QuCNN A Quantum Convolutional Neural Network with Entanglement Based Backpropagation Samuel Stein1 Ying Mao2 James Ang1 and Ang Li1

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QuCNN : A Quantum Convolutional Neural

Network with Entanglement Based Backpropagation

Samuel Stein1, Ying Mao2, James Ang1, and Ang Li1

1Paciﬁc Northwest National Laboratory

2Fordham University

Abstract—Quantum Machine Learning continues to be a highly

active area of interest within Quantum Computing. Many of these

approaches have adapted classical approaches to the quantum

settings, such as QuantumFlow, etc. We push forward this trend,

and demonstrate an adaption of the Classical Convolutional

Neural Networks to quantum systems - namely QuCNN. QuCNN

is a parameterised multi-quantum-state based neural network

layer computing similarities between each quantum ﬁlter state

and each quantum data state. With QuCNN, back propagation

can be achieved through a single-ancilla qubit quantum routine.

QuCNN is validated by applying a convolutional layer with a data

state and a ﬁlter state over a small subset of MNIST images,

comparing the backpropagated gradients, and training a ﬁlter

state against an ideal target state.

Index Terms—Quantum Computing, Quantum Machine

Learning, Convolutional Neural Networks

I. INTRODUCTION

Quantum computing is poised to provide computational

speedups that classical computing could never feasibly attain.

With continued quantum system development and applications

in domains such as Quantum Chemistry, Quantum Simulation,

and Quantum Machine Learning [1], [2], [17], [20], the

potential of Quantum Computing continues to grow. With

high level algorithm development and low level system de-

sign, we continue to improve the current state of the art.

Superconducting quantum processors of size 127 qubits have

been released by IBM [5], and 22 qubit trapped ion quantum

processors by IonQ [25], all with continued improvements in

overall qubit quality. Quantum Machine Learning has been a

ﬁeld of continued interest within quantum computing, with

hopes of applications to adapting classical machine learning

success to the quantum setting. However, to really demonstrate

quantum advantage for QML, the data that is being processed

should be quantum data. For example, through the use of

QRAM or Quantum Simulation [3], [8]. Quantum Machine

Learning [6], [12], [22], [23] has borrowed motivation from

classical machine learning over the last years, attempting to

mimic the classically successful techniques through a quantum

routine (QuClassi, QuantumFlow and Convolutional Quantum

network [6], [12], [23]). Classical machine learning requires

little motivation, and has seen wide spread and ubiquitious

success [7], [18]. Within deep learning, the extremely suc-

cessful design of a Convolutional Neural Network [19] saw

Samuel.stein@pnnl.gov

Ang.li@pnnl.gov

an explosion in performance within multiple domains [11],

[16], [24]. Convolutional neural networks perform high level

feature extraction using the inner product between ﬁlters and

subsections of data which convolutionally evolve traversing

the data. In this paper, we propose a natural adaption of

classical convolutional neural network techniques to quantum

computing through the use of the SWAP test, and demonstrate

entanglement style back propagation. We numerically demon-

strate the relation, and similarity between the implementation

of QuCNN and classically implemented convolutional layers,

and ﬁnally demonstrate the learning ability of parameterised

unitary layers in learning pre-trained solution ﬁlters. In this

work, a QuCNN layer is applied to a small training sample

on MNIST, illustrating the ability to perform a convolutional

operation in the quantum setting.

II. BACKGROUND

A. Quantum Computing

Quantum computing adopts classical computing techniques

such as bit representation and combines it with quantum me-

chanical phenomena such as entanglement and superposition

[15], [21]. In classical computing, data is represented as either

1 or 0, whereas in quantum computing we represent data as

|1i,|0i, or α|0i+β|1i, a superposition of both. The coefﬁcients

αand βrepresent the probability that the data is in the respec-

tive values state. Expanding on this, multiple bits are repre-

sented by the state |ψi=α|00...0i+β|00..1i+...+ω|11...1i,

where 2ncoefﬁcients represent the quantum state. Finally,

quantum computing exposes the computational potential of

quantum entanglement. Quantum entanglement is most easily

understood via the CNOT gate, which transforms the quantum

state of |00i+|01iinto |00i+|11i. In this transformation, the

ﬁrst bit is ﬂipped only if the second bit is in the |1istate.

This exposes computational potential to quantum computers

and has no classical analogue.

B. Convolutional Neural Networks

Convolutional Neural Networks perform high level feature

extraction from data that exhibits spatial relations [13]. Images

are a prime example of where spatially related data exists,

where pixels provide context to pixels around them, and

hence information. The convolutional layer is characterised by

having a set of ﬁlter banks, each of some tensor shape. Each

ﬁlter independently moves and performs the inner product

arXiv:2210.05443v1 [quant-ph] 11 Oct 2022

over the data in a pre-described way. This returns a single

value representing the similarity between the ﬁlter and the

data at that location. These layers are optimised similar to

all other neural network layers, through backpropagation over

some optimization function.

III. RELATED WORK

Adapting classical machine learning techniques to quantum

systems is an active area of research, with architectures such

as Quantum Convolutional Neural Networks, QuantumFlow,

QuGAN and QuClassi.

Quantum Convolutional Neural Networks [6] - QCNN -

adapts the classical notion of spatial data encoding, and adapts

it to quantum machine learning techniques. QCNN makes use

of dual qubit unitaries and mid-circuit measurement to perform

information down pooling, to which decisions can be inferred.

This is compared with an opposite direction traversal of a

MERA network.

QuClassi [23] proposes a state based detection scheme, bor-

rowing from classical machine learning approaches of training

”weights” to represent classiﬁer states. These states each

represent a probability of belonging to the states respective

class, which generates output layers synonymous with classical

classiﬁcation network outputs.

QuantumFlow [12] attempts to mimic the transformations

undergone in classical neural networks, and attempts to accom-

plish a similar transformation as the classical y=f(xTw+b).

This is accomplished via the usage of phase ﬂips, accumula-

tion via a hadamard gate accompanied with an entanglement

operation. QuantumFlow demonstrates the advantage of batch

normalisation, showing notable performance improvements

when normalising quantum data to reside around the XY

plane, rather than clustering around either the |1ior |0i

point. Furthermore, QuantumFlow demonstrates the reduced

parameter potential of quantum machine learning, illustrating

a quantum advantage.

Chen 2022 [?] makes use of a quantum convolutional

network to perform high energy physics data analysis. In

the paper, they present a framework for encoding localised

classical data, followed by a fully entangled parameterised

layer to perform spatial data analysis. Their numerical analysis

demonstrates the promise of quantum convolutional networks.

Notably, all of these works have taken a classical machine

learning technique, and adapted it in some form to the quan-

tum setting. We aim to accomplish the same with QuCNN,

adapting the convolutional ﬁlter operation.

IV. QUCNN

In this section, we walk through the adaption of a classi-

cal convolutional ﬁlter operation to QuCNN. We further go

on to demonstrate a quantum-implemented backpropagation

algorithm allowing for an almost entirely quantum routine to

compute the dL

dθigradient, where θiis the layer weight.

A. QuCNN Layer Architecture

Classical convolutional neural network’s learn a set of

feature maps for local pattern recognition over a trained data

set. This operation is characterised by the Convolution (Conv)

operation. One Convolution Operation comprises of a ﬁlter F,

and an input X, and performs Conv(X,F). This is described

by the convolution operation outlined in 1

yij =

k=1

W W

l=1

wklxsi+k−1,sj+l−1(1)

Importantly, wx0is equivalent to the dot product between

two vectors wand x. A comparable computation is realised in

quantum computing through the SWAP test algorithm, which

computes the equation outlined in Equation 2, with error

O(1

2). Given sufﬁcient samples, the SWAP test is an unbiased

estimator of the inner product squared.

SW AP (Q0,|ψi,|φi) = P(M|Q0i= 0) = 1

2+1

2|hψ|φi|2

(2)

Given this operation, we can perform similar convolutional

operations by performing the outlined Formula in 3, with i

ﬁlters:

yij =SW AP (|Ψii,|Xisi:si+k,sj:sj+l)(3)

where the statevector describing |Xisi:si+k,sj:sj+lhas the

same dimensionality, and hence number of qubits, of |Ψii.

The forward operation produces a similar output to a classi-

cal convolutional operation, whereby we are computing the

squared real inner product of two state vectors instead of the

inner product of two vectors.

In classical convolutional networks, the convolutional ﬁlter

is a tensor of varying activation, all of which are independently

optimised according to some loss function. With a quantum

state prepared via any ansatz, there is no way to control

one state amplitude’s magnitude without changing another

amplitude’s magnitude. This is due to the square norm re-

quirement of quantum states. Therefore, we optimise each

quantum state ﬁlter similar to the optimization procedure of a

variational quantum algorithm such as a variational quantum

eigensolver [4], [9], [10], [14]. This is visualised in Figure 1,

where n layers represents the number of parameterised layers

describing the quantum state. Each ﬁlter maintains its own

independent set of θ’s, where each ﬁlter attempts to learn its

own feature set.

The QuCNN architecture is applicable to purely quantum

data, that might be accessed via QRAM or other sources. How-

ever, QuCNN can operate on classical data, once the classical

data is translated into a quantum state prior to model induction

or training. Although computationally expensive, this is a pre-

processing step. Within this paper, we utilize a classical-to-

quantum encoding technique. Utilizing log2(n)encoding [12],

an input data point is broken up into spatially related data clus-

ters, with a pattern deﬁned by parameters such as stride, ﬁlter

size etc., and translated into a group of equivalent amplitude

encoded state vectors [|Xi1,|Xi2,|Xi3, ..., |Xin]. Each ﬁlter

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QuCNN:AQuantumConvolutionalNeuralNetworkwithEntanglementBasedBackpropagationSamuelStein1,YingMao2,JamesAng1,andAngLi11PacicNorthwestNationalLaboratory2FordhamUniversityAbstractQuantumMachineLearningcontinuestobeahighlyactiveareaofinterestwithinQuantumComputing.Manyoftheseapproacheshaveadaptedclass...

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QuCNN A Quantum Convolutional Neural Network with Entanglement Based Backpropagation Samuel Stein1 Ying Mao2 James Ang1 and Ang Li1

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