Machine learning method for simulating quantum computing system errors based on physical observations

Abstract

The application discloses a kind of machine learning methods based on physical observation analog quantum computing system error, belong to involve quantum computing field, including: step 1: from the one-dimensional p wave topological superconductor of belonging to BDI symmetry class constructs minimum extensible 2-MZM island;Step 2: according to the minimum extensible 2-MZM island established, the ground state hamiltonian of island is obtained;Step 3: in minimum extensible 2-MZM island, the hamiltonian of the interaction of boson heat bath is obtained by introducing boson heat bath interaction;Step 4: the state of minimum extensible 2-MZM island is described with time t changes by Pauli master equation;Step 5: the probability of different errors is obtained by using standard dwell time monte carlo algorithm to simulate Pauli master equation;Step 6: the final state probability of the i th MC event is predicted on minimum extensible 2-MZM island by training machine learning model through monte carlo event.This method can effectively predict error probability in the minimum extensible range, significantly improve efficiency.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of quantum computing, and in particular to a machine learning method for simulating quantum computing system errors based on physical observations. BACKGROUND

[0002] Since the early implementation of quantum computers, numerous studies on quantum computing have emerged. One of the most challenging problems in quantum computing is error correction. In quantum computing systems, the causes of errors include: imperfect gate operations; interactions between the system and the environment; differences between the implemented system and its design.

[0003] Currently, topological quantum computing methods for error correction have been proposed in the prior art, and this correction method can reduce the influence of the interaction between the system and the environment. In these topological quantum computing methods, it is found that Majorana zero modes (MZMs) with topological superconductor (TS) wires obey non-Abelian exchange statistics. Although MZM qubits are topologically protected by the non-locality of MZMs, they are still susceptible to quasi-particle localization. Quasi-particle contamination can cause two types of qubit errors: a loss of phase of MZMs, which causes MZM qubits to deviate from the computational subspace (X or Y error); and a phase error of MZM qubits (Z error).

[0004] The Pauli master equation is used to describe open systems in a thermal bath and can be considered in Monte Carlo (MC) simulations. It should be noted that MC consumes a large amount of computing power. Recently, machine learning methods have been introduced to study quantum devices. Compared with traditional methods such as MC, machine learning can help us reduce the computing power to simulate quantum computing system errors. SUMMARY

[0005] The purpose of the present application is to provide a machine learning method for simulating quantum computing system errors based on physical observations, which can reduce the computing power to simulate quantum computing system errors.

[0006] To achieve the above purpose, the present application provides a machine learning method for simulating quantum computing system errors based on physical observations, characterized by comprising the following steps:

[0007] Step 1: constructing a minimally extensible 2-MZM island from a one-dimensional p-wave topological superconductor belonging to the BDI symmetry class;

[0008] Step 2: obtaining the ground state Hamiltonian of the island according to the established minimally extensible 2-MZM island;

[0009] Step 3: introducing a bosonic thermal bath interaction in the minimally extensible 2-MZM island to obtain the Hamiltonian of the bosonic interaction;

[0010] Step 4: describe the state of the minimum extensible 2-MZM island changing with time t by the Pauli master equation;

[0011] Step 5: simulate the Pauli master equation by using the standard residence time Monte Carlo algorithm to obtain the probability of different errors;

[0012] Step 6: train the machine learning model by Monte Carlo event, and use the trained model to predict the final state probability of the i-th MC event on the minimum extensible 2-MZM island.

[0013] Further, the minimum extensible 2-MZM island constructed in step 1 includes two topological blocks and two MZMs, one non-topological backbone and one quantum dot.

[0014] Further, the ground state Hamiltonian of the minimum extensible 2-MZM island is:

[0015] H island =H1+H2+H3+H QD +H T , (3)

[0016]

[0017]

[0018] wherein H QD represents the Hamiltonian of the quantum dot, H T represents the tunneling Hamiltonian; H1, H2 and H3 are the Hamiltonians of different topological blocks, respectively; Δ1 is the superconducting pairing potential of the upper half of the 2-MZM island;

[0019] Δ2 is the superconducting pairing potential of the lower half of the 2-MZM island; h.c. is the overall charging energy of the island; t 13 and t 23 are the tunneling amplitudes between the blocks and the backbone; t QD 1 and t QD2 are the tunneling amplitudes between the blocks and the quantum dot.

[0020] Further, the Hamiltonian of the boson interaction obtained in step 3 is:

[0021]

[0022] wherein λ is the coupling strength, is the spinless electron operator located at j position in the block α, b α，j is the boson operator.

[0023] Further, the Pauli master equation described in step 4 is:

[0024]

[0025] where P(n,t) represents the sum of states |n> in the 2-MZM island; W(n|m) represents the transition rate from state |m> to |n>:

[0026]

[0027] where ΔE = E m - E n is the energy difference between state |m> and |n>; the hopping operator The behavior of quasi-particle excitation under the condition of sigma is described when η = {hopping, creation, annihilation}.

[0028] Further, the specific steps of step 5 include:

[0029] Step 1: initialize the relevant parameters, and initialize the 2-MZM island to state |m>;

[0030] Step 2: set the time δt = -ln(u)|[∑ n W(n|m)], where u is a random number uniformly distributed in the interval (0, 1);

[0031] Step 3: update the simulation time to t = t + δt, if t + δt < t sim , go to step 4, or go directly to step 5, where t sim represents the total time of the Monte Carlo simulation;

[0032] Step 4: randomly execute the hopping operator on the 2-MZM island according to the transition rate W(n|m);

[0033] Step 5: record the states of the two MZMs;

[0034] Step 6: repeat steps 3-5 to calculate the probability of different errors according to the obtained states of the MZMs.

[0035] Further, the machine learning model used in step 6 is an MLP multi-layer perceptron model.

[0036] Therefore, the machine learning method for simulating quantum computing system errors based on physical observation adopted by the present application has the following beneficial effects:

[0037] The present application utilizes the enhanced decision tree with gradient (BDTG) and multi-layer perceptron (MLP) model, which is trained on Monte Carlo simulation data and can effectively predict error probability within the smallest scalable range. Compared with traditional simulation technology, the efficiency of the present method is significantly improved.

[0038] The technical solutions of the present application are described in further detail below with reference to the accompanying drawings and examples. BRIEF DESCRIPTION OF DRAWINGS

[0039] Figure 1 Structure diagram of p-wave topological superconducting wire and minimum extensible MZM island.

[0040] Figure 2(a) is a flow chart of gradient boosting decision tree (BDTG) model training; Figure 2(b) is a multi-layer perceptron (MLP) method.

[0041] Figure 3(a) is a time evolution of error probability estimation using Monte Carlo (MC) machine learning method; Figure 3(b) is a time evolution of error probability estimated using BDTG machine learning method; Figure 3(c) is a time evolution of error probability estimated using MLP method machine learning method (ML-MLP). DETAILED DESCRIPTION

[0042] In the description of the present application, it should also be noted that, unless otherwise explicitly specified and limited, these examples are only used to illustrate the present application and not to limit the scope of the present application. In addition, it should be understood that after reading the content taught by the present application, those skilled in the art can make various modifications or modifications to the present application, and these equivalent forms also fall within the scope defined by the claims attached hereto.

[0043] 1. Minimum extensible 2-MZM island

[0044] The present application starts from the 1D p-wave topological superconductor of BDI symmetry class to construct a 2-MZM island. Such a wire model (1D p-wave topological superconductor) is also a spinless Kitaev model, and the Hamiltonian H in real space is:

[0045]

[0046] where, is the (annihilation) operator generated by the spinless electron at i, μ is the chemical potential, t is the tunneling strength, and Δ is the superconducting pairing potential;

[0047] When |μ|<2t, the model is topologically nontrivial. In particular, in the case of t=Δ and μ=0, the wire supports two Majorana zero-energy modes (MZMs) at both ends, and the model can be compactly represented as:

[0048]

[0049] where we use the Majorana operators and Decoupling Majorana operators γ1 and γ2N satisfies [H, γ1] = [H, γ2] = 0, and the Hamiltonian can be rewritten astop γ j =1,2N]=0, which are the MZMs at both ends of the wire. Quasi-particle operator d j =(γ) 2j +iγ 2j+1 The diagonal form of equation (2) can be visually represented by a box, such as... Figure 1 As shown, Figure 1 The red box in the image corresponds to MZMsγ 1,1 and γ 2,1 The upper (lower) blue boxes correspond to quasi-particles d. 1,i (d 2,i The green box corresponds to electron c. 3,i The yellow box corresponds to QD(quantumdot)f.

[0050] T. Karzig et al. proposed scalable designs for quantum computing. A key aspect of these designs is the configuration of MZM islands, which consist of p-wave topological superconducting wires and quantum dots (QDs). These MZMs are protected by a quasi-particle contamination mechanism, as the relatively large charging energy preserves the fermionic parity of the islands. However, parity-preserving quasi-particle excitations still lead to dephase in the MZMs.

[0051] Figure 1 Let represent a minimal scalable MZM island, decomposed into four parts: two topological blocks (blue), two MZMs (red), a non-topological backbone (green), and a quantum dot (yellow). The island's ground-state Hamiltonian is:

[0052] H island =H1+H2+H3+H QD +H T (3)

[0053] and:

[0054]

[0055] Among them, H QD H represents the Hamiltonian of a quantum dot. T H1 represents the tunneling Hamiltonian; H2 and H3 are the Hamiltonians of different topological blocks; two topological blocks support four MZMs. However, a clever design of the island is: close to the trunk ( Figure 1 The two MZMs on the blue side are hybridized, which requires the width of the island (i.e., the size of the trunk) to be shorter than the superconducting coherence length. Therefore, only γ remains. 1，1 and γ 2，1 Two MZMs. Hybridization produces a common, finite-energy Dirac fermion, such as Figure 1one of the middle green boxes. The Hamiltonian H3 contains a tunneling term (with tunneling amplitude t), which means that quasiparticles can tunnel on the backbone. Since the bulk coupled to the QD is spinless (or polarized), the spin of the non-interacting quantum dot Hamiltonian H QD can be ignored. H T The parameter t 13 and t 23 are the tunneling amplitudes between the bulk and the backbone; t QD 1 and t QD2 are the tunneling amplitudes between the bulk and the quantum dot; the overall charging energy h.c. of the island is relatively large and does not change if only parity conserving quasiparticle excitations are considered. Therefore, the energy change of the island is determined only by the ground state Hamiltonian of the island.

[0056] The present invention introduces the Hamiltonian of the interacting bosons:

[0057]

[0058] where λ is the coupling strength, is a spinless electron operator on site j in the bulk α, b α，j is a bosonic operator. The bosonic operator b α interacts locally with the electron on site j, which can be regarded as a quantum fluctuation of the chemical potential. The bosonic bath is trivial on the non-topological backbone, but will lead to quasiparticle excitations (α = 1, 2) that preserve parity on the bulk.

[0059] The time evolution of the physical system corresponding to equation (10) is described by the Pauli master equation:

[0060]

[0061] where P(n, t) is the overall population of the state |n> represented by each box in Figure 1 Each box in Figure 1 is either an occupied fermion state or an unoccupied fermion state. W(n|m) represents the transition rate from state |m> to |n>:

[0062]

[0063] where Δ ∈ = ∈ m - ∈ n is the energy difference between states |m> and |n>; the hopping operator describes the behavior of quasiparticle excitations under the σ condition when η = {hopping, creation, annihilation}.

[0064] It is worth noting that the MZMs near the backbone are hybridized, and the hopping operator The bath function γ(ω) in equation (12) is:

[0065]

[0066] where κ is the coupling constant, β = 1 / (k B T) is the inverse of the thermodynamic temperature and the Boltzmann constant k B .

[0067] 2. Monte Carlo simulation of 2-MZM island

[0068] The present application uses a standard residence time Monte Carlo algorithm to simulate the Pauli master equation (equation 11) to simulate the survival state of the physical model under the influence of the external environment, and the Monte Carlo simulation is based on the parameters shown in Table 1, and the range of the parameters used in the simulation is shown in Table 2, and the specific steps of the simulation are as follows:

[0069] Step 1: initialize the related parameters, and initialize the 2-MZM island to state |m>;

[0070] Step 2: set the time δt = -ln(u) / [∑ n W(n|m)], wherein u is a random number uniformly distributed in the interval (0, 1);

[0071] Step 3: update the simulation time to t = t + δt, if t + δt < t sim , go to step 4, or go to step 5 directly, wherein t sim represents the total time of the Monte Carlo simulation;

[0072] Step 4: randomly execute the jump operator on the 2-MZM island according to the transition rate W(n|m);

[0073] Step 5: record the states of the two MZMs (γ 1,1 and γ 2,1 ).

[0074] Repeat the above steps several times, so as to calculate the probability of different errors according to the obtained states of the MZMs.

[0075] Table 1: Monte Carlo simulation parameters

[0076]

[0077] Note: The parameters in the table are in units of Δ.

[0078] And, the inverse temperature β = 3 / Δ during simulation, the bulk length N = 3009, and the trunk length N3 = 6.

[0079] Table 2: Parameter range

[0080]

[0081] For simplicity, consider the 2-MZM island as a symmetric island, with its upper half identical to its lower half, Δ1 = Δ2 = Δ, N1 = N2 = N, t 13 = t 23 = t B , t QD1 = t QD2 = t QD .

[0082] Based on the obtained states of the two Majorana zero modes (MZMs), the specific steps to calculate the probabilities of different errors are as follows:

[0083] S1, Error detection and classification: In the final step of the simulation, the types of errors present in the system can be determined by detecting stabilizer operators and applying error correction protocols, including parity correction and the HDRG algorithm. This step can identify errors caused by the system's interaction with the thermal bath, as well as excitation state changes caused by weak photon pulse injection.

[0084] S2, Record error states: At the end of each simulation cycle, record the final state of the system. This includes recording any error states that were not corrected by the error correction protocol, as well as states that were successfully corrected by the correction protocol.

[0085] S3, Statistical analysis: After performing a large number of simulation cycles (as mentioned in the steps above, thousands of times), count the number of occurrences of each type of error. This includes counting the number of occurrences of various errors (such as bit flip errors, phase flip errors, etc.) and the number of occurrences of error-free states.

[0086] S4, Calculate error probabilities: Calculate the probability of each error based on the ratio of the number of occurrences of each error state to the total number of simulations. For example, if a particular error occurs 100 times in 1000 simulations, the probability of this error is 10%.

[0087] S5, Analyze results: By comparing the probabilities of different errors, evaluate the error correction capabilities and stability of the system. A high success rate of error correction indicates that the system has strong resistance to specific types of errors.

[0088] The quality of the MZM system can be characterized by the probability of obtaining the final state without errors, i.e., the correct state P C . The error probabilities of X / Y and Z in the final state are P XY and P Z , respectively. Obviously, P C + P XY + P Z= 1.

[0089] To simulate the effect of parameter values on the final probability of error, a large number of Monte Carlo events can be generated for each set of parameters. The uncertainty in the error probability calculated from the 68.3% confidence interval in a multinomial distribution is inversely related to the Monte Carlo statistics. To reduce the uncertainty in each set of parameters to the percent level, the Monte Carlo statistics should be 10 4 For example, in a total simulation of 2 x 10 4 out of 10 4 results have no error, then P C = 0.5 with a relative uncertainty of If one wants to determine a parameter range to obtain a high P C This requires a large amount of computation. In this example, there are 9 parameters that affect the error probability. When we scan the parameter space and assign 10 values to each parameter, we have 10 9 sets of parameters. This requires ~10 13 simulations.

[0090] 3. Machine learning method

[0091] Different parameter sets are correlated with each other in the same system. Modeling this correlation helps to reduce the computational load. Machine learning (ML) methods can provide us with these models. The probability of predicting the error of a quantum computing system can be described as a typical classification problem.

[0092] The present application uses gradient boosting decision trees (BDTG) and multi-layer perceptron (MLP) techniques in the toolkit for multivariate data analysis (TMVA) package. Before using the BDTG and MLP models, the BDTG and MLP models are trained by the method in the prior art using MC events, and the training process of BDTG and MLP is shown in FIG. 2. The i-th MC event with a set of parameters V undergoes quantum evolution after time τ and error correction process. The final state Stat can be represented by one of the three vectors:

[0093]

[0094] On the other hand, the BDTG and MLP models also predict the final state error probability of the i-th MC event with a set of parameters V:

[0095]

[0096] Define the loss function:

[0097]

[0098] A total of 2.5 x 10 7half of the total events, i.e. N = 1.25 x 10 7 The remaining half of the events were used as a test set to validate the effectiveness of the BDTG and MLP models.

[0099] To validate the effectiveness of the ML methods, the predicted results of the ML methods were compared with the MC calculation results. As can be seen from FIG. 3, both the BDTG and MLP methods can correctly predict the time evolution trend of the error probability of the 2-MZM island system. FIG. 3(a) includes the statistical uncertainty of the MC events following a multinomial distribution. Compared with the BDTG method, the results of the MLP method are smoother.

[0100] Finally, it should be noted that the above examples are only used to illustrate the technical solutions of the present application, but not to limit it. Although the present application has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present application can still be modified or replaced by equivalents, and these modifications or replacements cannot make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present application.

Claims

1. A machine learning method for simulating errors in a quantum computing system based on physical observations, characterized by: The method comprises the following steps: Step 1: constructing a minimum extensible 2-MZM island from a one-dimensional p-wave topological superconductor belonging to the BDI symmetry class; Step 2: obtaining a ground state Hamiltonian of the island according to the established minimum extensible 2-MZM island; Step 3: introducing a Bose bath interaction in the minimum extensible 2-MZM island to obtain a Hamiltonian of the Bose interaction; Step 4: describing a state of the minimum extensible 2-MZM island changing with time t by using a Pauli master equation; Step 5: simulating the Pauli master equation by using a standard residence time Monte Carlo algorithm to obtain probabilities of different errors; Step 6: training a machine learning model by using Monte Carlo event, and predicting a final state probability of an i-th Monte Carlo event on the minimum extensible 2-MZM island by using the trained model. The minimum extensible 2-MZM island constructed in step 1 comprises two topological blocks and two MZMs, a non-topological backbone and a quantum dot.

2. The machine learning method to analog quantum computing system error based on physical observations of claim 1, wherein, The ground state Hamiltonian of the minimum extensible 2-MZM island is: H island = H1+ H2+ H3+ H QD + H T , (3) where H QD represents the Hamiltonian of the quantum dot, H T represents the tunneling Hamiltonian; H1, H2 and H3 are the Hamiltonians of different topological blocks, respectively; Δ1 is the superconducting pairing potential of the upper half of the 2-MZM island; Δ2 is the superconducting pairing potential of the lower half of the 2-MZM island; h.c. is the overall charging energy of the island; t 13 and t 23 are the tunneling amplitudes between the body block and the main stem; t QD1 and t QD2 are the tunneling amplitudes between the body block and the quantum dot.

3. The machine learning method to simulate quantum computing system errors based on physical observations of claim 2, wherein, The Hamiltonian of the Bose interaction obtained in step 3 is: where λ is the coupling strength, is a spinless electron operator in the block α located at site j, b α,j is a bosonic operator.

4. The machine learning method to analog quantum computing system error based on physical observations of claim 3, wherein, The Pauli master equation described in step 4 is: Wherein, P(n, t) represents a state |n> of each module in the 2-MZM island; and W(n|m) represents a transition rate from a state |m> to |n>. where Δ ∈ = ∈ m - ∈ n is the energy difference between the states |m> and |n>; the jump operator The behavior of quasi-particle excitations under the σ condition is described when η = {jump, creation, annihilation}.

5. The machine learning method to simulate quantum computing system errors based on physical observations of claim 4, wherein, The specific steps of step 5 comprise: Step 1: initializing relevant parameters, and initializing the 2-MZM island to a state |m>; Step 2: Set time δt = -ln(u) / [∑ n W(n|m)], where u is a random number uniformly distributed in the interval (0,1); Step 3: Update the simulation time to t = t + δt, and if t + δt < t sim then go to Step 4, or go directly to Step 5, where t sim represents the total time of the Monte Carlo simulation. Step 4: randomly executing a jump operator on the 2-MZM island according to the transition rate W(n|m); Step 5: recording states of the two MZMs; Step 6: repeating steps 3-5, so as to calculate probabilities of different errors according to the obtained states of the MZMs.

6. The machine learning method to analog quantum computing system error based on physical observations of claim 1, wherein, The machine learning model used in step 6 is an MLP multi-layer perceptron model.

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