Beyond Circuit Connections: A Non-Message Passing Graph Transformer Approach for Quantum Error Mitigation
Authors: Tianyi Bao
Xinyu Ye
Hang Ruan
Chang Liu
Wenjie Wu
Junchi Yan
Venue: ICLR 2025 Poster (2025)
Keywords: Quantum Error Mitigation; Graph Transformer
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Despite the progress in quantum computing, one major bottleneck against the practical utility is its susceptibility to noise, which frequently occurs in current quantum systems. Existing quantum error mitigation (QEM) methods either lack generality to noise and circuit types or fail to capture the global dependencies of entire systems in addition to circuit structure. In this work, we first propose a unique circuit-to-graph encoding scheme with qubit-wise noisy measurement aggregated. Then, we introduce GTranQEM, a non-message passing graph transformer designed to mitigate errors in expected circuit measurement outcomes effectively. GTranQEM is equipped with a quantum-specific positional encoding, a structure matrix as attention bias guiding nonlocal aggregation, and a virtual quantum-representative node to further grasp graph representations, which guarantees to model the long-range entanglement. Experimental evaluations demonstrate that GTranQEM outperforms state-of-the-art QEM methods on both random and structured quantum circuits across noise types and scales among diverse settings.

Published as a conference paper at ICLR 2025
BEYOND CIRCUIT CONNECTIONS: A NON-MESSAGE PASSING GRAPH TRANSFORMER APPROACH TOWARDS QUANTUM ERROR MITIGATION
Tianyi Bao Xinyu Ye Hang Ruan', Chang Liu', Wenjie Wu', Junchi Yan
1. Sch. of Computer Science & Sch. of Artificial Intelligence, Shanghai Jiao Tong University Shanghai Innovation Institute [styl105, xinyu-ye, zzrhCi, only only-changer, wen jan jsjuu. edu
ABSTRACT
Despite the progress in quantum computing, one major bottleneck against the practical utility is its susceptibility to noise, which frequently occurs in current quantum systems. Existing quantum error mitigation (QEM) methods either lack generality noise and circuit types fail to capture the global dependencies of entire systems in addition to circuit structure. In this work, we first propose unique circuit-to-grape encoding scheme with qubit-wise noisy measurement aggregated. Then, we introduce GTranQEM, a non-message passing graph trans- former designed to mitigate errors in expected circuit measurement outcomes effectively. GTranQEM are equipped with a quantum-specific positional encoding, a structure matrix as attention bias guiding nonlocal aggregation, and virtual node to further grasp graph representations, which guaran- tees to model the long-range entanglement. Experimental evaluations demonstrate that GTranQEM outperforms -ath-offthhe-at QEM methods on both random and structured quantum circuits across noise types and scales among diverse settings.
quantum--eppesennian
INTRODUCTION
Quantum computing promises transform fields like cryptography (Gisin al., 2002) and machine learning (Biamonte et al. 2017) by solving problems intractable for classical computers, However, during the Noisy Intermechate-Scatt Quantum (NISQ) era (Brooks 2019), noises significantly impede its practical realization, limiting the performance and reliability of current quantum systems. While Quantum Error Correction (QEC) (Calderbank & Shor 1996. Gottesman 1997 Terhal 20155)
offers theoretical framework for correcting errors through syndrome extraction and adaptive state updates, it usually requires substantial number of additional qubits. This qubit overhead makes large-scale QFC implementations unfeasible with existing experimental hardware (Cai et al. 2023). Accordingly, Quantum Error Mitigation (QEM) has emerged as an alternative. Unlike QEC, QEM aims to produce near-noise-free results by algorithmically reducing noise-induced biases through post-processing, most without the need for extra qubits (Kandala et al.) 2019 [Bravyi et al. 2022
Daley al., 2022 Kim et al., 2023 Liao et al. 2024). These strategies are crucial steps toward achieving tear-term quantum utility and surpassing the capabilities of classical supercomputers.
Despite their promise, traditional non-learning-basee based QEM methods face practical limitations. For instance, Zero-Noise Extrapolation (ZNE) requires multiple executions of the same circuit at dif- ferent noise levels to estimate the zero-noise limit, leading to exponential sampling overheads (Cai) et al. 2023). Clifford Data Regression (CDR) (Czarnik et al. 2021) |) usually imposes restrictions on circuit structures by limiting the number of non-Clifford gates, and Probabilistic Error Cancellation (PEC) Van den Berg et al. 2023) can be susceptible to certain types of noise in the system.
Recent advances in machine learning-basee QEM methods have shown improved efficiency (Liao) et al. 2024) and insensitivity to both circuit structure and noise types (Kim et al. 2020). However, most of them fail to effectively encode the structural information of quantum circuits, such as the
*Correspondence author, Equal contribution. Work in part supported by NSFC 62222607.

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mitigators based on multilayer perceptrons (MLPs) (Kim al. 2020) ordinary least squares (OLS) (Liao et al., 2024). While those based on graph neural networks (GNNs) can model the topology of quantum circuits by representing them as graphs, they generally focus solely local interactions through message-passingg mechanisms that update node representations based on local neighborhoods but fail to capture the intrinsic nonlocality of global quantum systems (Bell, 1964 Brunner et al., 2014). Furthermore, a common drawback of these ML-based models is their strict reliance on ideal expectation values of training circuits as labels within the current paradigm, which can be challenging to obtain without classical simulations (Cai et al. 2023).
To address these challenges, we propose novel machine saaring-baseen QEM method beyond the circuit connections, capable of generalizing across diverse noise types and circuit structures. Unlike the approach of Kim al. (2020), we develop circuit-to-graph encoding scheme that effectively captures the structural information of quantum circuits. Compared to Liao et al. (2024), we introduce a non-message-pasee passing graph transformer to handle global context within quantum systems in addition to circuit structure, cireumventing the limitations of traditional GNNs. Furthermore, inspired by the mirror techniques for customized benchmarking (Proctor et al. 2022), we design a data augmentation scheme termed circuit inverse composition, which reduces dependence on ideal expectation values of training data and refines the learning framework. Overall, our main contributions are as follows:
1) Graph Encoding of Quantum Circuits: We introduce novel cireuit-to- -ggapp encoding scheme that effectively captures the structural information of quantum circuits. Circuit operations are rep- resented nodes with appropriately dimensioned features, qubit-wise noisy execution outcomes are embedded into edge features, and the overall system structure is encapsulated using a proposed quantum positional encoding.
2) Quantum-Native Graph Transformer: We tailor graph transformer for QEM that captures both the eircuit structure and intrinsic quantum nonlocality without relying on message passing. By leveraging multi-head attention mechanisms guided by structure matrix encoding the graph topology, it effectively captures long-range correlations within the quantum system. We also introduce a learnable Quantum Repressttereeeetatiivve (QCR) node that adaptively aggregates global graph information. After training, the QCR node's representation serves as the input to the mitigator.
3) Refinement of the Learning Framework: We devise data augmentation technique that constructs training data by composing circuits with their inverse circuits with random initial state assigned. It reduces the dependency on ideal expectation values in the training data, thereby refining the framework for Sarming-based models and extending their applicability for larger quantum systems.
4) Extensive Experimental Validation: We perform comprchensive experiments to assess the effectiveness of our proposed method. Our model demonstrates superior performance over both classical and machine soaming-basedd approaches random and Trotterized circuits. Additionally, demonstrate that GTranQEM preserves effectiveness while training without circuits' ideal expectation values, underscoring its practical applicability.
PRELIMINARIES
2.1
BACKGROUND ON QUANTUM ERROR MITIGATION
Here, we briefly describe the key background related to QEM. Readers are referred to the seminal textbook (Nielsen & Chuang 2002) for comprehensive details on quantum computing.
Quantum State Vector, is represented by a state vector (i.c., ket) in a complex Hilbert space, encapsulating all the information about quantum system. For single qubit, residing in 2- dimensional Hilbert space, the state vector is expressed as follows, with normalization |a|2+ [ = 1:
|v) = a00) + B1
where [0) and [1) are the computational basis states, and a, 3 E C. In an n-qubit system, the state vector (16) 2² is defined in a 2n dimensional Hilbert space and can be written as:
|+) ai J0- 0) + 02 10 00... 1) +
a2-
1),
where the normalization condition is given by Σ² |a,12 1, and 10 I0... 0) 10)00 0n represents basis state. Quantum states are manipulated using quantum circuits (QC), which are constructed by applying sequence of quantum gates.
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Expectation Value in Quantum Circuits. Consider a quantum circuit represented by the unitary operation U acting on an imitial quantum state (1)). To evaluate the performance and outcomes of quantum circuit, it is essential to compute the expectation value of an observable O. The ideal expectation value O is mathematically defined as:
(0) ideal = (1/ OU)))))
representing the expected measurement outcome of O when the system evolves under the unitary operation U without any noise. However, in practical scenarios, quantum systems are susceptible to various noise and errors due to decoherence, imperfect gate operations, and environmental dis- turbances. Consequently, the obtained expectation value, denoted (O)ndiy, deviates from the ideal value with a noise-induced bias (Cai et al. 2023). The primary objective of QEM is to recon- struct (0) ideal from the noisy measurements (0) nony, thereby enhancing the accuracy of quantum computations without the overhead of full quantum error correction.
2.2 QUANTUM DEVICE NOISE
Depending on whether the environment of the quantum system provides memory that retains cor- relations across system-environment interactions, the noise can be classified into Markovian and non-Markovian errors (Zhang et al. 2024).
A process that changes system's state P - p' is considered Markovian if p' depends only p. If the error associated with a particular gate is Markovian, then it is described by some map G p + p' that does not depend on the time, previous operations, any other context variable. Among Markovian errors, there are three key types: incoherent (stochastic) errors, coherent errors, and other physical errors. Incoherent errors contain bit-flip error, phase-flip егтоr, etc. A bit flip changes a qubit's state from (0) to [1] or vice versa, while a phase flip alters the relative phase between the states without changing the probability amplitudes. when depolarizing
and depolarizing
error occurs to set of qubits, random Pauli is applied to each qubit (Nielsen & Chuang 2002)
Coherent errors arise from unwanted or imperfect unitary rotations within quantum circuits (Huang et al. 2023). For slow noise processes, the error is often coherent error(Beale et al. [018)) Other physical errors, such as amplitude damping, where energy dissipates into the environment, leading to state decay(Blume- Kohout et al. 2022).
Mon-Maakovian errors have memory effects, where the noise depends on the system's 's history. One signature of non-M Mon-Markovian noise acting qubit can be obtained by analyzing the purity, of the qubit state, which is defined as p = "r[p"]. Oscillations of qubit coherence and purity over time indicate the presence of non-Markovian noise in idle qubits (Agarwal et al. 2024). The types of noise involved this paper are detailed in Appendix A10
2.3 GRAPH TRANSFORMER
Graph Transformers have emerged as a powerful tool for processing graph-structured data by com- bining the strengths of both Graph Neural Networks (GNNs) and Transformer architectures. To incorporate graph inductive biases into Transformers, many Graph Transformers integrate local message-passing mechanisms. For instance, some works incorporate sparse attention local neigh- borhoods (Dwivedi & Bresson 2020), or integrate various other types of Message-Passing Neural Networks (MPNNs) modules into their models (Chen al. 2022). However, such graph Trans- formers may at least partially inherit some of the limitations of MPNNs, such as over-smoothing, over-squashing, and expressive power limitations (Xu al. 2018). contrast, Graph Transformers without (local) message- message-passing (Ma et al., 2023) allow each node to attend to other nodes, even if they are far apart the graph structure. However, this case, learning meaningful attention scores typically requires capturing important positional structural relationships between nodes, and strong positional encodings are challenging to design for common graph data (Ma al. 2023). Nevertheless, for the quantum error mitigation problem, our proposed method only models quantum circuits as graphs to incorporate structural information but also generates effective positional encodings for each node based on the exccution order of the quantum circuit to address this challenge.
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Softhie
Gate Type
Operating Qubit
Parameters of Quantum Gatess
MutMull
Addi & Nomm
FPN
Quantum-native Positional Encoding
Linoar
Updated Node Embedding
Limer
Outiiiedio Holey
Embedding
Extre
Leamable
Structure Matrix
Edge Features
GCH Noode
Nods Postiooaa Embedding
Figure 1: Overview of the pipeline. This figure depicts the architecture of the proposed quantum error mitigation method. Left: The circuit-Ne to graph encoding process transforms input quantum circuit into directed acyclic graph (DAG) to capture the circuit structure. Each operation represented
with features including gate type, target qubit, and operation parameters, further enhanced by Quantun-Native Positional Encoding to capture the circuit's execution sequence. Right: The GTranQEM module processes the graph using multi-head attention mechanisms with structure matrix and incorporates qubit-wise noisy measurement outcomes from edge features. This updates the node representations with global dependencies built in the quantum systems.
3 APPROACH
Overview. In this section, we present our method for quantum error mitigation in detail. The pipeline is displayed in Fig. First, we introduce the problem formulation of the QEM adopted in this manuscript. Second, we propose
percuit-to-graple scheme tailored for the QEM task, which aggregates circuit information and dimensional noisy executions, with additional quantum-native positional encoding incorporated to capture circuits' structural and temporal aspects. Finally, we present non-message passing graph transformer GTranQEM to predict (O)
3.1 PROBLEM FORMULATION
aim to predict the ideal expectation value (O)ādeal of quantum circuit, formulated as a graph-level regression problem. Let D = [G X Ex, noicy y .yk)} k= be dataset of N samples, where each sample consists of quantum circuit Cje represented as directed acyelic graph (DAG) 9x = (Vk, &x), with node features X e R Ru and edge features Eg E IR & xd,, The corresponding noisy and aoisy
ideal expectation values are denoted by y/x (0) noisy E IR and yk = (0) ideal R, respectively.
Our objective is to learn function f. parameterized by graph transformer model, that maps each sample to a prediction its ideal expectation value which minimizes the mean squared error between the predicted and true ideal expectation values over the dataset:
min Σ
(y 92)
ON = (g
f((0 Xx, Es, Uk
noisy
For simplicity, we omit the subscript in the following sections and refer to a single sample with graph G (V,E), node features X, edge features E, noisy expectation value noiy,, and ideal expectation value y.
3.2 ENCODING SCHEME
We construct directed acyclic graph g (V,E), where the node set V represents the quantum operations (gates), including start and end operations for each qubit. The edge set E encodes the directed qubit flow between operations.
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Node Features. We denote N [V] the number of nodes. Each node e, V is associated with feature vector x E Rdd d' the dimension of the node features. Specifically, the node feature vector x is defined as:
x
g qc 0z),
where gi E {0, 1] Nove a one-hot encoding the gate type, with Ngyyp being the total number of unique gate types in the dataset. q, E {0. 1)%, Na is a multi-hot encoding of the qubits that the gate operates on, with Ng being the total number of qubits. 0, E IR is the normalized rotation angle for rotational gates (set to zero for non-rotational gates).
Edge Features. Each edge (oi, 0/) & represents the flow of qubits from operation e to operation 2/. The edge features etj E Rd capture the qubit-wise noisy expectation values of Pauli operators acting on the qubits carried by the edge. Formally, the edge connected to a two-qubit operation node acting qubits 91 and 42 is defined as
Cij = [Xx ) noisy (V /ooky, (Z nnossy (Xaz) noois,, (Y nooss
(Z noisy
where ) = (1b 10 @ Q & / / moiy, Q E X, Y, Z denoting a Pauli operator qubit q. If the edge carries only one qubit, the last three components are set to zero. Further details on the computation of qubet-wise expectation values provided in Appendix A9
This approach offers two main advantages. First, the method guarantees agnosticism to the noise types it encodes noisy expectation values with explicit knowledge of the noise type error rates required. Second, compared to the previous method (Liao et al. (024), which aggregates only the circuit-level noisy expectation value (7%0N.)noiss as scalar global feature, our approach includes multi-dimensional qubit-level noisy information, potentially enhancing GTranQEM robustness.
3.3
QUANTUM-NATIVE GRAPI TRANSFORMER
3.3.1
QUANTUM-NATIVE POSITIONAL ENCODING
Graphs, though mapping quantum circuits structure naturally, inherently lack sequential order. On the other hand, in quantum circuits, gates are applied sequentially along each qubit and may act concurrently across different qubits. To capture this temporality in addition to the structure, we introduce a quantum-native positional encoding scheme.
Specifically, for each node v; E V, we assign a positional encoding P: Rdd that reflects both the layer at which the gate operates and the qubits it acts upon. Specifically, let L denote the total number quantum layers in circuit C. We assign a layer index I, (0, 1, ... L + }} of each node us where 4 0 corresponds to the start operation, 4; L + 1 corresponds to the end operation. Let Qi S {1,2, .N denote the set qubits that gate i; acts upon. For single-qubit gates, Q, = {9}; for two-qubit gates, Q, {2).92).
We define the positional encoding P, by first flattening the pair (4,, Q.) to obtain a unique ID temporal position index of each 0; and then applying the positional encoding scheme from Vaswani al. (217)) to obtain a vector representation. As p; captures the execution order (via 4)) and the qubit interactions (via Q+) of each node, it allows the model leverage both the intrinsic properties of the quantum operations, the structural context within the circuit, and the related temporal information, enabling more accurate modeling of the entire quantum systems.
3.3.2 GTRANQEM ARCHITECTURE
With the quantum-nativ positional encoding, the final node features are then given by concatenation:
X pid ER, d d' + dp.
This enriched node feature matrix X E Nxd and edge feature matrix E together are utilized as input for the GEranQEM layer, as displayed in Fig. I Extending the standard Transformer encoder (Vaswani et al. 2017), each layer consists of malti-head sele-antention mechanisms with global attention bias, followed by feed-forward networks (FFNs) for nodes, with layer normalization and residual connections (Xiong et al. 2020).
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Multi-Head Attention. Specifically, given node features xf R and edge features of E R at layer l, the attention scores are computed as:
(Q (Q xK) (
)
R. aij
+ BBy .i. E
(I)
vdx
for attention head k E fl...., H. where Qti E Rd represents the Query vector from the central node i, and K&. E Rd xd represents the Key vector from all nodes in the graph instead of neighbor nodes. d2 = d/H as in (Vaswani et al. [201)), and is a learnable scalar parameter for the global attention bias M, (we shall introduce with details in the following section). The attention weights are accordingly obtained via the softmax function:
sollmaxjev
k.f.
(2)
For edge features, we first update E via ej oW + be). (ti 02) where We E Rdxd, and b. denotes projection weight matrix and bias. o-)) denotes an activation function. To employ qubit-wise noisy measurement outcomes built-in E to assist the ideal expectation value prediction, we browse to incorporate egde features towards its source nodes.Namely, the graph-wise edge attributes for node j is:
++1
1 Dj a((ug,))EE
Σ
i+1 is
where Dj denotes the degree of node j. The averaged edge feature e represents the mean of all edge features connected node j. which incorporates global edge information into each node's updated feature. Accordingly, node features are then updated as follows:
x-+1= = 0f
x
|| 1x
k.++1
Σ JEV
he ai
(Vkil
+ +1))
with k E IRdexd denotes the value vectors from all nodes, Of [dd((Hd)) represents a projection matrix, and denoting the concatenation operation.
Overall, this approach enables the model to integrate both node and edge information without relying on explicit message passing along the adjacency structure.
Global Attention Bias. Instead of relying on message passing, important to find way to include the graph topology for learning better representations. Previous studies demonstrate that structural information can be incorporated into attention score as a attention bias to guide the graph transformers without traditional message passing (Rampášek et al., 2023 Xia et al. 2023). Therefore, we compute a structure matrix M E IR N for the graph G, where Mij represents the shortest distance between nodes and 1 derived from the Floyd- Warshall algorithm (Floyd 1962). Namely,
distancee 0z) if distance(u;, 12)) S Maux
Mi
Maxx
otherwise,
where Mmxx denotes a maximum allowed step count.
The structure matrix M is accumulated the attention score as bias with learnable scalar weight in Eq. [ Given the prevalence quantum entanglement in most circuits, AM enables GTranQEM to grasp the global dependencies rather than focusing solely local neighborhoods. Overall, this mechanism is adaptive to QEM settings and can enhance the global awareness and the expressive power of graph transformers for the QEM problem.
The node representations will eventually pass through separate Fordard Networks with residual connections and layer normalization, followed Vaswani et al. (017).

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Quantum Cireuit-Representativ Node. To enhance the model's capacity to capture global patterns and dependencies within quantum circuits, we introduce a virtual Quantum Circuit- Representative Node (QCR node) into the graph-level regression model, which is a learnable node to capture the whole circuit representation, as shown in Fig. T The QCR node is designed to aggregate information from all other nodes via the attention mechanism, thereby preserving crucial information that might be lost through mean or max pooling operations.
Formally, we augment the node set V by adding the QCR node VOCR, resulting in an extended node set V VU {VoCR). The QCR node is connected to all other nodes by special directed edges:
= E U {(v, UOCR) by E V}.
There are no outgoing edges from the QCR node, and the feature vector of the QCR node, XOcR R2 is initialized as learnable parameter.
During the transformer layers, the QCR node participates in the self-attention mechanism, attending to all other nodes and aggregating information from the entire graph with the qubit-wise noisy information from edges. Eventually, we utilize the x of the QCR node as the input to the subsequent regression module (error mitigator) rather than applying a pooling operation overall nodes.
3.4 ERROR MITIGATOR
After passing through L. GTrannEM layers, concatenate the representation of QCR node, x Rdd with the noisy expectation value (Bncisy) obtained from the quantum circuit execution to form the input vector:
2 x DCR Uneisy
This vector z is then fed into regression module consisting of two fully connected layers with non- linear activation function in between. The regression module outputs the predicted noise-mitigated expectation value R.
EXPERIMENTS
In this section, we first propose the data augmentation schemes that refine the current framework of machine learning-basse -based QEM. We then elaborate on the experiment settings we utilized and display the performance of GFranQEM.
4.1 DATA AUGMENTATION STRATEGIES
4.1.1 REDUCING DEPENDENCE ON IDEAL EXPECTATION VALUES FOR TRAINING
As discussed in the introduction, a primary challenge in training learning-based QEM schemes is reliance ideal expectation values (EVs) as labels for training data, potentially limiting ML- based models' capability. To address this issue, we reduce the need for direct reliance on ideal EVs by constructing training circuits with known ideal outputs through inversed circuit composition. Specifically, for each circuit C' in the training set, we generate an augmented circuit C' by appending its inverse C-1 to itself, resulting in the composite circuit: Carann C-1
We initialize the circuit with a random state (wo,e). ) In the absence noise, the composite circuit Ō satisfies UU-1 = 1, leaving the state unchanged. Consequently, the ideal expectation value of the observable 7 (the Pauli-Z operator applied to all qubits) serves the training label is:
yk
ideal
(, 2 [(,,.)
It allows to obtain ideal expectation values without execut- ing the original circuits. Moreover, the augmented circuits retain structure similar to the original circuits, incorpo- rating the same types of gates. Such similarity ensures that the training data encapsulates noise characteristies comparable to those the original test circuits.
(1)
Figure 2: Visualization of the inversed circuit composition. The left half: original circuits.
Formally, the training dataset is constructed ts:Di
{ (Grs
Dk 1
noisy sideal
where each augmented circuit
k=

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20
10
Figure 3: Visualization Mitigation Performance via Trotter steps under three noise settings for 6-qubit Trotterized circuits. The symlog scale is used in the y-axis for clearer visualization.
All-t-Tratter
May-6 Max-6-Randen
Incoberent
Mived
Provider
Incobercnt
Mixed Noise
MAR
CIR
MAR
CIR
MAF.
GIR
MAR
CIR
MAR
CHR
MAR
GIR
Raw Noley
0.357450.1126
0.3237H0, IMI1
0.291 1697
00037-0007500
0.1405:40 1346
0.3538
15144
1071
0.8257
0.6567
0.9900
0:257610.2060
02208866.105
0.1501:0 2048
0.0611000423
0.4353
0LS
0.044500.0555
0.9657
004915000030
0,8000
0.84
0.5655
0.0835:06339
0.6900
0.034-2000255
0.683
0,6333
MI.P
10.0815
088500
100081
0.8867
019470-0000
0.8300
0.0091 H0,028
0.7.00
B18200 (10256
0.7300
0.0441 10000
B.6933
GNN
0,0049-0000005
0.8900
000975000618
0,3500
0.8767
0,0021-06033
0.6333
0,0933
00472
0.0733
GTeunEMM
01006000001100
0.001560:0222
022
0.9133
0.0066000226
0102000000235
0,044000003388
0.7433
Table Mean Absolute Error (MAE) and Circuit Improvement Ratio (CIR) for different methods under various noise settings and circuit configurations. The best values in cach column are highlighted in red, and the second- best values are highlighted in blue. Formulas for the metric caleulation are included in Appendix A12.2
CC Ci corresponds a graph Gk. and gldcal (Vo.. . (vo,s)) The test dataset consists of the original circuits, defined as: Diest = [ =1 This strategy enables the generation of training data for large-qubit circuits without the computational burden of simulating their ideal EVs, thereby effectively allowing the model to learn the underlying noise characteristics.
4.1.2 NOISE SCALING AMPLIFICATION
To enhance the model's 's robustness to varying noise levels, we adopt a noise amplification technique inspired by ZNE [Temme al. 2017 Li Benjamin 2017). Specifically, we scale the noise levels by factors À sampled from the interval max For each scaling factor X, the circuit is executed under
the scaled noise model obtain the corresponding noisy expectation value yi A.Boisy and qusit-wise expectation values used as edge features. This augmentation introduces diverse noise profiles into the dataset, enabling the model to generalize across different noise levels and enhancing its robustness to noise variability. Formally, for each circuit Cj in the training set, we generate augmented data points:
{ (g A.nssy
idcal)
.yk
where A (A1, A2..,, } represents the set of noise scaling factors.
By integrating these data augmentation strategies, reduce the dependence ideal expectation values for training and mitigate the model's 's sensitivity to noise with high variance, thereby better reflecting the conditions of real quantum devices.
4.2 SETUPS AND COMPETITORS
Experiment setups, computation resources, and model configurations are displayed in the Ap- pendix A12. Plenty of quantum error mitigation techniques have been developed for various circuit types and noise models (Cai et al., 2023). In this manuscript, we select baselines that are not only proven to be highly effective but also capable of handling different circuit types and noise models, similar to GTranQEM. Specifically, for classical-quantum methods, we use Zero-Noise Extrapolation (ZNE) ( (Temme et al. 2017 Li Benjamin 2017) and Probabilistic Error Can- cellation (Van den Berg et al. 2023) Additionally, we include leareing-based methods such as ML-GNN (Liao et al. 2024)) Clifford Data Regression (CDR) (Czamii et al., 202T). OLS (Liao
al. 2024). and ML.P (Kim al. 2020 Further details can be found in Appendix A11
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6 Random Provider
(b) 15 Qubit Trotber Cires Incoherent
(c) Qubit. Cire Extrapelation
Mean
Nean 9t Dev
Moon 9dd Dev
Ste Dev
[d] 4 Qubit Trotter- Noise Amplication
(e) Qubit Trotter- Coherent
(1) Qubit, No Ideal EV
Neas swi Dev
Mea
Mean
Std Dov
sui De
400 46 400
Figure 4: Performance of mitigation methods across various circuit configurations and noise settings. The x-axis displays different QEM methods, while the y-axis represents absolute eπror distribution. Panels (a) (e) use a symlog scale for clearer visualization, as error rates are low after mitigation. Each panel title indicates the specific setting. The width of each violin plot reflects data density at each value, with a broader shape near zero indicating hetter performance. Our method is shown in red, and raw noisy data is in blue.
4.3 QUANTUM CIRCUITS
The experiments mainly focus on two types of circuits: the Max-N-Random circuits and the All-N- Trotter circuits. Details for circuit and dataset configurations are provided in Appendix (A1244
Max-Qubit random circuits: We utilize the random-structure circuits to demonstrate the generality of GMranNEM. In this setting, circuits in a dataset consist of at most N, qubits and most N Ngate- The allowed gates chosen from (Pauli-X, Pauli-Y, Y. Pauli -Z, Hadamard, RX, RY, RZ, Controlled-No Controlled-7, SWAP, RZZ]. The types of, the orders of, and the qubits being operated by quantum gates are randomized within each eircuit.
Trotterized circuits: To benchmark our protocol's performance on structured circuits, we employ Trotterized brickwork circuits following the approach of Liao al. (2024). We focus on the Trotterized dynamics of the I-D transverse-field Ising model (TFIM) as described in Chakrabarti et al. (1996).
4.4 QUANTUM ERROR SETTINGS
We evaluate the effectiveness of GTranQEM using four types of noise models. (1) IBM's fake providers: We simulate noisy quantum circuit executions using several IBM fake providers, such as FakeHanoiV2,, which mimic the complex noise profiles of real quantum devices. This demonstrates the capability of GTranQEM in handling realistic quantum noise. (2) Incoherent noise setting: We construct a noise model with depolarization errors, which degrade the quantum unitary state into a mixed state, to showcase the efficacy of GTTranEEM in addressing a specific noise type. (3) Mixed noise setting: This model combines Pauli X errors, depolarization errors, and readout errors, aiming to evaluate QEM under customized, complex mixed-error scenario. (4) Coherent noise setting: This model combines coherent over-rotation with the Fake Lima Provider. (5) IBM's Brisbane We execute the 50-qubit circuits on IBM's Brisbane quantum device and obtain noisy expectation values. Configurations of each error setting are presented in Appendix A12.3.3
4.5 EXPERIMENTAL PERFORMANCE
Performance over Standard Settings. Trotterized Circuits. We first evaluated GTranQEM on 6-qubit Trotterized circuits with Trotter steps ranging from 1 to 20 with three different noises. As depicted in Fig. 3 GTranQEM consistently outperforms existing QEM techniques, maintaining lower MAE values across all Trotter steps, which highlights the robustness of our approach in mitigating emrors as circuit depth increases. Tab. demoonttrates that GTranQEMM is the only method that consistently achieves eircuit improvement ratio (CIR) over 90%. For 15-qubit circuits with incoherent noise,

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Published as a conference paper at ICLR 2025
Method
Cireuit-Seale Extrapolation
Nolse-Seale Amplitication
15-Qubit-Crreait
Coherent Noise
MAE
CIR 7
MAE 1
CIR
MAE
CIR
MAE 1
CIR
Raw Noiey
0.4
0.2958.1..6616
0.1900
0.132420.1238
ZNE & Benjamin 2017
0.352200.352
0,9300
0,090610.06200
0.9050
0,1933
0.4700
0.08
0.8667
CDK
0.08111000766
0,9500
0.066900.1036
0.018-001263
0.0826-0.1561
0.7933
PEC Vam den Berg or al, 2023
0,346420.2922
0.6000
0. 0.1966:0.1133 13888
0.7600
0.2076=0.2535
0.4100
0.1756=0.1827
0.4467
MLGNN
Line et al. 2024
0.1038==000727
0,9300
0.0425=0.0421
0.8700
0.0421=0.0299
0.6300
0.0699=0.1019
0.8167
OLS
0.120910.0009
0,9500
0.069610.0493
0.8550
0.0411=0.0312
0.6400
0.0813=0.1082
0.7633
2020
0.1:3810.8825
0.9600
0.016611.01162
0.8150
0.075110.0691
0.6700
0.0779:0.1012
0.8/67
GIranQEM
0.07881000611
0.9700
0.0325
0.8950
0.021620.0230
0.6800
0.061910.1017
0.8533
Table 2: Mean Absolute Error (MAE) and Circuit Improvement Ratio (CIR) for different methods under four advanced settings. Formulas for the metric calculation are included in Appendix A12.2
our method achieves an MAE of 0.0245, significantly smaller than other baselines, with the smallest standard deviation (std) of 0.0245, as shown in Tab. [ The error distribution of the 15-qubit circuits is displayed in Fig. 4b, where our method has the most significant concentration towards zero, showcasing its effectiveness in handling larger circuit configurations.
Random Circuits. Superior was also observed for random circuits. As presented in Tab. | the MAE and CIR indicate that GTranQEM achieves a noticeable error drop compared to baselines. For example, under incoherent noise, our approach yields an MAE of 0.0267, whereas the second-best GNN method achieves 0.0291. Fig. 4[a) visualizes the error distribution under provider noise, illustrating that our method greatly outperforms traditional QEM methods and also has notable improvement among other -baed methods.
Performance in Alternative Task Settings. Additionally, we evaluated our method under noise amplification, circuit extrapolation, and coherent noise conditions. As shown in Tab. aand Figs. 4 (c)-(e), GTranQEM consistently outperforms other mitigation strategies across these scenarios. For example, in the circuit-scale extrapolation task QEM methods are trained on shallow circuits with fewer than 15 Trotter steps and tested on deeper circuits with 16 to 20 Trotter steps GTraanEEM achieves an MAE of 0.0325 and a CIR of 0.895, These results demonstrate the robustness and generality of GTranQEM across various tasks and noise settings.
Performance with Refined Learning Paradigm. We further evaluated GTrranQEM in scenarios where ideal expectation values of original training circuits are excluded through inverse circuit composition. Two experiments were conducted: one using 6-qubit Trotter circuits subjected mixed noise and another involving large-scale 50-qubit circuits executed on a real quantum device (IBM Brisbane). As shown in Figs. (f) [ our method significantly outperforms baseline mitigation strategies and achieves a substantial reduction in error compared to raw noisy data. The error rates notably reduced towards zero, demonstrating the effectiveness of GTranQEM and our refined learning paradigm (see Appendix A12.4.2 for further analysis).
5 CONCLUSION
We have devised a wwwsaae-passssseeeee.. graph transformer, GTrranEM,, for quantum error mitigation that effectively captures both the graph structure of quantum circuits and the global dependencies in- herent in quantum systems. It takes quantum circuit graphs with built-in qubit-wise noisy information as input, generated through our devised encoding scheme. Unlike current machine learning-basee methods, our graph transformer aggregates information across the entire graph using structure ma- trix to guide the graph topology, incorporates a quantum-native positional encoding, and aggregates graph representations within a global context into a QCR node. The representation of the QCR node, which encapsulates both the circuit's 's structure and the global dependencies, then employed as input for the error mitigator, thereby reducing the impact of errors in circuit executions. The source code will be available at:
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A6 BROADER IMPACT
Our proposed method achieves nonlocal quantum error mitigation while preserving circuit topology through a non-messagge-assingg graph transformer. By modeling quantum circuits as graphs and em- ploying transformer architecture, we capture both their structural topology and global dependencies, such as Bell nonlocality (Bell, 1964; Brunner et al. (014)). This approach addresses limitations in existing machine learning-basc -baeed error mitigation methods (Kim al.) 2020 Liao et al. (024) that often overlook circuit structure or nonlocal characteristics. Enhancing the reliability of quantum computations the NISQ era (Brooks, 2019), our work contributes to advanced quantum algorithms and error mitigation strategies that fully exploit circuit topology and global entanglement. Ultimately, this advancement promotes practical quantum advantage and accelerates the integration of quantum computing into diverse scientific and industrial applications.
A7 LIMITATION
Compared to the traditional QEM methods, the ML-based QEM methods usually require the acqui- sition of the training set and training process. Offline training introduces additional time costs, but the advantage the ability to quickly infer unseen samples. On the other hand, our method uses measurement information from the circuit to construct the edge features, which may increase the measurement cost during circuit execution. However, fortunately, our circuit, we only use the results of Pauli measurements on each qubit, which are considered local observables. By employing the classical shadow algorithm (Huang et al. 2020), we can reduce the original M measurements to only log(A (M) measurements, and this reduction independent of the number of qubits in the quantum circuit. This approach allows us to address potential scalability issues as circuit sizes grow.
A8 DISCUSSION
We want to emphasize the key innovations of our paper lie in the targeted, quantum-reereesntttive design of GTraanEMM, including the following aspects:
1) Novel Encoding Scheme: We introduce a unique encoding approach for QEM. Following the standard approaches graph learning for quantum computing (Wang et al.] 2022b] Moflic al.. 2023 He et al. 2023), quantum circuits are represented as DAGss mapping each gate in the circuit to node the graph. Meanwhile, several works are designed utilize quantum circuits to learn graph data (Yan et al., [023 Wu et al.] 2024b), and these fall within the realm of quantum machine learning. However, our encoding scheme differs from previous work in several key aspects including node features with circuit attributes as well as quantum topological positional encoding, and edge features with noisy qubit-wise measurement outcomes 3.2 Overall, our encoding captures both the cireuit topology and the measurement information in a unified framework, strengthening the model's ability to perform mitigation tasks in complex quantum systems.
3) Customized Quantum-Native Non-Message Passing Graph Transformers: We are among the first to apply a non-message passing graph transformer architecture to the QEM problem. Traditional message-passing mechanisms focus on neighborhood aggregation, failing capture the long-range dependencies within the quantum system. In contrast, GTraQEM leverages the circuit's 's topology incorporated in the graph encoding, the non-message passing aggregation scheme, and a learnable QCR node to further facilitate graph-level feature aggregation, making GTraQEM particularly quantum-nan native and customized for QEM tasks.
4) Refinement of the Learning-Batt Based QEM Paradigm: Previous learning- based QEM meth- ods (Kim et al. 2020, Liao al., (2024) often rely on obtaining ideal expectation values of training circuits through classical simulation for mitigator tuning, which becomes impractical for larger circuits due to the exponential scaling of computational resources. We propose to refine the paradigm by constructing the training set with standard circuits composed with their inverse with random initial state assigned. This approach allows us to obtain ideal EVs as labels of the training circuits for supervision without circuit executions, as the initial state becomes the final quantum state. This signif- icantly reduces ML-based methods' reliance on the simulation of ideal EVs, achieving label-efficient training.
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5) Comprehensive ML Perspective and Effective Pipeline: Given the limited number of existing ML-based QEM methods, our work provides a fresh perspective by thoroughly exploring the intersec- tion of Graph learning and QEM. We propose effective and thorough pipeline for error mitigation across various circuit types and scales, highlighting its potential for wide applicability in the quantum computing community.
A9 EDGE FEATURE CONSTRUCTION
1. Construction of Pauli-Identity Operators for the Qubit. The Pauli-Identity operator 2g acting on the q-th qubit in -ubii system is constructed as:
Gg ( & Pg & Io(N.
where:
I is the 2 x identity matrix.
Pg E (X,, y. 2g} is one of the Pauli matrices acting on qubit q.
p denotes the n-fold tensor product of the identity operator.
2. Qubit-wise Noisy Expectation Values
For each qubit y carried by an edge (V,, vj), we compute the noisy expectation values set of the Pauli-Identig operators mentioned previously.
The qubit-wise wise noisy expectation value for Pauli operator 2g acting on qubit is defindd as:
(App ) poisy = Tr [panky Pg]
where moicy is the noisy state vector of the system.
Computation of Noisy Expectation Values
For each qubit q the edge (vi, Uÿ), compute:
"oisy noisy
Tr
p poisy
(700-1) X & (NNN
(Yg)
Tr
p
mnicy
83 I8 9) (100111)
noisy Tr moisy
(181011) & Z x NN -9)
4. Edge Feature Vector Assembly
Single-Qubit Edge:
the edge carries only one qubit y, the edge feature vector e:j E TR. is constructed as:
Cit [(X,) noisy, (Yg)) noisy
(Zg) yoloy 0, 0, 0]
The last three components are set to zero to maintain a consistent feature dimension.
Two- Cuvo-uuittio Edge:
If the edge carries two qubits qı and 42, the edge feature vector Cij R is constructed as:
eij [(
m2) poisy (Y,, noisy (Z noiss, (X mooss (Y /noisy (Z Jooisy
By following these steps, we construct edge features that capture essential quantum state information encoded in the noisy expectation values of Pauli operators for the qubits flowing between quantum operations. This representation is crucial for analyzing quantur circuits using graph learning techniques.
graph-based based machine
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A10 QUANTUM ERRORS
Quantum етгоr sources a critical challenge in quantum computing. These errors, ranging from coherent to incoherent and systematic to random, arise from imperfect qubit control, unwanted inter- actions, and measurement inaccuracies. In this paper, we conduct QEM experiments on incoherent enror,, coherent errors, and readout emror..
A10.1 INCOHERENT ERRORS
Incoherent errors are caused by random interactions with the environment, leading to non-unitary dynamics. These errors cause decoherence and loss of quantum information. Incoherent errors (from entanglement of the physical qubits with memoryless environment) can be modeled as random Pauli operators on the physical qubits. our paper, we involve two types of incoherent errors: bit-flip error and depolarizing error.
bit-flip channel transforms the quantum state from (0) to [1) (or vice versa) with a probability of 1 p, and has the following operators:
1 0
Eo - =o=
vD
0 1
0
E1 / pX /1
(8)
Its effect on a state is given by
T Br pp p) XpX.
The depolarizing channel is an important type of quantum noise. Imagine that we take a single qubit and depolarize it with probability p. meaning the qubit is replaced by a completely mixed state 7 7/2. The single qubit remains unchanged with probability 1 The state of the quantum system p after this noise given by:
pT
$ DE p1
(1 p)p.
(10)
we observe that for any p. the following holds:
p + YpY + 2p2
2
4
then we can obtain:
DDL[P = (1- 3p p f(XxX+ +YpY ZpZ).
(11)
4
This shows that the depolarizing channel has the set of operators 1p1-3p/41, vDX/2, /2, 2
A10.2 COHERENT ERRORS
Coherent errors arise from unwanted imperfect unitary rotations within quantum circuits, which can be modeled as:
U(0)= = 1:0
(12)
where = (01.... 0gr,) represents the magnitude of the coherent error on the N,, Pauli basis. These errors transform noiseless pure states into other pure states while preserving quantum coherence due to the unitary nature of the operations. Although they are purity-preservin coherent errors can significantly undermine the reliability of multi-qubit quantum computations.
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A10.3 READOUT ERRORS
Readout errors occur during the measurement process of quantum states. These errors manifest when the outcome of quantum measurement does not accurately reflect the actual state of the qubit. For instance, a qubit in the (0) state might be incorrectly measured as [1), and vice versa. Readout errors are particularly detrimental in quantum algorithms that rely heavily on accurate measurement outcomes for recision-making and result interpretation.
The probability of readout error can be characterized by a confusion matrix M, which quantifies the likelihood of each possible measurement outcome given the true state:
((O)0)
P(I))
M
P(O
where P(al6) denotes the probability of measuring outcome given the qubit was in state [6).
A11 RELATED WORKS
A11.1 QUANTUM ERROR MITIGATION
Traditionally, Zero-Noise Extrapolation (ZNE) widely implemented mitigation scheme. The essence of ZNE lies in deliberately introducing varying levels of noise into the quantum circuits and performing multiple exccutions of these circuits. Based on these noisy outcomes, ZNE employs extrapolation namely fitting techniques to estimate the ideal quantum state probability distribution in the absence of noise, whereby the Richardson extrapolation is a basic fitting method. However, ZNE requires high computational resources by necessitating multiple executions of quantum operations varying noise levels, leading to higher resource consumption. It also heavily relies on the accuracy of the assumed noise model for extrapolation.
Czarnik et al. (2021) proposes Clifford Data Regression (CDR) to predict the noise-free observable. The majority of con-Clifford gates in the target circuit are (approximately) replaced with nearby Clifford gates, such that classically simulable training data can be efficiently constructed, as the Clifford gates are more friendly for efficient simulation. Then linear regression model is employed that regresses the noisy expectation values of those circuits onto the ideal ones.
Kim al. (2020) employs an ordinary Neural Network (NN) and concatenated NN to learn the amount of error in the result measurement of quantum output states. However, the inputs of their NNs only consider the number of gates and ideal and noisy measurement information, [Liao et al. (2024) benchmarks four models: linear regression, random forests, multi-layer perceptrons, and graph neural networks, to minimize the sum squared error between the mitigated and ideal expectation values. They convert the transpiled circuit into acyclic graph and take it input to GNNs. However, they struggle to handle large-scale quantum circuits effectively because they train the model by mimicking other scalable QEM methods, such ZNE, which limits their model from outperforming the mimicked method. In contrast, our approach ensures scalability and leverages inverse circuits to handle large-scale problems effectively.
A11.2 GRAPH TRANSFORMER
In recent years, Graph Transformers have emerged as powerful tool for processing graph- structured data by combining the strengths of both Graph Neural Networks (GNNs) and Transformer architec- tures. This hybrid model has demonstrated strong performance across wide range of tasks, including node classification, graph generation, and molecular property prediction. Graph Transformers can be classified into message- saassaggg--sing---e. based and message-passing-ee..........s-.e approach to information propagation in graphs.
models, based on their
deessag-passsing based models integrate traditional GNN frameworks, where each node aggregates information from its neighbors. GAT /Veličković et al. 2018) is an early example, using attention mechanisms to assign weights neighboring nodes during aggregation. Graphormer (Ying et al.) 202I) extends this by adding global attention to enhance performance in tasks like molecular property prediction. GPS (Rampášek al. 2022), another variant, merges pesssng with self-attention
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for efficient local and global interaction handling. Liao & Smidt (2022) proposes equivariant graph attention transformer for 3D structures, which can be applied to tasks related to molecules and proteins (Yang al. 2022a Wu et al. 2024a).
However, graph transformers that use cressage-passingg inherit known issues of message-passingg such over-smoothing, over-squashing, and expressive power limitations, and differ significantly from Transformers used in other domains, thus making a transfer of research advances more difficult. models rely solely on the Transformer's self-attention mecha- nism, bypassing iterative neighbor aggregation. Graphi'T (Mialon et al. (02T), for instance, employs self-attention and positional encodings without traditional message- pastgggg.asing..... Graph- (Zhang
In contrast, con-message-passinng
et all.|22000) (2220) applies Transformer-styl attention directly to node embeddings, capturing long-range dependencies more effectively. GTNectee (Yun et al. 2019) focuses on hierarchical graph represen- tation using edge-wise attention, optimizing computational efficiency. For non-message-pasee paaasing learning meaningful attention scores typically requires capturing important positional or structural relationships between nodes, and strong positional encodings are challenging to design since the structure and symmetries of graph data are fundamentally different from that of other (Euclidean) domains. Nevertheless, in the field of quantum noise mitigation, we not only model quantum circuits as graphs to incorporate structural information but also generate effective positional encodings for each node based the execution order of the quantum eircuit. The proposed approach effectively addresses the challenge.
non-message-passingg
A11.3 GRAPH NEURAL NETWORKS FOR ELECTRONIC DESIGN AUTOMATION
Graph Neural Networks (GNNs) are specialized models designed to process graph-structured data through trainable parameters, which have served widespread graph applications like combinatorial
optimization (Li et al.) 2024 2025 2023) and electronic design automation (Xie et al.) 2021 Cheng jct al. 2022 Guo et al. 2022; Wang et al. 2022a). As GNN research progresses, novel architectures
are being introduced that target graph topological properties (Ye et al. 2023al b Zhao et al., 2023)
and scalability to large graphs (Wu et al. 2022 2023a b). Given the inherent (hyper-) structure of circuit netlists, GNNs are naturally suited for applications in Electronic Design Automation (EDA), particularly in the context of Power, Performance, and Area (PPA) estimation. Early GNN models, such as GCN (Defferrard et al. 2016 Kipf & Welling 2017), extend the concept of convolutions from low-dimensional regular grids to high-dimensional irregular domains, while GAT' (Velićkoićć et al., 2018) introduces attention mechanisms learn adaptive weights for neighboring nodes. GCNII (Chen et al. 2020) addresses the over-smoothing issue through initial residual connections and identity mapping. whereas GINE (Hu et al., 2020), an extension of GIN (Xu et al. 2018). incorporates edge features to enhance expressiveness and was originally designed to tackle graph isomorphism challenges. Despite their success, these models face limitations when applied to circuit graphs with long timing paths due to restricted local receptive fields, which could lead to over-smoothing problems.
To overcome these limitations, recent works have developed GNN variants specifically tailored for EDA tasks. Net² (Xie et al., 2022 leverages graph attention networks to estimate net length during the pre-placement phase. CircuiGNN (Yang et al. 2022b) represents circuits as heterogeneous graphs and introduces a topo-geom message-passingg paradigm to predict wire length and congestion. Timing- GCN (Guo et al.) 2022) models GNN inspired by timing engines, enabling effective slack prediction in pre-routing stages. LHNN (Wang et al. 2022a) employs a lattice hyper-graph neural network for routing congestion prediction. Finally, GTN (Fan al. 2024) proposes Seaahh-Tranffrmer
architecture as a surrogate model for circuit evaluation, achieving significant accelcration in circuit topology optimization without compromising performance.
A 12
EXPERIMENT SETTINGS
A 1
SETUPS
The quantum circuits in our experiments are executed on IBM's backend. Our graph-learning QEM algorithm is implemented system running Ubuntu 16.04, with CUDA 11.0. PyTorch 1.13.0. and PyTorch Geometric 2.3.1. Most experiments are conducted on an NVIDIA 4090 GPUs with 24GB of memory.
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A12.2 METRICS CALCULATION
To evaluate the performance of our proposed model, we employ two key metrics: Mean Absolute Error (MAE) and Circuit Improvement Ratio (CIR). These metrics provide comprehensive assessment of the model's effectiveness in mitigating errors within quantum circuits.
A12.2.1
MEAN ABSOLUTE ERROR (MAE)
The Mean Absolute Error (MAE) quantifies the average magnitude of errors between the predicted val- and the actual ground truth values, without considering their direction. It is defined mathematically as:
MAE-
D
Vil
(13)
where is the total number of quantum circuits evaluated. J. represents the predicted value for the i-th circuit. y/ denotes the actual ground truth value for the i-th circuit.
lower MAE indicates higher prediction accuracy, reflecting the model's ability to closely approxi- mate the true values of the quantum circuits.
A12.2.2 CIRCUIT IMPROVEMENT RATIO (CIR)
The Circuit Improvement Ratio (CIR) measures the proportion of quantum circuits that have been successfully improved through the error mitigation process. is calculated using the following formula:
Number of Improved Circuits
CIR
x 100%
(14)
Total Number of Circuits
where Number of Improved Circuits refers to the count circuits that exhibit enhanced performance post-error mitigation compared their original state. Total Number of Circuits is the overall number of circuits subjected to error mitigation.
A higher CIR signifies of circuits have benefited from the proposed enhancements.
effective error mitigation strategy, indicating that greater percentage
A A12.3 DATA PREPARATION
We provide detailed descriptions of the experimental settings used in our work to facilitate future reproduction. We cover the circuit datasets, noise models, experimental configurations across different quantum system scales, and the hyperparameters used in our method.
A12.3.1 CIRCUITS CONSTRUCTION
Trotterized Circuits Construction. We focus the Trotterized dynamics of the one-dimensional transverse-field Ising model (TFIM) as described in Chakrabarti et al.] (199)). The Trotterized circuits used in our experiments simulate the dynamics of spin chain governed by the Hamiltonian:
== > 2, + Σ X, = -fzz + hhix.
where J represents the exchange coupling between neighboring spins, and denotes the transverse magnetic field. We generated multiple circuit instances by randomly assigning the parameters J, h, and the evolution time t, with coupling strengths uniformly sampled from the paramagnetic phase (J h). The number of Trotter steps varied within specified ranges, determining the depth of the circuits.
Random Circuits Construction We utilize the ality of GTranQEM. In this setting, circuits in dataset consist of at most N, qubits and most N Ngate- The allowed gates are chosen from (Pauli-X, Pauli-Y. YY, Pauli-Z, -Z, Hadamard, RX, RY, RZ, Controlled- NOT, Controlled-Z, SWAP, RZZ]. The types of, the orders of, and the qubits being operated by quantum gates are randomized within each circuit. Rotation angle for rotation gates are randomly selected from [0, 2T]
circuits to demonstrate the gener-
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A12.3.2 CIRCUIT DATASET CONSTRUCTION
We constructed various circuit datasets to evaluate the performance of GTranQEMM across different system sizes and experimental settings. The datasets are categorized based on the circuit types and scales:
Max-6-Random and 11-6-Trotter Datasets For the max-6 max-6-random and all-6-trotter datasets, we generated circuits with up qubits. The training consists of 750 circuits with maximum length of 150 gates, randomly selected from the allowed gate list. We used an additional 150 circuits for validation and 300 circuits for testing. These datasets are used evaluate the performance of GTranQEM on small-scale quantum systems.
All-15-Trotter Dataset The all- 15-ttrotttr dataset comprises circuits with 15 qubits. We employed 260 circuits for training, 40 circuits for validation, and 100 circuits for testing. This dataset allows us to assess the scalability GTranQEM to medium-scale quantum systems.
All-S0-Trotter Dataset For large-scale systems, we constructed the all all-50-trotter dataset, consisting of 50-qubit circuits. We created 20 Trotter circuits with Trotter steps ranging from to 3. Each circuit was initialized with 10 random initial states, resulting in 200 training circuits. Two original circuits were used for validation, and the remaining 18 circuits were used for testing. This dataset was executed on real quantum device FFMM_rrisbane
evaluate the practical applicability of GTranQEM.
Circuit-Scale Extrapolation Dataset We constructed dataset with circuits having Trotter steps ranging from to 20. We used 260 circuits with Trotter steps from to 15 for training, 40 circuits for validation, and 100 circuits with Trotter steps from 16 to 20 for testing. This setting allows us to evaluate the ability of GTranQEM to extrapolate to deeper circuits beyond those seen during training.
Noise Scale Amplification Dataset We applied noise scaling factors A {1, 2,, to augment the training set without generating additional circuits. We used 200 circuits for training, resulting in 600 data points after noise scaling, and 200 circuits for testing. This dataset allows us assess the robustness of GTranQEM to varying noise levels.
6-Qubit No-Ideal Setting Dataset For the 6-qubit no-ideal setting, constructed training dataset of 400 inverse-composed circuits, each with one random initial state. We used 100 circuits for validation and 300 circuits for testing. This dataset was used to evaluate the performance of GTranQEM when ideal expectation values are not available.
Remarks. In all experimental settings, we ensured that there was no overlap between the training, validation, and testing datasets. This separation is crucial for obtaining unbiased evaluations of the model's performance.
A12.3.3 NOISE SETTINGS
We evaluated the performance of GTranQEM under various noise settings to best testify the perfor- mance of GTranQEM. The noise models used in our experiments are:
1) Fake Provider Setting FakeitanoiV2 is used, which emulates the noise characteristics of the IBM Hanoi quantum device.
2) Incoherent error-only setting Incoherent noise was simulated by applying a combination of Pauli X and depolarizing errors on all single-qubit and two-qubit gates. The error rates followed the statistics of the Sycamore quantum device (Arute et al. 2019). as listed in Table S
3) Mixed Noise Setting Mixed noise combines incoherent noise with additional error sources to simulate a more realistic noise environment.
4) Coherent Noise Setting Coherent noise introduced by adding systematic over-rotation errors to the double-qubit gates (CX, CY, CZ, and Swap gates) with an average over-rotation angle of 0.02T with combination of the noise model extracted from the Fake Lima provider.
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Figure 5: The topology of qubits of IBM Brisbane.
Qubit Trotter Circs via Incoherent
6 Qubit Titter Cies via Mix naises
6 Qubit Trotter Circs via Provideer
1e?
8
Figure 6: Visualization of the distribution of AEs for all-6-trotter circuits under three noise settings. Distributions closer represent better performance.
Table Error rates for Incoherent and Mixed noise settings based on the Sycamore quantum device Arute et al. 2019).
System
Qubits Single
Two-Quhit
Sycamore Anite
5) IBM's Brisbane For experiments real quantum devices, executed circuits IBM Bri sbane, 127-qubit quantum computer provided by IBM. The details of the quantum device follows:
1. Topology: IEM.Brisbare has 127-qubit architecture. We utilized the Qiskit quantum cloud platform, which optimizes the 50-qubit circuits and maps them onto the 127 qubits using the generate pwwset__ppaaae sasee___assmmaaaerreeeeee function. The topology of these qubits is illustrated in Fig.
2. Coherence Times:
The median relaxation time Ti is 213.21 μas and the median dephasing time T2 147.61 ps, indicating relatively long coherence times suitable for executing deep quantum circuits.
3. Fidelities:
TBM Bri sbane provides the metric 2Q Error (layered), representing the av- erage two-qubit error rate per layered gate (EPLG) for a 100-qubit chain. The 2Q Error (layered) for IBM.Brisbare is 2.06 x 10
4. Additional Parameters: The median echoed cross-resonance (ECR) error is 8.340 10-3 the median single-qubit gate (SX) enror is 2.345 x 10 10-4 and the median readout error is 1.370 x 1
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A12.4 EXPERIMENTAL CONFIGURATIONS
We conducted experiments across three scales of quantum systems -aaiiss-mmllllalllll (4-6 qubits), medium-scale (15 qubits), and large-scale (50 under both standard and non-standard settings.
A 112.4.1 STANDARD SETTINGS
We evaluated GTranQEM under three standard settings saaiing--mmaal-sceee small-scale, medium-scale, and large-scale systems- to demonstrate its effectiveness across different scales. For small-scale systems, we used datasets like max-6 max-6-random. all-6-trotter, and all-4-trotter under various noise conditions, including fake provider, incoherent noise only, mixed noise, and coherent noise. In medium-scale systems, assessed the scalability of GTranQEM using the all- all-/5-trotter dataset under incoherent noise only. For large-scale systems, we validated GTranQEM on real quantum devices by performing error mitigation on the all-50-trotter dataset executed on I3M.Bri sbane.
A12.4.2 NON-STANDARD EXPERIMENTAL SETTINGS
To further demonstrate the generality and robustness of GTranQEM, conducted experiments under non-standard settings.
Circuit Depth Generality We testify the generality of GTranQEM by performing mitigation tasks over the circuit-scale extrapolation datasets. Training conducted circuits with Trotter steps from to 15, and testing was performed on circuits with Trotter steps from 16 to 20. The results demonstrated in Tab. 2 and Fig. (c), demonstrate GTranQEM's capability of generalizing to deep circuits.
Evaluation Without Ideal Expectation Values We assessed the performance of GTranQEMM in the absence of ideal expectation values (EVs) by employing the 6-qubit no-ideal setting dataset to evaluate the effectiveness of our proposed circuit inverse composition scheme.
Initially, conducted experiments the 6-qubit dataset without ideal EVs provide insights into error mitigation for larger quantum circuits. The results, presented Fig indicate that GTrraQEM performs effectively under these conditions. Notably, GTranQEM achieves a superior performance over the traditional QEM methods showing the effectiveness of our circuit inverse composition paradigm; GTranQEM outperforms other machine learning-basee -ased models that operate under the same paradigm in this setting, highlighting the efficacy of our model architecture. It is important to mention that the 6-qubit setting serves primarily validate the proposed paradigm, as ideal EVs are readily obtainable for such small quantum systems. Indeed, GTranQEM demonstrates strong performance when trained with ideal EVs as labels (see Figs [n)
When extending the mitigation to larger qubit systems, the advantages of our paradigm become more pronounced. As depicted in Fig. (f), GTranQEM surpasses Zero-Noise Extrapolation (ZNE), which is among the few traditional QEM methods feasible within reasonable execution times for large-qubit circuits with general structures. Although multilayer perceptron (MLP) and graph neural network (GNN) models can also be utilized under paradigm for larger systems, they exhibit inferior performance compared to GTranQEM, underscoring the effectiveness of our model design.
A12.4.3 HYPERPARAMETERS
The hyperparameters of the GTranQEM model layers for each experimental setting are provided in Table 4. These parameters were chosen based preliminary experiments to optimize performance for each specific setting.
A13 TIME ANALYSIS
Comparing the computational time between machine learning (MLe- and classical quantum mitigation (QEM) methods is challenging due to inherent differences in their workflows.
Classical QEM approaches typically handle quantum circuits individually. potentially involving extensive mitigation time due to cireuit-specifie tuning, multiple eircuit executions, or additional
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Qubit Trotter No Ideal
Moon
Figure 7: Distributions of AFs for no ideal EV setting of all-6-trotter circuits under the mixed noise setting.
Table 4: Hyperparameters of GTranQEM layers for each experimental setting.
Experiment Name
Number of Layers
Hidden Dimension
All-6-Trotter, Mixed Noisc
24
-ll-t-rrotter, Incoherent Noisc
128
All-6-Trotter, Provider Noise
32
Max-6-Random, Mixed Noise
64
Max-6-Random, Incoherent Noise
64
Provider Noise
2
64
All-15-Trotte Incoherent Noise
256
Cireuit-Scale Extrapolation
64
Noise Amplification
64
6-Qubit, No Ideal Expectation Values
64
50-Qubit, No Ideal Expectation Values
512
Table Training and inference times for different methods over various datasets. The times for ZNE CDR correspond to the mitigation times applied to the same number of circuits during inference. All quantum circuit simulations executed via IBM Qiskit. GTraQEM is implemented on computing system running Ubuntu 18.04.6 with CUDA version 12.1. The majority of experiments performed NVIDIA GeForce RTX 4090 GPU with 24 GB of memory.
4-qubit, 300 circuits
6-qubit, 300 circuits
15-qubit, 100 circuits
Method
Training
Inference
Training
Inference
Training
Inference
GTraQEM
21.90
0.1760 s
38.70
0.1868
18.64 s
0.121 s
OLS
3.42 s
0.0734 s
5.53
0.0864
2.21
0.022 s
ML-GNN
31.57
0.1473
43.54
0.1916
26.41 s
0.104 s
MLP
25.57
0,0896
32.71
0.1037
15.94 s
0.098
ZNE
n/a
548.78
n/a
782.03
n/a
2166.46
CDR
n/a
1841.34
n/a
1796.65
n/a
17405.38
qubit overheads. Moreover, their performance is often significantly influenced by the number of qubits and the scale of the circuits.
In contrast, MI-based approaches consist of two primary stages: model training and inference. Once the model trained, inference (i.e., the actual error mitigation) can be performed quickly. For instance, after fine-tuning GTraQEM for 6-qubit Trotter circuits executed under the Fake Hanoi Provider, can mitigate the results new 6-qubit Trotter circuits on that device in negligible time. Consequently, direct and fair comparison between these two types of methods is not straightforward.
To provide comparative perspective, we present in Table 5 the training and inference times required by each ML-based method, alongside the mitigation times for Zero-Noiss Extrapolation (ZNE) and Clifford Data Regression (CDR) when applied to the same number of circuits during inference. The table demonstrates that GTraQEM achieves computational efficiency comparable to other MI--based
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Sid
0,037
0.233
0.0547
0.0467
0,9033
3
0.0322
0.0250
0,9100
0.0425
0:0364
0,9233
5
0.0444
0.0279
0,8433
6
0.0547
0.0467
0,9033
7
0.0224
0.3261
0,8907
8
0.0660
0.4252
0.9207
0.0488
0,0328
0,8867
10
0.0059
0,0305
0,9200
0.0373
0,030
0.8967
0.0256
14
0.4334
13
0.0272
Table 6: Ablation study results showing the effect of each module on the QEM performanaec The check marks indicate the inclusion of the respective module.
methods while delivering superior mitigation performance. While GTraQEM's inference time is marginally slower than that of the MLP and ML-GNN models, all three methods achieve inference times within one second, rendering the differences practically insignificant.
A14 ABLATION STUDIES
We modify the classical graph transformer architecture with quantum-native modules to construct graph transformer specialized for QEM tasks. To assess the impact of each proposed module, we conduct comprehensive ablation studies focusing on the quantum-nan -attiv positional encoding (PE), qubit-wise edge features, the Quantum Circuit Representation (QCR) node, and the message passing (MP) scheme.
For the experiments, we utilize 6-qubit Trotter circuits subjected to mixed noise damage to perform the error mitigation task. As a baseline, we employ the Message Passing Neural Network (MPNN) Gilmer et al. (2017) standard graph transformer model without any quantum adaptive modules for comparison. We systematically add or remove each module to observe its influence on performance.
When the QCR node is turned off, replace it with standard graph mean pooling layer to aggregate node features. Additionally, when the message passing scheme disabled, we use the standard adjacency matrix instead of the strueture matrix guide the training process.
Tab. summarizes the results of our ablation studies. Overall, the inclusion quantum- native modules significantly enhances the performance of our model, GTranQEM,
QEM tasks. Specifically, GTranQEM achieves a Mean Absolute Error (MAE) of 0.0315 + 0.0272. noticeably outperforming the baseline MPNN. which attains MAE of 0.0416 + 0.0347. We also report the highest Circuit Improvement Ratio (CIR), demonstrating the effectiveness of our approach.
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Reviews:
[{"review_id": "lVixPOGTRg", "reviewer": "Reviewer_6ert", "summary": "The authors put forward a new graph-transformer method for quantum error mitigation. By not using message passing, the approach has the potential to better capture the long-range correlations created by quantum circuits.", "questions": "Why are there no comparisons to other techniques in the 50-qubit data?\n\nWhat was the total wall clock time spent running each error mitigation method in your simulations?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "Strengths:\n   - It is a nice application of graph transformers. While it’s pretty straightforward, their approach works better than other GNN-based quantum error mitigation methods that I’ve seen.\n   - Pretty good results on simulated data. While a lot of the error bars overlap, the new method regularly outperforms competitors on many of the toy error models.", "weaknesses": "REVISION: Many of my concerns have been addressed.\n\nWeaknesses:\n   - Lots of misleading or slightly incorrect claims in the background section. See below.\n   - Simulations leave a fair bit to be desired. In particular, they don’t use any error models with coherent errors, which are particularly pernicious kinds of errors because they’re effects depend intimately on the unitary implemented by a quantum circuit (e.g., coherent errors can cancel). The IBM fake providers use an error model that models gate errors as a depolarizing channel followed by thermal relaxation (T1 decoherence). Their incoherence-only error model only features depolarizing noise, which is probably the easiest to mitigate\n   - No head-to-head comparisons on experimental data: They ran 50-qubit circuits on IBM Brisbane but then didn’t compare their new approach to other methods (if they did, I’m sorry! I missed it in the paper).\n\nMiscellaneous remarks:\n   - The circuit-to-DAG encoding (or a very similar encoding) used in this paper was used earlier in this paper: “QuEst: Graph Transformer for Quantum Circuit Reliability Estimation.” You should cite it.\n\nHere’s a list of the misleading claims in the paper: \n   - “Quantum Error Correction (QEC) (Calderbank & Shor, 1996; Gottesman, 1997; Terhal, 2015) first offers a theoretical solution by fully correcting quantum errors at the hardware level, but its implementation demands impractically qubit overheads and complex operations (Cai et al., 2023).” I don’t think that QEC corrects errors at the hardware level. Most QEC protocols (e.g., quantum error correcting codes) require syndrome data to be extracted and then processed by a classical co-processor. The quantum state is then adaptively updated (at least when running a circuit that uses a universal gate set) in response to the results of the syndrome data analysis. I’d change this statement to better reflect how QEC works.\n     I also don’t think it’s fair to say that QEC is “impractical.” Large-scale fault-tolerant quantum computation is certainly currently infeasible, but lots of teams are working on making it a reality! I’d re-phrase this to say that large-scale fault-tolerance quantum computation is currently well beyond the capabilities of our experimental hardware. \n   - “ Overall, QEM provides a feasible approach to enable imperfect quantum systems to produce reliable outcomes (Kandala et al., 2019; Bravyi et al., 2022; Cai et al., 2023), which is crucial for achieving practical quantum supremacy over classical supercomputers (Daley et al., 2022; Kim et al., 2023).” So, most quantum error mitigation methods also have an exponential-in-the-qubit-count overhead. So it’s not clear if they’ll offer a path to quantum computational supremacy. Past works, such as in the Kim et. Al. Paper, have only shown hints at quantum advantage, which is a much weaker claim. This sentence should be rewritten to reflect this.\n   - “ Machine learning-based QEM methods have recently been developed, offering greater generality across various settings.” This claim is questionable and requires citations. \n   - “ Bell nonlocality (Bell, 1964; Brunner et al., 2014) demonstrates that entangled particles exhibit correlations where the measurement outcome of one particle instantaneously influences that of another, regardless of the distance between them.” This is not an accurate description of Bell nonlocality. Entanglement does not allow particles to influence each other at a distance. No information is transmitted. Instead Bell nonlocality allows for measurement distributions with non-classical correlations. \n   - “ We introduce a data augmentation technique that constructs training data by composing circuits with their inverse circuits.” Composing circuits with their inverses in order to (ideally) create the identity circuit is not a new idea. It underpins almost all randomized benchmarking algorithms (Clifford RB, Magann et. Al.). You should at least mention this.\n   - Section 2.2 gives a very non-standard overview of the noise sources in quantum computers. For instance, I’ve never heard anyone use the term “real quantum device errors” before. I would re-write the section to first talk about Markovian vs. non-Markovian errors, then step through the differences between the kinds of Markovian errors a device can experience: incoherent/stochastic errors, coherent errors, and “other errors” like amplitude damping (see “A Taxonomy of Small Markovian Errors”). N.B. I’m pretty sure that all quantum error channels also have a Kraus decomposition (this is not clear from your claim in A.7.1 when you state that incoherent errors have a Kraus decomposition).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FwVWsxCBIx", "reviewer": "Reviewer_xXV6", "summary": "The paper proposes a novel approach, GTraQEM, for quantum error mitigation. This method is based on a graph transformer that employs multi-head attention mechanisms and a learnable Quantum Circuit-Representative node instead of traditional message-passing. The authors claim that this model captures both the circuit structure and its intrinsic nonlocality, which are limitations in existing learning-based or ML-based QEM methods. Experimental results demonstrate that GTraQEM outperforms ML-based approaches on both random quantum circuits and Trotterized circuits.", "questions": "My questions address the weaknesses of the paper:\n1. Why did you choose not to include classical QEM methods such as Probabilistic Error Cancellation, Symmetry Constraints, and Purity Constraints in your experimental comparisons?\n2. Can you provide experimental or theoretical comparison of the overhead associated with GTraQEM compared to other QEM methods?\n3. How do you plan to address potential scalability issues related to the need for expectation measurements in GTraQEM as circuit sizes and complexities increase?\n4. Could you explain why there is only ZNE compared in the no idea EV cases?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. GTraQEM introduces a unique non-message-passing graph transformer approach and a circuit-to-graph encoding scheme for quantum error mitigation.\n2. The authors present a data augmentation technique, called circuit inverse composition, which reduces dependency on ideal expectation values in the training data.\n3. The paper demonstrates GTraQEM’s error mitigation performance compared to learning- or ML-based approaches on both random and structured quantum circuits across various noise types.", "weaknesses": "1. While the authors compare GTraQEM with learning- or ML-based QEM approaches, showing many promising results, classical QEM methods such as Probabilistic Error Cancellation, Symmetry Constraints, and Purity Constraints are not included in the experimental comparisons or discussed anywhere in the paper.\n2. The authors clearly demonstrate that GTraQEM can achieve better performance in noise calibration compared to learning- or ML-based QEM methods. However, they do not compare the overheads of these methods, which is crucial for evaluating QEM methods.\n3. I am concerned about the scalability of GTraQEM due to its reliance on edge features that require expectation measurements. As quantum circuits grow in size and complexity, the need for extensive measurements could result in significant resource demands, potentially limiting the applicability of GTraQEM for larger quantum systems or more complex circuits.\n4. Typo: [Czarnik, 2021a] and [Czarnik, 2021b] refer to the same paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vUHHbrLX9D", "reviewer": "Reviewer_PHEy", "summary": "This work presents a novel quantum error mitigation (QEM) method, GTraQEM, which uses a machine-learning approach. GTraQEM encodes the quantum circuit into a graph, then uses a non-message-passing graph transformer to construct a QEM mitigator. The effect is proved by 1) numerical simulations using a quantum circuit simulator; 2) IBM's quantum computer, and the authors compares GTraQEM with many other existing QEM approaches.\n\nQEM is a popular way to handle the quantum error in near-term quantum computers which are subject to noise, before a practical quantum error correction can be realized.  ML-based QEM is of great importance in the NISQ era because of its practicality in the experiment.", "questions": "A few concerning points are listed as follows, and I hope the authors can clarify these before I change my mind about this paper's decision.\n\n1. The original data/code generated by this paper should be provided to improve reproducibility. Also, there should be an explanation about why ZNE does not have any effect in Fig. 3 and Table 1.\n\n2. The y-axis in Fig. 4 is the absolute error of EV. However, panel (f) implies there is some distribution over 2.0. In fact, it should be always in the range [0,2] (considering the observable should be within [-1,1]). \n\n3. Whether the type of the circuit matter in GTraQEM? How is the generalizability of the method?\n\n4. As this paper uses an IBM's quantum computer (Brisbane), it is recommended to articulate the hardware's physical parameters to improve the soundness, e.g. the topology (including the 50 used qubits), the coherence time, the single-qubit and two-qubit fidelities during the experiment.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "1. The technique used in this paper is novel, and the results are good. Inputting the circuit structure into the graph transformer to construct ML-based QEM is intuitive, which is also proved to be effective via numerical evidence.\n2. This paper uses a practical test case (transverse-field Ising model), which is considered to be one of the most possible applications of quantum computers in NISQ-era.", "weaknesses": "1. Unclear experiment settings.\n\nThe experiment settings used in this paper are not clearly stated in the experiment section. This highly weakens the paper's conclusion. Because QEM is an area that is mainly based on experimental/numerical evidence, it is vital to clarify the methods and the settings when the authors do not provide any codes.\n\nIn my opinion, the results shown in the figures and tables are not that convincing for me. For example, In Fig. 3, the results show that ZNE does not have any effect in all experiments. This clearly contradicts with the famous known results such as [Nature volume 618, pages500–505 (2023)], which also applies the transverse-field Ising model and trotterisation. \n\n2. The paper writing needs to be strengthened. For example, the 15-qubit circuits experiment in Table 2 seem to be confusing. Why do we need this?\n\n3. Figure 3 is unclear. I can hardly recognize the color.\n\n4. There are many typos. For example, 6 qubit should be 6-qubit (in Fig. 3), and the \nbroken sentence in line 408.\n\n5. The limitations of the proposed approach should be discussed, including the extra cost caused by the acquisition of the training set, and the cost of the training process.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "MGUTISfcbc", "reviewer": "Reviewer_KKBh", "summary": "This work presents a learning-based quantum error mitigation method by leveraging quantum-specific positional encoding and a structure information matrix for transformer to suppress the noise when estimate the expectation value.", "questions": "1. what does Fig.2. want to express? It is unclear the inverse quantum circuit used in here.\n2. In Fig1. does the model use DAG with learnable node(QCR Node) to construct parametrized structure matrix? does the output of GTraQEM module concatenates with $x_{QCR}^L$ to form the input of the regression module? and how to construct $x_{QCR}^{L}$ ?\n3. what's the sample complexity of such learning-base QEM model?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "this work exploits circuit topology in learning-based QEM empowered by multi-head attention and provides comprehensive experiments and shows the out-performance.", "weaknesses": "1. since it does not provide the related codes, the reproducibility is unknown.\n2. while this work applies the various exist techniques to enhance the QEM which may be interesting for the community of quantum computation, it does not propose some novel structure or interesting theoretical findings which have widely applications.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)