The double Semion model plays a principal role in the fields of gapped systems and new topological orders [26], and the Semion code is an error correction code that needs to be studied in depth in topological codes. Semion code is a QEC code with the characteristics of the double Semion model. The Semion code has a topological protection effect on quantum information and will not affect the global error due to local errors. Semion code is a non-CSS and non-Pauli topological code described as a hexagonal lattice Λ. We map the qubits in three-dimensional space and use the topology of the code to convert qubits into qubits in multi-dimensional space. The edges represent physical qubits and the vertices represent stabilizer operators. The vertex operator is represented by V _Q , and vertex Q is represented as shown in Figure 1(1), and the Pauli Z operator is represented as: V Q = Z i Z j Z k (1)

Plaquette operator is represented by P _G , and apply the Pauli X operator on the sides of the hexagon: P G = ∏ k ∈ ∂ G X k ∑ j ⃗ p G j ⃗ | j ⃗ 〉 〈 j ⃗ | (2) j ⃗ in the above formula represents the bit string of a state on the basis of calculation, ∂G belongs to the edge of the plaquette boundary, the value of p G ( j ⃗ ) is {±1, ±i}. The diagonal operator ∑ j ⃗ p G ( j ⃗ ) | j ⃗ 〉 〈 j ⃗ | acts on the twelve qubits in Figure 1(2).

RL problems consider an agent that interacts with the environment [27]. The agent can manipulate and observe parts and perform a sequence of actions to accomplish a particular problem. Through RL, we can find the optimal policy of the action subject in the system. The optimal policy is the policy that proxies the best return in the process of interacting with the system. Discrete problems are usually considered. At each time step t, the environment can be represented by a state s _t ∈ S, where S is the state space. Given a state, the agent can choose to perform an action a _t ∈ A, where A is the action space. According to the result after the agent selects the action, the state is updated accordingly, entering a new state s _t+1, and providing the agent with feedback on the action selection in the form of reward r _t+1, starting from time t, the return R _t = r _t+1 + λr _t+1 + λ ² r _t+1 + ⋯, where λ ≤ 1 is the discount factor that quantifies how one wants to value immediate and subsequent returns [28]. There will be a constant return r = 1 for each step. To formalize the agent’s decision-making process, we define the agent’s policy as π, and π(a, s) is the probability that the agent chooses a _t = a when the state is in s _t = s. By using a measure of discounted cumulative reward, the value of any given state depends not only on immediate rewards from that state following a particular policy but also on expected rewards in the future.

As shown in Figure 1(4), the subscript q runs over the vertices belonging to the plaquette G. β _q can be represented by twelve qubits as Σ q ∈ G β q = i n 7 + n 1 + n 6 − − n 1 − n 6 + i n 8 − n 1 − n 2 − − n 1 + n 2 + = i n 9 − n 2 + n 3 + − n 2 − n 3 − i n 10 + n 3 − n 4 + − n 3 + n 4 − = i n 11 − n 4 − n 5 − − n 4 + n 5 + i n 12 − n 5 + n 6 + − n 5 − n 6 − (3) and n i ± = 1 2 1 ± Z i (4)

According to the above analysis, p G ( j ⃗ ) can be clearly defined as ∑ j p G j ⃗ | j ⃗ 〉 〈 j ⃗ | = ∏ k ∈ ∂ G − 1 n k − 1 − n k + ∏ q ∈ G β q (5) Therefore, according to the above reasoning, we add β _q to P G ′ on each vertex, and we can obtain a P G ′ expression that conforms to the entire Hilbert space. P G ′ = ∏ k ∈ ∂ G X k ∏ k ∈ ∂ G − 1 n k − 1 − n k + (6) P G ′ satisfies the commutate [29] principle of the operator, and the plaquette operator allows the definition based on stability topological error-correcting code in agent form.

The same as the string operator in Kitaev toric code, T ^Z in Semion code is expressed as a string operator that generates grid excitation [30], that is, T ^Z is a string of Z operators. Each stabilizer commutes with these operators except the grid operator at the end of the string, and the string X produces the string operator of the vertex excitation, as shown in Figure 1(3). We commute the characters on the path G. The string is marked as T G + and is supported by coon. T G + can only act on the coon set of qubits non-trivially. According to the constraints: (1)The square of the string operator is 1. (2)It must be determined by exchanging with the stabilizer. The system of linear equations can be withdrawn from F ( j ⃗ ) , F j ⃗ = F j ⃗ ⊕ i ⃗ (7) the qubit of F ( j ⃗ ) in Coon(G) [11] is 0, and the ⊕ sign represents the sum of the remainder of the bit string to Z. The value of F ( j ⃗ ) is {±1, ±i}. So the string T G + is: T G + = ∏ k ∈ G X k ∑ j F j ⃗ | j ⃗ 〉 〈 j ⃗ | (8) The quasiparticle vertex excitation behavior generated by T G + is the same as that of anyons.

The positive chirality string is defined as T ⁺, the negative chirality string is T ⁻, and the negative chirality string can be got by calculating the T ^Z string operator, that is, T ⁻ = T ^Z T ⁺. The operator commutes with the Z operator, and the Z operator and T ^± do not commute. In conclusion, the commutation principle to be followed is: T ± , T ± = 0 , T ∓ , T ± = 0 , T Z , T ± = 0 (9)

The Hamiltonian is used as the coding space, Semion codes are alike to Kitaev toric codes, with vertex and plaquette operators. Embedding the Semion code in Kitaev toric code results in two quantum memories with logical qubits. The logical operator consists of T L + and T H + , H(L) is any homogeneous non-trivial path in the horizontal (vertical) direction, and the other pair logical operator is T L − and T H − , which are non-self-intersecting or overlapping the composition of each path is shown in Figure 2(1). Two logical qubits of the code require two pairs of logical operators, which are defined as X ₁ and X ₂: X 1 = T H − , Z 1 = T L Z , X 2 = T L + , Z 2 = T H Z (10)

The set of these operators satisfies the inverse relationship. The hexagonal lattice makes the distance of the X operator twice that of the Z operator, which can better avoid errors. To perform error correction, the stabilizers have to be measured periodically, and the excitations have to be annihilated by bringing them together using the string operators.

The error-correcting ability [31] of QEC codes depends on the type and strength of qubit manager errors [32–34]. In the context of topological codes, two error models have been extensively studied, namely depolarizing noise and independent bit-flip and phase errors. In the depolarizing noise model, each qubit has an error according to the following probability (1−p _error ) for no error, and p error 3 for X, Y, and Z errors. p _error is a parameter between 0 and 1. The model is symmetric between X, Y, Z.

In the independent bit-flip and phase errors, each qubit will be affected by the error, we record the probability of X error, Y error, and Z error as p _XYZ , so the probability of error is p error = 2 p XY Z − p XY Z 2 . As shown in Table 1.

Assuming that the X operator is applied to a qubit, for three possible edge orientations, the probability of a syndrome error can be obtained, with the “+” sign indicating the excitation on a given plaquette. Table 2 shows the probabilities of calculating a given flux pattern, corresponding to Figure 2(2).

Consider that the error operator of the n-qubit Pauli operator is E. In the stabilizer, errors are detected by measuring the stabilizer generator. If no errors occur, these measurements will output +1 eigenvalues. If an error E occurs, the same as The stabilizer generator against E commutation will output −1, and the output of the stabilizer measurement is the error syndrome. To correct the error, the inverse operator of the error is applied, and in the case of the self-inverse Pauli error, the same operator can be applied [35,36]. The main task of error correction is to determine the correction operator to apply to a given syndrome. The decoder is designed to give an error model and output a correction operator after analyzing the probabilities of all possible errors consistent with the observed syndrome. The optimal decoder is to choose the most suitable correction chain, and this choice will depend heavily on the specific error model.

Embed the Semion code into the torus. We improved it and used the Ref. [37] programming framework to map the hexagonal lattice of Semion code to square, Ref. [11] provided an idea for our conversion process. As shown in Figure 3, the left picture is a schematic diagram of a hexagonal lattice, the numbers with blue circles represent half calculations, there are sixteen symbols in total, the red numbers are vertex operators, and there are thirty-two in total. The data outside of the square is the period filling used, which shows the periodic boundary condition of the Semion code. The figure on the right is a converted square lattice, and the numbers with blue circles represent plaquette operators. The “∣” in the figure is to ensure hexagonal space structure. Its value is always recorded as zero and does not correspond to any element measured by the stabilizer. The numbers in the blue circles represent companion calculations. Letters were used to represent vertices and plaquette operators, when the code distance is d, there are 2d ² vertices and d ² plaquettes, so there are 3d ² stabilizers in total. Map the Semion code into a square and choose d = 4, so a square image of 8 × 8 is obtained. We assume that vertex and plaquette operators are marked from right to left and top to bottom. The syndrome of vertex s corresponds to the W _k,m of an image element, where k and m are expressed as follows: k = 2 × ⌊ q − 1 2 d ⌋ + 1 (11) m = mod q − 1 + 1 − 2 d ⌊ q − 1 2 d ⌋ , 2 d + 1 (12) The syndrome of plaquette G corresponds to the W _k,m of the image element, where k and m are expressed as follows: k = 2 × ⌊ G − 1 2 d ⌋ + 2 (13) m = mod 2 G + 1 − 4 d ⌊ G − 1 d ⌋ , 2 d + 1 (14)

Quantum computers are affected by the noise of the external environment, which makes the operations perform defects. Therefore, an error correction mechanism is needed to improve the defects. The decoding algorithm needs to count the homology of each particle to restore topological information [6,38]. Stabilizer code allows errors to be detected by measuring stable code operators without changing the encoding information and correcting errors by performing recovery operations [39]. If the encoding task has a specific structure, the decoding task can be easier to handle, and an efficient decoder with better performance can be obtained. The topological code stabilizer is geometrically local, and the abnormal return value indicates that some qubits have errors [40]. Local errors can be detected and corrected by encoding quantum information in a non-local manner. Error syndromes consist of measurements of non-trivial stabilizer operators, and syndrome analysis can infer what errors have occurred and how to correct them.

Using the Q function to represent the action-value function of a set of actions and the cumulative reward of the corresponding transition, update the estimate of Q using the formula: Q s , a + δ r + λ m a x a ′ Q s ′ , a ′ − Q s , a → Q s , a (15) where δ < 1 is the learning rate. The action-value function Q ( s , a ) represents the payoff of taking action a in state s and following a certain strategy at π. In the next step of Q -learning, Q ( s , a ) = r + λ m a x a ′ Q ( s ′ , a ′ ) is used to quantify Q , s → s′ is the optimal policy to follow for the current estimate of Q . The policy is given by max a Q ( s , a ) taking action a will eventually converge to the optimal policy, and it is quite useful to follow the ɛ-greedy policy, which takes the optimal action for the estimate of Q ( s , a ) with probability (1 − ɛ), but take a random action with probability ɛ. For a large state-action space, it is impossible to store a complete action-value function. In deep Q -learning, a deep neural network is used to represent the action-value function. The input layer is the representation of a certain state, and the output layer is possible the value of the action, using Q ( s , a , θ ) to denote the parameterization of the Q -function by the neural network, and θ to denote the network’s complete set of weights and biases.

The RL decoder used is the evaluation of the capability of generated action by an agent through reinforcement information provided by the environment, without telling the agent how to generate corrective action. Since the outer environment offers a little piece of information, an agent must learn through experience. It learns a mapping from the environment state to the behavior so that the selected behavior can get the maximum reward of the environment, and the system dynamically adjusts the parameters. To achieve the maximum enhancement signal. In a larger state-action space, it is impossible to save a complete action-value function, using depth Q -learning, a deep neural network represents the action-value function, and the input layer represents a specific state. The output layer is the value of some earthly actions. This simulation part uses a neural network-based decoder [41,42], uses RL methods to optimize the observation of Semion code syndrome, and gradually proposes the recovery chain of a syndrome.

Training of decoder adopts deep Q -network algorithm, which uses experience playback technology to store experience gained by an agent in the form of a conversion tuple in-memory buffer. A specific process is to first send syndrome to an agent in the action part, according to the defect of Q -network, select action store results in a buffer in the form of a tuple, and then enter the learning process, and use stochastic gradient descent algorithm to reduce Q -network prediction for a gap between target and sample target, according to the requirements of the target network, network parameters are optimized, and then a new training sequence is started, the weight of Q -network is synchronized with the weight of target network. In terms of sample selection, the samples are divided into three independent parts, namely the training set, validation set, and test set. The training set is used to estimate the model, the validation set is used to determine the network structure or parameters that control the complexity of the model, and the test set tests the performance of the final selected optimal model. 50% of the sample is the training set, 25% of the sample is the validation and test sets, and all three parts are randomly selected from the sample.

Taking depolarization noise as an example, through the training of the decoder, the data map shown in Figure 4 is obtained. It is found that the performance of the decoder is better, and the accuracy of error correction can reach 77.5%. The decoder in this paper is to calculate the threshold of the Semion code. The logical error rate is drawn in the range of the physical error rate for different code distances, and the threshold is generally determined as the physical error rate value at the intersection of the two. For physical error rates below the intersection of the two, the logical error rate will decrease as the code distance increases. For each physical error rate, the logical error rate is calculated as the average of multiple independent instances, and for experimental certainty, it must be determined that a certain number of logical errors are observed each time in an actual experiment. For the code distance d, the logical error rate p _logical should have the following correspondence: p logical = p error − p threshold × d 1 v o (16) where p _error is the physical error rate, p _threshold is the threshold, v _o is the scaling exponent. Based on the above formula, this paper obtains the data graph as shown in Figure 5. It can be observed in Figure 5(1) that when the logical error rate p _logical = 0.31257, the threshold p _threshold = 0.081574. Figure 5(2) can be observed, but when the logical error rate p _logical = 0.2642, the threshold p _threshold = 0.09542. Thresholds vary due to code distances and qubits. It is considered to compare the outcome of this paper with a series of previous estimates of thresholds, some small difference between estimates is reasonable due to not the same execution of decoding algorithms and numerical simulations. As can be seen from the two graphs in Figure 5, when the physical error rate is below the threshold, the greater the code distance the more errors can be corrected, so the logical error rate will be lower. When the physical error rate is above the threshold, although a larger code distance can correct more errors, the logical error rate will be greater as the code itself has more quantum bits and more errors will occur.

Our threshold is significantly lower than that of other papers, this difference seems to be related to the definition of logical error rate, some papers define logical error rate p _logical as the error rate measured per round [43–45], according to the analysis of Ref. [46], with the d increase, the p _error of continuous curve intersection will decrease, and this definition will lead to an overestimation of the threshold. This is roughly the same as the data of some articles. Therefore, it is difficult for this paper to make a conclusive statement on the difference in the results. Nonetheless, this paper achieves the feasibility of implementing Q -networks for Semion code decoders.

RL has a good effect on optimization problems. It can extract non-local laws from noise and perform transfer learning in various tasks. Applying this advantage to the cost of qubits passing through the quantum gate can reduce the cost of qubits. The qubits contain auxiliary qubits in the process of comprehensive measurement, and the logic overhead is the cost of auxiliary qubits in the process of comprehensive measurement. In this paper, the Q -network of RL is used to experiment, and the number of simulated qubits ranges from 3 × 10⁷ to 2.1 × 10⁸, and compare the original overhead under different thresholds and the optimized overhead of the Q -network, Figure 6(1) shows that when the threshold is p _threshold = 0.081574, as the number of qubits increases, both the original overhead and the Q -network overhead increase, but the Q -network optimized overhead is significantly lower than the original overhead. Figure 62) shows that when the threshold is p _threshold = 0.09542, as the number of qubits increases, the optimized overhead of the Q -network is also much lower than the original overhead, although when the number of qubits is 2.1 × 10⁸, the optimized Q -network has the overhead is slightly higher than the original, but this does not affect our overall results in the slightest. At the same time, comparing the results under different thresholds in Figure 6, we can find that the larger the threshold, the greater the overhead of the quantum circuit gate.