm{B}\in\mathbb{R}^{D_{2}\times D_{3}}$. In this way, the number of
parameters in the bilinear pooling-based fusion is reduced from $D_{1}\times
D_{2}\times D$ to $D_{3}\times(D_{1}+D_{2}+D)$. The entire process of LRBP is
shown in Fig. M8(c). In practice, a non-linear activation function is
introduced to enhance the capacity of the fusion model [24]. To clarify, Eq.
(18) is modified as follows:

$\bm{v}=\bm{P}^{T}(\sigma(\bm{A}^{T}\bm{x})\odot\sigma(\bm{B}^{T}\bm{y})),$
(19)

where $\sigma$ is a non-linear activation function, such as tanh in our work.

Attention-based fusion. Attention mechanisms have been widely used in deep
learning models, e.g. Transformer [37], to help the models focus on the most
relevant parts of the input data concerning the target. Typically, attention
mechanisms involve estimating the attention distribution across elements in a
set of feature vectors and then computing an attended vector based on this
distribution. In the context of attention-based fusion, attention mechanisms
are usually initially applied to unimodal features, and the attended features
from different modalities are fused. Here, we combine attention mechanisms
with low-rank bilinear pooling for feature fusion. Mathematically, the
attention distribution is inferred as

$\bm{a}={\rm
softmax}(\bm{Q}^{T}(\sigma(\bm{C}^{T}\bm{x}\cdot\mathbbm{1}^{T})\odot\sigma(\bm{D}^{T}\bm{Y}^{T}))),$
(20)

where $\bm{a}\in\mathbb{R}^{G\times S}$, $\bm{Q}\in\mathbb{R}^{D_{3}\times
G}$, $\bm{C}\in\mathbb{R}^{D_{1}\times D_{3}}$,
$\mathbbm{1}\in\mathbb{R}^{S}$, $\bm{D}\in\mathbb{R}^{D_{2}\times D_{3}}$,
$\bm{Y}\in\mathbb{R}^{S\times D_{2}}$. For every glimpse $g\in[G]$, we
calculate the attended feature vector $\bm{y}_{g}$ as a weighted sum:

$\bm{y}_{g}=\sum_{s=1}^{S}\bm{a}_{s,g}\cdot\bm{Y}_{s,g}.$ (21)

The overall attended feature vector $\bm{y}$ is constructed by concatenating
$\bm{y}_{g}$ in Eq. (21). The subsequent steps remain the same as those in the
fusion of $\bm{x}$ and $\bm{y}$. The entire process is depicted in Fig. M8(d).

###  C.3 Training details

The training phase of MC-Net consists of two stages. The first stage involves
separately training the measurement branch and the circuit branch of MC-Net.
The second stage focuses on fine-tuning the entire MC-Net. With this two-stage
training approach, MC-Net converges faster during training and achieves strong
generalization on the test data.

Training of the measurement branch. During the training of the measurement
branch of MC-Net, the circuit branch and the fusion module are temporarily
removed. This modified part is referred to as MeaNet. The feature vector
outputted by MeaNet serves as the final classical representation $\bm{v}$ of a
quantum state. Using the same objective function as described in Eq. (4) in
the main text, we update the parameters of MeaNet with the Adam optimizer
based on measurement data.

Training of the circuit branch. Training of the circuit branch of MC-Net
follows a similar procedure to that of the measurement branch. The measurement
branch and the fusion module are temporarily removed, and the remaining part
is named CircuitNet. The feature vector outputted by CircuitNet is considered
as the final classical representation $\bm{v}$ of a quantum state. Using the
same objective function as described in Eq. (4) in the main text, we update
the parameters of CircuitNet with the Adam optimizer based on circuit data.

Fine-tune of MC-Net. Once the MeaNet and CircuitNet are well trained, we begin
to fine-tune the entire MC-Net, with its measurement branch and circuit branch
inheriting parameters from MeaNet and CircuitNet, respectively. In comparison
to the independent training of single branches, a smaller learning rate is
used to update the parameters of the measurement branch, circuit branch, and
fusion module. This fine-tuning process ensures that the different components
of MC-Net work harmoniously and further improves the model’s performance.

##  Appendix D Extension of MC-Net for cross-platform verification with
multiple devices and the prediction of other properties of quantum states

In this section, we will explore the versatile extension of MC-Net for new
quantum system learning tasks and illustrate this with two examples. To
achieve this, we decouple the process of employing MC-Net for predicting
cross-platform fidelity into two distinct components, allowing for later
extensions. Subsequently, we will introduce methods for adapting MC-Net to
perform cross-platform verification on multiple devices and for predicting
quantum state purity.

The architecture of MC-Net can be divided into two main components: the
network before the fusion module, denoted as $\mathcal{B}$, responsible for
multi-modal learning to generate a general classical representation $\bm{v}$
of a quantum state; and the fusion module, denoted as $\mathcal{C}$,
responsible for predicting the target of the task. With this framework, the
predicted cross-platform fidelity $\mathcal{A}(\bm{x}_{i},\bm{x}_{j};\bm{w})$
in Eq. (2) can be derived as

$\mathcal{A}(\bm{x}_{i},\bm{x}_{j};\bm{w}=(\bm{w}_{\mathcal{B}},\bm{w}_{\mathcal{C}}))=\mathcal{C}(\bm{v}_{i},\bm{v}_{j};\bm{w}_{\mathcal{C}}),\text{with}\
\bm{v}_{i}=\mathcal{B}(\bm{x}_{i};\bm{w}_{\mathcal{B}}),\bm{v}_{j}=\mathcal{B}(\bm{x}_{j};\bm{w}_{\mathcal{B}}).$
(22)

Note that the network $\mathcal{B}$ is designed to be task-independent,
whereas $\mathcal{C}$ is customized for specific tasks. By fixing
$\mathcal{B}$ and adapting the task-specific module $\mathcal{C}$ to new
tasks, we can apply MC-Net to various tasks.

K-device ($K\geq 2$) cross-platform verification. K-device cross-platform
verification involves simultaneous fidelity estimation among multiple states
$\\{\rho_{i}\\}_{i=1}^{K}$. To achieve this, the network $\mathcal{B}$ is
applied to multiple states to generate a set of representations
$\\{\bm{v}_{i}\\}_{i=1}^{K}$. Subsequently, the average cross-platform
fidelity over $K$ devices can be calculated as follows:

$\mathcal{F}(\\{\bm{v}_{i}\\}_{i=1}^{K})=\mathcal{C}(\\{\bm{v}_{i}\\}_{i=1}^{K};\bm{w}_{\mathcal{C}})=\frac{1}{K}\sum_{i=1}^{K}\frac{\left\langle\bm{v}_{i},\bm{v}_{j}\right\rangle}{\left\|\bm{v}_{i}\right\|_{2}\left\|\bm{v}_{j}\right\|_{2}}.$
(23)

State purity estimation. Quantum state purity is a measure of the degree of
coherence and entanglement within a quantum state $\rho$, defined as

$\gamma=\text{Tr}(\rho^{2}).$ (24)

To predict the purity of a quantum state, we employ a fully connected layer as
the task-specific module $\mathcal{C}$. Let the weight matrix of the fully
connected layer be denoted as $\bm{w}_{\mathcal{C}}$ with a shape of $(D\times
1)$, where $D$ is the length of the representation $\bm{v}$ of $\rho$
extracted by $\mathcal{B}$. Then the estimated state purity is

$\hat{\gamma}=\mathcal{C}(\bm{v};\bm{w}_{\mathcal{C}})=\bm{w}_{\mathcal{C}}^{T}\bm{v}.$
(25)

Other quantum system properties, such as state entanglement and entropy, can
be inferred in the same manner in the framework of MC-Net.

##  Appendix E More numerical results

In this section, we present additional ablation studies on MC-Net.
Specifically, we begin with the scaling behavior of MC-Net with respect to the
system size. Following that, we compare various aggregation strategies for the
measurement data. Finally, we assess the performance of MC-Net using three
different fusion models.

###  E.1 Scaling behavior of MC-Net with respect to the system size

Considering that the number of measurements required by classical shadow-based
protocols scales exponentially with the number of qubits, we have
experimentally investigated the scaling behavior of MC-Net in relation to the
number of qubits to ascertain its resource efficiency compared to measurement-
based protocols. We examine multiple system sizes ranging from 4 to 10 qubits.
The test data for each system size are generated in the same manner as
described in the main text. To ensure a fair comparison of performance across
different system sizes, we employ the Relative Mean Square Error (RMSE) as our
metric, defined as
$\text{RMSE}=\frac{1}{S}\sum_{s}(\frac{\mathcal{F}^{(s)}-\hat{\mathcal{F}}^{(s)}}{\mathcal{F}^{(s)}})^{2}$.

The scaling behaviors of the Classical Shadow-based protocol (CS), MeaNet, and
MC-Net methods, as the number of qubits increases from 4 to 10 when fixing
$S=450$ and $M=500$, are depicted in Fig. M9. The CS method’s RMSE begins at
0.680 and escalates dramatically to 0.996, signaling a decrease in precision
with the rising qubit count. MeaNet, starting with a RMSE of 0.077, also sees
a significant increase, culminating in a RMSE of 0.674 at 10 qubits. In stark
contrast, MC-Net demonstrates remarkable stability and scalability. It starts
with an impressively low RMSE of 0.018 and only marginally increases to 0.032
as the number of qubits reaches 10. This distinct scaling behavior highlights
MC-Net’s superior performance in managing larger quantum systems, showcasing a
significantly more accurate performance than both CS and MeaNet and proving to
be the most dependable method in preserving precision as the system’s
complexity increases.

Figure M9: The performance of classical shadow based cross-platform
verification (CS), MeaNet and MC-Net concerning the number of qubits of the
system. The results are collected when fixing $S=450$ and $M=500$.

###  E.2 Ablation study on the measurement data aggregation

We discuss the effect of the architecture of the MC-Net’s measurement branch
on the performance. Recall that the measurement modality after $M$ snapshots
comprises a collection of $M$ measurement vectors
$\\{\bm{v}_{m}^{t}\\}_{t=1}^{M}$. To process this format of measurement data,
we introduce two strategies: early average and lazy average. In the early
average strategy, we initially calculate the average measurement vector
$\bm{\bar{v}}=\frac{1}{M}\sum_{t=1}^{M}\bm{v}_{m}^{t}$. This averaged vector
is subsequently fed into a neural network for further processing. This
approach is advantageous in terms of memory and computational complexity,
which remain at manageable levels even when dealing with a large number of
measurements. However, it focuses solely on the statistics of multiple
measurements, disregarding individual measurement results. Conversely, in the
lazy average strategy, we regard each measurement vector, $\bm{v}_{m}^{t}$, as
a point within the Hilbert space, offering a partial description of the
quantum state. It is important to note that a sufficient number of these
measurement vectors can collectively define a unique quantum state. With this
respect, each measurement vector is individually processed by a neural
network. Following this, we calculate the average over these processed
measurement vectors to obtain the measurement feature. The schematic
representation of these two schemes is depicted in Fig. M10.

Figure M10: Two strategies to process measurement modality. The label ‘NN’ is
the abbreviation of neural networks. (a) Early average. (b) Lazy average.

We examine the cross-platform fidelity prediction results of these two schemes
on platforms with real device noise data, which is the same as the dataset
introduced in the main text. To ensure a fair comparison, we maintain
consistent numbers and shapes of weight matrices for each layer in both
schemes. For the early average strategy, we selected a parameter shape for a
5-layer fully connected network as follows:
$[(32,64),(64,128),(128,256),(256,256),(256,256)]$. On the other hand, For
lazy average strategy, the lazy average strategy involved $3$ layers of 1D
convolution with the shape $[(32,64),(64,128),(128,256)]$ prior to the pooling
layer, and $2$ fully connected layers with shape $[(256,256),(256,256)]$ after
pooling. Throughout the training process, both schemes shared the same hyper-
parameters, including the learning rate, optimizer, and batch size, ensuring a
consistent evaluation and comparison.

The results are depicted in Fig. M11. In the left panel, we observe the
training and test performance of MeaNet using two strategies. While the early
average and lazy average strategies yield similar MSE levels on the training
dataset, the lazy average strategy results in a substantially lower MSE for
the test data, specifically $1.5\times 10^{-3}$ compared to the $2.7\times
10^{-3}$ achieved by the early average. Analyzing the dynamic training
process, it is evident that MeaNet with the early average strategy fails to
exhibit effective training, with training loss and test performance remaining
nearly constant. In contrast, the training loss and test error of MeaNet with
the lazy average strategy show a notable early decrease followed by a smoother
reduction. The right panel of Fig. M11 records the training and test behavior
when applying two strategies to MC-Net. It demonstrates a similar trend for
early average as observed in MeaNet. In the case of the lazy average strategy,
it achieves a smaller training loss and test MSE than MeaNet, underlining the
advantages of incorporating circuit layout data. These results emphasize that
the lazy average strategy used in this study effectively captures essential
measurement features for accurate cross-platform fidelity estimation.

Figure M11: The training and test performances of early average and lazy
average for platforms with real device noise model. The labels ‘EA’ and ‘LA’
are the abbreviations of early average and lazy average respectively. The left
panel records the MSE reached by MeaNet during training. The right panel shows
the MSE achieved by MC-Net.

###  E.3 Comparison of fusion module

The fusion module of MC-Net aims to integrate the measurement and circuit
features into a unified state representation. This integration is expected to
retain the essential information from both types of features, which
significantly contribute to the specific task, while filtering out any
unnecessary interference information present in each modality. To figure out
the impact of various fusion methods on the final performance of MC-Net, we
compare three kinds of fusion strategies: summation, concatenation, and low-
rank bilinear pooling. In the summation-based and concatenation-based fusion
methods, the trainable parameters are the weight matrix of fully-connected
layers with shapes $(256,256)$ and $(512,512)$ respectively. For the low-rank
bilinear pooling-based fusion, we set $D=D_{1}=D_{2}=D_{3}=256$ for the weight
matrices $\bm{P}$, $\bm{A}$ and $\bm{B}$, as defined in Eq. (19). The datasets
used in this ablation study are the same as those described in the main text,
including datasets with depolarizing noise and real device noise.

The numerical results are presented in Fig. M12. The left and right panels
show the performance on datasets simulated with depolarizing noise and real
device noise, respectively. It is evident that the low-rank bilinear pooling-
based fusion significantly outperforms the other two fusion models. It
achieves a one-order-of-magnitude reduction in both training and testing MSE
on both datasets. While concatenating the two feature vectors might
intuitively preserve more original information, the lack of sufficient
interaction among the elements of the feature vectors leads to poorer
performance in our task compared to low-rank bilinear pooling. These results
validate the effectiveness of the fusion model in MC-Net and inspire us to
design a more advanced fusion model to further enhance the performance of the
multi-modal learning model.

Figure M12: Comparison of various fusion models in MC-Net. The labels ‘sum’,
‘concat’, and ‘lrbp’ represent the fusion methods of summation, concatenation,
and low-rank bilinear pooling respectively. The left and right panels display
the performance on datasets simulated with depolarizing noise and real device
noise respectively.