Quantizing the smallest building block of classical NNs, namely, the neuron, is a key challenge in building quantum NNs. Our soft quantum neuron model is inspired by biological neurons (Fig. 1), which can be implemented on realistic quantum systems. The term “soft” utilized here highlights the ability of our model to handle realistic environments and evolutions, distinguishing it from the standard quantum computing models. It is worth noting that soft computing is a proprietary term that is conceptually opposite to hard computing. In our proposal, a quantum neuron is modeled by a noisy qubit, which can be coupled with its surrounding environment. The initial state of the jth neuron can be described by a density matrix {\rho }_j^{\mathit{\text{in}}} in the computational basis |0⟩ and |1⟩. The quantum neuron {\rho }_j^{\mathit{\text{in}}} accepts n_j outputs s_i (i = 1, 2, …, n_j) from the final states {\rho }_i^{\mathit{\text{out}}} (i = 1, 2, …, n_j) of the other possible n_j neurons. The output s_i is determined by a 2-outcome projective measurement on {\rho }_i^{\mathit{\text{out}}} in the computational basis. It is therefore a classical binary signal, namely, s_i = 0 or 1. When s_i = 1, corresponding to the case where {\rho }_i^{\mathit{\text{out}}} is measured and collapses to the state ∣1〉, the quantum neuron {\rho }_j^{\mathit{\text{in}}} is acted upon by an arbitrary superoperator {\mathcal{W}}_{\mathit{ij}}, while when s_i = 0 ({\rho }_i^{\mathit{\text{out}}} collapses to the state ∣0〉), nothing happens to {\rho }_j^{\mathit{\text{in}}}. Ideally, {\mathcal{W}}_{\mathit{ij}} can be replaced by a corresponding unitary operator W_ij. As a result, the evolution of the whole system from the state {\otimes }_{i=1}^{n_j}{\rho }_i^{\mathit{\text{out}}}\otimes {\rho }_j^{\mathit{\text{in}}} iswhere, {\mathcal{O}}_{\mathit{ij}} is a classically controlled single-qubit operation, the superprojectors {\mathcal{P}}_{|s⟩} are defined by {\mathcal{P}}_{|s⟩}\rho =|s⟩⟨s|\rho |s⟩⟨s|, \widehat{I} is the identity operator, and \mathcal{T} represents a time-ordering operation. All {\mathcal{W}}_{\mathit{ij}} act upon the target neuron {\rho }_j^{\mathit{\text{in}}} with specific temporal patterns. As different quantum operations {\mathcal{W}}_{\mathit{ij}} might be noncommutative, the time-ordering of these operations is important. The state of the target neuron after the evolution of Eq. 1 can be obtained by tracing out all the input neurons {\rho }_i^{\mathit{\text{out}}}, namely, {\rho }_j^{\mathit{mid}}=\mathrm{t}{\mathrm{r}}_{\left\{i\right\}}{\rho }_{\left\{i\right\}j}^{\mathit{mid}}=\mathcal{T}{\prod }_{i=1}^{n_j}‍\left[p_i{\widehat{I}}_j+\left(1-p_i\right){\mathcal{W}}_{\mathit{ij}}\right]{\rho }_j^{\mathit{\text{in}}}, where p_i\equiv p_i\left(0\right)= tr(|0⟩_i〈0|ρ_i^out).

After the evolution of Eq. 1, the target neuron {\rho }_j^{\mathit{mid}} is independently acted upon by a local bias superoperator {\mathcal{U}}_j. This operator is designed to improve the flexibility and learning ability of quantum neurons. Ideally, {\mathcal{U}}_j can be replaced by a corresponding unitary operator U_j. The action of {\mathcal{U}}_j is similar to adding bias to neurons in classical NNs [6,7]. The final state of the target neuron is thus {\rho }_j^{\mathit{\text{out}}}={\mathcal{U}}_j\left({\rho }_j^{\mathit{mid}}\right). Similarly, the output s_j of the target neuron is obtained by a 2-outcome projective measurement on {\rho }_j^{\mathit{\text{out}}} in the computational basis. The output signal s_j of the target neuron is

The output s_j can be accepted by all other connecting quantum neurons and affects the evolution of the quantum neurons that accept the output. This completes the specification of our proposed quantum neuron model. Strikingly, our model contains noisy cases, which allow our model to work under the conditions of noisy and faulty quantum evolutions and measurements. An elementary setup of our model is the soft quantum perceptron, which consists of a soft quantum neuron accepting inputs of n other soft quantum neurons and providing a single output, though in probability.

All final states of our quantum neurons are mixed states, as the evolution of these neurons depends on the measurements of their input neurons, thus introducing classical probability. Although such measurements make the neurons evolve into mixed states, the proposed quantum neurons can still develop quantum correlations arising from quantum discord. To make this clear, we consider the simplest 2-neuron case. For the 2 neurons in the states {\rho }_1^{\mathit{\text{out}}}= p₁|0⟩₁⟨0| + (1−p₁)|1⟩₁⟨1| (p₁ ≠ 0, 1) and {\rho }_2^{\mathit{\text{in}}}, the action of an operation {\mathcal{O}}_{12} results in the statewhere {\mathcal{W}}_{12} represents a specific quantum channel. Quantum correlations, if any, of {\rho }_{12}^{\mathit{mid}} can be quantified by the quantum discord [34]. Any bipartite state is called fully classically correlated if it is of the form [35] {\rho }_{12}^c={\sum }_{i,j}‍p_{\mathit{ij}}|i⟩₁⟨i|⊗|j⟩₂⟨j|; otherwise, it is quantum-correlated. Here, |i⟩₁ and |j⟩₂ are the orthonormal bases of the 2 parties, with nonnegative probabilities p_ij.

Obviously, for {\rho }_{12}^{\mathit{mid}} in Eq. 3, the first neuron becomes quantum-correlated with the second as long as {\mathcal{W}}_{12}\left({\rho }_2^{\mathit{\text{in}}}\right) and {\rho }_2^{\mathit{\text{in}}} are nonorthogonal [36–38]. In particular, Refs. [37,38] show the creation of discord, from classically correlated 2-qubit states, by applying an amplitude-damping process only on one of the qubits; for the phase-damping process, see Ref. [35]. Actually, {\rho }_{12}^{\mathit{mid}} in Eq. 3 is the classical-quantum state, as dubbed in Ref. [37]. While the discord is zero for measurements on neuron-1, measurements on neuron-2 in general lead to nonzero discord.

Thus, we reveal a crucial property of our quantum neuron model. Namely, quantum correlations arising from quantum discord can be developed between the proposed quantum neurons in our model, although these neurons are generally in mixed states. Note that the existing quantum NN models are mainly based on variational quantum circuits requiring 2-qubit gates.

More remarkably, our model can be equated to quantum circuits generating quantum entanglement. To illustrate this more clearly, we take 3 neurons in Fig. 2A as an example and represent their interactions by a quantum circuit model shown in Fig. 2B. To facilitate the demonstration, we consider the ideal case where {\rho }_2^{\mathit{\text{in}}} and {\rho }_3^{\mathit{\text{in}}} are pure states and {\mathcal{W}}_{\mathit{ij}} and {\mathcal{U}}_j are replaced by corresponding unitary operators W_ij and U_j, respectively. According to the principle of deferred measurement [3], measurements can always be moved from the middle step of a quantum circuit to the end of the circuit. Therefore, the circuit in Fig. 2C is equivalent to that in Fig. 2B. In the equivalent circuit, the unitaries that are conditional on the measurement results are replaced by controlled unitary operations on {\rho }_2^{\mathit{\text{in}}} and {\rho }_3^{\mathit{\text{in}}}. It is easy to verify that quantum entanglement can exist between neuron-1 and neuron-2 (as well as neuron-3) in Fig. 2C. Another example of the principle of deferred measurement can be found in teleportation [3]. Nonetheless, it remains unclear whether this equivalence can be effectively utilized in computing tasks. We leave this matter for future work.

Quantum neurons are connected together in various configurations to form quantum NNs with learning abilities, thus representing a quantum neural computing device obeying the evolution-measurement rules provided above. Our neurons can, in principle, be combined into quantum analogues of any classical network architecture that has proven effective in many applications. In this work, we present a fully connected soft quantum feedforward NN (SQFNN) for application to supervised learning.

Neurons are arranged in layers in a fully connected feedforward NN (FNN). Each neuron accepts all the signals sent by the neurons in the previous layer and outputs the integrated signal to each neuron in the next layer. Note that there is no signal transmission between neurons within the same layer. To date, there has been no satisfactory quantum version of this simple model. Because no neuron can perfectly copy its quantum state in multiple duplicates as an output to the next layer due to the quantum no-cloning theorem [39], the output is not perfectly shared by neurons in the next layer. Because of the same theorem, quantum neural computing and standard quantum computing have incompatible requirements that are difficult to reconcile [12]. Our quantum NN model resolves this incompatibility by measuring each soft quantum neuron to give classical information as the integrated signal. This feature is essential for our model to be a genuine quantum NN model, which, while incorporating a neural computing mechanism, uses quantum laws consistently throughout neural computing.

In fact, many studies have made bold attempts in this challenging area. For example, Ref. [14] introduces a general “fan-out” unit that distributes information about the input state into several output qubits. The quantum neuron in Ref. [15] is modeled as an arbitrary unitary operator with m input qubits and n output qubits. These attempts provide new perspectives for resolving the abovementioned incompatibility. Unfortunately, none of them directly confronts this incompatibility. The neurons in these schemes still cannot share the outputs of the neurons in the previous layer; conversely, each neuron can only send different signals to different neurons in the next layer. In that sense, our NN is quite different from these quantum NNs.

Figure 3 shows the concept of an SQFNN. Without loss of generality, we specify that signals propagate from top to bottom and from left to right. Therefore, the evolution equation of the jth neuron in the lth layer iswhere {\mathcal{O}}_{i^{\left(l-1\right)}{j}^{\left(l\right)}} acts on the ith neuron in the (l – 1)th layer and the jth neuron in the lth layer. The final state of the output layer of the network can be obtained by calculating the final state of each neuron layer by layer with Eq. 4 after considering the local bias superoperator acting upon each neuron. Note that due to the randomness introduced by the measurement operations, the result of a single run of the quantum NN is unstable, i.e., probabilistic. One way to prevent this instability is to obtain the average output of the network by resetting and rerunning the entire network multiple times. This average output is more representative of the prediction made by our quantum NN and is therefore defined as the final output of the network. For each neuron of the output layer, the average output includes the binary outputs in the computation basis and their corresponding probabilities. Although running the network multiple times seems to consume more time and resources, this increase is only equivalent to an additional constant factor on the original consumption [15] and has no serious consequences. Therefore, running the network multiple times is common for extracting the information of quantum NNs and is also widely adopted by other quantum NN models [15,20]. Strikingly, this repetitive operation is easy and fast for a quantum computer. For example, the “Sycamore” quantum computer executed an instance of a quantum circuit a million times in 200 s [40].

In supervised learning, the NN must output a value close to the label of the training point. The closeness between the output and the label is usually measured by defining a loss function. The loss function in our model can be defined in various ways, e.g., by the fidelity between the output and the expected output or by a certain distance measure. In the simulations shown below, a mean squared error (MSE) loss function is adopted, which can be written aswhere N represents the size of a training set, y^k represents the label of the kth training point, and {\tilde{y}}^k represents the predicted label of our network for the kth training point, which is the average value of the output layer of the network obtained by resetting and rerunning the entire network multiple times. This loss function can be driven to a very low value by updating the parameters of the network, thereby improving the network performance. However, the loss function is nonconvex and thus requires iterative, gradient-based optimizers. As information is forward-propagated in our network, we can use a backpropagation algorithm to update the parameters of the quantum operations. Moreover, since only single-qubit gates are involved in our model, the total number of parameters is not large and is approximately {\sum }_{l=1}^{L-1}‍3\left(n_l+1\right)\times n_{l+1}, where L is the total number of layers in a network and n_l is the number of neurons in the lth layer. This number is directly proportional to the length L of the network and the square of the average width of the network (i.e., the average number of neurons per layer). In particular, the state space involved in computing the gradients is always that of a single neuron, thus circumventing the problem that the state-space size grows exponentially with the number of neurons. Many optimization algorithms widely used in classical NNs are therefore effectively compatible with our quantum NN, such as Adagrad [41], RMSprop [42], and Adam [43].

Both classical and quantum samples are available for our network, which is similar to other quantum NNs. For classical data, the input features need to be encoded into qubits and fed to the input layer. For quantum data, the quantum states can be decomposed into a tensor product of the qubits in the input layer, as in quantum circuits.

In this section, we benchmark soft quantum perceptrons and SQFNNs with simple XOR gate learning, classifying nonlinear datasets and handwritten digit recognition. Our models show extraordinary generalization abilities and robustness to noise in the numerical simulations.

We now discuss the details of the simulation setting for XOR gate learning. Figure 4A shows the model structure for learning an XOR gate, where 2 neurons in the input layer receive and encode data points, and the neuron in the output layer predicts the outcome. We adopt a simpler and more efficient angle encoding method instead of the method adopted in Ref. [45] to encode the data (Fig. 4B) that accelerates the convergence of the training process. Specifically, for an input set {x^k}, we encode the ith feature x_i^k of the kth data point by applying a single-qubit rotation gate R^Y\left(x_i^k\pi \right) on the initial qubit ∣0〉, where Y represents the rotation along the Y axis and x_i^k\pi represents the rotation angle. Note that a common MSE loss function and the Adam algorithm [43] are used in the training processes for all tasks in this study. The soft quantum perceptron for learning XOR is optimized for 20 epochs, and the learning rate is set to 0.1. Table 1 shows how the test accuracy of our model for the XOR gate learning task varies as the flip probability p increases.

A 2-4-1 MLP structure is used for comparison in the task of classifying the “circles” dataset, as classical perceptrons are unable to classify nonlinear datasets. The reason for using this structure is that the 2-4-1 MLP needs to learn 17 parameters, which is approximately the same number of parameters that our model needs to learn. The structure of the PQC model used for comparison is adopted from Ref. [46]. This common layered PQC model is denoted aswhere \overline{\theta } represents the overall learnable parameters of the PQC, B_{\ell }\left({\overline{\theta }}_{\ell }\right) is a parameterized block consisting of a certain number of single-qubit gates and entangling controlled gates, and depth d represents the total number of such blocks. These qubits and controlled gates in the same block B_{\ell }\left({\overline{\theta }}_{\ell }\right) form a cyclic code. The control proximity range of a cyclic code, denoted as r, defines how the controlled gates work. For any qubit index j ∈ [0,  N − 1] of an N-qubit circuit, the entangling code clock has one controlled gate with the jth qubit as the target and the qubit with the index k = (j + r) mod (N) as the control qubit (see Ref. [46] for details). In each block B_{\ell }\left({\overline{\theta }}_{\ell }\right) of our setting, each qubit is acted on by a parameterized universal single-qubit gate. Then, the code block follows. One more optimizable single-qubit gate R^Y acts on each qubit in the final B_{\ell }\left({\overline{\theta }}_{\ell }\right). The control proximity range of a cyclic code r is fixed to 1. Specifically, a 2-qubit circuit of depth d = 1 and size s = 6 is used to classify the “circles” dataset, where d is the number of blocks B_{\ell }\left({\overline{\theta }}_{\ell }\right) and s is the total number of gates in the circuit other than in the encoding layer. In particular, the encoding method in Ref. [45] is also used in this PQC model for classifying nonlinear datasets. To enrich the expressivity of our model, we adopt the “parallel encodings” strategy mentioned in Ref. [47] when classifying the “moons” dataset, that is, using multiple neurons to repeatedly encode the same input in the input layer. In the task of classifying the “moons” dataset, we repeatedly encode each input with 3 neurons. For comparison, we also simulate the results of a 2-10-1 MLP and a 4-qubit PQC with d = 2 and s = 24. The MLP has 41 parameters to learn. The PQC model also adopts the “parallel encodings” strategy in this task. In particular, the soft quantum perceptron does not have hidden layers, so it is a simpler structure compared to the MLP. In fact, in addition to the results presented in the main text, we also found that a 4-qubit PQC with d = 10 and s = 40 could only achieve 94% accuracy when classifying the “moons” dataset. Table 2 shows how the test accuracy of our model for classifying the “circles” datasets varies as the flip probability p increases. Note that this effect is continuous, but the presentation of our results in a discrete table format may create an impression of discontinuity. Moreover, when the probability of bit flip or bit–phase flip reaches 0.50, the ∣0〉 and ∣1〉 components of the corresponding quantum state become indistinguishable in the computational basis, resulting in the inability to extract any relevant information. This causes a sudden drop in the probability of successful learning. This effect is particularly pronounced in close proximity to the 0.50 probability threshold.

The specific simulation seting is as follows. First, we extract several subdatasets from MNIST. For example, {3, 6} represents the subdataset containing 2 classes of the digits 3 and 6. After that, we apply the same downsampling size to all images from the same subdataset of MNIST. Specifically, we downsample the resolution of the original images from 28 × 28 to 4 × 4 for the datasets with 2 or 3 classes, and to 8 × 8 for datasets with 4 or 5 classes. Finally, we use the structure from Ref. [18] that contains a hidden layer for QuantumFlow, the classical MLP, and the SQFNN, where the hidden layer contains 4 neurons for 2-class datasets, 8 neurons for 3-class datasets, and 16 neurons for 4- and 5-class datasets. The input and output layers of these models (including SMP) are determined by the downsampling size and the number of digits in the subdatasets. Note that the PQC model is designed as a 4-qubit circuit of d = 10 and s = 120 due to the lack of the concept of neurons. The PQC model is usually used as a binary classifier in the current study. Therefore, the PQC model is only used to classify the datasets with 2 classes in this task. Other QuantumFlow settings, such as accuracy, are consistent with those in Ref. [18].