Quantum machine learning recently gained prominence due to the promise of quantum computers in solving machine learning problems that are intractable on a classical computer. Nevertheless, several studies on problems which remain challenging for classical computing algorithms are emerging. One of these is classifying continuously incoming data instances in online fashion, which is studied in this paper through hybrid computational solution that combines classical and quantum techniques. Hybrid approaches represents one of the current ways that opens to the quantum computation in practical applications. In this paper, we show how typical issues of online learning can be equally addressed with the properties of quantum mechanics, until to ofer better results. We aim at keeping the classification capabilities, which have learned on previously processed data instances, preserved as much as possible, and then acquiring new knowledge on new data instances. To this end, we propose a class of quantum neural networks, variational quantum circuits, that being adapted over time by exploiting techniques of model update used in classical neural networks. Experiments are performed on real-world datasets with quantum simulators.
Nowadays, Machine Learning models are employed in almost every possible task, including
medical diagnoses, fraud detection, and marketing. The widespread adoption of machine
learning in various fields can be attributed to the availability of relatively powerful computers in
recent times. The computational cost, especially for Deep Learning methods, can be extremely
high, necessitating training times of several hours, days, or even months on present-day
computers [
One of the categories of data-intensive problems in which the research on classical computing
has still struggling with is the one of learning from sequences of continuously incoming data.
Even the accurate solutions of Deep Learning find challenging working on that data scenario.
This is demonstrated by the recent research line addressing the so-called catastrophic forgetting
[
Due to the intrinsic dynamic nature of sequential data, it is not immediate designing efective
technologies to learn models that may provide accurate predictions [
QC has the potential and capabilities to reply to these issues and provide alternative
computational solutions. Indeed, classical data instances can be represented as quantum states
corresponding to points of the Hilbert space, which has more dimensions than the original space.
The features of classical data (for instance, a feature vector with three entries) can be blow up
into a representational space larger than the classical one (eight dimensions, resulting from
23, three is the feature vector size)[
We leverage these properties to design a quantum-classical framework which builds a classiifcation model in supervised setting and works in online manner by acquiring continuously incoming data instances. the framework adapts continually a classification model and keeps on learning over time. More precisely, it trains and updates a classifier on (sub-)sequences of incoming data instances (data blocks) marked as labelled, while uses it to estimate the class-value of unlabelled data instances. To do that, we resort to a form of quantum models able with learning capabilities, the so called variational circuits, which are quantum algorithms characterized by parameters that are varied and optimized as new data instances are acquired, in the same style of of the neural models training.
The framework has been tested on the binary classification task by using two real-world datasets. It has been also compared against a classical online data learner. The experimental results are encouraging and show the potential superiority in terms of accurate estimations over diferent experimental configurations.
In QC, data correspond to quantum states, described in terms of qubits and represented by
vectors of complex numbers. Computations are represented by quantum gates and performed
through matrices. The quantum states of a -qubits register are represented by a 2-dimensional
complex vectors of the Hilbert space = (C2)⊗ , while the quantum states of single qubits
are vectors of the Hilbert space = C2 and can be formulated as | ⟩ = |0⟩ + |1⟩ (in Dirac
notation), where |0⟩ and |1⟩ are the computational basis states of the space C2 that correspond
to the vectors [
A quantum state can be modified by gates, which are implemented by unitary matrices operating within . For instance, the gates which work on single qubits are 2× 2 matrices. In the following, we have the matrix representations of four single qubit gates, Hadamard (H), Rotation Y-gate (), Rotation Z-gate (), Pauli Z-gate ( ) respectively and the gate CNOT (CX) working on two qbits: : 1/√2 [︂ 11 − 1
1 ]︂
It should be noted that the gates and are parameterized on . This corresponds to
rotate the vector representing the quantum state, by degrees corresponding to , around the
y-axis and z-axis, within the tri-dimensional space of the Bloch sphere [
Algorithms are circuits with -qubits and consist of quantum gates which operate on the quantum state of the -qubits. By considering the qubits individually, the circuit consists of a sequence of quantum gates that leads the quantum state of the qubit to evolve. Intuitively, the matrix calculation of the first gate of the circuit is applied to the vector-representation of the qubit, then, the result becomes a (new) quantum state on which the second gate will be applied to. A concrete illustration is reported in Figure 1: the qubit |0⟩ at the top of the circuit is manipulated first by the gate H (we denote as |0⟩) and then by the gate R (we denote as RH|0⟩). While, by considering a n-qubit register, the circuit consists of the application in parallel of sequences of quantum gates on individual qubits. This corresponds to the application of tensor-products of gates on n-qubits, which leads the quantum state of the n-qubit register to evolve. For instance, in Figure 1, the input two-qubit register |00⟩ (|0⟩ ⊗ | 0⟩) is manipulated first by the tensor-product between gate H and gate Y (we denote as ( ⊗ )|00⟩) and then by the tensor-product between gate R(0) and gate R(1) (we denote as R(0)⊗ R(1))(H ⊗ )|00⟩).
At the end of a circuit, the final operation to perform on the qubits is the measurement
(the meter symbol in Figure 1), which, in this paper, is finalized to convert quantum states to
bitstrings and then to class labels. As result of the measurement, the qubits collapses to one of
two possible basis states, while the amplitudes indicate the likelihood to collapse in each of the
basis states. For instance, in the case of | ⟩ = |0⟩ + |1⟩, where the basis states are |0⟩ and
|1⟩, the amplitudes and indicate the square root of the probability that the qubit measures
as either |0⟩ or |1⟩: when the qubit collapses to |0⟩, it returns the quantum state [
A variational quantum circuit ( ) can be represented by a quantum circuit with adjustable
real-valued parameters (like those of the gate R( ) ). This kind of quantum circuits is similar
to neural networks and the gate parameters are optimized by classical optimization techniques,
such as gradient descent. Typically, a variational quantum circuit ( ) can be represented by a
has a layered architecture, where the layers are composed of single-qubit gates and two-qubit
gates. The same structure of gates is used for all the layers, which are repeated to engage the
parameterized gates in a better learning process. The parameters of the gates for all the layers
are updated based on the diference between the estimated class label and ground truth. For
further details the reader can refer to [
The overall framework (illustrated in Figure 2) combines classical computing techniques of feature selection, data sampling, normalization and model optimization with quantum neural models in the form of variational quantum circuits for the problem of binary classification. To distinguish the ones from the others, in Figure 2, they are tagged as either [CC] or [QC]. In the binary classification setting, we have data instances described by X ∪ , X are values of the set of descriptive attributes/features, while ∈ {− 1, 1}denotes the class label.
Feature selection. The component is used only at the beginning and selects the subset of descriptive features which will consider afterwards. It exploits a technique based on the mutual information between the class labels. This operation has been used to perceive the most relevant characteristics and alleviate the problem of choosing the dimension of the quantum circuits in terms of qubits. In fact, the number of the features of the input data determines the number of qubits.
Normalization. The component is used to scale the values of the previously selected features
within the range of [
Data sampling. The component is used to select a subset of the labelled data instances included within the previously processed data blocks. More precisely, when building the data block +1, the component takes a number of data instances equal to the sample size from the data block (see Figure 2), previously used for a training session. The samples will contain data instances of both the class labels and, for each class label, the component takes data instances with simple random techniques without replacement. The sample size is fixed
Classification model . The component includes two quantum circuits and presents a number of qubits determined by the number features selected by the component Feature selection. In the following, we report the technical details of the two circuits.
• The first quantum circuit takes the classical data and represents them as quantum states to be assigned to the qubits. Initially, all the qubits are set as |0⟩, which is the default where, |0⟩⊗ denotes the register with d-qubits at the state |0⟩ (|0⟩ ⊗ . . . ⊗ | 0⟩). In this work, ℱ has been implemented as follows | (X)⟩ = ℱ (X)|0⟩⊗ ⊗ (X)⊗ |0⟩⊗ where, the parameter for each gate is the normalized real-valued of of the feature (corresponding to the qubit on which works). The term ⊗ denotes the tensor product H ⊗ . . . ⊗ H over occurrences (that is, the number of selected features) of the gate H (the same holds for ).
This ensures each possible input X has a unique qubit encoding before being passed to
next gates. Clearly, to acquire all the data instances of the data block, the gates of the
feature mapping are not replicated along the circuit, instead the input data are enqueued
to the circuit as they arrive. We are aware that more sophisticated feature maps could be
used [
In this work, has been implemented as follows
⊗ ⊗ ⊗ where, each occurrence of the two-qubit gate CX takes one pair of qubits (over the d-qubit register) composed by the consecutive qubits indexed as and + 1. Finally, we perform measurements on the qubits and the measured state is recorded. Indeed, we record a collection of measured states because, in quantum machine learning [13], when training a classifier, the variational quantum circuit is run many times with the same input and parameter . So, we can estimate the expectation value of the circuit on and , over multiple runs, with the following:
|ℰ (X, ) = ⟨(X, )| ⊗ |(X, )⟩
where, ⊗ is the tensor product of the single qubit gate over occurrences. As
anticipated in Section 2, the notation |⟩ refers to a column vector of , while the
notation ⟨| refers to the row column calculated as the transpose conjugate of |⟩.
The gate has the interesting property that if the measured quantum state has odd
parity, it returns -1 (as eigenvalue), while, if the measured quantum state has even parity,
it returns 1. This implies that the expectation value of the circuit will always be within the
interval [
∑︁ ( (|X)) − =1 where, is the number of data instances of the data block.
The cost function is minimized by a classical computing optimizer based on gradient descent. The derivative concerns the expectation value ℰ () with respect to the current values of and is computed by means of the parameter shift rule [14]: ℰ = ℰ ( + ) − ℰ ( − ) (8) 2 The gradient value is the diference between the two output values of the circuit: the first value is the output of the circuit with the parameter increased by a value , and the second value is the parameter decreased by . Intuitively, the gradient is determined by running the circuit on the same input with two diferent automatically-computed configurations of the parameters.
Methodology of the framework. Learning classification models on continuously incoming data can be faced with time-windows models [15]. Time-windows models allows us to handle data instances by equally-sized blocks on which we train, update and apply the predictive capabilities of the classifier. Thus, we distinguish training sessions from prediction sessions. During a training session, the component Classification model is activated, which implies the execution the feature mapping ℱ on the data instances of the current data block and optimization process of the parameters of the variational circuit . During a prediction session, the component Classification model is activated only to estimate the class labels on the current data block by using the classifier up there updated. (5) (7)
To keep the classifier updated, we have to deal with the catastrophic forgetting efect raising when updating neural networks. In the literature, three alternatives are mainly suggested, replay methods, regularization-based methods, parameter isolation methods [16]. Shortly, replay methods store a limited set of data instances which are replayed while learning on a new training session. Regularization-based methods introduce regularization term in the loss function, consolidating for new training session the previously learned knowledge. Parameter isolation methods introduce new hyper-parameters for each new training session. Considering that the replay methods represent the solution which asks for less and leave unchanged the number of hyperparameters of the neural network, we lean for this approach when updating the classifier.
The framework operates in three steps, namely initialization, update, prediction. Training sessions are performed at the initialization and update steps. In the initialization step, the classifier is trained from scratch on the first data block 1 (Figure 2). The component Feature selection is used only at the initialization step. The other steps work on the feature here selected. Then, the framework prepares the update step and prediction step by collecting labelled data instances and unlabelled data instances for two diferent data blocks and respectively. Actually, the data block being already populated with the data instances provided by the component of Data sampling, while, those necessary to reach the size are taken from the incoming data instances, as they arrive. As explained above, this is done to mitigate the efect of catastrophic forgetting. As soon as one of the two data blocks has been filled (the number of collected data instances is equal to the predefined size), the respective step is performed. By supposing the data block of labelled data instances is completed for first, the update step will be performed, otherwise it will be the turn of prediction step working on . Clearly, the data block contains (unlabelled) data instances, as they arrive.
The succession of training sessions and prediction sessions is not predefined, coherently with the realistic assumption according to which the distribution of labelled and unlabelled data instances is not previously established and therefore not all the data instances are labelled.
We implemented the proposed framework in IBM Qiskit [17] and run experiments by using simulators on two real-world datasets, more precisely Ozone level detection 1 (having 2536 data instances, 73 features) and Spambase (having 4600 data instances, 57 features) 2. Data blocks have been partitioned so as having a portion of 75% of the dataset as labelled data instances (training sessions) and the remaining 25% as unlabelled data instances (prediction sessions and testing sets of the evaluation). The classical computing components described in Section 3 are those available in the toolkit Scikit-learn [18]. The number of runs of the classification model to estimate the expectation values is 1024, while the number of iterations (epochs) to optimize the parameters is 20. The number of layers for the variational quantum circuit is 3. The size of the sample (component Data sampling) is 20% the data-block size.
The experiments have been performed to emphasize the impact of the technical configuration of the framework on the accuracy, namely number of qubits (corresponding to the features 1https://archive.ics.uci.edu/ml/datasets/ozone+level+detection 2https://archive.ics.uci.edu/ml/datasets/Spambase selected) and size of the data blocks (number of data instances in each training/prediction session). In Table 1, we report the accuracy of the proposed framework ( ) compared to i) a classical computing solution (, originally designed for data stream learning) [19] and ii) a baseline of the framework that works on the whole dataset ( ). The values illustrated have been computed as the average computed over the data blocks of testing. As we can see, except two trials, HYQOL does not never worst than CC, even when the number of qubit is the higher (i.e., 10). Also, we note that the configurations of HYQOL with smallest set of qubits (i.e., 2) are better than those with largest set (i.e., 10), without, however, particular discrepancy between the two endpoints. The size of the data blocks seems not be determinant for the accuracies, but, it is evident that online learning can be beneficial for quantum-based classifiers compared to the version that works on the whole dataset ( ).
Hybrid quantum-classical machine learning methods have recently garnered significant attention due to their potential to leverage the strengths of both classical and quantum computing in order to solve computationally intractable problems in a more eficient manner. These methods aim to harness the power of QC to perform certain tasks while still relying on classical computing for other tasks, such as data preprocessing or postprocessing of results.
In [20], the authors proposed a hybrid quantum-classical convolutional Neural Networks model for prediction by on X-Ray images. The quantum part of the model consists of encoding, random quantum circuit, and decoding phases. The hybrid model delivers high accuracy and outperforms previous studies (classical machine learning models approaches) in sensitivity and F1-measure. A hybrid quantum CNN model was also introduced in [21], where the authors adopted a federated learning approach to protect models and avoid privacy failures attacks. Their experiment results show that the models with additional quantum convolution have slightly improved accuracy than the baseline classical models.
A hybrid quantum-classical model of Long short-term memory (LSTM), a kind of recurrent neural network (RNN), is proposed in [22]. The authors performed a study comparing their proposed hybrid model’s capability and performance with its classical counterpart. They found that the hybrid model converges faster and reaches a better accuracy than its classical counterpart. However, their simulations assumed conditions of absence of noise and decoherence. A hybrid quantum-classical approach was designed for generative adversarial learning in [23], in order to develop an anomaly/fraudulent transaction detection. The performances of their model were on par with the classical counterpart in terms of the F1 score.
One of the very few works which investigates the catastrophic forgetting with quantum algorithms has been recently published [24]. It focuses on the incremental learning, which, contrary to online learning, considers training sessions on diferent classification tasks. Inspired by the replay methods, they propose to constrains the model updated by projecting the gradient direction on the region outlined by previous task gradients. This is done also by storing a fraction of the training data of previous tasks on which the gradient descent is computed. A drawback is the necessity of computing gradients of previous tasks at each training iteration.
Overall, the recent works in the field of hybrid quantum-classical machine learning methods demonstrate the potential for these approaches to outperform classical machine learning algorithms on certain tasks, particularly those that involve large or complex datasets. As quantum computing technology continues to improve, it is likely that we will see further advancements in this field and the development of new, more powerful hybrid quantum-classical algorithms.
In this paper, we investigated the viability of quantum machine learning solutions to work on the realistic scenarios of changeability of the statistical properties of the data, which often implies the variability of the performances of the model. We conjecture this can be a machine learning problem in which the quantum solutions can lead innovation. On simulated quantum machines, the hybrid quantum-classical proposal ofers encouraging results, in terms of accuracy, often better than a classical computing solution working on data stream and hybrid solution working in batch mode (no online learning). As our opinion, three take-home messages can be identified from this paper. The first one is methodological, in that the online learning opens to practical applications able to combine quantum computing and classical computing techniques, which is likely the only way to concretely use current quantum technologies. The second one is experimental, in that it provides arguments on the fact that stable quantum devices could even do better in terms of performances and quality of the results, when used in predictive tasks. The third one tell us that, although the high-performance computation and tractability of hard problems are the promises of quantum computing which, with the current devices, often are not kept, the research on the lifelong computation can be a field in which quantum computing can already bring interesting results.. Corrado Loglisci acknowledges the financial support from the project "PNRR MUR project PE0000023-NQSTI" for this research.