Prediction of Compilation Options for Quantum Circuits via Adaptive Deep Residual Neural Network

• Quantum Bit, Qubit: Qubits are the basic unit of quantum computing. Unlike bits in classical computing, qubits can be in a superposition of 0 and 1. Classical bits can only take on the value of 0 or 1, while qubits can be represented by a state vector |ψ〉 whose states are: |ψ〉=α|0〉+β|1〉|\psi \rangle = \alpha |0\rangle + \beta |1\rangle where α and β are complex numbers, representing the probability that qubits are measured as |0〉 and |1〉, respectively, and they satisfy the equation |α|² + |β|² = 1. This superposition gives qubits great potential when dealing with complex problems.

• Quantum Gate: Quantum gates are the basic operation units in quantum computing that are used to transform qubits. Different from logic gates in classical computing, quantum gates are linear transformations of quantum states, commonly represented by unitary matrices. Common single-qubit quantum gates include Pauli-X gate (analogous to classical NOT gate), Hadamard gate (H-gate, used to create superposition states), and Controlled NOT gate (CNOT-gate). These gate operations can be represented as the following matrix:

  • X-gate (Pauli-X Gate): X=(0110)X = \left( {\matrix{ 0 \hfill & 1 \hfill \cr 1 \hfill & 0 \hfill \cr } } \right)

    It flips the state of the qubit, changing |0〉 to |1〉 and vice versa. Its symbol representation in a quantum circuit is shown in Figure 1a.

  • H-gate (Hadamard Gate) H=12(111−1)H = {1 \over {\sqrt 2 }}\left( {\matrix{ 1 & 1 \cr 1 & { - 1} \cr } } \right)

    It converts qubits from the ground state (|0〉 or |1〉) to a superposition state, whose symbol representation in a quantum circuit is shown in Figure 1b.

  • CNOT-gate (Controlled NOT Gate) CNOT=(1000010000010010)CNOT = \left( {\matrix{ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \cr 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \cr } } \right)

    This is a two-qubit quantum gate. Only when the control qubit is |1〉, the target qubit will be flipped. Its symbol representation in a quantum circuit is shown in Figure 1c.

• Quantum Circuit: A quantum circuit is a complex sequence of operations that implements quantum computing through (cascading) multiple quantum gates. It consists of qubits and quantum gates, where the quabits are used as input and are operated through a series of quantum gates to output the results. Each quantum gate acts on qubits from left to right, forming the process of quantum computing. Figure 2 shows a simple quantum circuit example, which includes an H-gate, two X-gates, and multiple CNOT-gates.

Figure 1.

Circuit symbol representation of common quantum gates.

Figure 2.

A simple quantum circuit example, including H-gate, X-gate, and CNOT-gate.

Quantum circuit compilation faces many choices (i.e., compilation options), which mainly include the selection of qubit technology, quantum device, quantum compiler, compilation steps, and optimization strategies. The combination of compilation options is crucial to the efficiency and performance of quantum circuit compilation.

• Qubit Technology: Qubit technology determines the design and implementation of different quantum devices. Currently, the main qubit technology include superconducting qubits, ion traps, topological qubits, etc. Each technology has its unique advantages and limitations. For example, superconducting qubits are excellent in scalability, but they require an extremely low temperature environment; whereas ion trap qubits have long coherence times and high operational precision, yet they are constrained by slower gate operation speeds.

• Quantum Device: A quantum device is a physical hardware that performs quantum computing, and its design is based on a specific qubit technology. Different quantum devices differ in qubit number, coupling topology, noise characteristics, and gate operation integration. Currently, the most widely adopted superconducting qubit-based devices in academic literature primarily include 27-qubits IBM Q Toronto, which adopts a two-dimensional planar coupling structure. The device coupling topology of IBM Q Toronto is shown in Figure 3a, where each qubit is connected through a coupling link. Currently, the most widely adopted ion trap-based devices in academic literature primarily include 11-qubits IonQ Harmony, whose device coupling topology is shown in Figure 3b. Unlike superconducting qubits, ion trap devices typically have a fully connected coupling graph, meaning that each qubit can operate directly with any other qubit. This fully connected topology greatly simplifies the compilation process of quantum circuits, but the technology is extremely demanding for control and prone to crosstalk.

• Quantum Compiler: Quantum compilers are tools that translate high-level quantum algorithms into quantum instructions that can be executed on specific quantum devices. Quantum circuits may be handled differently by different compilers, such as IBM’s Qiskit, Google’s Cirq, Quantinuum’s TKET, etc. They each provide compilation steps and optimization strategies.

• Compilation Steps and Optimization Strategies: Quantum circuit compilation mainly includes the following steps: quantum gate decomposition, qubit mapping, and optimization. The following examples explain the process of these main compilation steps.

Figure 3.

Common schematic diagram of quantum computing devices.

Figure 4.

Quantum circuit compilation.

The compilation steps differ across compilers, and each step involves multiple choices. These choices directly impact the execution performance of the final quantum circuit. For example, Qiskit has four optimization levels (i.e., O0, O1, O2, and O3). O0 (No Optimization) does not do any optimization and is suitable for fast compilation and debugging. O1 (Level 1 Optimization) carries out basic gate-level optimizations by removing unnecessary gate operations to moderately optimize the circuit structure. O2 (Level 2 Optimization) includes all optimizations of O1. Additional circuit transformations and optimizations are performed to balance compilation time and circuit performance. O3 (Level 3 Optimization) uses all available optimization techniques in addition to containing all the optimizations of O2. It minimizes the gate count and depth of the circuit to improve circuit performance, but may significantly increase compilation time. Different optimization levels provide distinct circuit optimization strategies, leading to variations in quantum gate decomposition, mapping, circuit optimization results, and computational complexity. The choice of optimization level depends on the quantum circuit itself and the target device. TKET offers two different qubit mapping strategies: Line Placement and Graph Placement. These strategies are used to map logical qubits from quantum circuits to physical qubits from quantum devices. Line Placement maps logical qubits into a linear arrangement of physical qubits, suitable for simple device topologies. Graph Placement maps logical qubits to a graph structure of physical qubits. It optimizes operations across qubits taking into account the topological structure of the device and is suitable for complex device architectures. Therefore, the choice of mapping strategy depends on the structural constraints of the quantum device in use and the specific requirements of the circuit.

All available compilation options are defined as compilation option combinations. A complete set of compilation option combinations can accomplish the compilation of quantum circuit, and different compilation option combinations yield different compilation effects. The currently available compilation options are shown in Figure 5. It mainly includes two qubit technologies (i.e., superconducting and ion trap), five devices (i.e., 8, 27, 80, and 127-qubits devices based on superconducting and an 11-qubits device based on ion trap), two compilers (i.e., Qiskit and TKET), as well as compilation steps and optimization strategies (i.e., Qiskit’s O0 to O3 optimization levels and TKET’s Line Placement and Graph Placement mapping strategies). These compilation options can form a total of 30 combinations, and the sequence of options highlighted in red is one combination.

Figure 5.

The currently available compilation options.

Starting from the quantum circuit itself, converting the quantum circuit into a graph form allows for the extraction of new quantum circuit features from a different dimension. Therefore, the Time-Weighted Interaction Graph (TWIG) and graph features are introduced. Based on the Interaction Graph (IG), the TWIG incorporates the execution order of two-qubit gates in the circuit by weighting them and adding them to the IG, thereby introducing dependencies and constraints on the execution order between two-qubit gates. This enables the graph to contain more information about the quantum circuit, which is beneficial for the graph features to better represent the quantum circuit. TWIG is designed to convert the execution semantics of quantum circuits (e.g., interaction strength) into a graph structure. The core motivation for its construction lies in modeling interaction intensity: the greater the number of two-qubit gate operations that two qubits jointly participate in, and the ‘earlier’ these operations occur (i.e., the shallower their logical layer), the more important their interaction should be deemed, and the greater its influence on the selection of compilation strategies (e.g., mapping). Notably, multi-qubit gates (i.e., three-qubit gates) are all converted into two-qubit gates during final execution. Since the specific gates generated by this conversion can only be determined once the target device is selected, only two-qubit gates are considered when constructing the TWIG in this paper.

In the above given quantum circuit LC, we consider all qubits Q = {q₁, q₂,…, q_NQ} and two-qubit gates TG = {tg₁, tg₂,…, tg_NTG} is used to construct the TWIG. Firstly, we establish N_Q one-to-one corresponding graph nodes (i.e., q_i→q_i) based on qubits Q. Then, to reflect the mutual influence between two-qubit gates that diminishes with logical topological distance–even if the gates are not optimized, their relative execution order can retain local ‘temporal proximity’ signals, which benefits the subsequent selection of mapping devices. To achieve this, a temporal level weighting is applied to each gate within the two-qubit gates. Specifically, the circuit undergoes topological levelization: gates that can be executed in parallel are grouped into the same level. Across different levels, weights decrease with level depth (gates within the same level have equal weights). It is assumed that the subscript of two-qubit gates TG = {tg₁, tg₂,…, tg_NTG} represents its execution level, we assign decreasing weights {N_TG, N_TG – 1,…, 1} to each gate sequentially (i.e., w_g1 = N_TG, wg₂ = N_TG – 1,…, w_gNTG = 1). Finally, for any gate tg_i in the two-qubit gates TG that acts on two qubits q_i and q_j, we determine an edge e_ij and assign the weight of the gate as the weight of the edge (i.e., w_ij = w_tg). If an edge e_ij already exists between q_i and q_j, the weight of the quantum gate is added to the edge weight (i.e., w_ij = w_ij + w_tg). This process is performed iteratively until all two-qubit gates have been traversed. The final result is the TWIG, in which each node represents a qubit, each edge represents the interaction between two qubits, and the weight of the edge reflects the importance of its interaction. The TWIG is actually an undirected weighted graph, whose properties are described as follows:

• (1)

  Nodes: Each qubit in the quantum circuit corresponds to a node in the graph.

• (2)

  Edges: If there are any number of quantum gate operations between two qubits, there are edges between the corresponding two nodes in the graph.

• (3)

  Weights: All two-qubit gate weights acting between a certain pair of qubits are accumulated to their corresponding edge weights, thus highlighting the importance of the qubit pairs. The weights can represent the number of two-qubit gate operations between qubits and the timing information of execution.

The following is an example of the construction of a TWIG.

Figure 9.

Transformation of TWIG.

After completing the construction of the TWIG, we need to further extract feature information from the graph that can fully represent the quantum circuit. There are many features of metric graphs, but not all features of metric graphs are relevant to our goals. After carefully studying Hernandez’s classification of graph features [27] and Bandic’s analysis of interactive graph features [15], we select some main graph features for extraction. The corresponding definition, meaning and calculation formula of each graph feature are as follows:

• (1)

  Average Weighted Shortest Path (AP): refers to the average weight of the shortest paths between all node pairs in the TWIG. The size of the average weighted shortest path can indicate the average interaction strength and importance between qubits. The calculation formula is as follows: 4AP=1NQ(NQ−1)∑i≠jdw(qi,qj)AP = {1 \over {{N_Q}\left( {{N_Q} - 1} \right)}}\sum\limits_{i \ne j} {{d_w}} \left( {{q_i},{q_j}} \right) where d_w (q_i, q_j) represents the shortest weighted path length between node q_i and node q_j, and N_Q is the number of nodes.

• (2)

  Max Degree (MX_d): refers to the maximum sum of connection weights of nodes in the graph. The maximum degree reveals the strength of the most important qubit interactions in a quantum circuit. The calculation formula is as follows: 5MXd=maxi∈NQ(deg(qi))M{X_d} = \mathop {\max }\limits_{i \in {N_Q}} \left( {deg\left( {{q_i}} \right)} \right) where deg(q_i) is the weighted degree with node q_i, and max() is a function to find the maximum value.

• (3)

  Min Degree (MI_d): refers to the minimum sum of connection weights of nodes in the graph. The minimum degree can reveal whether there are non-interacting qubits in a quantum circuit. The calculation formula is as follows: 6MId=mini∈NQ(deg(qi))M{I_d} = \mathop {\min }\limits_{i \in {N_Q}} \left( {deg\left( {{q_i}} \right)} \right)

  Its calculation method is similar to the maximum degree, but it takes the minimum value, and min() is a function to find the minimum value.

• (4)

  Average Clustering Coefficient (AC): refers to the average value of the clustering coefficients of all nodes in a graph. A high clustering coefficient indicates close interactions between qubits and helps to understand the density of global interactions in quantum circuits. The calculation formula is as follows: 7AC=1NQ∑i∈NQC(qi);C(qi)=∑i∈NQ2∑∃△qiqjqk(wijwjkwki)13eg(qi)(eg(qi)−1)MXdAC = {1 \over {{N_Q}}}\sum\limits_{i \in {N_Q}} C \left( {{q_i}} \right);C\left( {{q_i}} \right) = \sum\limits_{i \in {N_Q}} {{{2{{\sum\nolimits_{\exists {q_i}{q_j}{q_k}} {\left( {{w_{ij}}{w_{jk}}{w_{ki}}} \right)} }^{{1 \over 3}}}} \over {eg\left( {{q_i}} \right)\left( {eg\left( {{q_i}} \right) - 1} \right)M{X_d}}}} where C(q_i) is the weighted clustering coefficient of node q_i based on the strength of triangle subgraphs [28], eg(qi) is the number of edges directly connected with node q_i, and Δ is the triangle symbol.

• (5)

  Standard Deviation of Adjacency Matrix (SD): refers to the degree of dispersion in the distribution of connection strengths between nodes in the graph. A high standard deviation indicates diverse types of interactions between qubits and an uneven connection structure among qubits. The calculation formula is as follows: 8SD=1n2∑i=1n∑j=1n(Aij−μA)2;μA=1n2∑i=1n∑j=1nAijSD = \sqrt {{1 \over {{n^2}}}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{{\left( {{A_{ij}} - {\mu _A}} \right)}^2}} } } ;{\mu _A} = {1 \over {{n^2}}}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{A_{ij}}} } where A is the adjacency matrix of the graph, A_ij is an element of the adjacency matrix, and μ_A is the average value of all matrix elements.

  By extracting these features to form the N_Z = 5-dimensional feature vector Z = [AP, MX_d, MI_d, AC, SD], we can better understand the structural properties of quantum circuits and optimize the representation. These features play an important role in the performance analysis of quantum circuits and the prediction of compilation options. Here is an example of extracting features.

The DRAN-Net network model is divided into three parts: input layer, hidden layer, and output layer. The model structure is shown in Figure 10. Each of the three parts is described below:

• (1)

  Input layer. The input layer is the part that receives the representation vector F = [f₁, f₂,…, f₃₆] of the input quantum circuit and processes it into a high-dimensional vector. As shown in Figure 10, it first uses a linear layer to process the input representation vector, and maps the representation vector F from 36 dimensions to 256 dimensions F1inpF_1^{inp} for more complex processing. Because the relationships between features in the representation vector are complex and nonlinear, high dimensions help to capture these complex relationships. Later, using the ReLU (Rectified Linear Unit) activation function that helps the model learn nonlinear features and Dropout that randomly sets some activation values to zero. The final output is a 256-dimensional high-dimensional feature vector F3inpF_3^{inp}.

  Among them, the Liner Layer, also known as the fully connected Layer (FC Layer) or Dense layer, is a basic block in the neural network architecture. It is primarily used for linearly transforming input feature vector data to generate new feature representations. This transformation is realized through the learned weight matrix and bias term. As shown in the input layer in Figure 10, for input vector F, its mathematical expression is: 9F1inp=WinpF+binpF_1^{inp} = {W^{inp}}F + {b^{inp}} where W^inp is the weight matrix of the linear layer of the input layer, and b^inp is the bias term of the linear layer of the input layer.

• (2)

  Hidden layer. The hidden layer is the part of the network model that further learns deeper features from the high-dimensional feature vector output by the input layer. It is also the most important part of the network model learning feature data. As shown in Figure 10, the hidden layer adopts a combination of 4 residual blocks (RBs) and 3 attention residual blocks (ARBs) stacked alternately in sequence. Firstly, the 256-dimensional feature vector F3inpF_3^{inp} is initially learned through the residual block (RB) to obtain F1hidF_1^{hid}, and then its output is used as the input of the attention residual block (ARB) for deeper learning to obtain F2hidF_2^{hid}. By alternating input and output through feature mapping, the 256-dimensional high-level feature F7hidF_7^{hid} is finally extracted.

  Among them, the Residual Block (RB) makes the deep network easier to train and improves the performance of the model by introducing skip connections. The RB can prevent the loss of features in the network model, and a local residual network can be formed by stacking multiple RBs. It can solve the problems of gradient vanishing and gradient explosion in deep network training. Figure 11 illustrates the structure of the RB. To more effectively learn global feature information over a period of time, the input feature map F3inp∈R1×256F_3^{inp} \in {R^{1 \times 256}} is first processed through a trunk branch consisting of a linear layer and a ReLU activation function. Then the skip connection branch transfers the original input F3inpF_3^{inp} to the trunk branch for Element-wise Product operation. Finally, the ReLU activation function is applied to introduce nonlinearity and output: 10F1hid=Srb1(F3inp)=Relu((W2rb1Relu(W1rb1F3inp+b1rb1)+b2rb1)+F3inp)F_1^{hid} = {S^{r{b_1}}}\left( {F_3^{inp}} \right) = Relu\left( {\left( {W_2^{r{b_1}}Relu\left( {W_1^{r{b_1}}F_3^{inp} + b_1^{r{b_1}}} \right) + b_2^{r{b_1}}} \right) + F_3^{inp}} \right) where F1hidF_1^{hid} is the output of the first RB of the hidden layer, S^rb₁ () is its corresponding residual function. W1rb1W_1^{r{b_1}} and W2rb1W_2^{rb{b_1}} are the weight matrices of the first and second linear layers in the first RB, respectively, and b1rb1b_1^{r{b_1}} and b2rb1b_2^{r{b_1}} are their corresponding bias terms, respectively.

  Attention Residual Block (ARB) can make the deep prediction model pay more attention to important regions, concentrating on learning crucial features while weakening unimportant ones. As shown in Figure 12, the structural diagram of the ARB, it mainly consists of one attention block and three RBs. Three stacked RBs serve as the trunk branch, while the attention block acts as the mask branch. The output feature map F1hidF_1^{hid} of the previous RB is input to the trunk branch and the mask branch at the same time. Through the trunk branch, more comprehensive feature information can be learned using the RBs, resulting in S3arb1(S2arb1(S1arb1(F1hid)))S_3^{ar{b_1}}\left( {S_2^{ar{b_1}}\left( {S_1^{ar{b_1}}\left( {F_1^{hid}} \right)} \right)} \right). Through the mask branch, a feature weight value Aarb1(F1hid){A^{ar{b_1}}}\left( {F_1^{hid}} \right) is learned. Then, after adding 1 to each element of the feature weight value Aarb1(F1hid){A^{ar{b_1}}}\left( {F_1^{hid}} \right), an Element-wise Product operation is performed with the output of the three RBs. Finally, the output F2hidF_2^{hid} of the ARB is: 11F2hid=H(F1hid)=S3arb1(S2arb1(S1arb1(F1hid)))×(1+Aarb1(F1hid))F_2^{hid} = H\left( {F_1^{hid}} \right) = S_3^{ar{b_1}}\left( {S_2^{ar{b_1}}\left( {S_1^{ar{b_1}}\left( {F_1^{hid}} \right)} \right)} \right) \times \left( {1 + {A^{ar{b_1}}}\left( {F_1^{hid}} \right)} \right) where S1arb1()S_1^{ar{b_1}}(), S2arb1()S_2^{ar{b_1}}(), and S3arb1()S_3^{ar{b_1}}() are the first, second, and third RB in the first ARB, respectively. Each element of Aarb1(F1hid){A^{ar{b_1}}}\left( {F_1^{hid}} \right) has a value range of [0,1]. By applying 1+Aarb1(F1hid)F_2^{hid}, each element is assigned a weight within the range of [1,2], which makes the feature data become clearer as the depth increases. This can serve as a feature selector to enhance good features and suppress noise from the trunk branch features.

  Attention block (Attention) can adaptively adjust the importance of various features, so that more important features receive higher weights in subsequent computations. As shown in Figure 13, which illustrates the structure of the attention block. In order to explore the location of important target features in the input feature map, the input feature map F1hid{A^{ar{b_1}}}\left( {F_1^{hid}} \right) is simultaneously fed into three branches. Each branch first undergoes a linear transformation of the features through a linear layer, then introduces nonlinearity through the ReLU activation function to obtain preliminary feature relationships. Subsequently, a second linear layer further linearly transforms these features to obtain more expressive feature relationships. Immediately after, the elements from the three layers are multiplied together to obtain F^a₁, which assigns a score to each element. Finally, the Sigmoid activation function generates attention weights. This process assigns a larger weight to important features within the feature relationships, and the weight output is: 12Aarb1(F1hid)=Sigmoid(∏i=13W2a1(i)Relu(W1a1(i)(F1hid)+b1a1(i))+b2a1(i))1 + {A^{ar{b_1}}}\left( {F_1^{hid}} \right)

  Similarly, W1a1(i)F_1^{hid} and W2a1(i){A^{ar{b_1}}}\left( {F_1^{hid}} \right) = Sigmoid\left( {\prod\limits_{i = 1}^3 {W_2^{{a_1}(i)}} Relu\left( {W_1^{{a_1}(i)}\left( {F_1^{hid}} \right) + b_1^{{a_1}(i)}} \right) + b_2^{{a_1}(i)}} \right) are the weight matrices of the first and second linear layers of the i-th branch of the first attention block, respectively, b1a1(i)W_1^{{a_1}(i)} and b2a1(i)W_2^{{a_1}(i)} are their corresponding bias terms, respectively. For the input Fa1=[ f1(3),f2(3),…,f256(3) ]b_1^{{a_1}(i)}, the mathematical expression of the Sigmoid activation function is: 13Aarb1(F1hid)=Sigmoid(Fa1)=[ 11+e−f1(3),11+e−f2(3),…11+e−f256(3) ]b_2^{{a_1}(i)}

• (3)

  Output layer. The output layer is the part that performs final aggregation processing on the 256-dimensional high-level mapping features F7hid{F^{{a_1}}} = \left[ {f_1^{(3)},f_2^{(3)}, \ldots ,f_{256}^{(3)}} \right] outputted by the hidden layer. As shown in Figure 10, it first applies a Dropout to the high-level mapped features to prevent overfitting. Finally, a linear layer is used to map the dimensions to a 30-dimensional F1out=[ f1(4),f2(4),…,f30(4) ]{A^{ar{b_1}}}\left( {F_1^{hid}} \right) = Sigmoid\left( {{F^{{a_1}}}} \right) = \left[ {{1 \over {1 + {e^{ - f_1^{(3)}}}}},{1 \over {1 + {e^{ - f_2^{(3)}}}}}, \ldots {1 \over {1 + {e^{ - f_{256}^{(3)}}}}}} \right] that matches the number of label categories, and the Softmax activation function is used to calculate the probability distribution P = [p₁, p₂,…, p₃₀] corresponding to the prediction results of ϒ. The mathematical expression corresponding to the Softmax activation function is: 14P=[ p1,p2,…,p30 ]=Softmax(F1out)=[ ef1(4)∑i=130efi(4),ef2(4)∑i=130efi(4),…,ef30(4)∑i=130efi(4) ]F_7^{hid}

  The label with the highest probability p_i(p_iϵP) serves as the predicted label y_i(y_iϵϒ) for the quantum circuit. Each label corresponds to one compilation option combination in Figure 5. All compilation option combinations form a label set ϒ = {y₁, y₂,…, y₃₀}. The core task of this paper is precisely to find the compilation option combination with the highest fidelity score for a given quantum circuit. Thus, the probabilities of these classification prediction results are used to calculate the cross-entropy loss, which serves to evaluate the model’s performance.

Figure 10.

DRAN-Net.

Figure 11.

Residual block (RB).

Figure 12.

Attention residual block (ARB).

Figure 13.

Attention block.

In this study, in order to train the proposed DRAN-Net, we adopt a series of standardized steps to ensure the model’s effectiveness and generalization ability to different quantum circuits. The following is a detailed description of the model training process.

• (1)

  Data preprocessing. To accelerate convergence during model training and enhance model performance, we preprocess all quantum circuit representation vectors by standardizing them before training. This step can expedite the convergence of the gradient descent method, improve model training efficiency, and consequently better enhance model performance. The standardization formula is as follows: 15F=[ F1−μ1σ1,F2−μ2σ2,…,F36−μ36σ36 ]F_1^{out} = \left[ {f_1^{(4)},f_2^{(4)}, \ldots ,f_{30}^{(4)}} \right] where Fⁱ is a vector composed of all feature values of the i-th feature in all quantum circuit representation vectors, and μⁱ and σⁱ are the mean and standard deviation of the corresponding feature, respectively. Through this formula, the mean of each feature is transformed to 0, and the standard deviation to 1. This process ensures that features of varying scales are converted to a common scale, thereby preventing any single feature with a larger scale from disproportionately influencing the model training process.

• (2)

  Loss function. In forward propagation, the model uses the CrossEntropyLoss function to evaluate the discrepancy between the predicted results and the actual labels, thereby optimizing the model’s parameters. The formula is as follows: 16L*=−∑i=1Nbs∑j=130yilogpijNbsP = \left[ {{p_1},{p_2}, \ldots ,{p_{30}}} \right] = Softmax\left( {F_1^{out}} \right) = \left[ {{{{e^{f_1^{(4)}}}} \over {\sum\nolimits_{i = 1}^{30} {{e^{f_i^{(4)}}}} }},{{{e^{f_2^{(4)}}}} \over {\sum\nolimits_{i = 1}^{30} {{e^{f_i^{(4)}}}} }}, \ldots ,{{{e^{f_{30}^{(4)}}}} \over {\sum\nolimits_{i = 1}^{30} {{e^{f_i^{(4)}}}} }}} \right] where N_bs is the number of samples (batch_size), y_i is the real category label corresponding to the i-th sample, and P_ij is the predicted probability by the model that the i-th sample belongs to category j.

  To prevent model overfitting and ensure that the model’s parameters do not become excessively large, we introduce L1 and L2 regularization during the training process. This adds constraints on the model’s parameters, avoiding them from becoming too large or overly complex, thereby preventing overfitting. L1 regularization achieves feature selection by adding the sum of the absolute values of the weights to the loss function, with the formula as follows: 17L1=λ1∑j=1NW| wj |F = \left[ {{{{F^1} - {\mu ^1}} \over {{\sigma ^1}}},{{{F^2} - {\mu ^2}} \over {{\sigma ^2}}}, \ldots ,{{{F^{36}} - {\mu ^{36}}} \over {{\sigma ^{36}}}}} \right]

  L2 regularization reduces the values of the weights by adding the sum of the squares of the weights to the loss function, with the formula as follows: 18L2=λ2∑j=1NWwj2{L^*} = - {{\sum\nolimits_{i = 1}^{{N_{bs}}} {\sum\nolimits_{j = 1}^{30} {{y_i}} } log{p_{ij}}} \over {{N_{bs}}}} where λ₁ and λ₂ are hyperparameters representing the strength of the regularization, N_W is the total number of model weight parameters, w_j w_j(w_j ∈ W) is any parameter in any weight matrix. Therefore, the final loss function L of the model is: 19L=L*+L1+L2L = {L^*} + {L_1} + {L_2}

To build an efficient and interpretable representation set, we followed a systematic, data-driven feature engineering process. This subsection aims to achieve three goals: (1) verify the effectiveness of the newly proposed graph features; (2) identify and eliminate excessive redundancy between features; and (3) finally provide quantitative evidence for the effectiveness of our hybrid representation method.

Correlations between all features are unavoidable [27]. Multicollinearity (high correlation) between features not only increases model complexity and the risk of overfitting, but also indicates redundancy in the representation set. We calculated the Spearman’s rank correlation coefficients [30] between all 36 initial features (including 31-dimensional traditional features and 5 new graph features). We also plotted a heatmap of the correlation matrix, as shown in Figure 14 (-1/1 indicates perfect correlation, and 0 indicates no correlation).

Figure 14.

Feature correlation.

Analysis of feature correlations reveals that many feature pairs are correlated, with some showing perfect correlation–meaning their metric values are identical. For example, the count of t gates is perfectly correlated with that of tdg gates, while other features also exhibit some degree of correlation. Notably, the correlation between the proposed new features and other features does not exceed 95%. This indicates that the new features successfully capture novel structural information not described by traditional statistical features, statistically validating their unique value as an effective supplement. Based on this analysis, we established strict screening rules: for any highly correlated feature pair (with similarity ≥ 95%), only one is retained. We prioritize retaining features that are more commonly used in domain knowledge or have clearer physical meanings (e.g., retaining the count of t gates while removing the perfectly correlated count of tdg gates). Through this step, we removed 3 redundant traditional features (tdg, parallelism, rzz), reducing the feature dimensionality from 36 to 33.

To further quantify the actual contribution of each feature to model predictions, this paper employs the permutation feature importance method. The trained DRAN-Net model is used to generate a feature importance ranking chart based on the permutation feature importance in descending order, as shown in Figure 15. The results indicate significant differences in the influence of various features. Among them, the number of qubits has an importance greater than 0.6 in the model’s predictions, holding the highest significance. This strongly demonstrates the successful training of the model, as a quantum circuit cannot be executed if the device’s number of qubits is less than what the circuit requires. Among the new features proposed in this paper, the Average Clustering Coefficient has an importance greater than 0.3 in the model’s predictions, which convincingly proves the effectiveness of the proposed features. Moreover, features with a permutation importance of zero do not necessarily imply they are useless. Because redundancy among features and the model’s independence can result in a permutation importance of zero. Through this analysis, not only is the model’s decision-making basis revealed, but it also provides a clearer understanding of the role of each feature in the prediction task, further validating the effectiveness of the proposed new features.

Figure 15.

Feature importance.

To deeply verify the effectiveness of the proposed method, we design a series of ablation experiments to evaluate their impact on the overall performance. In the experimental results section, we detail the model performance metrics for each experimental configuration, including accuracy, Top-3 accuracy, and F1. All experiments are conducted under the same evaluation criteria, and results are reported as the mean ± standard deviation from five independent runs.

First, we apply the proposed quantum circuit representation method to part of traditional models for experimentation and analyze its performance results. The purpose of this step is to observe the impact of the proposed quantum circuit representation method on prediction performance, as well as its effectiveness and robustness, and obtain the specified evaluation results as shown in Table 2.

Figure 16 shows that the proposed new quantum circuit representation method (N-Pre) has improved the accuracy, Top3 and F1 of Random Forest, which performs best among traditional models. Compared with the traditional representation method (T-Pre) of quantum circuits, the proposed quantum circuit representation method improves the Random Forest accuracy by 1.49%, the Top-3 accuracy by 0.33%, and the F1 by 1.15%, proving the effectiveness of the proposed quantum circuit representation method. However, since the N-Pre method introduces more features than the T-Pre method, and the Gradient Boosting model, Decision Tree model, and Nearest Neighbor model are sensitive to feature scales, the computational complexity of these three models increases. This leads to a decrease in the F1 of Gradient Boosting, the F1 of Decision Tree, and the Top-3 of Nearest Neighbor. In addition, the N-Pre method improves the performance of other models, which demonstrates the robustness of the proposed method. In summary, although TWIG features demonstrate certain correlations with conventional features (Figure 14), ablation studies conclusively verify their ability to provide incremental predictive information. This indicates that the correlations primarily stem from these features describing the same underlying circuit complexity from different perspectives, while the unique temporal and structural dynamic information captured by TWIG features serves as the key driver for performance enhancement. Thus, the value of these novel features lies in providing complementary perspectives for a more comprehensive understanding of quantum circuits.

Figure 16.

Performance Comparison of T-Pre and N-Pre.

Notably, in the permutation importance analysis, only the new feature Average Clustering Coefficient demonstrated high importance, while the other four TWIG features ranked lower. To investigate the actual contribution of these features, we conducted an ablation experiment: we removed these four lower-importance features to form a combination of traditional features with the single Average_Clustering_Coefficient (T-Pre+AC). This feature set was subsequently evaluated using traditional models, with the designated assessment results presented in Table 3.

The experimental results in Figure 17 demonstrate that models using the condensed representation set (T-Pre+AC) achieve comprehensively lower accuracy than those utilizing the complete representation set (N-Pre). Apart from fluctuations observed in Decision Tree and Nearest Neighbor models due to their sensitivity to feature scaling, the same pattern holds for the other two metrics. This finding indicates that although these features individually exhibit limited impact on model output when perturbed in isolation, they collectively contribute complementary information that enhances the model’s overall representational capacity and generalization performance when integrated into the complete representation set. Consequently, we ultimately retained all TWIG features to achieve optimal performance.

Figure 17.

Performance Comparison of T-Pre+AC and N-Pre.

Next, we apply the proposed model DRAN-Net to traditional quantum circuit representation method and analyze its result performance. The core of this experimental step is to evaluate the effectiveness and robustness of DRAN-Net with traditional circuit representations. The specified evaluation results are shown in Table 4. By comparing and analyzing the performance differences between traditional models and DRAN-Net, we can gain insights into the advantages of DRAN-Net.

As shown in Figure 18, the proposed DRAN-Net model improves all metrics of the traditional model. Compared with the best performing Random Forest in the traditional model, DRAN-Net improves the accuracy by 4.93%, the Top-3 accuracy by 2.33%, and the F1 by 1.43%, which proves the effectiveness of DRAN-Net. Furthermore, the prediction results obtained by DRAN-Net using traditional representation are close to the best prediction performance currently achievable, thereby proving the robustness of DRAN-Net.

Figure 18.

Performance Comparison of model using T-Pre.

Through these ablation experiments, we can more clearly understand how the various components of the method work together and their specific impact on improving prediction accuracy. These supplementary experiments further demonstrate the contribution of these improvements to the overall performance improvement and verify the effectiveness, robustness and reliability of the proposed representation method and model method.

To comprehensively verify the superiority of the method proposed in this paper, we systematically compare it with two types of baseline methods. One type includes traditional machine learning models used in [11]: Random Forest, Gradient Boosting, Decision Tree, Nearest Neighbor, Multilayer Perceptron, Support Vector Machine, and Naive Bayes. The other type consists of more advanced graph neural network models: GIN (Graph Isomorphism Network), GCN (Graph Convolutional Network), and GraphSAGE. All comparative experiments are conducted under the same dataset and evaluation criteria. Results are reported as the mean / pm standard deviation from five independent runs, as shown in Table 5.

Figure 19 demonstrates that Ours (DRAN-Net+T-Pre) achieves the best performance in terms of accuracy, Top-3 accuracy and F1. The accuracy has reached 82.83%, the Top-3 accuracy has reached 98.14%, and the F1 has reached 82.80%. In particular, the Top-3 accuracy is close to error-free prediction. This can avoid a large number of invalid attempts for end users, which is of great significance. Compared with the current best Random Forest prediction results, the method proposed in this paper has been greatly improved in all metrics, including an increase of 5.44% in accuracy, 2.6% in Top-3 accuracy, and 2.05% in F1. The analysis of the experimental results verifies the advancement of our proposed method. Moreover, it shows significant advantages over graph neural network (GNN) baseline methods. Compared with the best-performing GNN model (GraphSAGE), it achieves an 11.73% improvement in accuracy, a 2.37% improvement in Top-3 accuracy, and a 22.39% improvement in F1. This significantly enhances the balance of classification. Even when compared with advanced GNNs designed specifically for graph structure processing, DRAM-Net+N-Pre still demonstrates clear advantages. Analysis of the experimental results fully verifies the superiority of the method proposed in this paper.

Figure 19.

Performance Comparison of various model methods.

To evaluate the applicability and scalability of the proposed method in broader quantum compilation decisionmaking scenarios, we migrate our approach to the device selection task defined in [13] and conduct direct comparisons with the baseline method (Random Forest) adopted in that study, with experimental results summarized in Table 6. It is worth noting that this task aims to select the optimal execution device for a given quantum circuit and has been formulated as a classification problem, with the original dataset comprising 600 circuits.

As shown in Figure 20, our DRAN-Net model comprehensively outperformed the original baseline method on this task introduced by [13]. Specifically, our model achieved statistically significant improvements in both accuracy (0.67% increase) and F1-score (1.7% increase). Particularly noteworthy is that while maintaining a perfect Top-3 accuracy of 100%, our model demonstrated more reliable performance on the more stringent accuracy. These results sufficiently indicate that the proposed method can effectively capture features applicable to various compilation decision scenarios, exhibiting excellent transferability and generalization potential. This provides empirical evidence for its integration as an intelligent component into more complex compilation frameworks such as [13]

Figure 20.

Performance comparison on the [13] device selection task.

In addition, when compared with the top-performing DRAN-Net+T-Pre in Table 4, DRAN-Net+N-Pre further delivers performance gains, with accuracy improved by 0.51%, Top-3 accuracy improved by 0.27%, and F1 improved by 0.62%. Since both configurations employ the same model method, this result further validates the advancedness of the representation method proposed in this paper. Separately, in comparison with the classification results of the top-performing Random Forest+N-Pre in Table 2, DRAN-Net+N-Pre exhibits performance improvements, including a 3.95% increase in accuracy, a 2.27% increase in Top-3 accuracy, and a 0.9% increase in F1. As both approaches utilize the same input representation set, this finding further confirms the advancedness of the DRAN-Net model.

Finally, to ensure the reliability of evidence, we conducted paired t-tests on all metrics of the proposed method and baseline methods on the test set. The goal was to evaluate the statistical significance of performance differences. For all comparisons, Welch’s corrected t-test was used when variances were unequal, while the independent samples t-test was applied when variances were equal. Additionally, the sample size was set to 5, which ensured the reliability of the statistical results.

Table 7 shows that for all performance metrics, the performance differences between Ours (DRAN-Net+N-Pre) and all baseline methods are statistically significant (p < 0.05). This indicates that this advantage does not only come from random errors, but more from the fusion advantages of the N-Pre representation and DRAN-Net’s modeling capability for feature fusion. These results fully validate the superiority and effectiveness of the overall method proposed in this paper for the quantum circuit compilation option prediction task.

To evaluate the practical application value of the proposed method, this paper randomly selected 100 representative circuits from MQT Bench (due to limited computational resources of the device, the selected circuits have qubit counts ranging from 5 to 50 and gate counts ranging from 50 to 5000). Exhaustive search (enumeration) of all 30 compilation options and DRAN-Net model prediction experiments were conducted separately to assess the execution efficiency (time) and effectiveness (quality) of exhaustive search versus the DRAN-Net model in single-circuit compilation decision-making. The experimental results are presented in Table 8.

As shown in Figure 21, the DRAN-Net model significantly reduces the compilation decision time from an average of 118 seconds to 0.27 seconds, achieving a 437-fold performance improvement. This indicates that our model accomplishes a leap from minute-level to sub-second response times, providing crucial technical support for interactive quantum program development. Furthermore, while achieving this dramatic speed improvement, the DRAN-Net model maintains competitive decision-making quality. The model’s Top-3 accuracy reaches 99.4%, meaning that for nearly all circuits, the optimal compilation option is included among the top three recommendations provided by the model. This ensures that in practical scenarios, users only need to try a very limited number of options to obtain near-optimal compilation results. The model’s accuracy is 90.4%, meaning that in approximately nine out of ten cases, its top recommendation is the globally optimal solution. Although there is a gap compared to the theoretical optimum achieved by exhaustive search, this level of accuracy is attained alongside a 437-fold acceleration. For the vast majority of application scenarios, the minimal suboptimal risk accepted in exchange for such a substantial efficiency gain is acceptable. In summary, the DRAN-Net model achieves an effective trade-off in the compilation decision-making problem, further demonstrating the validity of the DRAN-Net model.

Figure 21.

Exhaustive search vs. DRAN-Net prediction.