Coherent feed forward quantum neural network

Abstract

A coherent feed forward quantum neural network comprising: a first layer for encoding of input data onto at least one qubit; a second layer comprising a network of controlled parameterized rotation gates and uncontrolled parameterized bias gates; an output layer, wherein a measurement is performed on the final qubit or final set of qubits; and whereby the measurement is only performed once within the coherent feed forward quantum neural network in the output layer.

Description

COHERENT FEED FORWARD QUANTUM NEURAL NETWORK FIELD

[0001] Aspects of the disclosure relate to methods and systems for quantum computing. BACKGROUND

[0002] Over the past decade, Quantum Machine Learning (QML) has emerged as a dynamic field with promising potential for advancing machine learning techniques using quantum computing [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. While initial research focused primarily on adapting standard machine learning algorithms to quantum computing, such as Quantum Neural Networks (QNNs) and Quantum Support Vector Machines (QSVMs), progress has been hindered by the lack of a fault- tolerant quantum machine. Instead, researchers have focused on developing algorithms suitable for the current generation of noisy intermediate-scale quantum (NISQ) devices, resulting in a new wave of algorithms known as second-generation QML algorithms.

[0003] These recent QML algorithms are founded on Low Depth Parameterized Quantum Circuits (PQCs), which take a hybrid approach that combines the strengths of quantum processors with classical processors. This hybridization allows for the development of novel algorithms that have shown some advantages in specific use cases, although a genuine quantum advantage remains to be seen.

[0004] Most quantum machine learning (QML) models currently available employ complex encoding techniques, known as quantum feature maps, and use parameterized quantum circuits as models. Post-processing steps are similar to those used in classical machine learning (ML), and are performed on a classical computer. However, these feature mapping techniques are not always practical for real-world datasets, since the number of required qubits and the circuit depth increase with the number of features in the data. In addition, the term ”quantum neural network” is somewhat misleading for these models, since they do not adjust intermediate layers and neurons as a general neural network would.SUMMARY

[0005] In one of its aspects, a coherent feed-forward quantum neural network comprising: an encoding layer for encoding of input data onto at least one of a plurality of qubits; at least one intermediate layer comprising a network of controlled parameterized rotation gates acting on any of the plurality of qubits; an output layer, wherein a measurement is performed on at least one of a final qubit and a final set of qubits of the plurality of qubits; and whereby the measurement is only performed once within the coherent feed forward quantum neural network.

[0006] In another of its aspects, a method executed by a quantum apparatus comprising a quantum neural network, the method comprising the quantum apparatus executing operations comprising: at an encoding layer, encoding of input data onto at least one of a plurality of qubits; at at least one intermediate layer comprising a network of controlled parameterized rotation gates, applying a controlled operation on any of the plurality of qubits, and a biasing term on any of the plurality of qubits; performing a measurement on at least one of a final qubit and a final set of qubits of the plurality of qubits in the output layer; and whereby the measurement is only performed once within the coherent feed forward quantum neural network in the output layer.

[0007] In another of its aspects, a fixed coherent feed forward quantum neural network comprising: an encoding layer for encoding of input data onto at least one of a plurality of qubits, wherein the weights in the encoding layer remain untrained; at least one intermediate layer comprising a network of controlled parameterized rotation gates acting on any of the plurality of qubits; an output layer, wherein a measurement is performed on at least one of a final qubit and a final set of qubits of the of qubits; andwhereby the measurement is only performed once within the coherent feed forward quantum neural network.

[0008] Furthermore, there is provided a novel mixed circuit approach that incorporates entanglement layers, resembling a classical feed-forward neural network. This approach offers the advantage of adaptable intermediate layers, allowing for adjustment of the intermediate layers according to the characteristics of the data, which is particularly significant for classification tasks.

[0009] In the hybrid quantum–classical solution, all nodes are required to be implemented simultaneously on a dedicated quantum register, thus making the quantum computation more demanding. In the present application, a coherent feedforward quantum neural network (CFFQNN) reduces the necessity to store and process classical bits during intermediate stages. The CFFQNN defers measurements typically performed at an intermediate stage in a quantum circuit to the last computation stage or output layer. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Several exemplary embodiments of the present disclosure will now be described, by way of example only, with reference to the appended drawings in which:

[0011] Figure 1a shows an architecture of a quantum neural network;

[0012] Figure 1b shows a 3-qubit Feature Map circuit (ZZFeatureMap); and

[0013] Figure 1c shows a 3-qubit variational circuit with weight parameters;

[0014] Figure 2 shows a perceptron;

[0015] Figure 3 shows a successive encoding process of all data points performed on a single qubit;

[0016] Figure 4a an architecture of an artificial neural network with twohidden layers, in which:^^ ^^represents the weight parameters, ^^^^are data points, σ is a non-linear activation function, and ℎ^^ ^^= ^^^^ ^^^^^^;

[0017] Figure 4b shows an architecture of a CFFQNN with two hidden layers, where ^^^^are data points, and ^^^^ ^^s and ^^^^ ^^s represent the weight parameters;

[0018] Figure 4c shows a single intermediate node, other than the first hidden layer nodes, in the CFFQNN’s hidden layer;

[0019] Figure 5a shows a training curve for both CFFQNN and QNN on Iris dataset (Type-3 Vs. rest);

[0020] Figure 5b shows a performance comparison of CFFQNN and QNN based on evaluation metrics and the use of quantum resources on Iris dataset (Type-3 vs. rest);

[0021] Figure 6a shows a training curve for both CFFQNN and QNN on credit card fraud dataset;

[0022] Figure 6b shows a performance comparison of CFFQNN and QNN based on evaluation metrics on credit card fraud dataset;

[0023] Figure 6c shows a performance comparison of CFFQNN with QNN based on the use of quantum resources for credit card fraud dataset;

[0024] Figure 7 shows training curves for both CFFQNN and QNN for Iris dataset (Type-1 Vs rest);

[0025] Figure 8 shows a performance comparison of CFFQNN with QNN based on evaluation metrics and the use of quantum resources for Iris dataset (Type-1 Vs rest);

[0026] Figure 9 shows a training curve for both CFFQNN and QNN for Breast Cancer dataset;

[0027] Figure10 shows a performance comparison of CFFQNN with QNN based on evaluation metrics and the use of quantum resources for Breast Cancer dataset.

[0028] Figure 11a shows a performance comparison between CFFQNN, FixedCFFQNN, QNN, and a classical MLPClassifier for credit card fraud dataset;

[0029] Figure 11b shows quantum resources utilized by each of the four quantum models for credit card fraud dataset;

[0030] Figure 12a shows a performance comparison between CFFQNN, FixedCFFQNN, QNN, and a classical MLPClassifier for Breast Cancer dataset; and

[0031] Figure 12b provides a comparative analysis of the resources consumed by each of the four quantum models for Breast Cancer dataset. DESCRIPTION

[0032] The following detailed description refers to the accompanying drawings. Wherever possible, the same numbers are used in the drawings and thefollowing description to refer to the same or similar elements. While embodiments of the disclosure may be described, modifications, adaptations, and other implementations are possible. For example, substitutions, additions, or modifications may be made to the elements illustrated in the drawings, and the methods described herein may be modified by substituting, reordering, or adding stages to the disclosed methods. Accordingly, the following detailed description does not limit the disclosure. Instead, the proper scope of the disclosure is defined by the appended claims.

[0033] Moreover, it should be appreciated that the particular implementations shown and described herein are illustrative of the invention and are not intended to otherwise limit the scope of the present invention in any way. Indeed, for the sake of brevity, certain sub-components of the individual operating components, conventional data networking, application development and other functional aspects of the systems may not be described in detail herein. Furthermore, the connecting lines shown in the various figures contained herein are intended to represent exemplary functional relationships and / or physical couplings between the various elements. It should be noted that many alternative or additional functional relationships or physical connections may be present in a practical system.

[0034] Looking at Figure 1, there is shown an architecture of a quantum neural network 10. With the quantum circuit diagram 10, the qubits of the register are each initialized as |0^ 12, and then an initial state is prepared. The state preparation step also includes loading data via a feature map 14 that (as a result) is encoded in the state of the register. An example feature map 20 is shown in Figure 1b. Afterwards, a variational circuit 16 comprising quantum gates performs unitary transformations on the state of the register. A quantum gate is the basic logic unit to operate on one or more quantum bits (qubits). An example variational circuit 30 is shown in Figure 1c. Finally, one or multiple qubits are measured as the result of this computation indicated by measurement gates 18.

[0035] The functioning of the brain inspires the idea of a neural network 60 as shown in Figure 4a, in that the brain receives and processes information via a network of neurons, also known as biological axon-synapse-dendrite connections. A similar structure i.e. artificial neurons or or nodes 40 is used to process the dataand the information in a neural network 60. Each layer contains an arbitrary number of nodes which can individually be implemented on a quantum hardware. These nodes are connected to other nodes via links similar to those inside the brain, and each link has a weight, which determines the strength of one node’s influence on another.

[0036] Mathematically a perceptron is given by:

[0037] ^^ = ℎ = ^^(∑^^^^^^^^^^+ ^^) = ^^ ( ^^) (1)

[0038] where a is non-linear activation function, such as Logistic sigmoid or RELU function, b is the biasing term, and

[0039] ^^ = ∑^^^^^^^^^^+ ^^ (2)

[0040] For a binary classification task, the perceptron’s output ^^ ( ^^) is either 0 or 1, depending on the value of ^^. By adjusting the values of the weights ^^^^, a Neural network learns to classify the given data.

[0041] In general, an artificial neural network (ANN) is not just a single perceptron but a combination of multiple nodes in different layers. ANNs learn by optimizing the weights in each layer, which is done by using supervised learning methods such as backpropagation and gradient descent. These ANNs with sufficient layers can approximate any function of the input data with arbitrarily high accuracy.

[0042] Feedforward neural Networks (FFNNs) are a fundamental type of ANN in which information flows in only one direction from the input to the output layer via some hidden layers, as such, nodes in each layer are connected to the nodes in the next layer, as shown in Figure 4a. These FFNNs have demonstrated remarkable performance in various tasks, including classification, regression, etc.

[0043] Quantum Neural Network (QNN)

[0044] Quantum neural networks (QNNs) are among the most interesting algorithms in QML. Even though the field is still developing, it has started to be explored for its benefits in both academic and industrial setting. Almost all of the present QNNs are inspired by the variational approach used in many hybrid algorithms

[0015] .

[0045] In general, we start with encoding the classical data ^^ ∈ ℝ^^into the quantum circuit using a feature map ^^( ^^). Then we apply a variational circuit ^^^^ ^^ ^^( ^^) with multiple rotations ^^( ^^^^ ^^) and layers on our input state ^^(^^)|0〉⊗ ^^,where ^^ are the trainable weights. Finally, we measure the expectation value of some observable (e.g. ^^⊗ ^^) and do the classical post-processing to predict the class ^^̃ of the input data. Optimizing the weight parameters present in the variational circuit is done using some classical optimizers, eventually minimizing the cost function ^^( ^^ ( ^^, ^^), ^^̃) in consideration.

[0015] . A schematic diagram of a typical QNN is shown in Figure 1a.

[0046] Coherent Feedforward Quantum Neural Network (CFFQNN)

[0047] The CFFQNN model 70 presented herein uses a mixed approach. As can be seen in Figure 4b, the first layer 72 of CFFQNN model 70, or the encoding layer, is inspired from the input layer 62 of the ANN 60 of Figure 4a. In the second layer 74, there is a network of controlled parameterized rotation gates with trainable weights and biases inspired from the hidden layers of the ANN 60 of Figure 4a. Ultimately, only the qubits in the last layer 76 are measured. However, the measurement can be modified based on the data and the desired structure. For example, for classification tasks with more than two output dimensions, more output layers can be combined, where the number of outcomes distinguishable by measuring ^^⊗ ^^on ^^ output qubits is exponential in ^^: 2^^, if the scheme of

[0057] is used for each output qubit; even more possibilities can be distinguished by changing from

[0057] to greater than binary classification per qubit. This is akin to changing

[0033] -

[0037] from a perceptron to a multiclass perceptron. Accordingly, the qubits behave as a perceptron, and the data points with weights are encoded into the rotation angle of the ^^^^gate. i.e., a rotation through angle ^^ around the ^^-axis.

[0048] ) sin( ^^ / 2) cos( ^^ / 2)

[0049] In the computational basis, the encoding unitary in the first layer 72 may be represented as follows:

[0050] ^^( ^^^^^^^^) = ^^^^( ^^^^^^^^) )

[0051] The encoding of all data points is successively done on a single qubit one after another, as shown in encoding scheme 30 in Figure 3.

[0052] This single qubit operations can be written as: ^^ ^^ ^^ ⋯ ^^ ^^ ^^ (biasing term, and ^^0= 1, and ^^ = ∑^^^^^^^^^^+ ^^ .

[0055] Equation 3 resembles a single-layer perceptron. The only difference is, instead of the ^^ that appears in classical perceptrons, we have ^^. And, unlike ^^, the output of ^^ is not deterministic; it is probabilistic. Nevertheless, we know we will get 1 () 2^^ with probability ^^ 1 = cos 2 and 0 the rest of the time ^^(0)= 1 − ^^(1), depending on the value of ^^.

[0056] ( ^^(1) > ^^(0) if ^^ > ^^ { ^^(1) < ^^(0) if ^^ < ^^ (4)

[0057] where ^^ = ^^ / 2.

[0058] In the next layer 74 one biasing term is applied (Parameterized gate) on the qubit(s) and then other parameterized gates controlled by the qubits in the first layer 72, i.e., if the qubit in the first layer is in state 1, it will apply a parameterized ^^^^gate on the next layer qubit(s). The desired connectivity of the network is enforced by choosing which qubits in the first hidden layer control which qubits in the second hidden layer.

[0059] In the last layer 76, the expectation value of the ^^ operator on the qubit involved in the final layer operation is calculated, which is further used to decide the class of the data point, i.e.,

[0060] 1 if sign ^^ = 1 ^^̃ = {(^ ^)(^ ^) (5) 0 if sign ^^ ≤ 0

[0061] This method can create a complete feed-neural network (FFNN) similar to a standard one without doing intermediate measurements. The schematic diagram of Quantum FFNN is shown in the Figure 4b and may comprise additional hidden layers. However, any measurement operator may be chosen instead of ^ ^^^, in fact, other measurement strategies may be employed e.g. extra measurement qubits, or binning values of ^ ^^^ into more than two partitions, and so forth.

[0062] The second hidden layer’s operations can be written as:

[0063] ^^ )

[0064] where the variables ^^ and ^^ respectively denote the indices of the control layer (the previous layer) and the current layer, i.e., each term in the product over ^^ and ^^ represents the operation on the ^^th layer depending on the values of the ^^th layer. ^^ ⊗ ^^^^; ^^( ^^0; ^^) represents the biasing term on jth layer that is applied regardless of the configuration of qubits in the previous layer. ^^^^|1〉^^^1| ⊗ ^^the Y-rotation and Controlled Y-rotation gates, respectively, where the latter applies a rotation by angle ^^^^ ^^on the ^^thqubit conditioned in the ^^thqubit being in state |1^. Each parameter ^^0, ^^and ^^^^, ^^is trainable.

[0062] can be rewritten as ^^( ^^) =^^ ^^ ^^; ^^ 0+^^=1 ^^ ^^ ^^, the outer sum is over all bit strings ^^ that account for all possible states of the ^^ qubits; in the control layer and the product over ^^ over the current layer. The overall rotation angle on the jthqubit ^^0+∑^^=1^^^^^^^^ ^^resembles a classical neural network, while the quantum neural network creates an entangled state where each of these rotated target qubits are entangled with all of the possible states of the control qubits.

[0066] The aggregation of the hidden layers comprises a variational quantum circuit.

[0067] Figure 4c shows a single intermediate node 80, other than the input layer nodes, in the CFFQNN’s hidden layer. Figure 4c illustrates how the node applies weight values (denoted as ^^) to these nodes depending on the value of the last layer nodes. After this operation, the state |ϕi0^ changes to a new state |ψij^ that may continueto be acted upon by other control layers. One has the modify or remove these connections as per the desired network structure.

[0068] Based on observations, it is evident that the model described herein consistently demonstrates superior performance compared to the existing quantum neural network (QNN) model. This enhanced performance can be attributed to the incorporation of∑^^^^^^^^^^terms (inspired by classical neural networks) in the initial layer within a higher dimensional Hilbert space, as well as the integration of quantum entanglement in subsequent layers in a manner that puts into superposition many terms depending onsuperposed over ^^ values from which a classical perceptron would have to select only one. Figure 5a shows a training curve for both CFFQNN and QNN on Iris dataset (Type 3 Vs rest); Figure 5b shows a performance comparison of CFFQNN and QNN based on evaluation metrics and the use of quantum resources on Iris dataset (Type 3 Vs rest); Figure 6a shows a training curve for both CFFQNN and QNN on Credit Card Fraud dataset; Figure 6b shows a performance comparison of CFFQNN and QNN based on evaluation metrics on a Credit Card Fraud dataset; and Figure 6c is a performance comparison of CFFQNN with QNN based on the use of quantum resources for Credit card Fraud dataset; Figure 7 shows a training curve for both CFFQNN and QNN on Iris dataset (Type-1 Vs rest) dataset; Figure 8 shows a performance comparison of CFFQNN and QNNN based on evaluation metrics and the use of quantum resources on Iris dataset (Type-1 Vs rest) dataset; Figure 9 shows a training curve for both CFFQNN and QNN on Breast Cancer dataset; and Figure 10 shows a performance comparison of CFFQNN and QNNN based on evaluation metrics and the use of quantum resources on Breast Cancer dataset; Figure 11a shows a performance comparison between CFFQNN, FixedCFFQNN, QNN, and a classical MLPClassifier for credit card fraud Figure 11b shows quantum resourcesutilized by each of the four quantum models for credit card fraud dataset; Figure 12a shows a performance comparison between CFFQNN, FixedCFFQNN, QNN, and a classical MLPClassifier for Breast Cancer dataset; and Figure 12b provides a comparative analysis of the resources consumed by each of the four quantum models for Breast Cancer dataset.

[0069] In another example, a variant of the above-noted CFFQNN model, known herein as a FixedCFFQNN model, retains the architectural design of the CFFQNN with the exception that the weights in its initial layer remain untrained. This reduction in the number of parameters to be trained significantly speeds the training process while still outperforming previous QNNs.

[0070] The architecture of CFFQNN closely mirrors that of an ANN, sharing many of the same hyperparameters. This allows for customization in terms of the number of layers and nodes within each layer. Additionally, the measurement scheme can be tailored to fit specific data needs; even the parameter α in Eq. (5) is a hyperparameter. For instance, while a single measurement strategy for CFFQNN measures only the final qubit, for the FixedCFFQNN, a partial measurement approach, targeting all qubits except the qubits in the initial layer, is employed.

[0071] The corresponding results of the trained networks on test data are shown in Figures 5b (left-hand side), 6b, 8 (left-hand side), 10 (left-hand side), 11a and 12a. Additionally, the coherent feedforward quantum neural network (CFFQNN) does not necessitate measurements across all qubits; for all the datasets considered in this study, measurements were only performed on the last qubit. However, the measurement strategy can be adjusted based on the specific dataset under investigation. Notably, this model offers several advantages, such as the circuit’s number of C-Not gates and qubit count being independent from the number of features in the data. Moreover, it enables adjustments in intermediate layers, rendering it suitable for deep network architectures.

[0072] Results

[0073] The output of the CFFQNN 70 with the existing QNN model on different datasets is compared using the ZZFeatureMap[] to encode classical data into the quan- tum circuit and the RealAmplitudes is used as the variational circuit with thetrainable weights. The COBYLA [] and Gradient Descent optimizers were used to optimize the weights ^^. To predict the input data’s output class, a noiseless simulation using Qiskit’s Aer simulator is performed. A 2-qubit ZZFeatureMap circuit and a RealAmplitudes circuit with two repetition layers are shown in Figures 1b and 1c, respectively.

[0074] Data and Metrics

[0075] In one example, the Iris, Credit Card fraud, and Breast cancer datasets are each employed to independently evaluate the performance of the coherent feed forward quantum neural network model 70. These datasets are widely used as standard benchmarks for testing and comparing different models. To enhance the efficiency of the analysis, certain preprocessing techniques specific to each dataset were applied. For the Breast Cancer and Credit Card Fraud datasets, Principal Component Analysis (PCA) was used to reduce the dimensionality of the feature space, with the desire to capture the underlying patterns by extracting the most informative features and improv- ing the model’s performance. For the Credit Card Fraud dataset, the issue of class imbalance was also addressed. To ensure unbiased training and evaluation the excess class instances were eliminated, thereby balancing the dataset. This step enabled the model to learn from both the minority and majority classes more effectively, thereby enhancing its ability to detect fraudulent transactions accurately. By employing these preprocessing techniques on the selected datasets, a robust and reliable framework for evaluating the effectiveness of our quantum machine learning model can be created. More details about the datasets used to compare two models shown in Table 1.

[0076] Table 1: Details of the datasets [16,17] used to compare the two models Data-sets Features Features used labels Iris 4 4 3 Credit Card fraud 30 10 2 Breast cancer 30 10 2

[0077] Table 2 hows the metrics used to evaluate the two models. Metrics Equations Precision^^ ^^^^ ^^ + ^^ ^^ Recall^^ ^^^^ ^^ + ^^ ^^ Accuracy^^ ^^ + ^^ ^^^^ ^^ + ^^ ^^ + ^^ ^^ + ^^ ^^ F1-Score2 × Precision × RecallPrecision + Recall TP, TN, FP are FN are explained in Table 3 below: Table 3: Positive Prediction Negative Prediction Positive Class True Positive (TP) False Negative (FN) Negative Class False Positive (FP) True Negative (TN)

[0078] The training rates were all superior for the CFFQNN versus the QNN, as can be seen by the steeper slopes of the objective-function-versus-iterations curves for the CFFQNN versus the QNN and the reduced number of iterations required to achieve the minimum objective function value, in Figures 5a, 6a, 7, and 9. Note the differences in scale, such as the QNN achieving its minimum objective function value after approximately 200 iterations and the CFFQNN achieving its minimum objective function after approximately 20 iterations in Figure 5a using the gradient descent optimizer.

[0079] The number of iterations required to train the models are also compared in Figures 5b (right-hand side), 8 (right-hand side), and 10 (right-hand side), where the number is smaller and thus superior for the CFFQNN relative to the QNN.

[0080] Performance Comparison Results on Credit Card Dataset

[0081] In one example, Credit Card fraud datasets are employed to independently evaluate the perfor-mance of the CFFQNN and FixedCFFQNN, in which the PCA is employed to reduce the dimen-sion of Credit card dataset to seven features. For both CFFQNN and FixedCFFQNN, a structure with three nodes in the initiallayer, two in the subsequent layer, and a single node in the concluding layer, is employed. Figure 11a presents a performance comparison between CFFQNN, FixedCFFQNN, QNN, and the classical MLPClassifier. The bar chart clearly demonstrates the superior performance of both CFFQNN variants over the conventional QNN model. Despite augmenting the QNN with 35 parameters for a balanced comparison, there was no no-table enhancement in its results.

[0082] Figure 11b outlines the quantum resources utilized by each quantum model, with no leftmost bar for the MLP Clas-sifier due to the latter not being a quantum model. Evidently, the CFFQNN models require a reduced circuit depth, fewer Cnot gates, and a shorter simulator runtime compared to the QNN model. Additionally, the FixedCFFQNN model exhibits a reduced requirement on the number of pa-rameters.

[0083] Performance Comparison Results on Breast Cancer Dataset

[0084] For the breast cancer dataset, the feature space is narrowed down to 7 features and executed 100 iterations of COBYLA to enhance the quantum models. A network structure similar to that used for the credit card dataset in the proposed quantum models.

[0085] In Figure 12a, it is evident that both CFFQNN and FixedCFFQNN outperform the existing QNN models. The QNN’s Recall, F1, and Precision scores are all zero, indicating a failure of the model to effectively learn from the data, resulting in the misclassification of all test data into a single category.

[0086] Figure 12b provides a comparative analysis of the resources consumed by each of the four quantum models, with no leftmost bar for the MLP Classifier due to the latter not being a quantum model. While the resource utili-zation is largely consistent across all models, it is noteworthy that the QNN models demand a significantly higher number of C-NOT gates compared to the CFFQNN variants.

[0087] Through numerical experimentation, the superior performance of CFFQNN in classification tasks, including a notable variant: the FixedCFFQNN, is evident. In the FixedCFFQNN, the parameters in the first layer are not trained, yet the FixedCFFQNN still outperformed existing QNN models. This untrained variant underscores the inherent efficiency robustness of the CFFQNN architecture. Byachieving high accuracy while requiring minimal quantum resources, the standard CFFQNN and Fixed CFFQNN variant allow for a scalable and practical implementation of quantum machine learning applications.

[0088] The objective functions computed while training the models were all much smaller for the CFFQNN than the QNN. Note the differences in scale, such as the QNN’s minimum objective function value was approximately 27 and the CFFQNN’s minimum objective function value was approximately 20 in Figure 5a. The CFFQNN’s minimum objective function value is approximately 10 times smaller than the QNN’s minimum objective function value in Figure 9.

[0089] All of the performance metrics given in Table 2 were superior for the CFFQNN relative to the QNN when evaluated for each of the testing data sets. The Accuracy, Recall, F1 score, and Precision values in Figures 5b (left-hand side), 6b, 8 (left-hand side), and 10 (left-hand side), 11a and 12a are all closer to unity for the CFFQNN than the QNN. The training scores are also all superior for the CFFQNN because they are greater than those of the QNN.

[0090] The CFFQNN’s circuit depth was smaller than the QNN’s circuit depth for analyzing the same data set. This is seen in Figures 5b (right-hand side), 6b, 8 (right- hand side), and 10 (right-hand side), 11a and 12a. Smaller circuit depths reduce the number of chances for errors to occur in a quantum circuit, providing an advantage for the CFFQNN on error-prone quantum hardwares.

[0091] While this specification contains many specific implementation details, these should not be construed as limitations on the scope of any invention or on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations of particular inventions. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub- combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimedcombination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0092] Accordingly, the above description of example implementations does not define or constrain this disclosure. Other changes, substitutions, and alterations are also possible without departing from the spirit and scope of this disclosure.

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Claims

CLAIMS:

1. A coherent feed forward quantum neural network comprising: an encoding layer for encoding of input data onto at least one of a plurality of qubits; at least one intermediate layer comprising a network of controlled parameterized rotation gates acting on any of the plurality of qubits; an output layer, wherein a measurement is performed on at least one of a final qubit and a final set of qubits of the plurality of qubits; and whereby the measurement is only performed once within the coherent feed forward quantum neural network.

2. The quantum neural network of claim 1, wherein the data points with weights are encoded into a rotation angle ^^ of the ^^^^gate around a y-axis, such that ^^^^( ^^) = ( .

3. The quantum neural network of claim 2, wherein the input data is encoded using a feature map.

4. The quantum neural network of claim 3, wherein an encoding unitary in the first layer is. ^^cos^^5. The quantum neural network of claim 3, wherein the data points are successively encoded on a single qubit one after another.

6. The quantum neural network of claim 4, wherein the single qubit operation is represented as: ^^^^( ^^0^^0) ^^^^( ^^1^^1) ^^^^( ^^2^^2) ⋯ ^^^^( ^^^^^^^^)|0^where ^^ is the= biasing term, and ^^0= 1.

7. The quantum neural network of claim 3, wherein the output of ^^ is probabilistic.

8. The quantum neural network of claim 7, wherein one biasing term is applied on any of the plurality of qubits.

9. The quantum neural network of claim 7, wherein a controlled rotation term is applied on any of the plurality of qubits.

10. The quantum neural network of claims 8 and 9, wherein the measurement comprises calculating an expectation value of a ^^ operator on the at least one of the final qubit and the final set of qubits of the plurality of qubits in the output layer.

11. The quantum neural network of any one of claims 1 to 10, wherein the coherent feed forward quantum neural network is adaptable by adjusting the at least one intermediate layers based on the characteristics of the input data.

12. The quantum neural network of any one of claims 2 to 10, wherein the weights in the encoding layer remain untrained.

13. The quantum neural network of claim 12, wherein the qubits in the encoding layer are not measured.

14. The quantum neural network of claim 9, wherein a controlled two qubit operation is represented by ^^ ^^^^^^→ ^^^^^^= ⊗ ^^^^^^^^, where first subsystem ^^ is^^ is a target qubit, and ^^^^ ^^is a trainable weight that parameterizes a rotation on the target qubit.

15. The quantum neural network of claim 14, wherein a plurality of control qubits in one layer each perform parameterized controlled rotations on at least one of the target qubits in a subsequent layer, wherein a controlled multiple qubit unitary is represented by ,where index ^^ iterates over at least one of the control qubits and index ^^ iterates over at least one of the target qubits.

16. A method executed by a quantum apparatus comprising a quantum neural network, the method comprising the quantum apparatus executing operations comprising: at an encoding layer, encoding of input data onto at least one of a plurality of qubits; at at least one intermediate layer comprising a network of controlled parameterized rotation gates, applying a controlled operation on any of the plurality of qubits, and a biasing term on any of the plurality of qubits; performing a measurement on at least one of a final qubit and a final set of qubits of the plurality of qubits in the output layer; and whereby the measurement is only performed once within the coherent feed forward quantum neural network in the output layer.

17. The method of claim 16, wherein the data points with weights are encoded into a rotation angle ^^ of the ^^^^gate around a ^^-axis, such that ^^ ^^cos( ^^ / 2) −sin( ^^ / 2) .

18. The method of claim 17, wherein the input data is encoded using a feature map.

19. The method of claim 18, wherein an encoded unitary in the first layer is .

20. The method of claim 19, wherein the data points are successively encoded on a single qubit one after another.

21. The method of claim 20, wherein the single qubit operation is represented as: ^^^^( ^^0^^0) ^^^^( ^^1^^1) ^^^^( ^^2^^2) ⋯ ^^^^( ^^^^^^^^)|0^, ^^=1 where ^^ is the number of features in the data, ^^ = ^^0^^0is the biasing term, and ^^0= 1.

22. The method of claim 21, wherein the data are repeatedly encoded on a positive integer number of qubits is represented by^^(^^)= ^^(^^)|0〉⊗ ^^=(^^(^^)|0^)⊗ ^^23. The method of claim 21 or 22, wherein the output of ^^ is probabilistic.

24. The method of claim 15, wherein the data is manipulated using controlled parameterized rotation gates capable of generating entanglement and uncontrolled parameterized bias gates.

25. The method of claim 24, wherein a controlled two qubit operation is represented by ^^ ^^^^ ^^^^ ^^ ^^ ,where first subsystem ^^ is a ^^ is a target qubit, and ^^^^ ^^is a trainable weight that parameterizes a rotation on the target qubit.

26. The method of claim 25, wherein a plurality of control qubits in one layer each perform parameterized controlled rotations on at least one of the target qubits in a subsequent layer, wherein a controlled multiple qubit unitary is represented by ,where index ^^ iterates over at least one of the control qubits and index ^^ iterates over at least one of the target qubits.

27. The method of claim 21, wherein the measurement comprises calculating an expectation value of a ^^ operator on the at least one qubit in the output layer.

28. The method of any one of claims 15 to 27, comprising a further step of adapting the coherent feed forward quantum neural network by adjusting parameters of the at least one intermediate layer.

29. The method of claim 28, wherein the step of adapting the coherent feed forward quantum neural network is based on the characteristics of the input data.

30. The method of any one of claims 17 to 22, wherein the weights in the encoding layer remain untrained.

31. The method of claim 29, wherein the qubits in the encoding layer are not measured.

32. A fixed coherent feed forward quantum neural network comprising: an encoding layer for encoding of input data onto at least one of a plurality of qubits, wherein the weights in the encoding layer remain untrained; at least one intermediate layer comprising a network of controlled parameterized rotation gates acting on any of the plurality of qubits; an output layer, wherein a measurement is performed on at least one of a final qubit and a final set of qubits of the plurality of qubits; and whereby the measurement is only performed once within the coherent feed forward quantum neural network.

33. The fixed coherent feed quantum neural network of claim 32, wherein the qubits in the encoding layer are not measured.

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