HNN are multi-input, thresholded, binary nonlinear dynamic systems. The excitation function of the neuron is usually a step function, and the value of the neuron is −1, 1, or 0,1. When the value is 0 or −1, the current neuron is in the inhibited state, and when the value is 1, the current neuron is in the activated state. The HNN is a single layer neural network in which all neuron nodes are connected to other neuron nodes. There is no self-feedback between the nodes, forming a complete graph model. A neuron node in the inhibited state will enter the activated state when the stimulus exceeds a set threshold, i.e. it will jump from 0 or −1 to 1.

Each node in a HNN has the same function, and the output of a single node corresponds to the final state of that node, denoted by x _i , with the states of all nodes forming the state of the network X = x 1 , x 2 , x 3 , x 4 … x n − 1 , x n T . The topology and mode of operation is shown in Figure 2. The network enters a steady state and produces an output when the rate of change of the energy function of the network, ΔE = 0 or when a preset upper limit of iterations is reached. The energy function and the rate of change of the energy function are as follows. E ϵ = − 1 2 X T ϵ W X ϵ + X T ϵ θ Δ E = Δ E ϵ + 1 − Δ E ϵ (3) where W = x i j is the weight matrix, X = x i is the network state and θ = θ i is the threshold matrix.

The Hebbian rule describes the basic principle of synaptic plasticity, that is, continuous and repeated stimulation from presynaptic neurons to postsynaptic neurons can increase the efficiency of synaptic transmission.

The Hebbian rule is the oldest and simplest neuron learning rule. Here is the description equantion of the Hebbian rule: w i j = 1 p ∑ k = 1 p x i z x j z (4) Where w _ij is the connection weight from neuron j to neuron i, p is the number of training modes, and x i z is the i input of neuron k.

In the HNN, Hebbian rules can be used to design weight matrices: W = ∑ p = 1 P X p X p T (5) Here w _ii = 0, which means that there is no self-feedback between nodes. The equantion is rewritten as follows: W = ∑ p = 1 P X p X p T − I (6) Where I is the unit matrix and X is the system state of HNN.

Considering that the Hopfield network has M samples of X ^m , then: X m T X z = 0 , m ≠ z n , m = z (7) W X z = ∑ m = 1 m X m X m T − I X z = n − M X z (8) Because of n > M, therefore: f W X m = f n − M X m = sgn n − M X m = X m (9) According to Eq. 9: when a given sample, X ^m is the ideal attractor and produces a certain attractor domain around it, which will be “captured” by the attractor in the attractor domain. However, the condition that the given samples are orthogonal to each other is too harsh, which eventually leads to the attraction domain of some points outside the samples, which are regarded as pseudo attractors of the HNN.

The quantum Fourier transform is an efficient quantum algorithm for the Fourier transform of quantum amplitudes. The quantum Fourier transform is not the classical counterpart of the Fourier transform and does not speed up the Fourier transform process on classical data, but it can perform an important task-phase estimation, i.e. estimating the eigenvalues of the You operator under certain conditions. The matrix representation of the quantum Fourier transform is as follows: Q F T N = 1 N 1 1 1 ⋯ 1 1 ω ω 2 ⋯ ω N − 1 1 ω 2 ω 4 ⋯ ω 2 N − 1 ⋮ ⋮ ⋮ ⋮ 1 ω N − 1 ω N − 1 2 ⋯ ω N − 1 N − 1 (10)

Where, ω = e 2 π i N = cos 2 π N + i ⁡ sin 2 π N .

In the classical Fourier transform, the transformation takes the following form: y k = 1 N ∑ j = 0 N − 1 x j e 2 π j k / N (11)

The mathematical form of the quantum Fourier transform is similar to the mathematical representation of the discrete Fourier transform [34]. It is an operator defined on a set of standard orthogonal bases |0⟩, |1⟩⋯|N − 1⟩ with the following action: | j 〉 = 1 N ∑ j = 0 N − 1 e 2 π j j k / N | k 〉 (12) An arbitrary quantum state action can be expressed as: | ψ 〉 = ∑ j x ̃ j | j 〉 → ∑ j = 0 N − 1 x ̃ j Q F T | j 〉 = ∑ j = 0 N − 1 x ̃ j 1 N ∑ k = 0 N − 1 e i 2 π N j k | k 〉 = ∑ k = 0 N − 1 ∑ j = 0 N − 1 x ̃ j N e i 2 π N j k | k 〉 = ∑ k = 0 N − 1 y k | k 〉 (13) where the amplitude y k = 1 N ∑ j = 0 N − 1 x ̃ j e i 2 π N j k is the value of the discrete Fourier transform of the amplitude x ̃ j .

The transform itself does not have much obvious value, but it is an important component subalgorithm of the quantum phase estimation algorithm. The quantum Fourier transform corresponds to the quantum line diagram (omitting the SWAP gate), where R k = 1 0 0 e 2 π 2 k . Figure 3 illustrates the quantum circuit of the quantum Fourier transform.

The quantum phase estimation algorithm is the key to many quantum algorithms [6,35], and its role is to estimate the phase in the eigenvalues of the eigenvectors corresponding to the You matrix. The quantum circuit for quantum phase estimation is shown in Figure 4. The algorithm uses two registers, the first of which contains τ quantum bits with initial state |0⟩. The value of τ depends on the number of bits desired to be accurately estimated and the desired success rate. The second register has an initial state of x ̃ n . The essence of the process is the ability to perform the inverse Fourier transform: 1 2 τ 2 ∑ j = 0 2 τ − 1 e 2 π i φ j | j 〉 x ̃ n → | φ ̃ 〉 x ̃ n (14) where state | φ ̃ 〉 is the estimated value of φ.

Firstly, we will discuss HNN with the range restricted to cells with non-zero thresholds and a step function as the threshold function, which is by far the most common form of HNN. Secondly two consensus needs to be established: 1) the units in this network are perceptrons. 2) The perceptron can determine the weights and thresholds of the network for the problem to be learned. Focus on consensus i): Based on the definitions of HNN and perceptual machines above, it is clear that the unit in a HNN is a perceptual machine.

Focus on consensus ii): Consider a HNN with n cells, where W is the weight matrix of n × n, such that θ _i denotes the threshold of the cell i and the state of the network is X . If one wants this network to reach a steady state, it means that the following n inequalities must be satisfied: sign x 1 x 2 w 12 + x 3 w 13 + ⋯ + x n w 1 n − θ 1 > 0 sign x 2 x 1 w 21 + x 3 w 23 + ⋯ + x n w 2 n − θ 2 > 0 ⋮ sign x n x 1 w n 1 + x 2 w n 2 + ⋯ + x n − 1 w m n − 1 − θ n > 0 (15) Since it has no self-feedback, only the n (n − 1)/2 non-zero entries of the weight matrix W and the n thresholds of the cells appear in these inequalities. Let u denote the vector of n + n (n + 1)/2 dimension whose components are the non-diagonal elements of the weight matrix w _ij (i < j) and the n threshold minus signs. The vector u is given by the following equation: u = w 12 , w 13 , … , w 1 n , w 23 , w 24 , … , w 2 n , … , w n − 1 n , − θ 1 , … , − θ n (16) The vector x is transformed into n auxiliary vectors v ₁, v ₂, v ₃, … , v _n of dimension n + n (n + 1)/2 given by the expression: v 1 = x 2 , x 3 , … , x n ︸ n − 1 , 0,0 , … , 1,0 , … , 0 ︸ n v 2 = x 1 , 0 , … , 0 ︸ n − 1 , x 3 , … , x n ︸ n − 2 , 0,0 , … , 0,1 , … , 0 ︸ n v n = 0,0 , … , x 1 ︸ n − 1 , 0,0 , … , x 2 ︸ n − 2 , 0,0 , … , 0,0 , … , 1 ︸ n (17) Eq. 15 can be rewritten in the following form: sign x i v i ⋅ u > 0 (18) Eq. 18 shows that the solution to the original problem is found by computing the linear separation of vectors z _i . The vectors belonging to the positive half-space are those with sgn x i = 1 , and those belonging to the negative half-space are those with sgn x i = − 1 . This problem can be solved using perceptron learning, which allows us to calculate the weight vector v required for linear separation and from this to derive the weight matrix W with the threshold matrix θ. Figure 5 shows the correspondence between the HNN and the perceptron model.

First, t-qubit state|0⟩ are passed through the Hadmard gate, to obtain the superposition state | 0 〉 ⊗ τ → 1 2 τ ∑ J = 0 2 τ − 1 | J 〉 , where J is the integer form of the bit string j 1 , … , j τ , i.e. J = j ₁2 ⁿ⁻¹ + j ₂2 ⁿ⁻² + ⋯ + j _n 2⁰. Next, by an orcal operation O : O : 1 2 τ ∑ J = 0 2 τ − 1 | J 〉 ψ 0 → 1 2 τ ∑ J = 0 2 t − 1 | J 〉 U J ψ 0 | J 〉 U J ψ 0 = e 2 π i Δ ϕ h w , x ̃ J | J 〉 ψ 0 (19) Where U 0 = e i π , U = e i π ⊗ k = 1 n U k , U k = e − 2 π w k Δ ϕ 0 0 e 2 π i w k Δ ϕ , Δ ϕ = 1 / 2 n .

From Eqs. 13–19: 1 2 τ ∑ J = 0 2 τ − 1 | J 〉 U J ψ 0 = 1 2 τ ∑ J = 0 2 τ − 1 e 2 π i J φ | J 〉 ψ 0 (20) Finally the estimated phase can be obtained by inverse Fourier transform | φ ̃ 〉 : 1 2 τ ∑ J = 0 2 τ − 1 e 2 π i J φ | J 〉 ψ 0 → Q F T − 1 | φ ̃ 〉 ⊗ ψ 0

The connection between the HNN and the perceptron model was described above. It is now clarified how the design of the HNN weight matrix can be carried out using the quantum perceptron. Firstly, σ = ( v , u ) is input to the quantum perceptron model as an initial parameter and the model update rule for the quantum perceptron is as follows: U σ = ⊗ k = 1 n U k v k = ⊗ k = 1 n e 2 π i u k v k Δ ϕ v k = e 2 π i Δ ϕ ∑ k = 1 n u k v k ⊗ k = 1 n v k = e 2 π i Δ ϕ h u , v ⊗ k = 1 n v k = e 2 π i Δ ϕ h u , v σ (21) From the above equation, it can be deduced that |σ⟩ is an eigenvector of the matrix U and e ^{2πiΔϕh(u,v)} is the corresponding eigenvalue. By picking the appropriate value of t in the quantum perceptron, the inverse Fourier transform by: 1 2 τ ′ ∑ J ′ = 0 2 τ − 1 e 2 π i J ′ θ J ′ | σ 〉 → φ ̃ ′ ⊗ | σ 〉 (22) It is possible to obtain a value of, which is very close to the true phase, and also becomes closer to the true phase as the value of t becomes larger. Combining Eq. 19 gives: U ψ 0 = e 2 π i θ ψ 0 , θ = 0.5 + Δ ϕ h u , v ∈ 0,1 (23) Therefore the value of σ = ( v , u ) can be obtained by φ ̃ ′ .According to [], it can be known that in the perceptron model, its weight update rule: u i j ξ + 1 ≔ u j i ξ + 1 ≔ u i j ξ + η 2 σ i q − Y i q σ j q + σ j q − Y j q σ i q (24) where Y q = sgn u ( ξ ) σ q , η is the learning rate. However, when training with a perceptron, it is difficult to guarantee the separability of the data. Therefore, our perceptron model is trained using the delta rule, i.e. a gradient descent algorithm to search the space of possible weight vectors in order to find the best-fitting sample weight vector. The process is implemented with the aid of a classical computer. Its weight update rule is expressed in the same form as (Eq. 25), except that Y ^q = u (ξ) σ ^q .

We analyze the computational complexity of the HNN in two steps. 1) Analysis of the lift rate of the data to be trained after conversion of the HNN to the perceptron model. 2) The computational complexity required to complete the weight parameters by means of the quantum phase estimation algorithm. First we analyse i), any HNN with n nodes satisfying the requirements of Section 3.1 can be converted into a perceptron model with n (n − 1)/2 weight parameters. For ii), we analyze here two different algorithms for finding the weight parameters, namely the gradient descent-based algorithm and the Grover fast weight finding algorithm. The time complexity of the gradient descent-based algorithm is mainly controlled by the number of steps ɛ accuracy, i.e. O ∝ ɛ ²; the time complexity of finding the parameters using the Grover algorithm can reach O(n) under certain conditions. It is clear from this analysis that the final computational complexity is O n ϒ , regardless of the algorithm used. However, quantum machine learning is able to process information using quantum effects, as in this paper, where we input the training set as a superposition of feature vectors into a quantum perceptron model that can be processed simultaneously, and this process is not affected by the size of the model. The value of this process is small when the model size is small, and becomes more apparent as the model size increases and becomes the most important part of determining the computational complexity.

In the non-orthogonal simple matrix memory test, we demonstrated that our proposed solution can effectively cope with the memory confusion caused by non-orthogonal simple matrices; in the fragmented data recovery test, we demonstrated that our proposed QP-HNN has an average recovery rate improvement of 30.6% and a maximum of 49.1% in the effective interval compared with Hebbian rule-Hopfield network (HR-HNN), making it more practical. In the memory stress test based on QR code recognisability, our proposed QP-HNN is 2.25 times more effective than HR-HNN.

The non-orthogonal simple matrix memory test is set up for the Hebbian rule in the classical HNN, as one of the prerequisites for the design of the weight matrix using the Hebbian rule is that the input vectors must be orthogonal to each other, and if they do not satisfy orthogonality, the designed weight matrix may be incorrect. We demonstrate the impact of this deficiency using two non-orthogonal 3D row vectors X _v = [0, 1, 0] and X _ϑ = [1, 1, 1] as the input matrices for HR-HNN and QP-HNN, as shown in Figure 6 Where the trained weight matrix W _HR = [[0, 1, 1] [1, 0, 1] [1, 1, 0]] for HR-HNN, the weight matrix W _QP = [[0, 0.5, 0.3] [0.5, 0, 0] [0, 0, 0.2]] for QP − Hop and the threshold matrix θ _QP = [0.6, −0.1, 0.2].

In this subsection, we test and compare the recoverability of three different HNN: ClassicalPerceptron-Hopfield (CP-HNN), QP-HNN, and HR-HNN. Firstly, a random number generator was used to generate 100 60 × 60 binary matrices M = M b r 1 , M b r 2 … M b r i … M b r 99 , M b r 100 , and a different number of binary matrices M _ι , ι ∈ {1, 2 … 100} were randomly selected from M as the weight training matrices using QuantumPerceptron, ClassicalPerceptron, and HebbianRule to design the weight matrices respectively. A matrix M _bri was selected from M _ι and generated M b r i ′ = M b r i . 1 / 3 of the data in M b r i ′ was inverted to simulate data residuals, and this matrix was used as the input matrix for the network to test the recovery rate of the above three HNN. The example model is shown in Figure 7, and its recovery rate with different numbers of binary matrices memorised is shown in Figure 8.

From Figure 8, it can be seen that the resilience of the HR-HNN network decreases rapidly as ι becomes larger ι means that the orthogonality between the matrices in M _ι decreases, and consequently, memory confusion ensues. The resilience of the network basically fails at ι = 20 and is completely lost at ι = 30; the ClassicalPerceptron-Hopfield (CP-HNN) network is highly similar to the QP-HNN network in terms of resilience and has excellent robustness in the first and middle stages of ι growth because the network also trains the threshold This is equivalent to increasing the error tolerance space and mitigating errors due to the non-orthogonality of the vectors in the matrix. As can be seen from the diagram, the network is still very resilient at ι = 20. However, as ι increases, the fault tolerance space becomes saturated and the resilience decreases rapidly until it fails.

In order to visualise the memory capacity of the models, the differences between the models are presented using QR codes, which have different levels of fault tolerance and represent the number of error pixels that can be tolerated in the QR code. For our tests we have used the L level of fault tolerance, which allows for a maximum of 7% of incorrect pixels.

The QR code q ₁ is generated and stored in the “Successful Identification”, generating a QR code set Q = q n , n = 2,3,4,5 … the information in q _n is an irregular string of numbers generated by a random number generator, a randomly selected m - QR code from Q is used as the interfering QR code, and q ₁ is involved in the design work of HR-HNN and QP-HNN weight matrices. After 100 tests and statistical processing, the output matrix of HR-HNN can be successfully recognised when m ≤ 4; QP − Hop output matrix can be successfully recognised when, m ≤ 8. In Figure 9 we show a comparison of these two HNNs in terms of memory capacity.

The usability of HNN is also affected by the number of iterations required for the model to converge, which in turn is affected by the completeness of the weights, threshold information and input data. Therefore, building on the previous subsection, we further investigate the number of iterations required for q′ to recover to the state q ̂ where information can be correctly identified for different numbers of interfering QR codes, as shown in subplot a and subplot b in Figure 10. Subplot c shows the difference in the number of iterations required for q′ to recover to q ̂ with the same amount of information. As can be seen from the figure, QP-HNN possesses a significant advantage over HR-HNN for the q′ to q ̂ process, and this advantage becomes more pronounced as m grows.

Table 1 counts the recovery capacity limit of the HNN when the preset upper limit of 30,000 iterations is reached, where HR-HNN reaches the memory limit at m = 4, i.e. at m = 5, q′ cannot recover to q ̂ even if the number of iterations is increased, while in QP-HNN, the memory limit occurs at m = 8.