GenQu: A Hybrid System for Learning Classical Data in Quantum States

1 INTRODUCTION . In the past decade , machine learning and artificial intelligence powered applications dramatically changed our daily life . Many novel algorithms and models achieve widespread practical successes in a variety of domains such as autonomous cars , healthcare , manufacturing , etc . Despite the wide adoption of ML models , training the machine learning models such as DNNs requires a tremendous amount of computing resources to tune millions of hyper-parameters . Especially in the post Moore ’ s Law era , the limit of semiconductor fabrication technology can not satisfy the the rapidly increased data volume needed for training , which restricts the development of this field ( Thompson et al. , 2020 ) . Encouraged by the recent demonstration of quantum supremacy ( Arute et al. , 2019 ) , researchers are searching for a transition from the classical learning to the quantum learning , with the promise of providing a quantum speedup over the classical learning . The current state of quantum-based learning inspires alternative architectures to classical learning ‘ s sub-fields , such as Deep Learning ( DL ) or Support Vector Machine ( SVM ) ( Garg & Ramakrishnan , 2020 ; Beer et al. , 2020 ; Potok et al. , 2018 ; Levine et al. , 2019 ) , where the quantum algorithm provides improvements over their classical counterparts . For example , there are quite a number of adoptions of quantum learning algorithms in domains of expectation maximization solving ( QEM ) ( Kerenidis et al. , 2019 ) that speeds up the kernel methods to sub-linear time ( Li et al. , 2019 ) , Quantum-SVM ( Ding et al. , 2019 ) , and NLP ( Panahi et al. , 2019 ) . Employing quantum systems to train deep learning models is rather developed with a multitude of approaches to creating and mimicking aspects of classical deep learning systems ( Verdon et al. , 2019 ; Beer et al. , 2020 ; Chen et al. , 2020 ; Kerenidis et al. , 2019 ) , with the following challenges : ( i ) , such systems are held back by the low qubit count of current quantum computers . ( ii ) , learning in a quantum computer becomes even more difficult due to the lack of efficient classical-to-quantum data encoding methodology ( Zoufal et al. , 2019 ; Cortese & Braje , 2019 ) . ( iii ) , most of the existing studies are based on purely theoretical analysis or simulations , lacking practical usability on near-term quantum devices ( NISQ ) ( Preskill , 2018 ) . More importantly , the above challenges would presist even when the number of qubits supported in quantum machines get siginificantly increased : when the number of qubits in the quantum system increases , the computational complexity grows exponentially ( Kaye et al. , 2007 ) , which quickly leads to tasks that become completely infeasible for simulation and near-term quantum computers . Therefore , discovering the representative power of qubits in quantum based learning system is extremely important , as not only does it allow near-term devices to tackle more complex learning problems , but also it eases the complexity of the quantum state exponentially . However , to tackle the topic of low-qubit counts of current quantum machines is rather sparse : to the best of our knowledge , there is only one paper for the problem of the power of one qubit ( Ghobadi et al. , 2019 ) . Within this domain , the learning potential of qubits are under-investigated . In this paper , we propose GenQu , a general-purpose quantum-classic hybrid framework for learning classical data in quantum states . We demonstrate the power of qubits in machine learning by approaching the encoding of data onto a single qubit and accomplish tasks that are impossible for comparative data streams on classical machines , which addressing the challenges ( i ) and ( ii ) . Enabled by GenQU , we develop a deep neural network architecture for classification problems with only 2 qubits , and a quantum generative architecture for learning distributions with only 1 qubit , and , additionally , We evaluate GenQU with intensive experiments on both IBM-Q real quantum computers and simulators ( addressing the challenge ( iii ) ) . Our major contributions include : • We propose , GenQu , a hybrid and general-purpose quantum framework that works with near-term quantum computers and has the potential to fit in various learning models with a very low qubit count . • Based on GenQu , we propose three different quantum based learning models to demonstrate the potential of learning data in quantum state . • Through experiments on both simulators and IBM-Q real quantum computers , we show that models in GenQu are able to reduce parameters by up to 95.86 % but still achieves similar accuracy in classification with Principal Component Analysis ( PCA ) ( Hoffmann , 2007 ) MNIST dataset , and converge up to 66.67 % faster than traditional neural networks . 2 PRELIMINARIES . 2.1 THE QUANTUM BIT ( QUBIT ) . Quantum computers operate on a fundamentally different architecture compared to classical computers . Classical computers operate on binary digits ( bits ) , represented by a 1 or a 0 . Quantum computers however , operate on quantum bits ( qubits ) . Qubits can represent a 1 or a 0 , or can be placed into a probabilistic mixture of both 1 and 0 simultaneously , namely superposition . Superposition is one of the core principles that allows quantum computers to be able to perform certain tasks significantly faster than that of their traditional counterparts . When discussing a quantum framework , we make use of the 〈bra| and |ket〉 notation , where a 〈bra| indicates a horizontal quantum state vector ( 1×n ) and |ket〉 indicates a vertical quantum state vector ( n× 1 ) . A qubit , as it is some combination of both a |1〉 and |0〉 simultaneously , is described as a linear combination between of |0〉 and |1〉 . This combination is described in Equation 1 . |Ψ〉 = α|0〉+ β|1〉 , |Ψ〉 = [ α β ] , |0〉 = [ 1 0 ] , |1〉 = [ 0 1 ] ( 1 ) In Equation 1 , the state of |Φ〉 describes the probabilistic quantum state of one qubit , respectively |φ〉 . The values of α and β are the probability coefficients and what encode information regarding this qubit ’ s state . Although qubits can exist in both |1〉 and |0〉 at the same time , when they are measured for a definite output , they collapse to one of two possible value , where in the case above those values are |0〉 or |1〉 . The coefficients , α and β , indicate the square root of the probability that the qubit measures as a |1〉 or a |0〉 . The definite states we are measuring the qubit against are based on how we measure the qubit , measuring as one of two possible measurements . These two possible measurements are two orthogonal eigen-vectors , and can be in any 3-Dimensional direction . This is best visualized and understood by the Bloch Sphere representation of a qubit , as illustrated in Figure 1 . A qubit can be represented by the unit Bloch Sphere visualized in Figure 1 . In the case of |0〉 and |1〉 , we are measuring across the z axis . Although the qubit could be measured against the Y or X axis , once a qubit is measured in a direction and is observed as some vector , the qubit is in that state unless acted upon , therefore making a measurement in Z then X be fraught without further processing . A pure quantum state has data encoded and manipulated through rotations over the Bloch sphere surface . Relating to Equation 1 , the α and β can be thought of as the states |φ〉 distance to the state vectors |0〉 and |1〉 , where a high α indicates being relatively close to |0〉 and vice-versa.The power of quantum computing lies in the ability to sample the output repeatedly , thereby providing multiple ” answers ” for one question . 2.2 QUANTUM DATA MANIPULATION . To accomplish data transformation and data encoding , a qubit and its quantum state must be manipulated to encapsulate information onto it . Qubits are manipulated through quantum gates , which in turn manipulates the overall quantum state . These gates can allow for complete manipulation over the Bloch sphere in Figure 1 , and more specifically complete manipulation of the quantum state vector , which can describe the state of a mixture of more than 1 qubit . We introduce the few gates that we make use of in this paper in Equations 2 and 3 . RY ( θ ) = [ cos ( θ 2 ) − sin ( θ 2 ) sin ( θ 2 ) cos ( θ 2 ) ] RZ ( θ ) = [ e−iθ2 0 0 e −iθ 2 ] ( 2 ) CRY ( θ ) = 1 0 0 0 0 1 0 0 0 0 cos θ2 − sin θ 2 0 0 sin θ2 cos θ 2 CRZ ( θ ) = 1 0 0 0 0 1 0 0 0 0 e iθ 2 0 0 0 0 e iθ 2 ( 3 ) The gates above accomplish specific tasks of quantum state manipulation.Equation 2 allows for a single qubit to be manipulated to any position on a Bloch sphere ’ s surface , from any starting point on aforementioned sphere . Equation 3 accomplishes entangling two qubits with controlled rotations . Controlled rotations allow for a single qubits state to be entangled with another . In the case of a controlled rotation gate , a qubits state is manipulated based on whether the control qubit measures as a |1〉 . Although we brush over this for the sake of easier reading , quantum entanglement empowers quantum computers to accomplish phenomenal tasks . These two styles of gates , single qubit rotations and controlled qubit rotations , allow for complete manipulation of quantum states , be it 1 or more qubits . 2.3 QUANTUM DEEP LEARNING Quantum Deep Learning is a relatively new approach to Quantum Machine Learning that takes quantum circuits and applies similar training techniques and learning methods of how classical neural networks work Chen et al . ( 2020 ) ; Garg & Ramakrishnan ( 2020 ) ; Beer et al . ( 2020 ) . In traditional deep learning layers are often used , where a layer is some large transformation function that takes in a set of inputs , and outputs a set of outputs , where the number of inputs does not necessarily equal the output . These functions are connected in series , sometimes in parallel , and typically trained through the use of back propagation Goodfellow et al . ( 2016 ) ; Chen et al . ( 2020 ) . This data flow through layers to some output is similar to how quantum circuits operate . Similar to how classical deep learning works , the way this data flows through time is up to the practitioner , who chooses and designs their network according to their needs . Quantum deep learning is approached through the use of layering gates sequentially . For our paper , our layers are comprised of the gates in Equations 2 and 3 . Similar to how deep learning is parameterized by connection weights , these gates are parameterized through rotations ( θ ) . At the end of the quantum circuit , a loss function is described by the practitioner , and the quantum networks parameters θ are updated iteratively such that the circuits loss is minimized Beer et al . ( 2020 ) ; Crooks ( 2019 ) . In the case of binary classification , one can make use of quantum entanglement to pool data down to one qubit channel , which then can be used as the final classification output of the network . These layers are visualized in Figures 3 and 2 . In Figure 3 we visualize the grouping of these circuits to be reminiscent of quantum traditional deep learning layers with the oracle approach . Interpreting these operations can be seen as a qubit entering through the left starting in state |0〉 , passing through gates until it has been transformed into state |φ〉 . 3 GENQU FRAMEWORK AND LEARNING MODELS 3.1 GENQU FRAMEWORK Our proposed GenQu framework is illustrated in Figure 4 . Before any operation of the framework is performed , the data must be transformed from classical to quantum states . This is done by transforming classical data into applicable quantum rotations , and is described under section 3.2 . Following this , the rotations are loaded onto a quantum computer . The quantum circuit preparation section is where the circuit relating to a specific machine learning algorithm is designed . For example , this is where a deep neural network or a convolutional neu- ral networks architecture would be set up , initialized and prepared . This circuit is loaded onto the quantum computer after the quantum data loading section . Once the circuit is set up , it can be induced . Inducing the quantum circuit results in the quantum state transformation of the input data over the quantum machine learning model . From here , if the output of the model was a quantum state , one could end here and feed it to another quantum algorithm . However , in the case of updating learn-able parameters , the relevant qubits need to be measured . We feed the qubits measurements to a loss analysis section , where we update our parameters accordingly . Once the parameters have been updated , we repeat this process of circuit loading , circuit inducing , and measurement , updating parameters until a desired loss of the network is attained or a predefined number of epochs have run .

Summary: The paper claims to introduce a new quantum machine learning framework called GenQu. However, the description of the framework very vague (using classical computers to optimize the parameters of a fixed quantum circuit), and hardly novel. In fact, the same basic ideas are so well-known in the community that they are described in detail as usage examples for popular quantum computing platforms such as Qiskit and IBM Q.