A kernel-based quantum random forest for improved classication Maiyuren Srikumar1Charles D. Hill1 2yand Lloyd C.L. Hollenberg1z 1School of Physics University of Melbourne VIC Parkville 3010 Australia.

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A kernel-based quantum random forest for improved classiﬁcation

Maiyuren Srikumar,1, ∗Charles D. Hill,1, 2, †and Lloyd C.L. Hollenberg1, ‡

1School of Physics, University of Melbourne, VIC, Parkville, 3010, Australia.

2School of Mathematics and Statistics, University of Melbourne, VIC, Parkville, 3010, Australia.

(Dated: February 21, 2023)

The emergence of Quantum Machine Learning (QML) to enhance traditional classical learning

methods has seen various limitations to its realisation. There is therefore an imperative to develop

quantum models with unique model hypotheses to attain expressional and computational advantage.

In this work we extend the linear quantum support vector machine (QSVM) with kernel function

computed through quantum kernel estimation (QKE), to form a decision tree classiﬁer constructed

from a decision directed acyclic graph of QSVM nodes - the ensemble of which we term the quantum

random forest (QRF). To limit overﬁtting, we further extend the model to employ a low-rank

Nystr¨om approximation to the kernel matrix. We provide generalisation error bounds on the model

and theoretical guarantees to limit errors due to ﬁnite sampling on the Nystr¨om-QKE strategy. In

doing so, we show that we can achieve lower sampling complexity when compared to QKE. We

numerically illustrate the eﬀect of varying model hyperparameters and ﬁnally demonstrate that the

QRF is able obtain superior performance over QSVMs, while also requiring fewer kernel estimations.

I. INTRODUCTION

The ﬁeld of quantum machine learning (QML) is cur-

rently in its infancy and there exist, not only questions of

advantages over its classical counter-part, but also ques-

tions of its realisability in real-world applications. Many

early QML approaches [1,2] utilised the HHL algorithm

[3] to obtain speed-ups on the linear algebraic compu-

tations behind many learning methods. However, such

methods presume eﬃcient preparation of quantum states

or quantum access to data, limiting practicality [4].

Currently, the two leading contenders for near-term

supervised QML methods are quantum neural networks

(QNNs) [5,6] and quantum kernel methods [7,8]. QNNs

constructed from parameterised quantum circuits embed

data into a quantum feature space and train parameters

to minimise a loss function of some observable of the

circuit. However, such variational algorithms face prob-

lems of barren plateaus in optimisation [9–11] which hin-

der the feasibility of training the model. In contrast,

quantum kernel methods suggest a non-parametric ap-

proach whereby only the inner product of quantum em-

bedded data points are estimated on a quantum device

– a process known as quantum kernel estimation (QKE)

[7,8]. We refer to the model utilising QKE for con-

structing a support vector machine (SVM), as a quan-

tum-SVM (QSVM) – the most commonly explored quan-

tum kernel method. Despite rigorous proofs of quan-

tum advantage using a QSVM [12], they are not without

their limitations for arbitrary real-world datasets. Van-

ishing kernel elements [13] subsequently require a large

number of quantum circuit samples (suggesting a deep

connection with barren plateaus [14]) and indications

∗srikumarm@unimelb.edu.au

†cdhill@unimelb.edu.au

‡lloydch@unimelb.edu.au

that quantum kernels fail to generalise with the addi-

tion of qubits without the careful encoding of the correct

problem-dependent inductive bias [14]. Furthermore, the

kernel approach quadratically grows in complexity with

the number of training samples, as opposed to the lin-

ear scaling of QNNs. This becomes untenable for large

datasets. These limitations indicate that further work is

required to practically deploy QML techniques.

The close similarity of QNNs and quantum kernel

methods [15] highlight the importance of developing al-

ternate QML models that present distinct model hy-

potheses. In this work, we propose a quantum random

forest (QRF) that fundamentally cannot be expressed

through a kernel machine due to its discontinuous de-

cision tree structure. Such a model is inspired by the

classical random forest (CRF) [16].

The CRF has shown to be an extremely successful

model in machine learning applications. However, there

exist many barriers permitting the entry of an analogous

QRF. The main challenge is the determination of an algo-

rithm that is not limited by either the intrinsically quan-

tum complication of the no-cloning theorem [17], or the

hard problem of eﬃciently loading data in superposition

[4]. As opposed to previous works [18,19] that employ

Grover’s algorithm with input states of speciﬁc form, the

QRF proposed in this work is ultimately an ensemble

of weak classiﬁers referred to as quantum decision trees

(QDTs) with an approximate QSVM forming the split

function – shown in Figure 1. Crucially, each QDT has

random components that distinguishes it from the en-

semble. The aim is to form a collection of uncorrelated

weak QDT classiﬁers so that each classiﬁer provides dis-

tinct additional information. The possible beneﬁts of an

ensemble of quantum classiﬁers have been expressed [20],

with recent suggestions of boosting techniques for ensem-

bles of QSVMs [21]. However, the distinctive tree struc-

ture has not yet been explored. Not only does this allow

for discontinuous model hypotheses, we will see that the

structure naturally accommodates multi-class supervised

arXiv:2210.02355v2 [quant-ph] 19 Feb 2023

problems without further computation. In comparison,

QSVMs would require one-against-all (OAA) and one-

against-one (OAO) [22] strategies which would respec-

tively require constructing |C| and |C|(|C|−1)/2 separate

QSVMs, where Cis the set of classes in the problem.

The expressive advantage of the QRF is predicated on

the distinguishability of data when mapped to a higher

dimensional feature space. This proposition is supported

by Cover’s Theorem [23], which states that data is more

likely to be separable when projected to a sparse higher

dimensional space. Cover’s result is also a principal mo-

tivator for developing both quantum and classical kernel

methods. However, it is often the case that generalisa-

tion suﬀers at the expense of a more expressive model – a

phenomena commonly known through the bias-variance

trade-oﬀ in statistical learning. To counter the expressiv-

ity of the (already quite expressive) quantum model, we

make a randomised low-rank Nystr¨om approximation [24]

of the kernel matrix supplied to the SVM, limiting the

ability of the model to overﬁt. We refer to this strategy

as Nystr¨om-QKE (NQKE). We show that the inclusion

of this approximation further allows for a reduced circuit

sampling complexity.

The paper is structured as follows: in Section II we ad-

dress the exact form of the QRF model, before providing

theoretical results in III. This is followed by numerical re-

sults and discussion in Section IV. Background to many

of the ML techniques are presented in Appendix A.

II. THE QUANTUM RANDOM FOREST

MODEL

The Quantum Random Forest (QRF) is composed

of a set of Tdistinct quantum decision trees (QDTs),

{Qt}T

t=1, which form the weak independent classiﬁers of

the QRF ensemble. This is illustrated in Figure 1. As

with all supervised ML methods, the QDTs are given the

set of Ntraining instances, xtrain

i∈RD, and their associ-

ated ncclass labels, ytrain

i∈ C ={0,1, ..., nc−1}, to learn

from the annotated data set, S={(xtrain

i, ytrain

i)}N

i=1

sampled from some underlying distribution D. Each tree

is trained using Np≤N– which we will refer to as

the partition size – instances sampled from the original

dataset S. This acts as both a regularisation strategy, as

well as a way to reduce the time complexity of the model.

Such a technique is known as bagging [25] or bootstrap

aggregation and it has the ability to greatly minimise the

probability of overﬁtting.

Once each classiﬁer is trained, the model is tested with

a separate set, Stest of size N0, from which we compare

the predicted class distribution with the real. To obtain

the overall prediction of the QRF model, predictions of

each QDT is averaged across the ensemble. Speciﬁcally,

each QDT returns a probability distribution, Qt(~x;c) =

Pr(c|~x) for c∈ C, with the overall prediction as the class

with the greatest mean across the ensemble. Hence, given

an input, ~x, we have prediction of the form,

ey(~x) = arg max

c∈C (1

t=1

Qt(~x;c)).(1)

The accuracy of the model can be obtained with

the comparison of predictions with the real labels,

1/N0P(~x,y)∈Stest δy,ey(~x)where δi,j is the Kronecker delta.

The distinction between training and testing data is gen-

erally critical to ensure that the model is not overﬁtting

to the trained dataset and hence generalises to the pre-

diction of previously unknown data points. We now elab-

orate on the structure of the QDT model.

A. A Quantum Decision Tree

The Quantum Decision Tree (QDT) proposed in this

work is an independent classiﬁcation algorithm that has a

directed graph structure identical to that of a binary de-

cision tree – illustrated in Figure 1and elaborated further

in Appendix A 1. Each vertex, also referred to as a node,

can either be a split or leaf node, distinguished by colour

in Figure 1. A leaf node determines the classiﬁcation

output of the QDT, while a split node separates inserted

data points into two partitions that subsequently con-

tinue down the tree. The split eﬀectiveness is measured

by the reduction in entropy, referred to as the informa-

tion gain (IG). Given a labelled data set, S, that is split

into partitions SLand SRby the split function, we have,

IG(S;SL,SR) = H(S)−X

i∈{L,R}

|Si|

|S| H(Si),(2)

where His the entropy over the class distribution deﬁned

as H(S) = −Pc∈C Pr(c) log2Pr(c) where class c∈ C oc-

curs with probability Pr(c) in the set S. Clearly the IG

will increase as the splitting more distinctly separates in-

stances of diﬀerent classes. Using information gain is ad-

vantageous especially when more than two classes are in-

volved and an accuracy-based eﬀectiveness measure fails.

Given a partitioned training set at the root node, the

QDT carries out this splitting process to the leaves of

the tree where a prediction of class distribution is made

from the proportions of remaining data points in each

class. Mathematically, with a subset of data points S(l)

supplied at training to leaf `(l)indexed by l, we set its

prediction as the probability distribution,

`(l)(S(l);c) = 1

|S(l)|X

(~x,y)∈S(l)y=c,(3)

where [p] is the Iversion bracket that returns 1 when the

proposition pis true and 0 otherwise. When training the

model, a node is deﬁned a leaf if any of the following con-

ditions are met, (i) the training instances supplied to the

node are of a single class – in which case further splitting

is unnecessary, (ii) when the number of data points left

FIG. 1. The quantum random forest (QRF) constitutes an ensemble of classiﬁers, referred to as quantum decision trees (QDTs).

This tree structure is in turn a directed graph structure of nodes deﬁned by the split function NΦ,L :RD−→ {−1,+1}and Φ, L

labels the type of function. The split function is an SVM that admits a Nystr¨om approximated quantum kernel,

K. The kernel

The randomly selected Lpoints for Nystr¨om approximation determine the hyperplane generated in the feature space, HΦ.

for splitting is smaller than a user deﬁned value, ms, ie.

|S(l)| ≤ ms, or (iii) the node is at the maximum depth

of the tree, d, deﬁned by the user. Once trained, predic-

tion occurs by following an instance down the tree until

it reaches a leaf node. The unique path from root to leaf

is referred to as the evaluation path. At this end point,

the prediction of the QDT is the probability distribution

held within the leaf, deﬁned at training in equation (3).

The ensemble of predictions made by distinct QDTs are

subsequently combined with Eq. (1).

The crux of the QDT model lies at the performance

of split nodes, with the lth node deﬁned through a split

function, N(l)

θ:X −→ {−1,+1}, where θare (hyper-)

parameters of the split function, that are either man-

ually selected prior to, or algorithmically tuned dur-

ing, the training phase of the model – the speciﬁcs are

elaborated in the next section. Instances are divided

into partitions S−={(~x, y)∈ S|N(l)

θ(~x) = −1}and

S+={(~x, y)∈ S|N(l)

θ(~x) = 1}. At this point it is nat-

ural to refer to the information gain of a split function

as, IG(S|N(l)

θ) := IG(S;S−,S+) with the aim being to

maximise IG(S|Nθ) at each node. In essence, we desire

a tree that is able to separate points so that instances of

diﬀerent classes terminate at diﬀerent leaf nodes. The in-

tention being that a test instance that follows the unique

evaluation path to a particular node is most likely to be

similar to the instances that followed the same path dur-

ing training. Hence, eﬀective split functions amplify the

separation of classes down the tree. However, it remains

important that the split function generalises, as repeated

splitting over a deep tree can easily lead to overﬁtting.

It is for this reason decision trees often employ simple

split functions to avoid having a highly expressive model.

Namely, the commonly used CART algorithm [26] has a

threshold function for a single attribute at each node.

In the next section we specify the split function to

have the form of a support vector machine (SVM) that

employs a quantum kernel. The core motivation being to

generate a separating hyperplane in a higher dimensional

quantum feature space that can more readily distinguish

instances of diﬀerent classes. However, this is a complex

method that will inevitably lead to overﬁtting. Hence, we

take an approximate approach to reduce its eﬀectiveness

without stiﬂing possible quantum advantage.

B. Nystr¨om Quantum Kernel Estimation

The proposed split function generates a separating hy-

perplane through a kernelised SVM. The kernel matrix

(or Gram matrix) is partially computed using Quantum

Kernel Estimation (QKE) [27] and completed using the

Nystr¨om approximation method. We therefore refer to

this combined process as Nystr¨om Quantum Kernel Es-

timation (NQKE) with its parameters determining the

core performance of a splitting node. Background on

SVMs and a deeper discussion on the Nystr¨om method

are supplied in Appendix A 2.

Given a dataset S(i)={(xj, yj)}N(i)

j=1 to the ith split

node, the Nystr¨om method randomly selects a set of L

landmarks points S(i)

L⊆ S(i). Without loss of generality

we may assume S(i)

L={(xj, yj)}L

j=1. The inner product

between elements in S(i)and S(i)

Lare computed using

a quantum kernel deﬁned through the inner product of

parameterised density matrices,

kΦ(x0, x00) = Tr[ρΦ(x0)ρΦ(x00)],(4)

where we have the spectral decomposition ρΦ(x) =

Pjλj|Φj(x)ihΦj(x)|with parameterised pure states

{|Φj(x)i}jthat determine the quantum feature map,

x−→ ρ(x). In practice, ρΦare pure states re-

ducing the trace to k(x0, x00) = |hΦ(x00)|Φ(x0)i|2=

|h0|U(x00)†U(x0)|0i|2, allowing one to obtain an estimate

of the kernel by sampling the probability of measuring |0i

on the U(x00)†U(x0)|0istate.

The Nystr¨om method can be used for the matrix com-

pletion of positive semi-deﬁnite matrices. Using the sub-

set of columns measured through the inner product of

elements between Sand SLfor an arbitrary node (drop-

ping the the node label for brevity), which we deﬁne as

the N×Lmatrix G:= [W, B]>where Gij =k(xi, xj),

i≤N,j≤Land W∈RL×L,B∈RL×(N−L), we

complete the N×Nkernel matrix by making the ap-

proximation K≈GW −1G>. Expanding, we have,

K≈b

K:= "W B

B>B>W−1B#,(5)

where, in general, W−1is the Moore-Penrose gneralised

inverse of matrix W. This becomes important in cases

where Wis singular. Intuitively, the Nystr¨om method

utilises correlations between sampled columns of the ker-

nel matrix to form a low-rank approximation of the full

matrix. This means that the approximation suﬀers when

the underlying matrix Kis close to full-rank. How-

ever, the manifold hypothesis [25] suggests that choos-

ing L << N is not entirely futile as data generally does

not explore all degrees of freedom and often lie on a sub-

manifold, where a low-rank approximation is reasonable.

The Nystr¨om approximation does not change the

positive semi-deﬁniteness of the kernel matrix, b

K

0, and the SVM optimisation problem remains con-

vex. Furthermore, the reproducing kernel Hilbert

space (RKHS), induced by the adjusted kernel, ˆ

k, is

a nonparametric class of functions of the form, HΦ=

nfΦfΦ(·) = PN

i=1 α0

iˆ

kΦ(·, xi), α0

i∈Ro, which holds as a

result of the celebrated representer theorem [28,29]. The

speciﬁc set of {α0

i}N

i=1 for a given dataset is obtained

through solving the SVM quadratic program. This con-

structs a split function that has the form,

NΦ;α(x) = sign "N

i=1

αieyiˆ

kΦ(x, xi)#,(6)

where α0

i=αieyi,αi≥0 and eyi=F(yi) with the function

F:C −→ {−1,1}mapping the original class labels, yi∈ C

– which in general can be from a set of many classes,

i.e. |C| >2, to a binary class problem. In Section IV,

we provide numerical comparisons between two possible

approaches of deﬁning the function F, namely, (i) one-

against-all (OAA), and (ii) even-split (ES) strategies –

see Appendix B 3 for more details.

The construction of a split function of this form adopts

arandomised node optimisation (RNO) [30] approach to

ensuring that the correlation between QDTs are kept to

a minimum. The randomness is injected through the

selection of landmark data points, as the optimised hy-

perplane will diﬀer depending on the subset chosen. Fur-

thermore, there exist possibilities to vary hyperparam-

eters, Φ and Lboth, down and across trees. This is

depicted in Figure 1with split functions notated with

a separate tuple (Φi, Li) for depths i= 1, ..., D −1 of

the tree. The speciﬁc kernel deﬁned by an embedding Φ

implies a unique Hilbert space of functions for which kΦ

is the reproducing kernel [31]. The two types of embed-

dings numerically explored are deﬁned in Appendix B 2.

The QRF approach of employing distinct kernels at each

depth of the tree, gives a more expressive enhancement

to the QSVM method. A simple ensemble of independent

QSVMs would require greater computational eﬀort and

would also lose the tree structure present in the QRF.

III. THEORETICAL RESULTS

QML literature has seen a push towards understanding

the performance of models through the well established

perspective of statistical learning theory. There has re-

cently been a great deal of work determining the gener-

alisation error for various quantum models [32–35] that

illuminate optimal strategies for constructing quantum

learning algorithms. This is often achieved by bounding

the generalisation error (or risk) of a model producing

some hypothesis h∈H,R(h) = Pr(x,y)∼D[h(x)6=y],

where Dindicates the underlying distribution of the

dataset. However, Dis unknown and one simply has

a set of samples S ∼ DN. We therefore deﬁne the empir-

ical error of a particular hypothesis has the error over

known points, b

R(h) = 1

NP(xi,yi)∈S 1h(xi)6=yi. This is

often referred to as the training error associated with a

model and is equivalent to (1 −accuracy). Achieving a

small b

R(h) however does not guarantee a model capable

of predicting new instances. Hence providing an upper

bound to the generalisation error R(h) is imperative in

understanding the performance of a learning model.

The QDT proposed in this work can be shown to gener-

alise well in the case where the margins at split nodes are

large. This is guaranteed by the following Lemma which

builds on the generalisation error of perceptron decision

trees [36], with proof supplied in Appendix B 5.

Lemma 1. (Generalisation error of Quantum Decision

Trees) Let HJbe the hypothesis set of all QDTs composed

of Jsplit nodes. Suppose we have a QDT, h∈HJ, with

kernel matrix K(i)and labels {y(i)

j}N(i)

j=1 given at the ith

node. Given that minstances are correctly classiﬁed, with

high probability we can bound the generalisation error,

R(h)≤e

O 1

m"Jlog 4mJ2

+ log(4m)2

i=1 

N(i)

j,k=1

y(i)

jy(i)

k(K(i)+)jk#!,

(7)

where K(i)

jk =|hΦ(x(i)

j)|Φ(x(i)

k)i|2and {y(i)

j}N(i)

j=1 are re-

spectively the Gram matrix and binary labels given to the

ith node in the tree, and K+denotes the Moore-Penrose

generalised inverse of matrix K.

The term sK=PN(i)

j,k=1 y(i)

jy(i)

k(K(i)+)jk is referred to as

the model complexity and appears in the generalisation

error bounds for kernel methods [33]. The term is both

inversely related to the kernel target alignment [37] mea-

sure that indicates how well a speciﬁc kernel function

aligns with a given dataset, as well as being an upper

bound to the reciprocal of the geometric margin of the

SVM. Therefore, Lemma 1highlights the fact that, for a

given training error, a QDT with split nodes exhibiting

larger margins (and smaller model complexities) are more

likely to generalise. Crucially, the generalisation bound is

not dependent on the dimension of the feature space but

rather the margins produced. This is in fact a common

strategy for bounding kernel methods [38], as there are

cases in which kernels represent inner products of vectors

in inﬁnite dimensional spaces [22]. In the case of quan-

tum kernels, bounds based on the Vapnik-Chervonenkis

(VC) dimension would grow exponentially with the num-

ber of qubits. The result of Lemma 1therefore suggests

that large margins are analogous to working in a lower

VC class. However, the disadvantage is that the bound

is a function of the dataset and there are no a priori

guarantees that the problem exhibits large margins.

The Nystr¨om approximated quantum kernel matrix

used in the optimisation of the SVM is also a low rank

approximation, with estimated elements that are subject

to ﬁnite sampling error. This prompts an analysis on the

error introduced – stated in the following Lemma.

Lemma 2. Given a set of data points S={xi}N

i=1 and

a quantum kernel function kwith an associated Gram

matrix Kij =k(xi, xj)for i, j = 1, ..., N, we choose a

random subset SL⊆ S of size L≤Nwhich we deﬁne as

SL={xi}L

i=1 without loss of generality. Estimating each

matrix element with MBernoulli trials such that e

Kij =

(1/M)PM

p=1 e

K(p)

ij with e

K(p)

ij ∼Bernoulli(k(xi, xj)) for

i≤N,j≤L, the error in the Nystr¨om completed N×N

matrix, e

K, can be bounded with high probability,

||K−e

K||2≤e

O NL

M+N

√L!,(8)

where e

Ohides the log terms.

The proof is given in Appendix B 5. This Lemma in-

dicates that there are two competing interests when it

comes to reducing the error with respect to the expected

kernel. Increasing the number of landmark points Lre-

duces the number of elements estimated and hence the

error introduced by ﬁnite sampling noise. On the other

hand, the Nystr¨om approximation becomes less accurate

as expressed through the second term in Eq. (8). How-

ever, it is important to realise that the approximation

error of the Nystr¨om method was by design, in an at-

tempt to weaken the eﬀectiveness of the split function.

To understand the eﬀect of ﬁnite sampling error on the

SVM model produced, we employ Lemma 2to show that

we can bound error in the model output – proof in B 5.

Lemma 3. Let f(·) = Piαik(·, xi)be the ideal Nystr¨om

approximated model and e

f(·) = Piα0

k(·, xi)be the equiv-

alent perturbed solution as a result of additive ﬁnite sam-

pling error on kernel estimations. With high probability,

we can bound the error as,

|f(x)−e

f(x)|≤OM N4/3√L

√M!(9)

where OMexpresses the term hardest to suppress by M.

Lemma 3indicates that M∼ O(N3L) will suﬃce to

suppress errors from sampling. This is in comparison

to M∼ O(N4) for without the Nystr¨om approximation

[12]. However, note that this is to approximate the linear

function ffor which the SVM has some robustness with

only sign[f(·)] (6) required. A caveat to also consider is

that it is not necessarily true that a smaller ||K−e

K||2

or |f(x)−e

f(x)|will give greater model performance [39].

To address the complexity of the QRF model, for an er-

ror of =O(1/√M) on kernel estimations, we can show

that training requires O(T L(d−1)N−2) samples and

single instance prediction complexity of O(T L(d−1)−2)

circuit samples – discussed further in Appendix B 4. The

main result here is that we are no longer required to

estimate O(N2) elements, as is the case for QSVMs.

Though this is a profound reduction for larger datasets

with L << N, it should be noted that datasets may re-

quire L=O(N). Nonetheless, the number of estimations

will never be greater than N2, on the basis that kernel

estimations are stored in memory across trees.

Finally, to show that the QRF contains hypotheses un-

learnable by both classical learners and linear quantum

models, we extend the concept class generated with the

discrete logarithm problem (DLP) in [12], from a single

dimensional clustering problem to one in two dimensions.

We construct concepts that separate the 2D log-space

(torus) into four regions that can not be diﬀerentiated

by a single hyperplane. Hence, the class is unlearnable

by linear QNNs and quantum kernel machines – even

with an appropriate DLP quantum feature map. Classi-

cal learners are also unable to learn such concepts due to

the assumed hardness of the DLP problem. Since QDTs

essentially create multiple hyerplanes in feature space,

there exists f∈HQDT to emulate a concept in this class.

Further details are presented in Appendix B 7.

IV. NUMERICAL RESULTS & DISCUSSION

The broad structure of the QRF allows for models to

be designed speciﬁcally for certain problems. This in-

cludes the selection of a set of embeddings at each level

of the tree that distinguish data points based on diﬀerent

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Akernel-basedquantumrandomforestforimprovedclassicationMaiyurenSrikumar,1,CharlesD.Hill,1,2,yandLloydC.L.Hollenberg1,z1SchoolofPhysics,UniversityofMelbourne,VIC,Parkville,3010,Australia.2SchoolofMathematicsandStatistics,UniversityofMelbourne,VIC,Parkville,3010,Australia.(Dated:February21,2023)Thee...

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A kernel-based quantum random forest for improved classication Maiyuren Srikumar1Charles D. Hill1 2yand Lloyd C.L. Hollenberg1z 1School of Physics University of Melbourne VIC Parkville 3010 Australia.

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