Image Compression and Classification Using Qubits and Quantum Deep Learning
Authors: Ali Mohsen
Mo Tiwari
Venue: ICLR 2022 Submitted (2022)
Keywords: quantum machine learning
flexible representation of quantum images
quantum neural network

Abstract: Recent work suggests that quantum machine learning techniques can be used for classical image classification by encoding the images in quantum states and using a quantum neural network for inference. However, such work has been restricted to very small input images, at most $4 \times 4$, that are unrealistic and cannot even be accurately labeled by humans. The primary difficulties in using larger input images is that hitherto-proposed encoding schemes necessitate more qubits than are physically realizable. We propose a framework to classify larger, realistic images using quantum systems. Our approach relies on a novel encoding mechanism that embeds images in quantum states while necessitating fewer qubits than prior work. Our framework is able to classify images that are larger than previously possible, up to $16 \times 16$ for the MNIST dataset on a personal laptop, and obtains accuracy comparable to classical neural networks with the same number of learnable parameters. We also propose a technique for further reducing the number of qubits needed to represent images that may result in an easier physical implementation at the expense of final performance. Our work enables quantum machine learning and classification on classical datasets of dimensions that were previously intractable by physically realizable quantum computers or classical simulation.


Under review as a conference paper at ICLR 2022
IMAGE COMPRESSION AND CLASSIFICATION USING QUBITS AND QUANTUM DEEP LEARNING
Anonymous authors Paper under double-blind review
ABSTRACT
Recent work suggests that quantum machine learning techniques can be used for classical image classification by encoding the images quantum states and using a quantum neural network for inference. However, such work has been restricted to very small input images, most 4, that are unrealistic and cannot even be accurately labeled by humans. The primary difficulties in using larger input images is that bitherto-propost qubits than physically realizable. We propose a framework to classify larger, realistic images using quantum systems. Our approach relies on a novel encoding mechanism that embeds images in quantum states while necessitating fewer qubits than prior work. Our framework is able to classify images that are larger than previously possible, up to 16 x 16 for the MNIST dataset a personal laptop, and obtains accuracy comparable to classical neural networks with the same number of learnable parameters. We also propose a technique for further reducing the number of qubits needed to represent images that may result in an easier physical implementation the expense of final performance. Our work enables quantum machine learning and classification on classical datasets of dimensions that were previously intractable by physically realizable quantum computers or classical simulation.
encoding schemes necessitate more
INTRODUCTION
In the past decade, deep learning has been remarkably successful on a wide variety of classical learning tasks (Gil (2016); LeCun et al. (2015)). In parallel, quantum computing (QC) has long promised dramatic increases computational power over classical computers, culminating recent demonstration of quantum supremacy in machine with 53 programmable qubits in Arute al. (209)). However, even these quantum systems are already approaching the limits of classical simulability by the world's largest traditional supercomputers (Boixo et al. (018);; 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 1: Beer et al. (2020): E. Farhi (2018)).
In this paper, digits (LeCun & Cortes 2). Prior work has been restricted to only highly-compressed rather unrealistic input images due to their inefficient encoding schemes that injective maps from classical images to their corresponding pure quantum states. These frameworks have used input images of maximum resolution of 4 x 4, which is too coarse even for humans to provide accurate labels (see Figure For larger images, injectivity would necessitate the same number of qubits as bits that are present in the original image. However, recent advances quantum machine learning (QML) on classical data, such as the Flexible Representation of Quantum Images (FRQI) in P.Q. Le Hirota (201T), 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 fewer-qubit systems. This approach necessitates novel QNN architecture for classification, which we describe in Section S) The input to our model are images of resolution up to 16 x 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.
quantum neural network (QNN) classify the MNIST dataset of handwritten

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Our main contributions are follows:
1. We provide novel construction to compress images and encode them in their FRQI states. Our construction uses only 2-qubit gates, which permits its use common quantum machine learning packages such as Cirq and Tensorflow Quantum (TFQ) and may be of independent interest (Cing (02T); GoogleAl (2020)).
2. We propose a new QNN layers, CRADL and CRAMI, which we use in 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 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 quantum neural network obtain 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 and description of future work in Appendix g
PRELIMINARIES
2.1
QUANTUM COMPUTING
Here we provide brief review of quantum computation. For detailed reference and an interactive coding tutorial, we refer the reader to Nielsen & Chuang (001I) and Qiskit (2017)
respectively.
In quantum computation, the basic unit of information is a two-state quantum mechanical (QM) system called qubit; the 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 (x) ao [0) + ai [1] Eie ag (i), where each a; E C. When qubit measured, the wavefunction collapses and the result of the measurement is state (0) with probability |ao| and state
[1) with probability |a 2 The space of all possible states of the qubit called the Hilbert space
Hi; the states ()) and 11) provide a basis for this Hilbert space. [
Multi-qubit systems are represented mathematically by the tensor product of multiple single-qubit systems. Notationally, we write
[NN)=
Σ JE
dinda 11,, (2....in) iN)
where the states 162, f2....... .x) provide a basis for the multi-qubit Hilbert space Hy and (v/x) is generally superposition of these basis states. A two-qubit state (ı)) H2, cannot necessarily be factorized into two single-qubit states (4/1). (x2)) E Hi:
|+> = Σ
ai 1 12) * (eb))
X [122 = (ap 0) + a, (1)) @ (ab 10) + (1)
we call a state which cannot be so factored a mixed state. In particular, we notice that H2 > H2 8 H1. and 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
We require | 2 + |a1|2=1 Formally, a Hilbert space is inner product vector space that is also complete metric space with respect the distance function induced by that inner product.
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on set of qubits can be represented 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 abilitt to be entangled. Informally, entanglement refers to the property of quantum mechanical systems whereby the state one qubit cannot be described independently of the other qubits' states. For example, the state (00) is maximally entangled as knowledge of one qubit's 's state complete specifies the other's.
In general, quantum algorithms are procedures whereby 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 (197);; Shor 1997); Grover (1966).
2.2 COMMON QUANTUM GATES
Several common quantum gates defined below. These gates are defined by their action 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 00) |+) = / (10) + (1))
H [1) 1-) J v2 (10) (1))
(1)
x 0) = [1)
x [1) = (0)
(2)
common 2-qubit gate is CNOT, which flips the second qubit if the first qubit state [1):
CNOT [00} 100)
CNOT = 001> CNOT = (10)
(3)
CNOT 100 = 111)
3 RELATED WORK
Many studies use QNNs to model either inherently quantum quantum-encoded classical data but are generally restricted to very small images (Li et al. (2020); Henderson et al. (2020); Oh 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 FRO E. Farhi (2118); Aïmeur et al. (2013): ; Paparo et al. (2014); Schuld et al. (2014); Kapporr et al. (016 Amongst this line of work, [.. 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) because they use inherently quantum
data (Sasaki & Carlini 2002)); Gambs (2008); Sentís et al. (2012); y Dunjko et al. (2016) Monrås et al.
(2017;: Envarez-Rodriguer et al. (2017): Du et al. (2020): Sentis al. (019)) Beer et al. (2020); Caro & Datta (2020)) Amongst these papers, Schuld et al. (2000) is perhaps closest to this work. The authors, however, take the mixed-state encodings of images given for input QNN and do not describe how to construct the quantum states, Other work, such as Beer et al. (220)), assumes the wavefunctions as given proposes a generalization of the perceptron to the quantum setting, which provides 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, third line of work uses quantum convolutional neural networks via semi-ellmsesseeee classccassical simulations meant to model the noise introduced by quantum effects, as in Kerenidis et al.] (2019). These approaches do not provide fully quantum simulation to evolve the quantum states, which would require construction the actual data's wavefunctions as in our work.
For demo of standard approach following F. Farhi 2018 using TFQ, which could also serve as preliminary for this work, see https//www..tnnsoflowwwwwws http://wwwwww...ooogoooaaaannnaa
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Throughout prior work, encoding classical data in quantum states efficiently appears to be common open problem.
FORMAL SETTING
4.1 PROBLEM STATEMENT
In classical image classification, the imput model is an n x n-pixel image E {0, 1).2 and our goal is to learn a classification function with binary outputs felassical parametrized by weights wt
Selassical (w) 10, + {0,1}
(4)
In the quantum setting, the input to our classification function is still an n X n-pixel image but must be encoded a [log2n] + dimensional Hilbert space by an encoding function F, where the +1 is for the readout qubit. The quantum neural network is sequence of unitary operations = UU2..... UN parametrized by angles 01, ....... . Oy. To obtain classification prediction,
measurement is performed the readout qubit:
f fquantum (0) (0, - +00,13
(5)
: :00, 1] 2m
F
14(0), H mensure {0, 1}
That is Squantum ( measure e (I()) o F.
In Sections and propose implementation of the FRQI algorithm P..rithhmP........ Le & Hirota 22011 to construct F, propose a construction of (0), describe how to learn the parameters B via standard backpropagation, and describe the final measurement step.
4.2 DATASET AND QUANTUM ENCODING
Crucial to our approach is the encoding of a classical datapoint (e.g. an image) in a quantum state. In our experiments, we use the MNIST dataset of handwritten digits[LeCun Cortes (2010). Following E. Farhi (018)), we restrict our dataset to those of only two ground truth labels: 3 and 6. We downsample image resolutions either x 8 or x using bilinear interpolation. The remaining dataset is approximately 12, 000 training images and 1, 100 validation images for each resolution. Finally, we transform the images to black and white by thresholding the pixel color.
In Figure D we present MNIST image downsampled to different resolutions. Prior work uses resolutions of only x 4, but loses many important features of the original data EE. Farhi (2018). With the FRQI encoding and the further compression
able to encode higher-resolution images current quantum hardware; this insight motivates our study different downsampled resolutions.
After preprocessing, each image is black and white 2n 2" 2nd dimensional binary vector. Our objective is to encode the image as a wavefunction |V/dara):
(daata)
Σ
1go, ......, 92n-1) @ (cos 0, (0) + sin 0g (1)
1a.....99n-11
(6)
q= 90 -tw..aa.....-..... 1ef01122
In Equation 6 each basis state Iqu. Q...... ... 1mn-1) of the "pixel qubits" represents a possible bitstring of length 2 with the strength of the superposition component and color determined by 0g in the "color qubit". In our dataset, each 0,, either or -.
In some experiments in Section 5
we also consider allocating more qubits to encode the color angle
fg instead of the pixel locations:
|Waata)
Σ
Igo, Q1 ..... ,.. 3) s (cos θ 10) + sin 0g 11)
9= 9== (20099991...... (0,1)
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Original Image
16x16
Bxl
4x4
3
3
0.8
0.5
Original Image (BW)
16x16 (BW)
8x8 (BW)
4x4 (BW)
0.4
3
E
15
25
0.0
10
20
Figure 1: A digit from MNIST presented different downsampled resolutions (downscaled resolutions indicated top of each image. The top consists of grayscale images with 0 color 1 in the original dataset. The bottom row are black and white images obtained by thresholding the pixel color from respective images above.
To do this, we exploit the observation that the color qubit is always either [0) or [1) and map into the last two pixel qubits according to the transformation:
92n-22 92n @ (q0) - (ğc) = cos 0g 10) + sin B, (1)
(7)
where each q, {0, 1} and
8g
9c +
92n-2
92n-1 1
0g + (2n-2
92n-1 1
(8)
4
METHODS
5.1 ENCODING THE IMAGES IN WAVEFUNCTIONS
In our approach, we must first encode the image in a quantum wavefunction. We pass an initial state of O) through quantum circuit with given structure, demonstrated by Figure 2 for 4-qubit state. Initially, a Hadamard operation 2 performed on the 2n pixel qubits. This followed by series of n--onnrolledd X X-gates (also known generalized TOFFOLI gates Toffoli (980)) Rasmussen eet a (2020); Shende & Markov (2008) with alternating X gates that determine the color qubits which will be transformed. The re-uubit circuit is constructed recursively from smaller-qubit circuits by observing the symmetries in the construction.
X
Figure 2: An example circuit to construct the superposition state for 4 qubits, using 4 q-qubit generalized TOFFOLI gate. We follow standard quantum circuit diagram conventions in which the dots represent the control of gate by the given qubits.
To construct this circuit we need to define the generalized TOFFOLI gate for qubits from basic two-qubit gates; this also enables our implementation in standard packages such as Cirq (Cirg (202)) that only support backpropagation through two-qubit gates. We the following lemma, first shown in Barenco et al.) (1995). to recursively decompose the n-qubit generalized TOFFOLI gate as sequence of (n 1)-qubit generalized TOFFOLI gates and CNOT gates:
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Lemma 1. (Barenco al. (1995), Lemma 7.3): For rotation matrix R(t), n-controlled rotation gate can be decomposed into a circuit of the form shown in Figure 3
(9)
R00
K102)
R(-12)
RU000
Figure 3: n-qubit controlled gate in terms of (n 1)-qubit controlled gates
From the recursive properties of the diagram above we see that an n-qubit controlled rotation decomposes into (2 3 - 1) one-qubit controlled rotations. Since we flip at most 2 pixels for
given image, we see that for each image
# one-qubit controlled gates S 2n (2 (2"3" - 1)
in addition to the standalone X X-gates to encode the 2n x 2n image in the amplitudes of the input wavefunction. Though this bound is exponential in n, find this acceptable as it is still classically simulable for larger images that are primarily constrained by the size of their qubit representations.
5.2 READOUT QUBIT AND PREDICTED LABELS
Our wavefunction must also contain a readout qubit on which we perform measurements that will be the model's predicted labels. As such, we prepare the wavefunction (pm.) (&aata) @ |readout). We choose the Z-gate for measurement and thus initialize the readout qubit in the |+) state, which is common practice to produce an initially unbiased output:
(
Ipann)) @ H (0) |Dddtta) X |+)
(10)
where (dlataa is prepared in Section 5.1
The model which is used to transform (Wi) a QNN with L layers. Following E. Farhi (2018), each layer is represented by parametrized unitary matrix. The model's output state is:
ouo (0)) = (O2) (01) (IPasta) @ (+))
(11)
where 0 = (0)...., 0z). The final measurement is performed with Z-gate on the readout qubit; the predicted label is (Pout (0) 1 & Z (0)). We train the model's parameters, 0, Oz, via stochastic gradient descent (SGD) using these predictions and the hinge loss:
loss (t) (0) ()) (wl (0) 2n+1 Z ( (0))
(12)
where the superscript (i) is used to refer to the 7th training example.
5.3 IMPLEMENTATION DETAILS
We use Cirq CCig 2021) to encode the images into their respective wavefunctions and TensorFlow Quantum (TFQ) A 22200) to train the model via the paradigm described in Section [..2 TFQ permits the use of Parametrized Quantum Circuits (PQCs), which describe the unitary operations of the QNN, single Keras layer Chollet et al. 20155) within the standard TensorFlow framework.

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Backpropagation through quantum layers is nontrivial. We recall that, for any layer, the 2" x 2" unitary operator can be expressed in terms of the exponential of a 2" x 2" Hermitian operator H called the Hamiltonian, which in turn be decomposed into its Pauli decomposition (a tensor product of n Pauli matrices, which form an orthonormal basis over the Hilbert space of Hermitian matrices over IR):
U(+) exp
Σ
(la.)
(13)
o.E(Lo1.02.09)>
When the layer is restricted to unitary operations whose Hamiltonian has single term in the Pauli decomposition, gradients with respect to the layer's parameters can be computed analytically using the parameter shift techniques first introduced in Mitarai al. (2018) and Schuld et al. (2019) (Harrow & Napp (202T)). These techniques provide way to calculate partial derivatives of parameterized quantum circuits in terms of other functions that use the same circuit architecture with shifted parameters. For this reason, restrict the gates in quantum layers to be multi-qubit exponential Pauli gates. For these gates, analytic gradients can also be computed because they are rotations of operations whose Hamiltonians contain single type of term. For example, the X.X X-aate can be written:
(X (X = cxp (0 1) (X 1))] = -1500x -(x-1) -20000
5.4 NETWORK ARCHITECTURE
A general 2 x 2" learnable unitary operation would consist of 2 trainable real parameters and the standard representation of this parameters follows equation equation 13 Note that, as described in Section 5.3, this does not necessarily permit analytic gradient computation in the backpropagation step.
To permit analytical gradient computations, we construct layers having specific structure; an example such layer is demonstrated Figure Each layer consists of either X ZZ operators applied in succession to each pixel qubit and the readout qubit, followed by the same operation on the pixel qubit and the color qubit. Empirically, the Color-Readout-A -orroo----eeenatnn--oub--l------ architecture (CRADL) presented in Figure 4 resulted in the best performance.
Each consecutive pixel-readout and pixel-color pair of gates share the same learning parameter, a rotation angle. One double layer of the type shown in Figure 4 will have 2n learnable parameters, where n is the number of pixel qubits. A circuit with L layers will therefore have nL parameters. In the experiments in Section 6, choose the number of layers such that the number of parameters n. comparable to those the classical benchmark, given fixed number of qubits.
X
x
3:
Iwsdou:
Figure 4: A "CRADL" network double layer with pixel qubits, consisting of consecutive pixcl-readout pixel-color X gates, followed by analogous ZZ gates
We note that there are equivalent network architectures that lead to comparable results, such as the architecture shown in Figure
Color-Readout-ANe Aeeadduut-Attemmtinn--iiii
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X
X
Z
(d01 gAtl
1esded:
x
Figure 5: A "CRAML" network layer with 6 pixel qubits, consisting of consecutive pixel-readout pixel-color XX and 2Z gates
RESULTS
We benchmark our quantum learning framework against a classical neural network with two hidden layers with ReLU activations and a single-neuron output layer; in this setting, the quantum and classical models have a comparable number of parameters. We trained the quantum neural network for 10 epochs, which is the same number of epochs after which the classical neural network began to overfit (as determined by cross- cross- validation). All experiments were conducted on personal laptop with GPU (Machook Pro, 2.4 GHz 8-Core Intel Core 19 CPU, 64 GB 2667 MHz DDR4 RAM).
Network
x 8 Image
16 x 16 Image
Classical CNN
94 +: 1%
98.9 + 0.3%
Quantum CRADL
92 + 1%
Quantum CRADL 2Q
88 = 1%
90 + 1%
Table 1: Test accuracies after the 10th training epoch for classical and quantum networks.
MNIST Bx8 Image
MNIST 16x16 Image
100
100
Validation Accuracy
Validation Accurecy
70
80
Layer
Training Epoch
Training Fpach
Figure 6: Test accuracy versus training epoch for classical and quantum models, with and without the extra 2 qubit compression described in Section 4.2 When images embedded qubits (green curve, left), achieve performance comparable classical networks with the same number of parameters. When images are further compressed (orange curves), performance degrades.
Table demonstrates our final results. On 8 8 images, the QNN without the extra two-qubit compression achieves performance comparable to the classical network, whereas the network with the extra two-qubit compression (denoted -2Q) performs worse. For 16 x 16 images, we add more layers to the QNN so that the the number of parameters remains comparable to the classical dense network and must use the extra two-qubit compression due the computational cost of the experiment. We
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observe that the QNN is unable achieve the same performance the classical network, likely due to the extensive compression of the images in the quantum states.
Figure 6 shows the validation accuracy of quantum and baseline classical models versus training epoch. We observe that the classical neural network and both quantum neural networks demonstrate similar validation performance curves.
7 DISCUSSION AND CONCLUSION
We note that the method we describe in Equation g to reduce the number of required quhits results in worse performance. This degradation test accuracy may be palatable in applications that attempt to minimize their qubit usage. When we attempted to lower the number of necessary qubits further, we observed unstable learning behavior. In such settings with reduced feature dimensionality, it may be necessary to redesign the network architecture.
In this paper, we developed proof of concept for recently proposed QNN models. In the process, we proposed methodology to map classical images quantum states that may be of independent interest to the community. We also propose new form of quantum neural network layers motivated by the highly entangled input states, the CRADL and CRAML layers in Section 5.3 and demonstrate that model consisting of these layers achieves performance competitive with classical neural networks with comparable number of parameters. Furthermore, our work is evidence that quantum machine learning algorithms can scale to data of dimensions larger than those previously tractable by classical simulation or available quantum hardware and classify MNIST images size 16 16 on a personal laptop.
8 FUTURE WORK
We did not do comprehensive survey of the space of all possibly unitary operations that could be used for each hidden layer, but we could imagine that invoking or more qubit gates to the layer circuits would improve the learning outcomes, that would give direct access to the correlation structure. The circuit above Figure composed entirely of two-qubit gates is trying to work around this limitation.
FigureHcomposee
We would like in the future to conduct a more systematic survey of network architectures to enhance the learning outcomes with possibly more involved quantum gates, either within TensorFlow Quantum or implemented separately, and study the cost benefit of forgoing analytic gradients for some of these gates for more flexibility in construction. Another lower hanging fruit optimizing the encoded input circuits in terms of memory. As long as we write and simulate quantum algorithms on digital computers, representing the wavefunction in this manner seems inevitable and we should find better optimizations to describe the encoding procedure through circuits with much less gates than the nice recursive algorithms discussed in this paper.
We observed in Section 6 that compressing the images into larger quantum systems resulted in better performance at the expensive of greater complexity in physical realization. leave a more thorough analysis of this tradeoff, including the understanding of the interplay between the data qubits and the color qubit, to future work.
We also note that may view the encoding with limited circuit gates limited qubits form of implicit regularization, which has been observed to improve generalization performance as in Smith et al. (2016). We leave investigation of these connections future work.
ETHICS STATEMENT
We do not foresee any ethical concerns with our work.

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REPRODUCIBILITY STATEMENT
The authors have submitted all the code necessary to reproduce their results in the Supplementary Materials as a zip. The code contains a README md file with instructions on how to reproduce the experimental results. The only dataset required is the MNIST dataset, which is easily obtained from http:: /ttt:://yann. 1...leecc. lecun. com/ hxdb/mmnise The preprocessing applied to the MNIST dataset is described in detail in Section S
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541534 0032--2
SUPPLEMENTARY MATERIAL AND APPENDIX
Supplementary code submitted zip.
13


Reviews:
[{"review_id": "dxJnZaG8M70", "reviewer": "Reviewer_snts", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 2, "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": "\n#### Positive\n- The related work and formal problem description sections are well written and organized.\n- The experimental validation of the proposed approach goes beyond single runs.\n\n#### Negative\n- The introduction is extremely vague. Why would quantum computers outperform their classic counterparts? What does it mean for data to be \"inherently quantum\"?\n\n- Authors claim to have the ability to 8x8 or 16x16 encode images on current quantum hardware. A least one major company now offers free access to their machines.\n    - Are any experiments on actual quantum hardware available?\n\n- If \"these quantum systems are already approaching the limits of classical\nsimulability by the world’s largest traditional supercomputer\" (page 1), why would simulation experiments on a consumer laptop suffice in the paper's case? \n\n- Page 5 observes that the color qbit is either $\\braket{0}$ or $\\braket{1}$.\n  MNIST digits are black and white. Would this observation hold for color images?\n\n#### Minor remarks\npage 3, top: .. completely specifies the other's.", "summary_of_the_paper": "The paper presents an image coding approach based on 2qubit gates.\nDownsampled MNIST digits are classified using standard components.\nAdditionally, the experimental section studies the use of reduced codes.", "summary_of_the_review": "The paper presents an interesting idea, however, since the new contributions appear to be limited to equations seven and eight, I am not convinced the contributions are sufficient for inclusion in the proceedings.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "cOXpTIprygq", "reviewer": "Reviewer_N7sU", "summary": "", "questions": "", "limitations": "", "rating": 1, "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": "Strength: \n1. Use FRQI to embed images with higher resolution.\n2. Propose CRADL and CRAML as parameterized circuits to perform the classification tasks.\n\nWeakness:\n1. Major: The novelty seems limited, the embedding method which encodes the images to the state vector of the qubits, is a well-established one and has already been used in previous QNN work such as [1]. The proposed parameterized circuit part is also similar to that in [2]. \n2. Major: Although the image resolution can be large, from 4x4 to 8x8 or 16x16, they are binarized, which will obviously lose a considerable amount of information. Why binarized 8x8 is guaranteed to be better than int 4x4 such as that used in [3]? \n3. Major: since the paper claims to have a good embedding method, comparisons with other embedding methods are necessary, such as phase-based embedding [3][4]; state-vector embedding[2], etc. \n4. Major: The proposed embedding circuit seems to be very deep (figure 2), which will be significantly impacted by noise in NISQ device. Some experiments on real quantum devices, such as on publicly available IBMQ machines can help justify the results.\n5. Major: The experiment is only performed on a single MNIST dataset and there are no ablation studies on the hyperparameters it used. Comparisons with previous QNN work are completely missing.\n6. Minor: the claim of \"our work is the first to propose a data encoding scheme and QNN that can be used to classify realistic\nimages.\" is overclaimed, [2][3][5] all perform classification on the realistic MNIST dataset. [3] even perform 10-class and 4-class classifications on real quantum machines.\n7. Minor: As in figure 6, the empirical results seem not good enough as the classical NN model can consistently outperform the quantum models.\n8. Minor: More justifications on why using the CRAML and CRADL should be provided. The design space is very large, why choose those kinds of ansatz?\n\n\n\n\n[1] Jiang, Weiwen, Jinjun Xiong, and Yiyu Shi. \"A co-design framework of neural networks and quantum circuits towards quantum advantage.\" Nature communications 12.1 (2021): 1-13.\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] Wang, Hanrui, et al. \"Quantumnas: Noise-adaptive search for robust quantum circuits.\" arXiv preprint arXiv:2107.10845 (2021).\n\n[4] Lloyd, Seth, et al. \"Quantum embeddings for machine learning.\" arXiv preprint arXiv:2001.03622 (2020).\n\n[5] Henderson, Maxwell, et al. \"Quanvolutional neural networks: powering image recognition with quantum circuits.\" Quantum Machine Intelligence 2.1 (2020): 1-9.\n", "summary_of_the_paper": "The paper study the problem of quantum neural networks for classical image classification. It leverages the FRQI framework to enable larger size of the input image instead of using very small 4x4 input image. They construct parameterized quantum circuits using XX and ZZ gates (called \"CRADL\" and \"CRAML\") to perform the transformations of quantum state and perform classification.", "summary_of_the_review": "The novelty is insufficient as the embedding part and parameterized circuit part is not new. the evaluation is insufficient: no comparisons with state-of-the-art; no evaluation on real quantum devices; no ablation studies; no justifications of why the proposed embedding can outperform others.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "ATRw4_CZ99F", "reviewer": "Reviewer_PiSA", "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": "From my perspective, the topic of image classification with practical implementation studied by this paper is interesting and can be of general impact. However, after taking a closer look at this paper, I feel that the contribution in results as well as the novelty in algorithm design are not significant for this paper to be accepted by ICLR 2022. Please find my detailed comments below:\n\n- To me, the technical contribution of this paper is minor. In particular, although the encoding from Eq. (6)-(8) was not claimed in previous papers on image classification, the math is standard with only one extra qubit added. For the circuit synthesis in Section 5, it is mostly the standard trick in Lemma 1 proposed by Barenco et al. more than 20 years ago, and for the circuit implementation the authors simply call the standard package TensorFlow Quantum from Google AI. The Flexible Representation of Quantum Images (FRQI) was also originally proposed by Le and Hirota back in 2011. In general, compared to the previous work of Farhi and Neven which tests 4*4 input images on quantum neural networks, this work is more like running experiments with moderately larger encodings and scales, but the methods are routine.\n\n- I feel that the abstract has significant overclaims compared to what the paper achieves. In particular, the abstract claimed that their result “obtains accuracy comparable to classical neural networks with the same number of learnable parameters”. However, in Section 6 of the paper, the authors admit that “we observe that the QNN is unable to achieve the same performance as the classical network”. Looking at Table 1, the classical success probability is 98.9%+-0.3% while quantum is 90+-1%, both in 10 epochs; the quantum result is actually much worse than the classical counterpart, not even comparable to me. In addition, it is also claimed that “our work enables quantum machine learning and classification on classical datasets of dimensions that were previously intractable by physically realizable quantum computers or classical simulation”, which is not wrong but does not make much sense to me because classical computers can run images with resolution much much better than 16*16. In addition, the authors’ experiments are restricted to only numbers 3 and 6. In all, also with my first point above, this work is not only still a toy model compared to the current scale of classical machine learning methods, but also does not offer much value to the design of quantum machine learning algorithms.\n\n- Although the paper is not long, it has various typos:\n\n- - At the end of Section 1: “a description of future work in Appendix 8” -> “a description of future work in Section 8”\n\n- - Section 3, related work: The reference to E. Farhi (2018) should be changed to Farhi and Neven (2018) as in https://arxiv.org/pdf/1802.06002.pdf\n\n- - Eq. (4) and the text beforehand: I believe the definition domain should change from {0,1}^{2n} to {0,1}^{n^2}\n\n- - End of Section 4.1, Sections 5 and 5 should be fixed; for the reading of the same sentence, P.Q. Le & Hirota (2011) -> (Le & Hirota (2011)). In many places of the paper there should be brackets for references.", "summary_of_the_paper": "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.", "summary_of_the_review": "Based on the concerns I mentioned above, I recommend rejection for this paper at ICLR 2022.\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "-fY8VzUu-75", "reviewer": "Reviewer_MiCP", "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": "The paper provides a novel quantum algorithm for loading and classifying images inspired from classical machine\nlearning models. The authors try to incorporate several properties of quantum computing into their algorithm. Given an\nimage, the authors start by downsampling it to a lower resolution and build a quantum circuit that loads this image into a\nquantum state. This data-loading circuit is made with elementary gates that build a superposition of pixel colors. An\nadditional readout register is added to the circuit where trainable gates by gradient descent are added to the overall circuit\nin order to map the input to its label. Moreover, they also run small experiments on two different resolutions (8x8,16x16)\nand show how their model compares to a classical CNN.\nHowever, the paper has several drawbacks for acceptance at ICLR. The contributions are very marginal compared to\nstate-of-art results in quantum machine learning. First, the data-loading procedure assumes that a binary representation\nof the image is enough for classification which may not be true for datasets with low-contrast images. The number of\nqubits is logarithmic with respect to the image dimensions but the circuit seems to have a large depth. The authors do not\ncompare their procedure to other data-loaders used in quantum machine learning. Second, the quantum circuit used for\nclassification is neither novel nor close to state-of-art quantum neural networks. It is not clear what this circuit is able to\nlearn and what are its strengths and limits. Last, the paper lacks a theoretical analysis of the proposed quantum algorithm\nexcept some arguments provided in section 5.1. It is not clear what are the gate and time complexities and how does it\ncompare to existing approaches", "summary_of_the_paper": "In this paper, the authors propose a certain type of trainable quantum procedures used as machine learning models for\nthe classification of images. They explicitly show how to implement the circuits for loading compressed images into\nquantum states that are fed to a trainable unitary that writes the predicted label into a readout register. They conclude on\nnumerical simulation performed on the MNIST dataset.", "summary_of_the_review": "The authors proposed a new algorithm for classification problems combining a new quantum data loader for images and\nan existing trainable quantum circuit. However, the paper lacks arguments that motivate using their new approach\nbecause the authors did not provide neither empirical evidence that their quantum algorithm outperforms existing\napproaches nor improvement of the running time to train their circuit", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This submission proposes a new encoding mechanism, i.e. a new quantum data loader for images with a reduced number of qubits, which is then used for image classification with off-the-shelf quantum neural networks (from TensorFlow Quantum). There are two major concerns raised by most reviewers. The first concern regards the novelty in the design of the quantum data loader and the use of off-the-shelf quantum neural networks (QNNs), the latter of which is neither novel nor close to state-of-art QNNs for the same purpose.  The quantum data loading procedure also assumes a binary representation of images that might not be enough for low-contrast images. Although the number of qubits is reduced, the circuit tends to have a large depth which makes it hard for practical implementations.  The second concern regards the overall performance of the proposed solution for image classification, where a clear quantum benefit is missing, or in some cases, a quantum disadvantage shows up.  Based on these discussions, we believe that the submission requires substantial improvements before its publication.
Decision: Reject