1. Introduction
Currently, quantum systems promise faster computing, more secure communication, and more precise sensing [1–3]. However, all quantum systems face a fundamental problem: quantum bits (qubits) are highly prone to errors. Therefore, it is essential to incorporate quantum error-correcting codes (QECC) into fault-tolerant quantum systems [4, 5]. QECCs typically assume that qubits are subject to depolarizing noise, where the bit-flip (X) errors, the phase-flip (Z) errors, and the bit/phase (Y) Pauli errors occur with equal probabilities. However, in reality, physical qubits are often affected by biased noise, where one type of error is more likely to occur than others. For instance, phase noise predominates in superconducting qubit architectures, as discussed in the literature [6, 7]. As a result, bias-tailored QECCs [8, 9] have been designed to exploit the asymmetry in noise to improve the error correction performance of quantum codes. Research by Bonilla Ataides et al [8] has demonstrated that a variant of the surface code, named XZZX code, achieves significant performance under biased noise conditions. In a recent study by Joschka Roffe et al [9], a bias-tailored lifted product code structure was introduced, providing a framework for extending bias-tailored methods beyond two-dimensional topological codes. They presented examples of bias-tailored lifted product codes based on classical quasi-cyclic codes and conducted numerical evaluations of their performance using belief propagation (BP) with ordered statistics decoding (BP + OSD).
In the decoding algorithms for sparse graph codes, the belief propagation algorithm exhibits large advantages in low decoding overhead and excellent decoding performance [10, 11]. Quantum low-density parity-check (QLDPC) codes can also use BP algorithms for decoding. It should be noted that due to the presence of quantum short cycles and quantum degeneracy [12, 13] in QLDPC codes, traditional BP algorithms often fail to decode successfully. In recent years, various improvements to BP algorithms have been proposed, such as BP with additional memory effects (MBP) [12], BP with posterior adjustment [14], BP + OSD [15, 16], minimum-weight perfect matching (MWPM) [17–20], and neural BP (NBP) [21–23]. However, most of these improvements are designed for symmetric noise decoding systems. Nowadays, the development of machine learning (ML) [24–26] is very rapid, and NBP is constrained by the code length in this problem. The rapid advancement and iteration of GPU technology are expected to significantly improve its scalability and efficiency. We chose to build a neural network decoder, which is a major trend in ML development [27–29], even though it has a complex and time-consuming training process and tends to have a linear increase in computational complexity [30] compared to traditional BP. However, the performance of a trained decoder is often unsurpassed by other methods [31–33]. The main objective of this paper is also to demonstrate the advantages of neural networks in the field of bias-tailored quantum codes.
In this paper, we design a new neural network-based decoding algorithm, NBP + CNN + BIAS, for biased quantum codes to effectively handle quantum information under biased noise. The name reflects the core components of our approach: NBP as the primary decoding framework, convolutional neural network (CNN) to accelerate convergence, and bias-tailored optimization to adapt to asymmetric noise. This algorithm leverages the asymmetry of the noise to enhance the error correction performance of QECCs. The contributions of this paper are as follows:
A biased noise model is developed to generate asymmetric noise vectors, which serve as the input dataset for the neural network.
A neural network decoding algorithm based on the belief propagation framework is proposed. By utilizing a customized multi-objective loss function, the algorithm performs back error correction and mitigates the issue of quantum error degeneracy during quantum information transmission. The decoding threshold on surface codes reaches 20%.
Several convolutional optimization schemes are discussed, which accelerate the convergence speed of the proposed algorithm.
Numerical results show that for bias-tailored quantum codes, NBP + CNN + BIAS performs better than BP + OSD. Our proposed NBP + CNN + BIAS decoder achieves an order of magnitude improvement in their error suppression relative to higher-order BP + OSD.
2. Preliminaries
2.1. Calderbank-shor-steane codes
The Calderbank-Shor-Steane (CSS) code family is a special type of QECCs that has independent X-stabilizers and Z-stabilizers so that each stabilizer is independently composed of X-type Pauli operators and Z-type Pauli operators [34]. A CSS code can be constructed by the parity-check matrices HX and HZ corresponding to the two classical codes CX and CZ, which can be expressed in the following matrix form:
$\begin{eqnarray}{H}_{{\rm{CSS}}}=\left[\left.\begin{array}{c}0\\ {H}_{X}\end{array}\right|\begin{array}{c}{H}_{Z}\\ 0\end{array}\right],\end{eqnarray}$
where HCSS fulfills the symplectic criterion ${H}_{Z}\cdot {H}_{X}^{{\rm{T}}}=0$, and T is a transposition operator.2.2. Bias-tailored XZZX codes
The hypergraph product code is a special method for constructing CSS codes that allows the construction of quantum codes from any two classical codes [35]. Let C1 = [n1, k1, d1] and C2 = [n2, k2, d2] be two linear codes whose parity-check matrices are given as H1 and H2, respectively. Then, the HX and HZ of the hypergraph product (HGP) code are given as ${H}_{X}=\left({H}_{1}\otimes {{\mathbb{1}}}_{{n}_{2}}| {{\mathbb{1}}}_{{m}_{1}}\otimes {{H}_{2}}^{T}\right)$ and ${H}_{Z}=\left({{\mathbb{1}}}_{{n}_{1}}\otimes {H}_{2}| {{H}_{1}}^{T}\otimes {{\mathbb{1}}}_{{m}_{2}}\right)$, respectively. Based on the preceding discussion, the structure of the HCSS matrix can be expressed as follows:
$\begin{eqnarray}{H}_{{\rm{CSS}}}=\left[\left.\begin{array}{c}0\\ {H}_{X}\end{array}\right|\begin{array}{c}{H}_{Z}\\ 0\end{array}\right]=\left[\left.\begin{array}{cc}0 & 0\\ {H}_{X1} & {H}_{X2}\end{array}\right|\begin{array}{cc}{H}_{Z1} & {H}_{Z2}\\ 0 & 0\end{array}\right],\end{eqnarray}$
where we denote by HX = [HX1, HX2] and HZ = [HZ1, HZ2], respectively. Corresponding to the HGP code structure, these submatrices are defined as follows: ${H}_{X1}={H}_{1}\otimes {{\mathbb{1}}}_{{n}_{2}}$, ${H}_{X2}={{\mathbb{1}}}_{{m}_{1}}\otimes {{H}_{2}}^{T}$, ${H}_{Z1}={{\mathbb{1}}}_{{n}_{1}}\otimes {H}_{2}$, and ${H}_{Z2}={{H}_{1}}^{T}\otimes {{\mathbb{1}}}_{{m}_{2}}$.We apply the Hadamard rotation operation to the HCSS matrix [9], yielding the matrix HHR as follows: 2 ) as the parity-check matrix of a QECC in the standard CSS form and equation (3 ) as the parity-check matrix of a bias-tailored QECC in the XZZX form.
$\begin{eqnarray}{H}_{{\rm{HR}}}=\left[\left.\begin{array}{cc}0 & {H}_{Z2}\\ {H}_{X1} & 0\end{array}\right|\begin{array}{cc}{H}_{Z1} & 0\\ 0 & {H}_{X2}\end{array}\right].\end{eqnarray}$
At this stage, we have obtained the parity-check matrix HHR for the bias-tailored XZZX code [9]. Here, we take the HGP code as an example. In fact, as long as the parity-check matrix of a quantum code can be decomposed into four submatrices, the Hadamard rotation transformation can be applied to construct a bias-tailored code. Then, we refer to equation (It is worth noting that the method of equations (2 ) and (3 ) achieves an improvement in the decoding effect only in the form of specific constructions [9, 36]. An example is the HGP construction. After Hadamard rotation, the symmetric structure of the HGP code is destroyed, and the HX matrix of the HGP code is completely dependent on the H1 matrix. In this case, if H1 has better parameters compared to HCSS. Then, the enhancement of its decoding effect is obvious under the bias condition favoring HX. We will discuss this further in the experimental section.
3. Neural belief propagation decoder
According to figure 1, the specific process of the decoder is as follows:
Figure 1. The structure of NBP + BIAS decoder. The decoder mainly consists of a biased noise model and a neural belief propagation (NBP) decoder using a multi-objective loss function. |
The quantum channel generates simulated asymmetric noise, which is received by the quantum biased noise model. This model produces the rotated asymmetric noise errors X and Z required by the decoder and applies them to the HZ and HX matrices (see equations (
The input information to NBP consists of equation (
The marginalized probability λv is used to infer errors through the equation
$\begin{eqnarray*}\hat{{\boldsymbol{e}}}=\,\rm{sigmoid}\,({\lambda }_{v})=\frac{1}{1+{{\rm{e}}}^{-{\lambda }_{v}}},\end{eqnarray*}$
where e is the natural constant. If these values fall within the range [0, 0.5), the BP hard decisions assign $\widehat{{\boldsymbol{e}}}=0$, while these values fall within [0.5, 1], the BP hard decisions assign $\widehat{{\boldsymbol{e}}}=1$. Verify whether the results $\widehat{{\boldsymbol{e}}}$ satisfy $({\boldsymbol{e}}+\widehat{{\boldsymbol{e}}})\cdot {{\boldsymbol{H}}}_{i}^{\perp }=0$. If the condition is not met, logical errors have occurred.
Additionally, the convolution operation mentioned in section
Figure 2. Convolutional optimization schemes. The symbol ‘⋯' represents a repeating structure and ‘T' is the number of repetitions. |
3.1. Biased noise model
The characteristics of quantum codes require that their decoding methods depend on their syndrome [37, 38]. First, the syndrome s of the XZZX code is calculated by applying the Hadamard rotation stabilizers to the error vector e. Denote HR(x) by the Hadamard matrix applied from equation (3 ) to the vector x, i.e.4 ) decouples into two problems,
$\begin{eqnarray}\begin{array}{rcl}\left({{\boldsymbol{s}}}_{X}| {{\boldsymbol{s}}}_{Z}\right) & = & \,\rm{HR}\,({\boldsymbol{e}})={H}_{{\rm{HR}}}\cdot \left({{\boldsymbol{e}}}_{X1}{{\boldsymbol{e}}}_{X2}| {{\boldsymbol{e}}}_{Z1}{{\boldsymbol{e}}}_{Z2}\right)\\ & = & \left({H}_{Z1}\cdot {{\boldsymbol{e}}}_{X1}+{H}_{Z2}\cdot {{\boldsymbol{e}}}_{Z2}| {H}_{X1}\cdot {{\boldsymbol{e}}}_{Z1}+{H}_{X2}\cdot {{\boldsymbol{e}}}_{X2}\right).\end{array}\end{eqnarray}$
Then, the equation ( $\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{s}}}_{X} & = & {H}_{Z}\cdot \left({{\boldsymbol{e}}}_{X1}| {{\boldsymbol{e}}}_{Z2}\right),\\ {{\boldsymbol{s}}}_{Z} & = & {H}_{X}\cdot \left({{\boldsymbol{e}}}_{Z1}| {{\boldsymbol{e}}}_{X2}\right).\end{array}\end{eqnarray}$
Biased noise model
| Input: CSS parity check matrices HX and HZ; code length N; column length N1 of HX1; Pauli error rates εX, εY, εZ |
| Output: errors eX and eZ |
| 1: eX = 0, eZ = 0 |
| 2: pX = 0, pZ = 0 |
| 3: Hadamard rotation |
| 4: for i = 1 to N1 |
| 5: pX[i] = εX + εY |
| 6: pZ[i] = εZ + εY |
| 7: end for |
| 8: for i = N1 + 1 to N |
| 9: pX[i] = εZ + εY |
| 10: pZ[i] = εX + εY |
| 11: end for |
| 12: generate random errors |
| 13: for i = 1 to N |
| 14: ξ = random(0, 1) |
| 15: if ξ < pX[i] |
| 16: eX[i] = 1, eZ[i] = 0 |
| 17: else if pX[i] < ξ < pX[i] + pZ[i] |
| 18: eX[i] = 0, eZ[i] = 1 |
| 19: else if pX[i] + pZ[i] < ξ < pX[i] + pZ[i] + εY |
| 20: eX[i] = 1, eZ[i] = 1 |
| 21: end if |
| 22: end for |
| 23: return eX, eZ |
In equations (4 ) and (5 ), the HX and HZ matrices are the parity-check matrices of the standard CSS quantum code before the Hadamard rotation operation. This step demonstrates that, for simulation purposes, it is sufficient to simply apply the Hadamard rotation operation to the error vector while retaining the original CSS form of the stabilizer matrices. This operation is illustrated in the yellow box in figure 1.
We initialize the error distribution by considering an error model in which each qubit is independently affected by Pauli noise. If εX, εY, and εZ are the physical error rates for X, Y, and Z errors, respectively, then the total physical error rate is ε = εX + εY + εZ. Under this condition, the error bias is ${B}_{i}=\frac{{\epsilon }_{i}}{{\sum }_{j\ne i}{\epsilon }_{j}}$, where εi, εj ∈ {εX, εY, εZ}. In general, the symmetric channel has BX = 0.5, i.e. εX = εY = εZ. The specific details of the noise model corresponding to this section can be found in algorithm 1 .
3.2. Neural belief propagation
Neural networks soften the information during the decoding process, effectively avoiding the quantum degeneracy problem often encountered in quantum decoding. For highly degenerate quantum codes, there are many low-weight stabilizers corresponding to local minima in the energy topology, making traditional BP easily trapped in these local minima near the origin. Weight updates in neural networks can enhance the network's perception ability of network by adjusting the step size, allowing the energy minimization process to converge. Belief propagation is an iterative algorithm used to approximate the mean of each variable node ev.
Note that the weight parameter setting follows the rule: it is used to construct some kind of weight input that removes the influence of its own elements.
For example, for the matrix
$\begin{eqnarray}\left[\begin{array}{ccc}1 & 1 & 0\\ 0 & 1 & 1\end{array}\right].\end{eqnarray}$
Construct the weight matrix w:
At position (0,0), the matrix entry is 1. We duplicate the row [1,1,0], set the 0th element to 0, yielding [0,1,0], and store it as the 0th column of w.
At position (0,1), the entry is 1. We duplicate the row [1,1,0], set the 1st element to 0, yielding [1,0,0], and store it as the 1st column of w.
At position (0,2), the entry is 0. This entry is skipped.
At position (1,0), the entry is 0. This entry is skipped.
At position (1,1), the entry is 1. We duplicate the row [0,1,1], set the 1st element to 0, yielding [0,0,1], and store it as the 2nd column of w.
At position (1,2), the entry is 1. We duplicate the row [0,1,1], set the 2nd element to 0, yielding [0,1,0], and store it as the 3rd column of w.
Then, steps 1–4 below provide a structured overview of the process for enhancing the BP algorithm with a neural network. The original BP algorithm is extended by incorporating additional trainable weights wi, enabling adaptive optimization. The neural network updates these weights through a loss function, adjusting their values after each complete training cycle to improve decoding performance.
Step 1: Initialize the variable nodes
$\begin{eqnarray}{\lambda }_{v}^{(1)}={w}_{in}{l}_{v}={w}_{in}\,{\mathrm{log}}\,\frac{P({e}_{v}=0)}{P({e}_{v}=1)}={w}_{in}\,{\mathrm{ln}}\,\frac{1-\epsilon }{\epsilon },\end{eqnarray}$
where ε represents the estimated physical error probability of the channel, lv denotes the prior log-likelihood ratio (LLR) information of the variable ev, and win is the weight of the initialized input information.Step 2: Obtain the messages passed from the variable nodes to the check nodes
$\begin{eqnarray}{\lambda }_{v\to c}^{(t+1)}=\tanh \left(\displaystyle \frac{1}{2}\left({w}_{lv}^{(t)}{l}_{v}+\displaystyle \sum _{c^{\prime} \in { \mathcal N }(v)\backslash c}{w}_{cv}^{(t)}{\lambda }_{c^{\prime} \to v}^{(t)}\right)\right),\end{eqnarray}$
where ${\lambda }_{c\to v}^{(t)}$ represents the message passed from the check node to the variable node in the t-th iteration of the BP process, and ${ \mathcal N }(x)\backslash y$ denotes the set of all neighboring nodes of x except y. Additionally, ${w}_{lv}^{(t)}$ denotes the weight of the LLR information in the t-th iteration, while ${w}_{cv}^{(t)}$ represents the weight of the message passed from the check node to the variable node in the t-th iteration of BP.Step 3: Obtain the messages passed from the check nodes to the variable nodes 4 , 5 ). Additionally,${w}_{vc}^{(t)}$ represents the weight of the message passed from the variable node to the check node in the t-th iteration of the BP process.
$\begin{eqnarray}{\lambda }_{c\to v}^{(t+1)}={(-1)}^{{s}_{c}}2{\tanh }^{-1}\displaystyle \prod _{v^{\prime} \in { \mathcal N }(c)\backslash v}{w}_{vc}^{(t)}{\lambda }_{v^{\prime} \to c}^{(t)},\end{eqnarray}$
where sc is the error syndrome passed to the check node, and its calculation method is given in equations (Steps 2 and 3 are iteratively performed until the number of iterations reaches T.
Step 4: Marginalization
$\begin{eqnarray}{\lambda }_{v}={w}_{lv}^{(\,T\,)}\left({l}_{v}\right)+\displaystyle \sum _{c\in { \mathcal N }(v)}{w}_{\mathrm{out}}{\lambda }_{c\to v}^{(\,T\,)},\end{eqnarray}$
where T represents the maximum number of BP iterations set, and wout represents the weight of the output layer.Note that we intentionally separate $\tanh $ and ${\tanh }^{-1}$ into the variable layer and check layer, respectively, to avoid the gradient vanishing problem during the neural network's information transmission process. This separation ensures smoother message passing and facilitates the insertion of convolutional layers in subsequent convolutional optimization steps.
3.3. Multi-objective loss function
There are three types of decoding failures in quantum error correction [12, 22]. Denote the parameters of a QECC by [[N, K, D]]. We mainly consider two types of errors. The first type is a classical error, which occurs when the inferred error $\hat{e}$ is inconsistent with the actual error e, i.e. $\hat{e}\ne e$. The second type is a quantum logical error, defined by the condition that the inferred error $\hat{e}$ does not satisfy $({\boldsymbol{e}}+\widehat{{\boldsymbol{e}}})\cdot {{\boldsymbol{H}}}_{i}^{\perp }=0$, where $HM{({H}^{\perp })}^{{\rm{T}}}=0\,\mathrm{mod}\,2$, $M=\left[\begin{array}{cc}0 & {I}_{N}\\ {I}_{N} & 0\end{array}\right]$, and H⊥ is referred to as the symplectic dual of H with respect to the symplectic inner product M. Note that H includes both HX and HZ, meaning H is a 2(N − K) × 2N binary matrix. In quantum error correction, inferred errors that don't satisfy classical error conditions can still meet logical error conditions.
Traditional BP is limited by the difficulty of accurately locating the target error from a given syndrome, and instead reduces the error rate by utilizing a second error type [12]. Neural networks, on the other hand, have an advantage in solving the first type of error by softening the information in the transmission process and avoiding falling into local minima. Due to the unique error patterns of quantum codes, relying solely on the cross-entropy loss from traditional neural networks is insufficient. Unlike the NBP method in [21], the NBP + BIAS employs a multi-objective strategy that combines cross-entropy loss with logical error loss. The multi-objective loss function is $L=a{{ \mathcal L }}_{c}({\boldsymbol{\lambda }};{\boldsymbol{e}})+(1-a){{ \mathcal L }}_{l}({\boldsymbol{\lambda }};{\boldsymbol{e}})$, where the hyperparameter a is used to control the weight of the two losses. For the classical errors, we have 10 ). OPT represents the logical operator, which is typically computed in the CSS form for the logical operator LZ. It should be ensured that LZ is not in the row space of HZ and is within the null space of HX and similarly for LX. Additionally, it is necessary to ensure that LX and LZ anticommute.
$\begin{eqnarray}{{ \mathcal L }}_{c}({\boldsymbol{\lambda }};{\boldsymbol{e}})=-\frac{1}{N}\displaystyle \sum _{i=1}^{N}{e}_{i}\,{\mathrm{log}}\,{\hat{e}}_{i}+\left(1-{e}_{i}\right)\,{\mathrm{log}}\,\left(1-{\hat{e}}_{i}\right).\end{eqnarray}$
For the logical errors, we have $\begin{eqnarray}{{ \mathcal L }}_{l}({\boldsymbol{\lambda }};{\boldsymbol{e}})=\frac{1}{K}\displaystyle \sum _{j=1}^{K}\displaystyle \sum _{i=1}^{N}\left({\mathrm{OPT}}_{ji}f\left({e}_{i}+{\hat{e}}_{i}\right)\right).\end{eqnarray}$
The function $f(x)=\left|\sin \frac{\pi x}{2}\right|$ is used to convert discrete integer mod 2 operations into continuous mod 2 operations. $\hat{e}=\,\rm{sigmoid}\,({\lambda }_{v})=\frac{1}{1+{{\rm{e}}}^{-{\lambda }_{v}}}$, where e is the natural constant. T denotes the transpose operation. The calculation of λv is given by equation (4. Convolutional optimization schemes
CNNs have demonstrated successful applications in low-level tasks such as image denoising and super-resolution, where the primary objective is to recover the original image pixels from noisy observations. This principle shares similarities with channel decoding, inspiring this study to explore whether this capability can be transferred to the decoding task of QECCs. In this work, we investigate the role of convolutional layers in NBP through extensive experiments. These layers assist in decoding by extracting local features, particularly during the early stages, helping the model identify local error patterns. Combined with the information update rules in recursive NBP, convolutional layers can smooth and enhance the update of local confidence values, thereby accelerating convergence and reducing the bit error rate. Several proposed schemes are shown in figure 2. In these schemes, each convolutional step consists of two layers: the first employs a 3 × 3 convolutional kernel with 16 filters, followed by a 1 × 1 convolutional kernel with 8 filters. Both layers utilize the ReLU activation function. The structure of this convolutional layer is illustrated in figure 3.
Figure 3. Structure of convolutional step. |
5. Numerical results
The dataset for neural network training is randomly generated by an asymmetric noise channel, as shown in algorithm 1 . Note that for numerical results without specified Bi values, the default is Bi = 0.5.
5.1. Experimental environment and parameter definitions
For convenience, the decoder proposed in this paper is denoted as NBP+BIAS, and its convolutional optimization scheme is referred to as the NBP + CNN + BIAS scheme. The experiments in this paper were conducted on an NVIDIA RTX 3060 Laptop GPU. The NBP + CNN + BIAS decoder is implemented based on the TensorFlow framework, utilizing the Adam optimizer to minimize the loss function. The experimental setup is shown in table 1.
Table 1. Experimental setup. |
| Parameter | value |
|---|---|
| learning rate | 0.0001 |
| number of hidden layers | 20 |
| number of BP iterations | 10 |
| total number of batches | 40000 |
| number of batches per epoch | 1000 |
| physical error rate ε | 0.01 ∼ 0.26 |
| batch size for each physical error rate | 30 |
| error bias Bi | {0.5, 5, 10, 30, 50} |
| convolutional layer parameters | {3 × 3, 16;1 × 1, 8} |
| batch size for validation set | 50 |
In a multi-objective loss function, the choice of hyperparameters is a complex issue. Table 2 provides a reasonable explanation for the value of the hyperparameter selected in this paper. Therefore, we choose the value of the hyperparameter a = 0.3.
Table 2. Statistics of classical error rate and logical error rate for bias-tailored quantum code [[12, 2, 3]] under different hyperparameters a. The physical error rate ε is 0.06. |
| The value of a | classic error rate | logical error rate |
|---|---|---|
| 0.1 | 0.020292 | 0.016583 |
| 0.2 | 0.018125 | 0.014854 |
| 0.3 | 0.015646 | 0.011083 |
| 0.4 | 0.015875 | 0.012583 |
| 0.5 | 0.015854 | 0.013354 |
| 0.8 | 0.018937 | 0.015229 |
| 1.0 | 0.020958 | 0.020667 |
5.2. Decoder performance optimization
As an example, consider scheme 1 in figure 2, which is applied to the quantum code [[58,16,3]]. This code is constructed via the hypergraph product of the [7,4,3] code. We examine the spatial structure of the convolutional model in figure 4 and the weight distribution histogram in figure 5. This convolutional model, composed of 3 × 3 and 1 × 1 convolutional kernels, demonstrates significant advantages in the NBP + CNN + BIAS decoding of QECCs.
Figure 4. Visualization of the spatial structure of convolutional kernels. All the data results are obtained with a physical error rate of ε = 0.1. |
Figure 5. Histogram of the weight distribution of convolutional kernels. All the data results are obtained with a physical error rate of ε = 0.1. |
As shown in figure 4, the 3 × 3 convolutional kernel consists of 16 filters with diverse weight distributions, effectively capturing local spatial information such as local error patterns and noise structures in QECCs. This enhances the ability to locate and correct errors. The 1 × 1 convolutional kernel, with 8 filters, performs feature fusion and optimization across channels, integrating local features and highlighting critical information, which further improves the accuracy and efficiency of message passing.
From the weight distribution in figure 5, the weights of the 3 × 3 convolutional kernel are concentrated and smooth, reflecting its stability and robustness in BP decoding, effectively suppressing gradient instabilities caused by noise. In contrast, the 1 × 1 convolutional kernel exhibits a wider weight distribution, indicating stronger capabilities in global information representation and adjustment, ensuring the efficiency and accuracy of iterative updates.
Then, we analyze four partially iterative convolutional schemes and three fully iterative convolutional schemes, with their framework details illustrated in figure 2. According to figure 6, the optimization process of iterative scheme 1 is relatively slow, as its logical error rate decreases only slightly during the first 20 epochs and aligns with other schemes only after the 20th epoch, indicating lower optimization efficiency. In contrast, schemes 1 and 2 exhibit performance comparable to the NBP + CNN + BIAS method, achieving rapid convergence within 5–10 epochs and stabilizing thereafter, demonstrating similar optimization speed and effectiveness. Notably, schemes 3 and 4, along with iterative scheme 3, perform the best. These schemes significantly reduce the logical error rate during the early training stages (1–2 epochs) and converge quickly, showcasing higher optimization efficiency and faster convergence speed, achieving ideal performance in a shorter time. In scenarios with strict time requirements, we recommend selecting scheme 3, as it has relatively low resource consumption.
Figure 6. Comparison of convergence speeds for different convolutional schemes on [[58,16,3]]. All the data results are obtained with a physical error rate of ε = 0.1. |
At the end of this section, we consider the threshold of QECCs. The threshold is a key metric for evaluating the error correction capability in quantum computing, defined as the maximum physical error rate εth that a system can tolerate under a specific noise model. If the physical error rate ε satisfies ε < εth, the logical qubit error rate will decrease exponentially with the depth of the error correction cascade. Conversely, when ε ≥ εth, error correction may fail to suppress errors effectively, potentially leading to the accumulation of logical qubit errors. The threshold is a necessary condition for fault-tolerant quantum computation, and its value directly determines the noise tolerance range of the quantum computing system. The decoding thresholds of different decoders were tested under Surface codes with d = 3, 5, 7 to demonstrate the decoding advantages of NBP + CNN + BIAS. See Table 3 and figure 7.
Table 3. Observed thresholds for numerical simulations of the decoders applied to Surface codes (d = 3, 5, 7). ‘*' represents the case of circuit-level noise. |
Figure 7. Decoding Surface codes with NBP + CNN + BIAS. |
Note that a quaternary neural network combined with overcomplete check matrices is proposed to decode QECCs in the literature [22]. As shown in figure 8, the decoder in the literature [22] is named NBP4, where the larger numbers represent more rows of overcomplete matrices. We compared with it and found that their decoder focuses more on adding reasonable check rows to improve decoding performance. However, we also get good performance while preserving the original matrices.
Figure 8. Performance comparison of two decoders on [[46,2,9]]. |
5.3. Decoding performance on biased noise channels
Figure 9(a) shows the decoding curves for the CSS code and the XZZX code under different X-biases with a fixed physical error rate of ε = 0.06. The [[12,2,3]] code provided in [9] is tested here in both the CSS standard form and the XZZX form. Compared to the BP + OSD in [9], our NBP + CNN + BIAS demonstrates significantly better performance, with an order of magnitude improvement in decoding performance. Moreover, under the same quantum noise condition, specifically BX = 0.5, the logical error rates for both codes are equal. However, as the bias increases, decoding performance begins to diverge between the two codes. For the CSS code, the logical error rate starts to rise and eventually converges to a reachable upper bound. In contrast, increasing the bias of the XZZX code results in an exponential decrease in the logical error rate. This result can be attributed to the fact that the XZZX code decouples to its submatrix under high-bias conditions. The code distance of this quantum code is 3, while the distance dX under infinite bias for X errors is 6. Thus, the improvement in code performance with increasing bias is as expected.
Figure 9. Standard CSS quantum codes versus bias-tailored XZZX quantum codes. All legends with the suffix ‘BIAS' represent the decoding of the bias-tailored codes using the biased noise model. All the data results are obtained with a physical error rate of ε = 0.06. The OSD orders of [[12,2,3]] and [[129,28,3]] are 7 and 30, respectively. |
The standard CSS quantum code [[129,28,3]] can be constructed using the hypergraph product method, which involves two classical BCH codes [7,4,3] and [15,7,5]. Now, according to the algorithm 1 , we simulate the data results for the XZZX code [[129,8,3]]. Observing this quantum code, its code distance is 3, and as the X-bias increases infinitely, the error correction performance of the code increasingly relies on its subcode [15,7,5]. In other words, with infinite bias in X errors, the code distance dX of this code is 5. The test results shown in figure 9(b) demonstrate the significant advantages of the XZZX code. The Hadamard rotation operation reasonably breaks the symmetry of the hypergraph product code, allowing the XZZX code to utilize noise bias to improve decoding performance. Additionally, the NBP + CNN + BIAS algorithm presented in this paper has a notable decoding advantage due to the fact that [[129,28,3]] is a degenerate quantum code [12, 16].
Figure 10 shows the decoding curves of bias-tailored codes [[12,2,3]] and [[24,2]] (provided in [9]) with physical error rates ranging from 0.06 to 0.1. We observe that as the X-bias BX increases, the error rate decreases exponentially. Additionally, our NBP + CNN + BIAS decoder outperforms BP + OSD, with even the worst-performing curve of our decoder being better than the best-performing BP + OSD curve. This demonstrates that our decoder has a stronger noise resistance capability.
Figure 10. Simulation curves of bias-tailored quantum codes [[12,2,3]] and [[24,2]] under different physical error rates. The OSD orders of [[12,2,3]] and [[24,2]] are 7 and 13, respectively. |
6. Conclusions and discussions
Physical qubits are often influenced by biased noise, where one type of error is more likely to occur than others. In this paper, we have proposed a neural network-based decoding algorithm NBP + CNN + BIAS for bias-tailored quantum codes. Extensive simulations and analyses have revealed the following advantages of NBP + CNN + BIAS. We have employed the asymmetry of noise to enhance the decoding performance of bias-tailored QECCs. We have shown that the NBP + CNN + BIAS decoder demonstrates a higher decoding threshold on Surface codes. At last, we have shown that the considered convolutional schemes improve the convergence rate during the training phase of the decoder.
Future research could focus on decoding non-binary quantum codes or longer codes over asymmetric channels, emphasizing efficient algorithms that leverage noise asymmetry and ensure scalability. Investigating the impact of noise bias and exploring advanced optimization strategies, such as neural networks or hybrid methods, may enhance decoding performance which need further researched.