Image Compression and Classification Using Qubits and Quantum Deep Learning

1 INTRODUCTION . In the past decade , deep learning has been remarkably successful on a wide variety of classical learning tasks ( Gi ( 2016 ) ; LeCun et al . ( 2015 ) ) . In parallel , quantum computing ( QC ) has long promised dramatic increases in computational power over classical computers , culminating in a recent demonstration of quantum supremacy in a machine with 53 programmable qubits in Arute et al . ( 2019 ) . However , even these quantum systems are already approaching the limits of classical simulability by the world ’ s largest traditional supercomputers ( Boixo et al . ( 2018 ) ; Preskill ( 2012 ) ) . The power of quantum computation suggests that quantum analogues of deep learning models like feedforward neural network may outperform their classical counterparts , especially when the data is inherently quantum ( Wan et al . ( 2017 ) ; Beer et al . ( 2020 ) ; E. Farhi ( 2018 ) ) . In this paper , we use a quantum neural network ( QNN ) to classify the MNIST dataset of handwritten digits ( LeCun & Cortes ( 2010 ) ) . Prior work has been restricted to only highly-compressed , rather unrealistic input images due to their inefficient encoding schemes that are injective maps from classical images to their corresponding pure quantum states . These frameworks have used input images of a maximum resolution of 4× 4 , which is too coarse even for humans to provide accurate labels ( see Figure 1 ) . For larger images , injectivity would necessitate the same number of qubits as bits that are present in the original image . However , recent advances in quantum machine learning ( QML ) on classical data , such as the Flexible Representation of Quantum Images ( FRQI ) in P.Q . Le & Hirota ( 2011 ) , have demonstrated that quantum wavefunctions can utilize quantum entanglement to encode classical data using exponentially less qubits than their corresponding classical representation in bits . In this paper , we use the FRQI method to embed the input images in fewer-qubit systems . This approach necessitates a novel QNN architecture for classification , which we describe in Section 5 . The input to our model are images of resolution up to 16 × 16 whose quantum encoding only requires 8 qubits ( 6 for pixel locations , 1 for color , and 1 for readout ) . To the best of our knowledge , our work is the first to propose a data encoding scheme and QNN that can be used to classify realistic images . Our main contributions are as follows : 1 . We provide a novel construction to compress images and encode them in their FRQI states . Our construction uses only 2-qubit gates , which permits its use in common quantum machine learning packages such as Cirq and Tensorflow Quantum ( TFQ ) and may be of independent interest ( Cirq ( 2021 ) ; GoogleAI ( 2020 ) ) . 2 . We propose a new QNN layers , CRADL and CRAML , which we use in a model trained with the images ’ FRQI states as input . 3 . We show that our trained QNN achieves accuracy comparable to classical models with the same number of parameters . 4 . We propose a novel technique to further compress black and white images , and study the scaling behavior of our model with the extent of image compression . Organization : In Section 2 , we provide a brief review of the formalism of quantum computation . In Section 3 , we provide an overview of related work and motivate our study . In Section 4 , we describe our dataset and how each image is encoded in a quantum state . In Section 5 , we describe the prescription for using a quantum neural network to obtain a classification prediction for a given image and describe the model we use for classification . In Section 6 , we present our results . We conclude with Section 7 and a description of future work in Appendix 8 . 2 PRELIMINARIES . 2.1 QUANTUM COMPUTING . Here we provide a brief review of quantum computation . For a more detailed reference and an interactive coding tutorial , we refer the reader to Nielsen & Chuang ( 2011 ) and Qiskit ( 2017 ) respectively . In quantum computation , the basic unit of information is a two-state quantum mechanical ( QM ) system called a qubit ; the two states are traditionally written |0〉 and |1〉 . A qubit can be in either of these two states , as well as a quantum superposition of these states , formally written as a wavefunction |ψ〉 = a0 |0〉 + a1 |1〉 = ∑ i∈ { 0,1 } ai |i〉 , where each ai ∈ C. When a qubit is measured , the wavefunction collapses and the result of the measurement is state |0〉 with probability |a0|2 and state |1〉 with probability |a1|2 1 . The space of all possible states of the qubit is called the Hilbert space H1 ; the states |0〉 and |1〉 provide a basis for this Hilbert space . 2 Multi-qubit systems are represented mathematically by the tensor product of multiple single-qubit systems . Notationally , we write |ψN 〉 = ∑ { i1 , i2 , ... iN } ∈ { 0,1 } N ai1 , i2 , ... iN |i1 , i2 , . . . iN 〉 where the states |i1 , i2 , . . . iN 〉 provide a basis for the multi-qubit Hilbert space HN and |ψN 〉 is generally a superposition of these basis states . A two-qubit state |ψ〉 ∈ H2 , can not necessarily be factorized into two single-qubit states |ψ1〉 , |ψ2〉 ∈ H1 : |ψ〉 = ∑ { i1 , i2 } ∈ { 0,1 } 2 ai1 , i2 |i1 , i2〉 6= |ψ1〉 ⊗ |ψ2〉 = ( a0 |0〉+ a1 |1〉 ) ⊗ ( a′0 |0〉+ a′1 |1〉 ) we call a state which can not be so factored a mixed state . In particular , we notice thatH2 > H1⊗H1 , and a similar result holds for multi-qubit systems with N > 2 . Under the laws of quantum mechanics , these wavefunctions – or states – evolve in time as determined by linear unitary transformations . Furthermore , any operation that is physically possible to perform 1We require |a0|2 + |a1|2 = 1 2Formally , a Hilbert space is an inner product vector space that is also a complete metric space with respect to the distance function induced by that inner product . on a set of qubits can be represented as a unitary operator . We refer to such unitary operators as quantum gates and note that unitary operators can be viewed as rotations in Hilbert space . A remarkable property of quantum states is their ability to be entangled . Informally , entanglement refers to the property of quantum mechanical systems whereby the state of one qubit can not be described independently of the other qubits ’ states . For example , the state |00〉 is maximally entangled as knowledge of one qubit ’ s state complete specifies the other ’ s . In general , quantum algorithms are procedures whereby an initial wavefunction is transformed under a sequence of unitary operations , or quantum gates , and a measurement is made of the transformed state ; this measurement is often performed on a readout qubit and is the output of the algorithm . Many quantum computation algorithms are designed to exploit properties of entangled systems ( Bernstein & Vazirani ( 1997 ) ; David & Richard ( 1992 ) ; Simon ( 1997 ) ; Shor ( 1997 ) ; Grover ( 1996 ) ) . 2.2 COMMON QUANTUM GATES . Several common quantum gates are defined below . These gates are defined by their action on the basis states of the Hilbert space , since they extend linearly to superpositions of the basis states . The definitions for the single-qubit Hadamard gate H and the Pauli-X gate X are : H |0〉 = |+〉 : = 1√ 2 ( |0〉+ |1〉 ) H |1〉 = |−〉 : = 1√ 2 ( |0〉 − |1〉 ) ( 1 ) X |0〉 = |1〉 X |1〉 = |0〉 ( 2 ) A common 2-qubit gate is CNOT , which flips the second qubit if the first qubit is in state |1〉 : CNOT |00〉 = |00〉 CNOT |01〉 = |01〉 ( 3 ) CNOT |10〉 = |11〉 CNOT |11〉 = |10〉 3 RELATED WORK . Many studies use QNNs to model either inherently quantum or quantum-encoded classical data but are generally restricted to very small images ( Li et al . ( 2020 ) ; Henderson et al . ( 2020 ) ; Oh et al . ( 2021 ) ) . One line of work encodes classical data in quantum systems and focuses on learning the classifier ’ s circuit architecture . These approaches require an injective map from the input image to a corresponding pure quantum state , which forgoes the exponential compression advantages afforded by methods such as FRQI3 ( E. Farhi ( 2018 ) ; Aïmeur et al . ( 2013 ) ; Paparo et al . ( 2014 ) ; Schuld et al . ( 2014 ) ; Kapoor et al . ( 2016 ) ) . Amongst this line of work , E. Farhi ( 2018 ) propose the general setup that we follow in this paper . In contrast with their work , however , we use the FRQI technique to exploit the dimensionality of the multi-qubit Hilbert space and need much fewer qubits . Other studies take the quantum wavefunction as given , either by assuming the classical data is already provided in its quantum-encoded form Schuld et al . ( 2020 ) or because they use inherently quantum data ( Sasaki & Carlini ( 2002 ) ; Gambs ( 2008 ) ; Sentís et al . ( 2012 ) ; Dunjko et al . ( 2016 ) ; Monràs et al . ( 2017 ) ; Alvarez-Rodriguez et al . ( 2017 ) ; Du et al . ( 2020 ) ; Sentís et al . ( 2019 ) ; Beer et al . ( 2020 ) ; Caro & Datta ( 2020 ) ) . Amongst these papers , Schuld et al . ( 2020 ) is perhaps closest to this work . The authors , however , take the mixed-state encodings of images as given for input to a QNN and do not describe how to construct the quantum states . Other work , such as Beer et al . ( 2020 ) , assumes the wavefunctions as given proposes a generalization of the perceptron to the quantum setting , which provides a more generalized framework than in E. Farhi ( 2018 ) These authors use inherently quantum data , in contrast with our work . We use classical data and explicitly construct quantum circuits to encode classical images into their wavefunctions . Our approach lends itself to direct experimentation and is usable with modern quantum machine learning packages . Finally , a third line of work uses quantum convolutional neural networks via semi-classical simulations meant to model the noise introduced by quantum effects , as in Kerenidis et al . ( 2019 ) . These approaches do not provide a fully quantum simulation to evolve the quantum states , which would require construction of the actual data ’ s wavefunctions as in our work . 3For a demo of a standard approach following E. Farhi ( 2018 ) using TFQ , which could also serve as a preliminary for this work , see https : //www.tensorflow.org/quantum/tutorials/mnist Throughout prior work , encoding classical data in quantum states efficiently appears to be a common open problem .

Summary: This paper studies the image classification problem on quantum computers. The prior of image classification by Farhi et al. can only work on 4-by-4 input images, while this paper conducts experiments on 16-by-16 images for the MNIST dataset.