GenQu: A Hybrid System for Learning Classical Data in Quantum States
Authors: Samuel A. Stein
Ray Marie Tischio
Betis Baheri
Yiwen Chen
Ying Mao
Qiang Guan
Ang Li
Bo Fang
Venue:  (2021)
Keywords: Quantum Machine Learning
Qubits
Kernel Methods
Deep Neural Network

Abstract: Deep neural network-powered artificial intelligence has rapidly changed our daily life with various applications. However, as one of the essential steps of deep neural networks, training a heavily-weighted network requires a tremendous amount of computing resources. Especially in the post Moore's Law era, the limit of semiconductor fabrication technology has restricted the development of learning algorithms to cope with the increasing high-intensity training data. Meanwhile, quantum computing has exhibited its significant potential in terms of speeding up the traditionally compute-intensive workloads. For example, Google illustrates quantum supremacy by completing a sampling calculation task in 200 seconds, which is otherwise impracticable on the world's largest supercomputers. To this end, quantum-based learning becomes an area of interest, with the promising of a quantum speedup. In this paper, we propose GenQu, a hybrid and general-purpose quantum framework for learning classical data through quantum states. We evaluate GenQu with real datasets and conduct experiments on both simulations and real quantum computer IBM-Q. Our evaluation demonstrates that, comparing with classical solutions, the proposed models running on GenQu framework achieve similar accuracy with a much smaller number of qubits, while significantly reducing the parameter size by up to 95.8\% and converging speedup by 66.67% faster. 

Under review as a conference paper at ICLR 2021
GENQU: A HYBRID SYSTEM FOR LEARNING CLASSI- CAL DATA IN QUANTUM STATES
Anonymous authors Paper under double-blind review
ABSTRACT
Deep neural network-powered artificial intelligence has rapidly changed daily life with various applications. However, as one of the essential steps of deep neu- ral networks, training heavily-weighted network requires tremendous amount of computing resources. Especially in the post Moore's Law era, the limit semiconductor fabrication technology has restricted the development of learning algorithms to cope with the increasing high intensity training data. Meanwhile, quantum computing has exhibited its significant potential in terms of speeding up the traditionally compute-intensive workloads. For example, Google illustrates quantum supremacy by completing a sampling calculation task 200 seconds, which is otherwise impracticable on the world's largest supercomputers. To this end, quantum saantummbbasdd learning becomes an area of interest, with the promising of a quantum speedup. In this paper, propose GenQu, hybrid and general-purpose quantum framework for learning classical data through quantum states. We eval- uate GenQu with real datasets and conduct experiments on both simulations and real quantum computer IBM-Q. Our evaluation demonstrates that, comparing with classical solutions, the proposed models running on GenQu framework achieve similar accuracy with a much smaller number qubits, while significantly re- ducing the parameter size by up 95.86% and converging speedup by 66.67% faster.
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 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 hyper-parameters. Especially in the post Moore's Law era, the limit of semiconductor fabrication technology cannot 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 transition from the classical learning to the quantum learning, with the promise of providing quantum speedup over the classical learning. The current state of quantum-based learn- ing inspires alternative architectures classical learning's 's sub-fields, such Deep Learning (DL) or Support Vector Machine (SVM) (Garg & Ramakrishnan) 2020) [Ber et al. 2020) Potok et al., 2018 Levine al.) 2019), where the quantum algorithm provides improvements over their classical counterparts. For example. there are quite 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 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
held back by the low qubit count of current quantum comput- ers. (ii), learning in a quantum computer becomes even more difficult due to the lack efficient classical-to-quantue data encoding methodology (Zoufal et al. 2019 Cortese & Braje 2019). (iii),

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most of the existing studies are based purely theoretical analysis 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 in- creases, 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. There- fore, 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 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 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 purpose crantum-classiiy hybrid framework for learn- ing 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). En- abled by GenQU, we develop a deep neural network architecture for classification problems with only 2 qubits, and quantum generative architecture for learning distributions with only I 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, hybrid and general- purpose quantum framework that works with near-term quantum computers and has the potential to in various learning models with a very low qubit count.
Based on GenQu, we propose three different quantum based learning models to demon- strate the potential of learning data in quantum state.
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)(Hoflmann) 2007 MNIST dataset, and converge up to 66.67% faster than traditional neural networks.
Through experiments
2 PRELIMINARIES
2.1 THE QUANTUM BIT (QUBIT)
Quantum computers operate on a fundamentally different architecture compared classical com- puters. Classical computers operate on binary digits (bits), represented by a I or a O. Quantum computers however, operate on quantum bits (qubits). Qubits can represent a or a O, or can be placed into probabilistic mixture of both and 0 simultaneously, namely superposition. Superpo- sition 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 quantum frame- work, we make use of the (bra) and (ket) notation, where a (bral indicates a horizontal quantum state vector (1 x n) and [ket)) indicates vertical quantum state vector (n x 1). A qubit, as it is some combination of both [1) and [0) simultaneously, is described as a linear combination between of (0) and [1). This combination is described in Equation [
) = a0)) + B11) 1v) =
0
10) = 1
11)
0 1
(I)
3
In Equation the state of (b) describes the probabilistic quantum state of one qubit, respectively (ø). The values of a and ,3 are the probability coefficients and what encode information regarding this qubit's state. Although qubits can exist in both [1) and 10) at the same time, when they are measured for a definite output, they col- lapse one of two possible value, where in the case above those values are [0) or [1). The coefficients, a and 8, indicate the square root of the probability that the qubit measures [1) or (0). The
Figure 1: Bloch Sphere
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definite states measuring the qubit against are based on how we measure the qubit, measuring as one two possible measurements. These two possible mea- surements are two orthogomal 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 D
A qubit can be represented by the unit Bloch Sphere visualized in Figure [ 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 measured in direction and observed 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 [] the a and 3 can be thought of as the states lo) distance to the state vectors [0) and [1), where a high a 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 manip- ulated 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 | and more specifically complete manipulation of the quantum state vector, which can describe the state of mixture of more than I qubit. We introduce the few gates that we make use of in this paper in Equations and
sin (g)
(0)
CO8 sin
7
Rz)
20
0
e
2
0
0
0
0
1
0
0
CRY
COS
sin 0
CR2(0)= =
(3)
sin
cos
18
The gates above accomplish specific tasks of quantum state ieriipuulation.Euuatii 2 allows for a single qubit to be manipulated to any position on a Bloch sphere's is 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 controlled rotation gate, qubits state is manipulated based on whether the control qubit measures as [1). Although we brush over this for the sake of easier reading, quantum entanglement empowers quantum computers accomplish phenomenal tasks. These two styles of gates, single qubit rotations and controlled qubit rotations, allow for complete manipulation of quantum states, be it or more qubits.
2.3 QUANTUM DEEP LEARNING
Quantum Deep Learning is 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).
10>
Ry
Rz(0)
|/>
10> 10>
CRy(8)
CRz(6)
10> (4)
(o) Single Qubit Unitary
Entanglement
Figure 2: Gate Deep Learning Gate Design
In traditional deep learning layers are often used, where layer is some large transformation func- tion that takes in a set of inputs, and outputs a set of outputs, where the number 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. (022)). This data flow through layers some output is similar to how quantum eircuits 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
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through the of layering gates sequentially. For paper, layers 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 (8). At the end of the quantum circuit, a loss function is described by the practitioner, and the quantum networks parameters 0 are updated iteratively such that the circuits loss is minimized [Berr al. (2020); Crooks 2019).
In the case of binary classification, one can make use of quantum entanglement to pool data down 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 visu- alize the grouping of these circuits to reminiscent of quantum traditional deep learning layers with the oracle approach. Interpreting these operations can be seen qubit entering through the left starting in state (0), passing through gates until it has been transformed into state (+).
10)
14)
10)
U,(Og)
Ug(4
(ba)
[4)
10>
(u>
Figure 3: Quantum Deep Learning Layers
GENQU FRAMEWORK AND LEARNING MODELS
3.1 GENQU FRAMEWORK
Our proposed GenQu framework is illustrated in Figure Before any operation of the frame- work is performed, the data must be trans- formed from classical quantum states. This is done by transforming classical data into ap- plicable quantum rotations, and is described un- der section 3.2. Following this, the rotations are loaded onto a quantum computer. The quan- tum circuit preparation section is where the cir- cuit relating to specific machine learning al- gorithm is designed. For example, this is where a deep neural network or a convolutional neu- ral networks architecture would be up. initialized and prepared. This circuit is loaded onto the quantum computer after the quantum data loading section.
Classic Computer
Quantum Computer
Classic Data
Data
Quantum Data
Transformation
Orcut/Loading
Quantum Circuit
Inducing Quantum
Preparation
Circuit
Output Data
Loss Analyysi in Classic States
Qubits
Measurement
Figure 4: GenQu: A Hybrid Framework
Once the circuit is up, it be induced. Inducing the quantum eircuit results in the quantum state transformation of the input data over the quantum machine learning model. From here, the output of the model was 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 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 desired loss of the network is attained or predefined number of epochs have run.
3.2 DATA QUBITIZATION
Prior to discussing our methods of illustrating the learning power of qubits, we introduce our ap- proach to encoding classical data into quantum states, We encode two dimensions of data per qubit, by the simple two step process outlined in Equations 5
Encoded onto
RY (2sin (Vr1)
(4)
(d)
Encoded onto
RZ(2sin 1 (var))
(5)
())
The value of is encoded along the Z-axis, followed by T2 being encoded along the Y-axis. For these rotations to work, the data along each dimension must be normalized to be in the range of (0,1). This reduction in qubit count is pertinent for the case of quantum machine learning as the state space vector of a quantum system is tensor rank 2n values, and therefore halving the qubit count provides 2 reduction in state space.
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3.3 SINGLE QUBIT KERNELIZED CLASSIFICATION
When tackling a classification task on a quantum system, we want to encode our data such that the probability of measuring a (1) is comparative with the probability of classifying data point as Class 1. Therefore, in the case of classical data sets, we can wrap 2 dimensions of data around qubit such that we maximize the ability of the qubit to classify the data. In the case of the circles data set, a data set comprised of points non-linearly separable, we can wrap the qubit with data points such that the rotation around the axis is correlated to the distance from the circle center. This is visualized in Figure This encoding accomplishes on one qubit the encapsulation of 2 dimensions of non-linearly separable data, whilst accomplishing a separation task. For this to be done, two parameters per qubits are used to transfer between Classical data to Quantum state. These parameters are the rotations around the Y axis, proceeded by a rotation around the Z axis.
Translating the classification problem of the data outlined in Figure 5 to GenQu framework, we follow the following ap- proach. We begin by translating our data into rotations according to the functions outlined in 3.2, however the values en- coded vector distances from the cir- ele center. This can be considered both the Quantum Circuit Preparation and Data Transformation components, as the ex- pected measurement under Qubits Mea- surement is equivalent to the classification of a data point. There is no updating of parameters in this case, therefore we do not iterate and update circuit parameters.
Figure 5: Circles dataset and its qubit representation
3.4 QUANTUM DEEP LEARNING ARCHITECTURE
In this paper we make use of the data qubitization techniques outlined above, along side current quantum machine learning techniques to enable highly performent quantum deep learning. Through encoding 2 dimensions of data per qubit, the number of neurons in our network input is half the dimensionality of the data set. Quantum deep learning layers comprised of single qubit operations are namely called single qubit unitary layers. In these layers each qubit has RY and RZ gate appended, thereby adding 2n parameters, where n iWs the number of qubits. Another type of layer consists of operations acting on two qubits per gate, where operations are control operations (CRY or CRZ). These are namely Entanglement layers. Entanglement layers entangle all qubits by some learnable amount, performing CRY and CRZ gates qubits i and i + 1 until there no qubits left to pair. Entanglement layers require 2(n - 1) parameters. These gates are visualized in Figure Through the use of the entanglement layer, time point across a circuit dynamically. In our case of illustrating binary classification, we make use of the entanglement layer to pool down data from the other qubits onto qubit, which is measured and used as the classification qubit. The probability of the qubit measuring 1) is thought of as the similar how
can reduce and grow the number of qubits at any
probability of labelling the quantum data that fed to the circuit Class single output neuron operates of activation function Sigmoid operates in classical neural networks.
Fitting this Quantum Deep Learning model to our GenQu framework, we begin by translating our data into rotations described under 3.2. From here, the practitioner can describe their full quantum deep learning architecture and initialize the parameters. The data is loaded onto a quantum computer in series, with the quantum data loading circuit being appended with the quantum deep learning architecture. The quantum circuit is induced and the classification qubit measured. We feed this result back to classical computer, calculate our loss and update our parameters accordingly. This repeated until convergence occurs or sufficient accuracy is attained.
3.5 QUANTUM GENERATIVE NATURE
Another powerful use of qubits is in the use of representing data. Through using trainable circuits as discussed above, we can measure two values from one qubit. Therefore, quantum deep neural network can be trained mimic some data it is fed by defining some loss function such as the
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(C. by:
Ry(Pamm-I)
Re(Param-1)
X
(c. 1y:
Ry(Param-2)
Rz(Param-3)
@ (Param--))
@ (Paaamm-))
Ry(Parmm-6)
Ra(Param-7)
Figure 6: Quantum Deep Learning Circuit
Mean Squared Error, and generate new samples that are close to what the qubit was trained on. This is similar to Generative Adversarial Networks Goodfflloo et al. (2014), however does not take any noise as an input, nor does it require two networks to be used. We do not claim that ours is better, however it is one of the side effects of qubits being used to represent data, and quantum deep learning models. Therefore, we illustrate through the use of quantum deep learning how quantum deep learning architecture with a tuned loss function can generate data similar to that of the data it was fed, and generative diversity significantly greater that is unattainable using similar architectures within its classical counterparts.
Translating quantum generative state to GenQu framework, repeat the steps outlined in the Quantum Deep Learning architecture above. However, the only change would be the loss function such that the quantum state instead of loss function such cross entropy, could be mean squared eπror or some other applicable loss function. Furthermore, no data loading for input is necessary. and instead are just loaded as qubits in the state of [0). When generating data, the qubits are measured and sent to the Output Data stream.
4 RESULTS
We implement GenQu with IBM Qiskit and Tensorflow Quantum. It is evaluated with the above mentioned three applications, kernelized classification, quantum deep learning and quantum gen- erative nature. We evaluate GenQu on both simulators and IBM-Q quantum computers (mainly Rome). We compare our results with traditional convolutional neural networks with different num- bers of parameters. In the rest of the evaluation, we denote CNN XP to be classical neural networks with X parameters and QNN XP is quantum based neural networks with x parameters.
4.1 THE KERNELIZED CLASSIFICATION
As proof of single qubit natural machine learning, we employ the encoding of circles dataset illustrated in Figure 5 onto one qubit through the radial kernel method. A single qubit has data points encoded the vector distance from the center of the circle in Figure The qubit is then measured, and the P(lØ)) [0) is equivalent to P(Class = 1). Through doing so, we attain 100% accuracy on separating the non-linearly separable data set, whilst maintaining both dimensions of information. Furthermore, when our experiment is run on IBM-Q'ss Quantum Computer Rome 100% accuracy is attained, thereby confirming our model architecture works both on simulators and real quantum computers. This approach, although not novel, is done to illustrate that certain problems can be tackled very efficiently with qubits and how the solution can be successfully run on real quantum computers.
4.2 QUANTUM DEEP LEARNING
To evaluate the learning potential of quantum deep learning, and the ability to use fewer data- channels than its classical counterpart, we make use of the MNIST data set. The MNIST data set is an image data set comprised of gray scale hand -dadn digits of resolution 28 by 28. It is in- feasible represent these images on current near-term quantum devices, and hence we make use of PCA (Hoffmann) 2007) to reduce dimensionality from 784 to 4. In this case, we only need to make use 2 2 qubits to feed our data to our quantum deep neural network. We provide the deep learning circuit visualized in Figure 6 As can be seen in the circuit, there is total of 8 parameters. This network is comprised one single qubit unitary layer (Parameters through 3), one entanglement layer which accomplishes data pooling onto one qubit (Parameters 4 and 5), and finally single qubit unitary on the final output qubit (Parameters 6 and 7). We compare our architecture to classi- cal deep learning architectures and compare parameter counts when using the gradient descent

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102
Strvcator
1F-2
0.3
Accuracy
(a) vs 6
0.75
1.4
0602
96
1-
Numbers
(c) IBM-Q vs Simulations
Operts
(b)
Figure 7: QNN (simulation), QNN (IBM-Q Rome), TFQ and CNN Results
approach (Adam Optimizer), same epochs and same data set. The quantum network is trained to perform binary classification of two numbers from the MNIST dataset, where the classification is measured by the Qubit (0, in Figure 6 As for the classical networks, make use of network comprised of a middle layer of tensor size 2, 8, 16 and 32. The comparative training results are vi- sualized in Figure 7 In Figure aa), the numbers and 6 are used train the data set. The quantum network outperforms all other comparative solutions, with close to equivocal performance of a 193 parameter deep neural network, thereby attaining 95.86% parameter count reduction, and verging 33.33% faster than said network. However, in the case difference between parameter counts than what observed in other cases, such and In Figure 7bb,, we observe how there is substantial learning ability to be gained from increasing the classical parameter count. However, similarly in this case, the 8 parameter QNN's performance is matched by the 97 parameter CNN, 11.76% reduction in parameters. We also compare our model to the Ten- sorflow Quantum (TFQ) MNIST classification example on the same number pair Broughtn et al. 2020). We illustrate that our network outperforms Tensorflow Quantums MNIST classification task in Figures (a) (a) and (b). This illustrates the significant learning potential of quantum networks and specifically the architecture used in this paper. These architectures are able to, in certain cases, reduce parameter counts significantly with no sacrifice to performance. Furthermore, in our case have encoded two dimensions of data per qubit. Feeding dimensions of data to deep neural network through 2 neurons is impractical, and is a further example of how powerful qubits are in deep learning.
and 6, there is a less significant
We validate our results by running similar experiments on real quantum computer using the IBM- Q platform, comparing the accuracy's attained simulator that of on quantum computer. These results are visualized in Fi. As can be seen, for numbers 4-3 and 9-6, actual quantum computer performance was extremely similar to that of the simulator, with a difference of less than 5%, and 9-6 having measured difference of 0.2%. However, in the case 3-1 we observe more significant differences between actual Quantum Computing implementation. The largest difference between simulators and actual quantum computers was 7.25% on the 3-1 dataset, which is due to the noise on the quantum computer that depends on the computer itself and the rotated workloads on it (random factors). We further validate our results by comparing them to the TFQ MNIST classification task, and show how in some cases our architecture and network attaining significantly higher accuracy. Specifically, in the case of 4-3 with 20% improvement and 3-1 with 22% improvement. However, in certain cases our network attained the same accuracy as TFQ, in the cases of 2-9 and 9-6.
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64
(b) Epoch 25 CNN 196P
Epoch CNN 2144P
(a) Epoch 25 QNN 2P
Figure Single qubit generative model learn the distribution PCA MNIST digit 0
3
7
7
7
J
7
7
7
7
7
0
9
3
7
7
7
9
1
9
7
7
(a) Epoch 0
(b) Epoch
(c) Epoch 50
Figure 9: Images Generated by GenQu
4.3 QUANTUM GENERATIVE NATURE
Another point of interest is how powerful qubits are representing data sets. This has significant implications in Generative Adversarial Networks (GANs) and loading data sets. We illustrate this potential by minimizing the distance between a single qubit's quantum state and the MNIST data set PCA ed to 2 dimensions and of class O. We illustrate in Figure & how single qubit, visualized by the blue shading, using only parameters (an RY and RZ gate in series on one qubit), can com- pletely mimic the data it was fed (2 dimensions). If sampled, the qubit will generate all samples it was fed as well as generate new unique samples similar to that of which it was fed. We make use of the architecture of Generative Adversarial Network Goodfellow et al. (014) compare this to a classical neural network, and observe how poorly the classical counterpart performs. When given 9800% more parameters (2 vs 196), visualized in Figure M)) the network was still unable to mimic the data fed to the network. The classical network was able to converge when provided with 2144 parameters, visualized in Figure d)). d). Furthermore, we illustrate the generative potential by sampling from the aforementioned quantum circuits with a shot count of 15, performing reconstruc- tive PCA on the data and plotting the images. As seen in the images the qubit is able learn from almost nothing in a(a) to reasonable results after 50 epochs in c).. This further goes to illustrate the significant machine learning potential and parameter reduction potential of quantum machine learning.
5 CONCLUSION
This paper proposes GenQu, hybrid and general- purpose quantum framework for learning classical data. It demonstrates the significant expressibility of qubits, and their extensive applications in machine and deep learning.
Based on GenQu, we propose three different learning models that make use a low-qubit count in near-term quantum computers. In the model for kernelized classification. GenQu is able to encode the circle dataset onto a single qubit achieve 100% accuracy in the experiment on real quantum computer. With quantum deep learning model in GenQu, when encoding dimensions of data per one qubit, is able to show reductions in parameters equivalent to 95.86% whilst still attaining a similar accuracy better than that of classical deep learning models. Finally with respect qubits learning potential, a single qubit generative model is proposed. generate 2 dimensions of data PCA' ed from the MNIST's 0 class.
able to completely learn to
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Christa Zoufal, Aurélien Lucchi, and Stefan Woemner Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information, 5(1):1-9, 2019.
A APPENDIX: EXPERIMENTS ON IBM-Q QUANTUM COMPUTERS
To justify proposed model, conducted experiments multiple IBM-Q sites including Vigo, Ourense, Rome, Bogota and Valencia. Figure [0 is the setup for Kemelized classification on the circle dataset. We ran this circuit with 40 shots (repetition) 20 times on different locations and achieved accuracy of 100%.
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Figure 10: Experiment setup for Kernerlized classification
In PCA MNIST dataset, we train the model on 913 samples of two specific digits (9 and 6) on different IBM-Q locations to justify our founding. Figure 12 and 13 present a sample of each iteration over actual IBM-Q machine which ran for 913 times to get accuracy of 95% on average.
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Figure 11: Quantum eircuit for binary classification on MNIST dataset (T1)
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Under review as a conference paper at ICLR 2021
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Figure 12: Quantum circuit for binary classification on MNIST dataset (T2)
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Figure 13: Quantum circuit for binary classification on MNIST dataset (T3)
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Reviews:
[{"review_id": "jPC3pMqbpA", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "I agree that promoting experiments for real quantum hardware is important. But I don't think the team has yet to create a working platform that could be accepted to ICLR. If the open-sourced platform is the main contribution (rather than a new understanding of how quantum computers could be useful for machine learning problems), then the authors should submit the manuscript after having the open-sourced software available. Furthermore, a lot of the wording should be changed (the current version sounds like they are proposing a new quantum machine learning framework, while they are creating an open-sourced platform).\n\n##########################################################################\n\nSummary:\n\nThe authors propose a framework, GenQu, for learning classical data using quantum computation. The classical computer would encode the classical data into quantum circuits. The quantum computer would then run the quantum circuit, measure the resulting quantum state, and feed the measurement data back to the classical machine. This process would repeat until the classical computer output the final result.\n\n##########################################################################\n\nReasons for score: \n\nThis framework is not new and has been widely adopted in the quantum machine learning community. It is unclear to me what is being proposed by this work. The framework is known as a variational quantum-classical algorithm and there is extensive literature for different applications in quantum computing, such as quantum chemistry, simulating quantum field theory, optimization, and machine learning. For example see references [1, 2] for existing proposals for machine learning applications. Due to the lack of meaningful contributions, I would not recommend acceptance.\n\n##########################################################################Pros: \n\nCons: \n\n1. The framework is not new. It is not scientifically correct to claim the proposal of a new framework \"GenQu\" when this has already been widely adopted in the quantum machine learning community.\n\n2.  The authors did not provide any new theoretical insights into how quantum computation can learn classical data better.\n\n3. The numerical experiments were not strong enough to justify any form of advantage using the quantum computer. Furthermore, these numerical experiments have already been presented in the literature. For example, a tutorial in Tensorflow Quantum [3] has also included such an experiment.\n\n#########################################################################\n\n\n[1] Havlíček, Vojtěch, et al. \"Supervised learning with quantum-enhanced feature spaces.\" Nature 567.7747 (2019): 209-212.\n\n[2] Farhi, Edward, and Hartmut Neven. \"Classification with quantum neural networks on near term processors.\" arXiv preprint arXiv:1802.06002 (2018).\n\n[3] Peruzzo, Alberto, et al. \"A variational eigenvalue solver on a photonic quantum processor.\" Nature communications 5 (2014): 4213.\n\n[4] https://www.tensorflow.org/quantum/tutorials/mnist", "extra_scores": {}, "extra_text": {}}, {"review_id": "DLpuw85e9mh", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 2, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "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.\nThe only remotely nontrivial part of the paper is contained in Section 4.2 about Quantum Deep Learning, where the authors consider the MNIST data set. Upon closer inspection it turns out that they use PCA to reduce the dataset to 4 dimensions, which is in turn used to train a \"quantum neural network\" to perform binary classification (i.e. to discriminate between '0'-instances and '5'-instances). The authors claim that such a quantum classifier provides an advantage versus a convolutional neural network in terms of\n 1. the number of training epochs (while ignoring the time needed to perform PCA), and\n 2. the number of parameters (while ignoring the parameters needed to describe the principal components).\nAdditionally, no confidence intervals are visible on Fig. 7, which suggests that the data might have been obtained from a single experimental run. Finally, there are several instances of sloppy writing, such as the inconsistent usage of math mode for variables, the statement P(|\\phi>) = |0>, the typo \"iWs\" instead of is, etc.", "extra_scores": {}, "extra_text": {}}, {"review_id": "9TVKFP72JPo", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors identify that “classical” learning is running into limitations due to power and scale of computing systems. The authors suggest “quantum” learning might supplant classical learning and solve these fundamental challenges.  And, the authors suggest a method that is efficient in the number of QuBits, which can be quite precious.  \nThe authors believe this work is well motivated by the fact that little work has been done in finding ways to combine the virtues of quantum methods with the power of deep neural networks.\nThe authors propose an encoding scheme to map data efficiently to a qubit, to exploit multiple qubit samples. The ability to entangle multiple qubits to represent points on the Bloch Sphere is a key concept for efficiently using the qubit space.  I believe the biggest contribution this paper makes is this efficient mapping of data to qubit space.\n\nOverall, this paper demonstrates that some ASPECTS of neural computing can be mapped to a quantum, qubit system.  This is a timely topic as the scale and energy usage of classical computers for deep learning training is becoming untenable, and many are thinking about ways to make this problem tractable.  The most novel concept in the paper is the symmetry described between the nature of how entangled qubits can encode information, and how a neural network progressively makes data more separatable.  The actual learning process appears to be done on a classical computer (backprop and update) and then mapped back into the quantum system.  To me, a much more interesting result would be exploiting the quantum nature of the computer to solve this non-convex optimization problem of reducing the loss via layers of features. The generative aspect described is a nice side-effect (as the authors indicate) but isn’t really a novel concept.  \nThe results of the paper really are proof that the basic functionality is there, but not really a proof of concept in a general way.  The dimensionality of a small neural network for MNIST had to be reduced to be able to be realized on an actual quantum system; this weakens the claims of these results as it may be untenable as complexity is scaled-up.  Nevertheless, I believe this result does demonstrate an existence proof, and the community will benefit.\n\nIn general, the language should be cleaned up.  There are just a few sentences with a slightly odd structure.\n\nSection 3: I can see how the data is encoded to make it linearly separable.  But how does this relate to backprop? I can see that backdrop isn't really handled by the quantum system, so the separable mapping of the data is really just a general data concept.\n\nSection 3.1: It looks like the actual backprop is done on a classical computer.  A much more interesting result would be to do the actual gradient descent in the quantum domain.\n\nSection 3.2: Is the data encoding done on a classical computer?\n\nSection 3.5: Looks like the generative concept is really just exploiting the stochastic nature of qubits as a random generator.  I think this points to a strong parallel between neural networks and quantum systems but isn’t really a new concept.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "5c64BZwISl2", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper presents GenQu, a hybrid and general-purpose quantum framework for learning classical data through quantum states. By encoding two dimensions of data per one qubit. they demonstrate the effectiveness of their framework via two classical classification tasks, where 1 and 2 qubits are used, respectively. This paper is more like an entry-level tutorial, rather than a technical paper.  More technical contributions are needed towards a paper.\n\n\n1. One of the key contributions claimed by the author is that \"they show the power of encoding two dimensions in one qubit ...This thereby reduces the quantum state dimensionality by 2^(n/2).\" I am super surprised by this claim. What is your baseline? One qubit per one dimension of classical information? Could you refer to the paper from which you get this baseline? There are just too many papers[1] about how to encode classical data into quantum states.  The coding scheme proposed in the paper is not novel and not even state-of-the-art.  \n\n2. What is the key difference between your framework and TensorFlow Quantum[2]? For me, TensorFlow Quantum is a much stronger framework. For example, the SINGLE QUBIT KERNELIZED CLASSIFICATION case study in section 3.3 is just an illustrative example in [2].\n\n[1] Biamonte, Jacob, et al. \"Quantum machine learning.\" Nature 549.7671 (2017): 195-202.\n\n[2] Broughton, Michael, et al. \"Tensorflow quantum: A software framework for quantum machine learning.\" arXiv preprint arXiv:2003.02989 (2020).\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a new quantum machine learning framework which is evaluated on the MNIST dataset. While the paper was relatively well written, reviewers noted that most of the ideas are already well established and used in quantum machine learning community. Thus it was not clear what novelty is provided relative to related work.
Decision: Reject