Distributed and Secure Kernel-Based Quantum Machine Learning
Authors: Arjhun Swaminathan
Mete Akgün
Venue: Accepted by TMLR ()

Abstract: Quantum computing promises to revolutionize machine learning, offering significant efficiency gains for tasks such as clustering and distance estimation. Additionally, it provides enhanced security through fundamental principles like the measurement postulate and the no-cloning theorem, enabling secure protocols such as quantum teleportation and quantum key distribution. While advancements in secure quantum machine learning are notable, the development of secure and distributed quantum analogs of kernel-based machine learning techniques remains underexplored.

In this work, we present a novel approach for securely computing three commonly used kernels: the polynomial, radial basis function (RBF), and Laplacian kernels, when data is distributed, using quantum feature maps. Our methodology formalizes a robust framework that leverages quantum teleportation to enable secure and distributed kernel learning. The proposed architecture is validated using IBM’s Qiskit Aer Simulator on various public datasets.

Published in Transactions on Machine Learning Research (04/2025)
Distributed and Secure Kernel-Based Quantum Machine Learning
Arjhun Swaminathan
argjon...ooiinnshannnnnnkuuunnne de
Mete Akgün1.2
mette.akguen@uui--tebbeeg de
'Medical Dala Privucy and Privacy-preserene Machine Learning (MDPPML), University of Tübingen Institute for Bioinformatics and Medical Informatics (TBMT), University of Tübingen.
"Corresponding Author
Reviewed on OpenReview: https: //tps://owe opwneevee......e...mmee net/ form?? id- 3jdIOaEW3k
Abstract
Quantum computing promises revolutionize machine learning, offering significant effi- ciency gains for tasks such clustering and distance estimation. Additionally, provides enhanced security through fundamental principles like the measurement postulate and the no-cloning theorem, enabling secure protocols such quantum leleportation and quantum key distribution. While advancements in secure quantum machine learning are notable, the development of secure and distributed quantum analogs of kernel-based machine lcarning techniques remains underexplored.
In this work, we present novel approach for securely computing three commonly used kernels: the polynomial, radial basis function (RBF), and Laplacian kernels, when data distributed, using quantum feature maps. Our methodology formalizes a robust frame- work that leverages quantum teleportation enable secure and distributed kernel learning. The proposed architecture is validated using IBM's Qiskit Aer Simulator on various public datascts.
1 Introduction
Quantum computing is set to revolutionize machine learning (ML) by leveraging its capability to encode higl-dimensional data into quantum bits, or qubits. These qubits exist in a superposition of states, enabling quantum data represent data exponentially more efficiently than classical computing data represented using N classical bits equivalently be represented by log2N qubits. Further, in addition to superposition, quantum entanglement enables qubits to exhibit strong correlations that persist regardless of spatial sepa- ration. These correlations are essential for achieving exponential speedups in specific quantum algorithms (Jozsa & Linden 2003). Although practical quantum computers are still in their infancy, various quantum machine learning (QML) techniques have been proposed (Looyd et al. 2013 2014 Biamonte et al. 2017
Notably, quantum computing has exhibited substantial efficiency gains some computational tasks com- pared to classical computing (Schuld & Petruccione) 20018] Zhao et al. 2021): the estimation of distances and inner products between post-processed N N-dimensional vectors is achieved in O(log(N)) compared to (NN). Similarly, clustering N-dimensional vectors into M clusters is expedited to O(log using quantum data, compared to O(poly(MMN).
Quantum computers are not only efficient at handling high-dimensional data, but are inherently secure. This stems from two fundamental principles of quantum mechanics: the measurement postulate and the no- cloning theorem (Wootters & Zurek 1982). Quantum data collapse upon measurement and cannot be copied without restng the original data, offering absolutely secure communication. Secure quantum computing

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is well studied and includes protocols such as quantum teleportation Bennett et al.
1993
Bouwmeester
et al. 1997), quantum key distribution (Bennett & Brassard 2014 Bennett et al.
1992]. and quantum
secure direct communication Long & Liu 2002 Deng et al.) 2003 Wang et al. 2005
Sheng et al. 2022
Additionally, quantum computers utilizing various technologies, such as trapped ions, photons, supercon- ducting circuits, and so forth, are actively being developed, enhancing the practical implementation of these secure communication protocols.
In parallel, kernel-based ML methods have emerged as effective tools for classification and regression tasks. These methods compute similarities between data points in high-dimensional spaces, making them partic- ularly suited for problems where data is not trivially separable. In contrast to more advanced counter- parts, such as deep learning, many kernel-based ML. techniques often offer greater interpretability (Morocho- Cayamcela et al.) 2019 Ponte & Melko 2017) and better accuracy when high-dimensional data is limited, which is often the case in many weal-world applications (Ding ct al. 2021
Montesinos-Lóópe et al. 2021
Although much of the focus in QML has been on developing quantum or hybrid (quantun-classicc)
deep
learning and neural networks (Garg & Ramakrishnan, 2020 Kwak et al.
2023 Dang al.) 2018 Basheer
et al. 2020), quantum analogs of kernel-based ML techniques
important
alternatives, with landmark
studies focusing on centralized data (Havlíčck et al. 2019 Schuld & Killoran
2019).
However, real-world scenarios often involve distributed data (Yu et al. 2006 Hannemann et al. 2023 in
which multiple parties wish to collaboratively train a model while ensuring the privacy of their datasets. Designing QML techniques that work securely in such distributed settings remains a critical challenge, and current research on distributed kernel-based QMT. methods is still sparse. In particular, /Sheng & Zhou (2017) introduced a distributed framework that computes the distance between two data points for classification tasks. Later, [Schuld & Killoran (2019) demonstrated that this approach is specific instance of a more general principle: encoding data into an dufnite-dimenssion to mapping data into a nigher-dimensional feature space for kernel computation. This insight establishes a fundamental link between quantum encoding and kernel feature maps, revealing that [Sheng & Zhou (2017) 's distance computation is the quantum analog of the linear kernel within a broader theoretical context.
Hilbert space as quantum data is equivalent
Building on the works of [Sheng & Zhou (017)) and Schuld & Killoran 2019)) our rescarch investigates this intriguing relationship between quantum feature maps and kernel functions in the context of distributed kernel computations. Specifically, we identify appropriate quantum encodings for commonly used kernels and explore their applications in distributed learning settings. Our approach encodes classical data into quantum states using Random Fourier Features (RFF) (Rahimi & Recht 2007) and uses a robust architecture for secure and distributed kernel-based learning. This archilecture leverages quantum teleportation to ensure data sccurity and employs a semi-honest server to compute kernels and train ML. models.
We used publicly available datasets and Qiskit's Aer Simulator (Wille et al. 2019 to validate our architecture. Our evaluation demonstrates that the proposed approach ensures data security and achieves performance comparable to centralized classical and quantum methods. However, it is important to acknowledge that achieving identical or superior accuracy to classical methods is not anticipated in our study. This is not only due to widely recognized challenges in quantum computing, such as quantum noise (Bharti et al., 2022), approximations in quantum state preparation, and hardware constraints, but also due to the inherent probabilistic nature of quantum states. We make the following three contributions:
Introduction of Quantum Feature Maps: We introd quantum feature maps for the polyno- mial, Radial Basis Function (RBF), and Laplacian kernels and theoretically prove the correctness of these feature maps.
2. Architecture for Secure Kernel Computation: We formalize secure architecture to compute linear, polynomial, RBF, and Laplacian kernels using quantum encoding in a federated manner on distributed datasets.
3. Implementation and Validation: We theoretically validate our architecture, and for empirical validation, we implement it for lincar kernels on publicly available datasets using Qiskit's Aer Sim- ulator. Due to the limitations of the simulator and our lack of access to a real quantum computer, we were unable to test. other kernels at this stage.
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2 Background
2.1 Kernel-based Machine Learning
In machine learning, one typically works with a dataset X consisting of data points f 22,...,......................... .NJ E X, where the goal is to identify patterns to evaluate previously unseen data. Kernel-based techniques employ a similarity measure, called the kernel function, between two inputs to construct. models that effectively capture the underlying properties of the data distribution. This kernel function is often an inner product in feature space, typically of higher dimensionality,, where nonlinear relationships between data points become more apparent. Various kernel functions used in practice, such as linear, RBF, polynomial, and Laplacian kernels. These functions are designed to accommodate diverse data characteristics, making them suitable for various applications. Beyond their many applications, these methods have a rich theoretical foundation, which we briefly explore below.
Definition Kernel function (Aizermant 1964)
A kernel function K is map K X x map from the input space X' to Hilbert space ())).
C that satisfies K(x v) = (0(x),(())) where Φ :.:² X - H is
One refers to ø feature map. Since for any unitary operator U H - H. (o(x), o(v) (Uő(2), Ud(v))
kernel can be related to many different feature maps. However, kernel theory defines a unique Hilbert space associated with a kernel, called the Reproducing Kernel Hilbert Space (RKHS) as follows.
Definition 2. Reproducing Kernel Hilbert Space (RKHS) [Aronszajn] 1950 Let X be an input space and (R, (),))) the Hilbert space of functions f C. Then R an RKHS if there erists a function K X X such that for all X and R, the following holds:
f(x) (f,
Alternatively, considering an associated feature map, X H. then R is the space of functions f:X - C such that for all X' and e H. f(x) = (v, ((x))
Typically, large family of machine learning problems aims to compute a prediction function f X' C that takes training or test data and predicts the corresponding label. This is often formulated as the solution to the following optimization problem:
Σ L(y, +X
min FER
(1)
where L loss function, Ty are training data points, y5 the corresponding labels, À regularization parameter controlling the trade-off between the loss and the complexity of the function f. and ?(f) a general regularization term that penalizes the complexity of This prediction function generally lives in an RKHS. The representer theorem (Schölkopf & Smola 2002) then states that the solution to this optimization problem can be formulated as follows.
f(x)= = Σ a ((,
where K the corresponding kernel in the RKHS. Hence, optimization in an infiniteddimensional space is reduced to -iite-dimensional problem of solving for a by computing the kernel values at the training data points.
In summary, kernel functions are fundamental in machine learning, as they enable the transformation of data into digher-dimensional spaces where nontrivial relationships between data can be studied. By leveraging the thcoretical framework of RKHS and the representer theorem, kernels facilitate efficient computations for large class of machine learning models, such as Support Vector Machines (SVM), Gaussian Processes, and Principal Component Analysis (PCA) (Shawe-Tay 2004
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2.2 Random Fourier Features
Kernel moth often face signilicant compulational challenges, particularly with large datasets. To address this issue, Rahimi & Recht (2007) introduced Random Fourier Features (RFF) as an effective approach to estimate kernel functions using finite-dimensional feature maps. RFF enables efficient computation kernel approximations by leveraging the Fourier transform properties of shift-invariant kernels. One defines RFF as below:
Definition Rondom Fourier Features (RFF) (Rahimi & Rechl, 2007
Given stift-inwariant kernel k(æ y) that is the fourier transform of probability distribution x, the corresponding lower dimensional feature map 2: RD - IR defined by
=xx) (V2cos(wir + bi)...., /220sswaaz + ba)).
with wi x((w1,..... Wa)) and b; independent samples from the uniform distribution U|0 2m), satisfies the following inequality for all
-
De²
P (12=(z)22V) -
k(x,0)1 2 26) S 2xpp
The feature map is called an RFF.
2.3 Quantum Encoding
Quantum encoding techniques are crucial for translating classical data into quantum states. There are various methods to do so, such basis encoding, angle encoding, amplitude encoding, and Hamiltonian evolution ansatz encoding, each with its own distinct advantages and disadvantages. For example, one such encoding is defined below.
Definition Amplitude Encoding (Schuld el al, 2015 Given classical data x (11, 22 ...., , IN)", where N 2n, the amplitude encoding of the data is defined the quantum state:
(+(x) Σ
())
(2)
where lj) represents the compulational basis states of an n-qubil system.
The computational basis here refers the set of basis states that span the state space of an n-qubit quantum system. These states are represented with (j) where j E 10.1....,2 2" 1}. and are expressed tensor products of individual qubit states (0) and [1). For example, the computational basis in 2-qubit system consists of states:
(100) (01), (10)
In the context of work, will adopt RFF determine the quantum encodings necossary for the computation of different kernels.
3
Related Work
3.1 Quantum Feature Maps
Building on the concept encoding classical data into quantum states, (Sculld & Killoran 2019 formalized the idea a quantum feature map by noting that any quantum encoding x + (vx))) behaves like a feature map and maps complex Hilbert space H. hence naturally inducing a kernel. This opens up the possibility of utilizing the rich theory of kernel methods alongside quantum computing. Of partieular interest when two input vectors and y embedded in N-dimensional Hilbert space via amplitude encoding, The resulting inner product of the encodings corresponds to the linear kernel.
(v(x)) of k( y)-
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Beyond discussing the linear kernel, the work introduced additional quantum feature maps that correspond to other well-known kernels. For instance, the Copies of Quantum States map given by
3d
x (1), (f1,..,, (N) (x()))
Σ
I
(i)
|
is associated with the homogeneous polynomial kernel, expressed as k(a, = (x y)²
under the inner product (v<x)))(()) Similarly, the work proposed the following feature map-
x (Fi....., 2N) + (xxx) (cos(x2)-22 12j 2) sin(12)-1) [2j 1)).
which is associated with the cosine kernel, expressed as
k(x,y) II cos(x;: y:),
under the inner product (b(x))((y))
As discussed earlier, large family of ML algorithms optimize the functional in ① to obtain a prediction function. There are two primary approaches to this optimization in the context of QML: the implicit approach and the explicit approach. The implicit approach uses the representer theorem and computes kernels while offloading the remaining tasks to classical computing, as demonstrated by Rebennrost al., (2014); [chuld & Killoran (2019); (chuld (2021). The explicit approach uses variational circuits to solve the optimization RKHS, discussed by Havliček et al. (2019); Schuld & Killoran (2019) Cerezo et al. (2021). Our work follows the implicit approach in a distributed setting where quantum states
problem in the infinite-dimensionn
are used for kernel computation, and the modeling is offloaded to classical computing.
3.2 Distributed Secure Quantum Machine Learning (DSQML)
To the best of our knowledge, only one study, Sheng & Zhou (017)), has implemented kernel-based tech- niques using quantum computing within distributed framework. Their work introduced the Distributed Secure Quantun Machine Learning (DSQML) algorithm, which facilitates distance computation using a polarization-based quantum system. The sctup is designed to ensure security against potential cavesdrop- ping or interference during the computation, as any disturbance by an adversary can be delected.
The DSQML framework offers two operational modes: Client-Server and Blient-Server-Databbe Client-Server model, a client with basic quantum technology aims to classify a single data point into one of two clusters, A and B by computing its distance from the reference vectors va and уp- The client uses amplitude encoding to quantum encode the data point and employs quantum teleportation to delegate the inner product computation to the server. This can be interpreted as a computation of the linear kernel between the data point and the reference vectors as though in a distributed setting.
In the
In the more complex Clint-Server-Databbss model, the client lacks significant quantum resources and can only perform single-qubit preparation and measurement. In this setup, multiple databases encode their respective data points using amplitude encoding and transfer them to the server via quantum teleportation. The server utilizes Fredkin gate with the client-prepared ancilla qubit as control, performs Hadamard gate, and returns the ancilla qubit to the client. The client then measures the qubit to extract the inner product, which is computed over multiple repetitions.
Although DSQML essentially computes the lincar kernel in a distributed sctting using amplitude encoding, it does not explicitly acknowledge or leverage the intrinsic relationship between quantum encoding and kernel methods proposed later by [Schuld & Killoran 2019). Consequently, it overlooked the broader kernel
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framework that can be leveraged for various supervised and unsupervised machine learning tasks across different types of data, including images, text, and numeric data.
In contrast, our research substantially broadens these initial concepts by facilitating the computation of srooding-inducces kernels and other standard kernels such as polynomial, RBF, and Laplacian kernels for data of any dimensionality. This generalization to other widely used kernels much stronger and is important for future study, Further, our method follows simple Client-Server model and can easily be adapted by relabeling to Seantt-erver-Databbae model.
4 Quantum Feature Maps
Although (achuld Schuld & Killoran (2019) pointed out the implicit connection between quantum encoding techniques and feature maps, they devised feature maps only for the linear kernel, the homogeneous polynomial kernel, and the cosine kernel. We extend this by defining the quantum feature maps associated with three widely used kernels in ML: the polynomial, RBF, and Laplacian kernels.
4.1 Polynomial Kernel
Given classical data x = (I1, x2 ,,,) we define the following quantum feature map:
Vavd
11 b1 by VehNaa 1),
(3)
lkg!..... KN+1!
where the multi-index k (ki,..., kN41) runs over all combinations such that E kn d. and c c= 1- if d N, c= -1---alllc if E 2N.
Theorem 1. The quantum feature map above a well-defined quantum state.
Proof. To be well defined, we require the map to be normalizable. Consider the map x + (xx).. Then, using the multinomial theorem Aizerman 1964
Boser et al. 1992). we have that
2
Vavd dl Vhilk!! KN+1!
[+(x]|= Σ
(a/11) (x Pe
ki=d
= (a||=]] +c)d=1 =1
This completes the proof.
Theorem The quantum feature map defined above yields the polynomial kernel 2002).
Schöökopp 8 Smola
Kynty (ax" y + c)a
under an inner product.
Proof. This follows directly from the multinomial theorem. Let ((x) and @(y) be two quantum feature maps of classical data x and y, defined as above. Then,
Javd!
(d(z))o(v))
E
x KN ks ... B kNn1
KN+1
ki (ax y + c) Kpoly(2, 3).
This completes the proof.

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4.2 RBF Kernel
Using RFF, given classical data (21:22,-.
22............ ................................ ,2N"", define the following quantum feature map:
x x+ x())
VD Σ cos (1 [2j 2) sin sin(wf>) [2j 11)
(4)
where [2Dg determines the number of qubits used and the approximation quality, and wy independent samples from the normal distribution (N0,,
Theorem 3. The quantum feature map above well-defined quantum state.
Proof. To be well defined, we require the map to be normalizable. Consider the map + vxxx. Then, we have that
||v())] E (us sin?(wfar x) 1.
This completes the proof.
Theorem 4. The quantum feature map defined above yields the RBF kernel (Broomhead & Lowe 1988
KRBF(I,8) = exp (-11=- -11--y1i y11²
2n
under on inner product.
Proof. Let d(x) and ((y) be two quantum feature maps of classical data and y, defined above. It follows that
[(((()))(())) =
Σ
[cos(w3 (w) y) + sin(wz swsin v)
D
j=1
1
Elcos (w} (x)1
(5)
D
Using Euler's formula, this can be rewritten as
((((r)))(w))
1
Σ
(Blexp(iw]] --)) + Elexp(-iww) (x u))))
(6)
2D
j=1
Since normal distributions remain closed under linear transformations Waakerly et al. 2008
1 w (x y) Wyr( Uk) N ( 1 Σ o un)2) ~N (0. 000221l 112 u12
Hence wif (x y) is a normal distribution. Then G can be rewritten as
D Σ
B(()())))())
M (uu + M MM (o)(e-v(-ii))
2D
5=1
where (t) E[exp (tZ) is the moment generating function of random variable Z. Hence, since the
moment generating function of normal distribution Z ~ N(u,22) is given by Mz(() exp (4p 22).

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have
D
((((())
-
2D
Σ
exp
20
x - 111²) + exp (-zoel 2002 o
-
w11²)
3=1
1
exp
(22²
2a
|
w12
KRRF(E,,))
(7)
This completes the proof.
4.3 Laplacian Kernel
Using RFF, given classical data x (01, 22,.......... "2,,,.,,N",, we define the following quantum feature map:
x (T
Σ
(cos(wf a;) 12j-2] 2) sin sin(w x 12j 1)),,
(8)
VD
where (2D) determines the number of qubits used and the approximation quality, wy are independent samples from the Cauchy distribution C(0,a=1T), and 0j are independent samples from the uniform distribution U(0,2m0 2m).
Theorem 5. The quantum feature map above is well-defined quantum state.
Proof. To be well defined, we require the map to be normalizable. Consider the map x ++ 10xx). Then, we have that
1 [+)())] = D Σ cos (w] + +a)) sin? + aa)) = 1.
This completes the proof.
Theorem 6. The quantum feature map defined above yields the Laplacion kernel (Smola & Kondor 2003
||
2112
K2 v) = exp
a
under an inner product.
Proof. Let Φ(x) and Φ(y) be two quantum feature maps of classical data x and y, defined above. follows like in Theorem 4 that.
Σ (iw] (z v))1 + Elexp(-iux) (x y))1)
[((())))(())
(9)
2D
j=1
Since Cauchy distributions remain closed under linear transformations
Nolan 2012
N
0.
1
N
I wel C (0, r 8lla
w (z y) (0 yu) C
Σ
Hence, w²' (I -) 4) is Cauchy distribution. We rewrite 99
as
[(((())((())) = 2D
Σ
(e)()=))(1) + D (-1))
1=1
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where (t( E[exp (itZ)] is the characteristic function of random variable Z. Hence, since the charac-
teristic function of Cauchy distribution Z ~ C(N,7) is given by Φzl(1) = exp (ily - 7/4)) we have
D
|| -
U11 =
+ exp (- 2111
B[((())l(()))
= 2D
Σ
exp
Ct
j=1
yl11
exp
(2(o,v)
(10)
This completes the proof.
5
Computational Complexity
5.1
Classical Setting
In classical computation, kernel evaluation involves performing pairwise operations N-dimensional vectors. For example:
The polynomial kernel requires computing the inner product O(N) and raising the result to the power - (O(d) lcading to a total complexity of O(N + d).
The RBF kernel requires computing the squared norm of the difference between two vectors O(N) and applying the exponential function O(1) leading to a total complexity of O(N).
The Laplacian kernel involves the Lı-norm computation (O(N) and applying the exponential function O(1) leading to a total complexity of O(N).
This indicates that classical methods face challenges when and d are large.
5.2 Quantum Setting
The quantum approach involves two main steps: (1) state preparation and (2) computing inner products in the corresponding Hilbert space.
Preparing the quantum feature map for an N-dimensional vector scales with O(N). This step a bottleneck in the quantum pipeline. However, using quantum feature maps offers implicit security guarantees through the no-cloning theorem. Secure classical systems require additional overhead, such as homomorphic encryption or secure multi-party computation, to achieve comparable privacy, which can be computationally more expensive Fan & Vercauteren 2012).
State Preparation:
Inner Product Computation: Once states are prepared, computing the inner product scales logarith- mically with the dimension of the computational basis, D, which determines the number of qubits required. Specifically, the number of qubits used [log, D1, and the computational complexity is therefore O[[lgg2 D1) for one shot.. The dimension D depends the kernel being computed:
For polynomial kernels: D= (N+)), as seen in (3). Using Stirling's approximation (Stirling 1730 the complexity can be approximated as O( ) for large N and d. In Appendix B.1 we show that to achieve an additive approximation error of e, one must use M = 0(1/e2) number of shots, and hence the overall complexity becomes O(d N) M) = O(d N)/e).
For RBF and Laplacian kernels: D = 2D, where D is the number of random Fourier fcatures used lo approximate the kernel. The number of qubits required in this case scales as [2D)], leading to a complexity of O[[oo, (2D)1). In Appendices B.2 and B.3 we show that to achieve an additive approximation error of with high probability, one must choose D O(1/2) and use M O(1/2) shots. Consequently, the overall computational complexity for the estimation of the inner product in these cases becomes O(log, (2D) M) ((Togg(2/)))

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6 Distributed Secure Computation of Kernels
6.1 Architecture
Our architecture builds on foundational concepts in quantum computing, including quantum teleportation
(Bennett et al. 1993) and Fredkin gales (controlled-SWAP) (Fredkin Toffoli 1982), to enable secure and distributed kernel computation. While prior work in the field (Sheng & Zhou 2017 explored similar ideas within saaile-uuuuissiiigllliieeee framework, we extend and generalize these principles to an n-qubit system, k-client system, and thus capable of handling high-dimensional data in diverse settings.
Our architecture comprises multiple clients, central server, and helper entity. The clients hold sensitive data from which they want to learn privately and collaboratively. The central server is tasked with computing the kernel securely and privately. The helper prepares entangled quantum states to facilitate quantum communication. All entities in this setup are capable of performing the necessary quantum operations. The architecture is depicted in Figure]
Helper
Server
Communication Channel
Classical
Quantum
Alice
Bob
Figure 1: Visualization of our architecture consisting of helper, server, and multiple clients.
We employ an infrastructure in which clients are initially provided with their shared seeds securely, using established cryptographic primitives. The helper party ensures the fair distribution of seeds, adhering to standard wrivacy-preeevvigg protocols.
6.2 Protocol Description
Without loss generality, we describe our protocol with two participants. Our method naturally extends any number of participants. To begin the protocol, Alice and Bob declare the size of their classical data in bits, denoted by N. The helper entity then computes the number of qubits, n, needed encode the data for single participant based on the chosen encoding technique. The protocol is established within total system of (6n + 1) qubits.
The protocol's circuit diagram is detailed in Figure below. The correctness of the protocol is theoretically shown in Appendix A
6.2.1 Helper: Quantum State Preparation for Teleportation
The helper generates 2n Bell states, which are maximally entangled two-qubit states, to facilitate quantum teleportation. The helper begins by distributing the qubits between Alice, Bob, and the server as follows:
1. In the first set of Bell states, one qubit from each entangled pair represented by 10) HA is sent to Alice, and the other represented by (0) to the server.
2. In the remaining n Bell states, one qubit from each entangled pair represented by [0) HB is sent to Bob, and the other represented by [0) to the server.
This enables the quantum teleportation of Alice's and Bob's encoded data to the server for secure computa- tion.
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14t
H
*
10),
H
(0)
X
Helper
10)
X
Z
<ply)
10>
X
|0),n
X
Quesaa Gar
Chanst
Bob
X Prikke Gite
Figure 2: Quantum circuit diagram associated with our secure and distributed quantum-bant based kernel com- putation architecture.
6.2.2 Clients: Data Encoding and Measurement
Alice and Bob determine the encoding their data represented by (w) and (o) respectively, with multiple encodings for every data point based on the required model accuracy. The encoding sequence derived from the initial shared seed. Subsequently, Alice and Bob execute the following steps:
1. Apply a Controlled-X gate to the qubits they received from the helper using their original quantum data as control.
Apply a Hadamard gate to their original data.
3. Measure their data and the received qubits in the computational basis.
4. Communicate the results to the server through an encrypted classical communication channel.
The server applies the appropriate and Z gales to adjust the qubits holds.
6.2.3 Server: Inner Product Measurement
The server then executes standard swap test (Barenco et al. 1997). It prepares ancilla qubit in the zero state, applies a Hadamard gate, and uses it to conditionally swap the two sets of qubits received from Alice and Bob. After reverting the ancilla qubit with another Hadamard and measuring it, the output helps determine the required inner product (Buhrman et al. 2001). This measurement process is repeated times enhance accuracy.
6.3 Security of Protocol
In our proposed adversarial model, clients, including Alice and Bob, as well as the server, are semi-honest. They adhere to the defined protocol, yet may attempt to infer additional information from the data they handle. A helper entity, deemed a semi-honest third-party, guarantees the integrity of the quantum states used communication, similar to protocols implementing secure multi-party computation (SMPC) (Yao) 1982). The server is explicitly characterized non-colluding with the clients, consistent with established norms in distributed and federated architectures (Swaminathan al. 2024 Hannemann al. 2024
In setup, each client processes exclusively their data and cannot access information from other clients, offectively mitigating the risk of adversarial clients learning the data from honest input partics. The non-colluding server does not know the series of encodings applied to the original data and hence cannot reconstruct the original classical data from the quantum data it receives since it does not know how to measure it. It only learns the kernel matrix reflecting similaritics between participants' data. However, since the labels are obfuscated and are irrelevant to model training, the server gains no knowledge beyond the similarity distribution pertaining to obscured labels. Further, following the same argument, we note
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that in the case of the presence of colluding malicious clients and non-colluding malicious server, the non-adversarial clients' data remain private. However, malicious participants can affect the accuracy of the model.
An adversarial hairy-patty attempting to eavesdrop on the quantum data would face significant challenges due to the no-cloning theorem WWooters& Zurek, 1982), which prohibits the duplication of quantum in information without destroying the original information. In the event of interception, the adversarial entity would need to generate and transmit its own quantum data to the server. This can be effectively detected if clients and the server periodically exchange predetermined random quantum states, allowing the server to check for any discrepancies indicative of interference (Sheng & Zhou, 2017). Additionally, the utility of intercepted data is limited for the third-party as the encoding of data for transmission is randomized and only known to the clients through the pre-shared seed.
7 Experimental Evaluation
All the proof-of-concept experiments in our evaluation were carried out using classical computing resources in a High-Parformance Computing (HPC) cluster. Each node within this HPC environment was equipped with an Intel XEON CPU E5-2650 v4, complemented by 256 GB of memory and 2 TB of SSD storage capacity. We used the Qiskit Aer Simulator to run the program offline due to limited access to IBM's quantum resources. As result, were restricted to simulating only 31 qubits in our environment. Notc that we do not report any timings since the experiments are run a simulator.
Our experiments focused on computing the linear kernel. Given the limitation of simulating only 31 qubits, which confines us to 2 features, we adopted this approach and assigned n = 7 qubits to cach party in our distributed setup. While implementing other kernels, such as encoding-indun induced kernels, RBF kernels, polynomial kernels, and Laplacian kernels, would require more qubits than available, our primary goal is to validate the architecture rather than exhaustively test every encoding. Since the validity of these encodings has already been theoretically established, our focus is demonstrating that our architecture functions expected within this framework.
In addition, we tested our methodology in a two-party configuration. This can be easily expanded, as the data can be redistributed lo additional participants while maintaining consistent results. Due to the constraints on qubit simulation, a two-party configuration is employed, allocating 14 qubits for the two data providers, 14 for the helper, and 1 for the server.
7.1 Accuracy Analysis
We present a comparative analysis of our distributed quantum kernel learning setup against centralized quantum kernel computation and ralized classical kernel computation. Centralized quantum kernel computation only performs the swap test and does not constitute quantun teleportation. The datasets used for this analysis are widely used and publicly available. These include the Wine dataset (178 samples, 13 features) (Asuncion et al. 2007), the Parkinson's disease dataset (197 samples, 23 features) Sakar et al, 2019), and the Framingham Heart Study dataset (4238 samples, 15 features) [Bhardwaj] 2022). Kernel- based training was performed using SVM for all datasets, and PCA was applied to the binary datasets (Parkinson's 's and Framingham Heart Study) to reduce dimensionality. After applying PCA, SVM was used on the transformed data to obtain accuracy metries. All SVM training and evaluation were performed using stratified 5-fold cross- vols-validation ensure unbiased accuracy metrics. The accuracics of the different models on the datasets are summarized in Table 1 below.
As expected, centralized classical methods generally achieve the highest accuracy, serving as baseline. Centralized quantum methods show competitive performance, although slightly lower than their classical counterparts, due to the inherent characteristics of quantum data and quantum simulators. Our distributed quantum architecture exhibits comparable but not the same accuracy as the centralized quantum architecture because of the complexity introduced by additional gates in the quantum circuit. All experiments were conducted with 1024 shots of the quantum circuit ensure reliable accuracy. Here, shots refers to the number of times, p, a circuit is repeated.
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Table 1: Comparison of accuracies across different methods and datasets.
Accuracy
Dataset
Method
Centralised
Centralised
Distributed
(Samples x Features)
Classical
Quantum
Quantun
Wine (178 x 13)
kernel-SVM
0.9860 + 0.0172
0.8805
0.8874
0.0259
Parkinsons
kernel-SVM
0.8196 + 0.0644
0.7875
0.7983
0.0798
(197 x 23)
kernel-PCA
0.7872 + 0.0716
0.7451
0.7660
0.0744
Framingham Heart Study (4238 15)
kernel-SVA.
0.6788 + 0.0108
0.6308
0.6340
0.0143
kernel-PCA
0.6788 + 0.0095
0.6249
0.6422
0.0092
7.2 Effect of Noise
Quantum computing is susceptible to various types of errors due to environmental interactions and im- perfections in quanlum gate implementations. Our objective to evaluate the performance of distributed kernel-based QML under different noise conditions on Qiskit and compare with a classical SVM. We employed three noise models:
No Noise: This model assumes an idcal environment without noisc. It serves as a bascline for evaluating the performance of our protocol in the absence of errors.
Noise Level 1: This model introduces depolarizing error with an error rate of 0.1% for single-qubit gates and two-qubit gates. The depolarizing error is type of quantum error in which qubit with certain probability is replaced by completely mixed state, losing all of its original information.
Noise Level 2: This model simulates more challenging environment with a depolarizing error rate of 1%.
Our results reported in Figure show that increasing noise had a negative impact on model performance.
a Classical
Quantum (No Noise)
Quantum (Noise Level
(Quantum Noise Level
Accuracy
0.8
0.6
Heart Study kSVM
Parkinsons kSVM
Parkinsons 5-PPAA
Parkinsons 6-PPCA
Parkinsons 7-KPCA
Figure 3: Comparison of accuracy scores across different noise levels. The baseline includes centralized clas- sical kernel computation and our distributed quantum kernel computation with no noise. We incrementally introduce noise, using depolarizing error Level and Level 2, to evaluate and report the corresponding accuracy loss.
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7.3 Effect of Shots
Here, detail the impact of varying the number of shots used in Qiskit to repeat. quantum circuit on the performance of our proposed algorithm. We used a subset of the Digits dataset containing 100 samples [Pedregosa et al. 2011). The objective was to classify these samples into 10 labels (0-9) and to evaluate the classification accuracy using linear sernel-based SVM.
We varied the number of shots, specifically using 128, 256, 512, and 1024 shots, to observe the effect on the classification accuracy. The results, depicted in Figure indicate improved performance with increased number of shots.
0.95
Accuracy
0.9
0.88
Contralized Classical SVM
0.8
0.75
0.7
128
256
512
1,024
Nunber of Shots
Figure 4: Accuracy of linear kernel-based SVM on subset of the Digits dataset leaturing 100 samples and 10 labels, compared against the number shots run by the simulator.
8 Conclusion and Future Work
In this study, introduced novel kernel-based QML algorithm that operates within a distributed and secure framework. Using the implicit connection between quantum encoding and kernel computation, our method supports the calculation of encoding-induced and traditional kernels. To that end, we extended the existing body of knowledge by introducing three novel quantum feature maps designed to compute the polynomial, RBF, and Laplacian kernels, building upon previous studies that primarily focused on linear and homogeneous polynomial kernels. Furthermore, we have theoretically validated that the proposed quantum feature maps help compute the concerned kernels.
Using hybrid quantum-classical architecture, our approach functions in distributed environment where central server assists data providers in processing their data collaboratively, akin to centralized model. This setup primarily addresses the case of a semi-honest. scenario a common consideration in studies involving distributed architectures. We have demonstrated that our proposed framework upholds security against semi-honest partics as well as external eavesdroppers.
The application of our architecture to compute the linear kernel for publicly available datasets using Qiskit's distributed framework, yielding accuracies comparable to those achieved in
Aer Simulator validates centralized classical and quantum frameworks.
In the future, our aim is to adapt our methodology to scenarios that involve actively malicions entities. Ad- ditionally, given the rich potential of kernel theory, further theoretical and practical exploration of quantum feature maps represents a promising direction for future research the field.
Supplementary information
Our code and results are available at the following URL: https://github: httpe///githuub..cmm/mmnngg cot.n/ppm1 distributed-secure kew.re--eeeeeebbeeeee
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Acknowledgments
This research supported by the German Federal Ministry of Education and Research (BMBF) under project number 01ZZ2010 and partially funded through grant 01ZZ2316D (PrivateAMM). The authors ac- knowledge the usage of the Training Center for Machine Learning (TCML) cluster at the University of Tübingen.
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A Correctness of Architecture
In this section, we provide a theoretical proof correctness for our architecture, demonstrating that it accurately computes the kernel matrix for given input data. Without loss of generality, consider an n-qubit system, where Alice's encoded data represented by (f) and Bob's data by (v) As illustrated in Figure the Helper initializes the system by preparing 2n Bell states. The initial state consists of [0) ( 10)
(0) on sn. and (0) Hn: where the superscript denotes the qubits in each subsystem, c.g., the i-th qubit of Alice's data is represented as |
Let (v) be written as (0) + B 1) and (Ø) as 00) 7 [1) in the computational basis. Initially, the entire
system is in the stale:
()) / @ (0) 8m
HA @ 10) @ 10) @ &n @ 10)
For simplicity, we track only the i-th qubit of each n-qubit state:
1 2 @ 10)s 10) & 10) 10)
After applying Hadamard gates to the Helper qubits, the system evolves to the following state:
( 80 V2 (10) + (A))))) s 10)sA s 0) &
(10) + 11) @ 10)
Upon applying the Controlled-X gates, the system is entangled, preparing it for quantum teleportation:
1
(1) & (100) +111) & (100)s + (11) & l
2
Next, Alice and Bob perform Controlled-X gales, resulting in the state:
1
2 a ( A.HASA + (a11)) + B (101)
(7 /000) + (011) SB. + [10) SB.BB. + 101) SB.HBBB $
8 2
After applying Hadamard gates, the system evolves
1
(a |000) + 1100
(0100) A.010
(11100 A.H.SA
4
+a ( + +
(001)A,
(|101) (1101)A.HA.AA)
(7 (0100))) + 1001)
- +6 (010)
8(011)
+y |110)] SB.HBBB +7 11l 1010
1001)
61100)
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Once the classical bits are communicated to the server, the server applies the appropriate X and Z gates, resulting in the system state:
10), (a s) + +1)))s @ (z sp) + s8)
After applying Hadamard gate, the server obtains:
(10) + (1))))s @ (8))
The final step involves applying Fredkin gates, with the ancilla qubit as the control:
V2 (D00)) (10) a.S.S)
1
Upon applying Hadamard gate to the ancilla qubit, we obtain:
2 (10 @ (WO) + (A, + (1), @ (116) SA,SB - ((bb) sA,Sa)
Measuring the ancilla qubit along the computational basis yields:
1 + 11000))
Pr(0), ((vl (øl (øl (+1) (Iv) (0) (0) wv)
2
where, P(O)a is the probability that the anscilla qubit is in the (0} state. Rearranging, this then determines the inner product of (e) and (Ø).
[K v (Pr(()) -I
(11)
B Error Analysis and Measurement Complexity
B.1 Shot Complexity for Inner Product Estimation
In this section, we derive the relationship between the number of measurement shots, M, and the additive error in estimating the inner product of quantum states.
As discussed earlier in 11 the probability of obtaining the outcome (0) in the inner product measurement
is
2 [W
For each shot, define the Bernoulli random variable X, by
X; 1, if the outcome is 10),
0. if the outcome is (1).
We know E[X.] = p and Var(X,) = pp- After M independent shots, the sample mean is X = Em X,/M, with E[X] p and Var(X) (p(1 p)/M). Labeling (1 + [[<)))))) /2 = I, estimator for / is i= 2X -1 Its variance is Vfr 4 Var(X) (4p(1 - p))/M), and the standard deviation
p(1-p) M
aa = 2
The standard deviation attains its maxinum at p 1/2. In that case, p(1-p) = 1/4, and hence an 1/VMM. To achieve additive error in estimating I, we thus require 1/VM which implies
M M 0
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B.2 Determining Number of Qubits for the RBF Kernel
In this section, derive the relationship between the number of qubits required, given by [log)(2D)] and the additive error e in approximating the RBF kernel. We consider the quantum feature map corresponding to the RBF kernel defined in 4
x ZER (())=
VD Σ
cos (cos(wfz)) [2j 2) sin Sin(wfz) [2j 1))
Here, wy are drawn independently from the multivariate normal distribution NNO, The associated kernel approximation is then given by
(xx,))= = (v(x). = 1
Σ cos (wf (x 4)
D
1=1
As shown 7 the expectation of this approximation
[(, = exp (-
1 20 (-
10 u1? = KRBF(E,
Define for each j = 1....... D, the random variables X, = cos (ww (x y)) Then, the kernel approximation be written as kx,g) Because W/ ~ NN o 2I), follows that Zj N (0, | y|l²/0²) Standard results for the cosine of a
X Next, compute the variance of each Xj. Let Z, = (x y).
Gaussian random variable yield
E|cos( (2,) = exp ( || - y/1²) )
(1 1 + (xep 0²
y1 ))
E|cos² (Z,))
20
2
Thus, the variance of X, is
( (-1x- )
Var(X,) = Bicos 2 (Zz)) (E[cos (2,)))
1-cxp cxp
0²
Since the Xj are i.i.d., the variance of the average Rxx v) is Var (k(x,)) = v/D.
We then define the centered random variables Y; := Xj - E[X,]- Each Yj then has zero mean and variance
v. Since cos(-) bounded in [-1, 1], it follows that [X,] S 1 and [X, -[[x, = |Y,| s 2,. We then apply
Bernstein's s inequality (Bernstein 1924h to the sum ΣØ Y,, whose summands are bounded by M (here
M 2), and whose total variance o² vD. We have for any 0,
D Σ j=1
12 2vD + 2 M1
Pr
Y
2t
s
2exp
Setting = De, we see
De²
Pr - E( 20)= s eexp
2r
+ 3/2
For sufficiently small (i.e. when the term f is negligible relative to 2v), this simplifies to
Pr ([
KREF(E, w\ 2r) < 2exp
De²
2v
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To ensure that the additive error is bounded by with probability at least - 8, require
DE
exp
- 6.
2c
Taking natural logarithms and rearranging, we obtain
2u
D>
m
Thus, to guarantee an additive error of at most with confidence 1-5, 6, the number of random features must satisfy
D=
For large ||-- / we have v = 1/2, and for small || 3112/02, we can approximate the exponential term in with the first order of the Taylor expansion. Hence
for large || - 012 |ل= for small || - /lz-y1//0² 0/2//02
D
(12)
10
It is important to note that the constant v depends on the ratio | For a fixed pair (z, y), increasing
a results in a smaller v. Consequently, for larger a (i.e., for a wider kernel), a smaller number of random features is required to achieve a given error tolerance e. Conversely, for smaller a (i.e., for narrower kernel), the ratio || - /ol increases, leading to a larger v and thus a larger is necessary. This can be seen in Figure 5 a).
Moreover, when becomes very small (for example, when < 0.5) when || yll is very large (i.e., v, E Rd, and d is very large), the exponential term in rapidly approaches zero, and asymplotically approaches its maximum value of 1/2. In this regime, the relationship between D) and e becomes independent of a. Consequently, the error curves will converge as we observe in Figure 5(c).
B.3 Determining Number of Qubits for the Laplacian Kernel
In this section, we derive the relationship between the number of qubits required, given by [log((DD) and the additive error e approximating the Laplacian kernel. We consider the quantum feature map corresponding to the Laplacian kernel defined in { as
x IE (z)))
VD Σ (cos (wf a,) 12j-22
- 2) sin (wz x + az) [2j 1))
where wy are drawn independently from the Cauchy distribution C(O, (1) and the phase shifts ay are independent samples from the uniform distribution U(0,22 2T). The associated kernel approximation then given by
k(x,y) (v(z), (()))
Σ cos (wf (a 4))
j=l
As shown in 10 the expectation of this approximation is
E [(xx,v) vo] = exp (- || [-1x-~y]]]
y/1 =llt) )
(-
(L(1,y).
a
As before, define for each j = 1,.... D. X, = cos (w](x v)). Then, the kernel approximation can be written as (xx y) = Σリ D. Next, we compute the variance of cach Xj. Let Z, = w - -). Since my is drawn from C(O, a -1) and Cauchy distributions remain closed under linear transformations, we have
|| vll2
~ ~0,
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The characteristic function of Cauchy random variable Zj C(O, is given by Φ2, (t) exp (->(t)) Thus, setting / = we obtain
E|cos( (2)) = cxp(->) = exp (- yll1 =
Morcover, using the trigonometric identity cos) (1 cos cos(22)))/2, we have
1 + exp(-2x) + exp (-
--
E [coos [[co) (2j)]=
2
2
Thus, the variance of X, is
-11z-v91
V v:= Var(X,) E|cos (Z,)) -B/coos( (2;)))² - 1-exp
2
Now, following the same procedure in B.2 by defining Y, X3 - -[[x,] and applying Bernstein's inequality on Ej=1 Yj, we find that
for large || ylli/a,
D
- ac
for small || yll1/a.
It is important note that the constant depends on the ratio ||2 ylayyl For a fixed pair (x. y), increasing a results in a small v. Consequently, for larger (i.e., for a wider Laplacian kernel), the required D achieve given error tolerance is smaller. Conversely, for smaller (i.e., for narrower kernel), increases. Thus larger D is necessary. This can be seen in Figure 5 (b. Similar to before, when becomes very small (for example, when a < 1) or when ||. yll is very large (i.e., 2, R and d is very large), the exponential term steadily approaches zero, and v asymptotically approaches its maximum value of 1/2. In this regime, the relationship between D and c becomes independent of a as we observe in Figure d(d).
B.4 Classical Simulation of Kernel Approximation Error
In this section, describe experiments rum classical computer to validate the theoretical claims made in Sections B.2 and B.3 Our goal is to assess the quality of the kernel approximations obtained via our feature maps for both the RBF and Laplacian kernels. To quantify the approximation quality, we compute the relative Frobenius norm error between the estimated kernel matrix and the exact kernel matrix. The error metric is defined as
Koxact Kappeox
Kexact
where ||l is the Frobenius norm.
Our empirical results as seen in Figure 5 confirm that the kernel approximation error decreases as the number of random features increases. In both the RBF and Laplacian kernel experiments, the observed decay rate aligns with the theoretical O(1/VD) behavior predicted earlier. These findings, albeit using classical resources provide strong evidence that the proposed feature maps produce reliable approximations to exact kernel matrices.
C Broader Impact Statement and Ethical Concerns
Our work focuses on computing commonly used kernels in machine learning using quantum computing. As a piece of fundamental rescarch, our contribution theoretical in nature and does not, itsolf, pose any direct negative socictal impacts or ethical concerns. The methods and datasets involved do not contain sensitive personal data, nor do they target adversely affect any vulnerable populations.
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Published in Transactions on Machine Learning Research (04/2025)
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(a) RBF Kernel, d = 10. Different curves correspond to various values of σ, with the x-axis showing the number of random features D (and corresponding qubits, [2Dg).2D)]).
(b) Laplacian Kernel, == 10. Different curves corre- spond to various values of a, with the x-axis showing the number of random features D (and correspond- ing qubits, [log)(2D)]).
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(c) RBF Kernel, d = 100, 000. In this regime the | y/I term dominates, causing the variance to approach 1/2 and all curves coincide.
(d) Laplacian Kernel, = 100, 000. Again, the || - y/11 term dominates, causing the variance approach 1/2 and all curves to coincide.
Figure Kernel approximation error curves for the RBF and Laplacian kernels under two data regimes. All experiments were performed on simulated datasets with 100 samples, and each experiment was repeated times with the errors reported as averages. Top: Experiments with small feature dimension (d 8) show clear dependence on (RBF) a (Taplacian). Bottom: Experiments with large feature dimension (d= = 100, 000) exhibit nearly overlapping curves regardless of kernel parameters, as the large || - yll values force the variance to saturate at 1/2, resulting = O11ee).
While it acknowledged that kernel-based machine learning methods can, in certain applications, be associ- ated with issues such bias amplification or limited interpretability, especially when applied to non-curatod datasets in safety-critical systems, our work builds upon established techniques without introducing fun- damentally new methods that would exacerbate these concerns. We recognize that downstream applications of kernel methods may encounter ethical challenges. However, such considerations are intrinsic to the broader field rather than a consequence of our specific contribution.
Given the theoretical scope of this work, no further mitigation strategies are necessary. Nevertheless, we urge practitioners to consider the ethical implications any concrete application of these methods.
23


Reviews:
[{"review_id": "a2y21HBA8L", "reviewer": "Reviewer_vwmN", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n1. The paper is easy to follow and understand, even for readers without a strong background in quantum computing. \n2.  The paper addresses the important problem of secure and distributed quantum machine learning. \n3. The proposed approach is potentially practical and could be implemented on real quantum devices.\n\n\n\n**Weaknesses**\n1. While the paper introduces quantum feature maps for polynomial, RBF, and Laplacian kernels, the underlying principle of encoding classical data into quantum states to implicitly define a kernel is not novel.  This work extends the concept to new kernels, but the novelty lies in the specific construction of each map, not a fundamental shift in how quantum feature maps are obtained.    \nFor example, the polynomial kernel's feature map utilizes the multinomial theorem to ensure the resulting quantum state is properly normalized, which is a direct translation of the classical feature space into a quantum state space.  Similarly, the RBF and Laplacian kernel feature maps leverage Random Fourier Features (RFF) to approximate the kernel function in a quantum state.  These constructions are technically sound but primarily apply existing principles.    \n2. The building blocks for security in this paper are not new.  They use existing tools like quantum teleportation, which is a known method for transferring quantum information securely.  They also rely on a \"helper\" to set things up, which is a common approach in secure quantum systems.   While these tools are effective, the paper doesn't introduce new ways of achieving security.   \n3. The system is designed for a specific scenario where everyone mostly follows the rules.  It assumes that the different parties involved might try to peek at information they shouldn't have access to, but they won't try to completely break the system or cheat in a major way. \nThis is a limited type of security because it doesn't protect against more serious attacks where someone might intentionally try to disrupt the system or steal information.\n4. The authors have mentioned \"previous work did not identify or utilize the implicit relationship between quantum encoding and kernels\". However, there are many works starting from Maria’s work that use squeeze states or other nonlinear coherent states to define quantum kernel methods, to several other works that investigated by proposing new quantum kernels. For example, recent work on Non-Linear Optical Reproducing Kernels, operates by nonlinearly transforming data into feature space constructed with quantum states. They present a novel feature space constructed using Kerr coherent states, which generalize su(2), su(1, 1) coherent states, and squeezed states. \n5. he empirical validation is restricted to linear kernels due to limitations in simulating large-scale quantum systems.  This limitation raises concerns about the architecture's practicality for more complex kernels with higher qubit requirements. \n6. The experiments involve only two parties due to qubit constraints.  While the approach is theoretically extendable, the lack of empirical evidence for multi-party computation raises questions about scalability and performance in larger, more complex distributed settings.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a method for performing kernel-based quantum machine learning on distributed datasets in a secure manner.  The authors introduce quantum feature maps for polynomial, radial basis function (RBF), and Laplacian kernels, proving their correctness theoretically.  They propose a secure architecture utilizing quantum teleportation for distributed kernel computation.  The architecture is validated for linear kernels on public datasets using Qiskit's Aer Simulator.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Refer Weaknesses", "broader_impact_concerns": "No statement found about Broader Impact Statement in the paper.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "41YoCrVSao", "reviewer": "Reviewer_ec1v", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n1. The work is a novel contribution on an active area of research\n2. The experiments are theoretically quite interesting\n\n**Weaknesses**\n1. The work is poorly motivated and poorly cited, making it hard to place the contribution within the field. Additionally, for a non-expert in the field, it can be challenging in spots to follow the work.\n2. The work is in rough shape in spots, and I would recommend that the authors give the work an additional proof read. I've highlighted a non-exhaustive set of concerns in the requested changes section.\n3. The related work section is quite brief. I would recommend that the authors highlight some of the seminal quantum ML papers, as well as quantum linear algebra papers, and some of the seminal papers for kernel methods. I'd also recommend extending the literature review to include other novel computational paradigms such as analog computers [1]. This isn't exhaustive, but I would invite the authors to perform a more thorough review of the literature.\n4. The mathematics is hard to follow in spots. For example, the choice of computational basis states in the kernels (eg why $| 2 j -2 \\rangle$ vs $| 2 j -1 \\rangle$)\n\n[1] https://www.nature.com/articles/s42256-024-00943-2\n\n**Questions**\n\n1. Are there practical considerations associated with Thm 1? How much precision would it require as a function of feature complexity?\n2. In Thm 2, what assumptions are required to make the assumption that $w^t (x - y)$ is normally distributed?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work introduces quantum extensions to multiple standard kernel based machine learning methods. In doing so, the authors provide a way to perform machine learning on quantum computers in situations where the data is distributed using quantum resources such as quantum teleportation. These results are validated with a popular quantum simulation software, IBM Qiskit.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Could you provide citations for the relevant algorithms in the introduction. For example, references to the quantum knn algorithm [1] or basic linear algebra [2] would make the work easier to follow for members of the community not familiar with quantum machine learning.\n2. The introduction is imprecise in spots. For example, the introduction states that quantum computers yield representational benefits due to quantum superpositions, but it's been shown that multipartite entanglement is the only quantum resource necessary to achieve an exponential speedup over classical algorithms [3]. It would be nice to see the introduction cleaned up and made more precise.\n3. There are multiple runon sentences in the introduction. `counterparts such as deep-learning, many kernel-based ML techniques often offer greater interpretability (Morocho-Cayamcela et al., 2019; Ponte & Melko, 2017), and provide better accuracy and models when the high-dimensional data is limited which is often the case in many real-world applications (Ding et al., 2021; Montesinos-López et al., 2021).` for example\n4. The work could stand to be better motivated through a clearer invitation in the introduction\n5. The last paragraph of the introduction is unclear. In particular, the two sentences that follow `although` are difficult to understand how they fit with the rest of the work.\n6. Please expand the definition in eq1 to consider more than just $l_2$ regularization.\n7. Please define and clarify what a computational basis is (bottom of page 3).\n8. At the top of page 4, please clarify what `Copies of Quantum States` is? It's hard to follow.\n9. Please clarify whether the architecture in 5.1 is a contribution of this work. if it is not, please cite it\n10. Please clarify in 5.2.1 what a bell state is, and why they have to be bell states. Do you see performance drops when you have less than maximal entanglement?\n\n\n[1] https://arxiv.org/abs/2003.09187v1\n[2] https://link.springer.com/article/10.1007/s42484-021-00048-8\n[3] https://arxiv.org/abs/quant-ph/0201143", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "z0Xbjb3NTz", "reviewer": "Reviewer_ThpE", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n* Novelty: The paper addresses a gap in the literature by extending quantum kernel-based machine learning to a distributed and secure setting.  The introduction of new quantum feature maps for commonly used kernels is a great contribution.\n* Validation: The authors validate their approach through theoretical analysis and empirical evaluation using Qiskit Aer Simulator on publicly available datasets.\n\nWeaknesses: \n* Limited Evaluation: Due to limitations in the simulator, the evaluation focuses on linear kernels and a two-party configuration.  Testing other kernels and more complex configurations would provide a more comprehensive assessment of the approach.\n* Semi-honest Adversary Model: The security analysis assumes semi-honest participants.  Exploring the framework's robustness against actively malicious entities would be beneficial.\n* Lack of Clarity on Scalability: The paper does not explicitly discuss the scalability of the proposed approach for large datasets and complex machine learning tasks.  Providing insights into the computational and communication complexity as the number of parties and data dimensions increase would be valuable.\n* Limited Discussion of Quantum Advantage: The paper briefly mentions the potential efficiency gains of quantum machine learning but does not provide a detailed analysis of the quantum advantage offered by the proposed approach.  A more in-depth discussion of the conditions under which the quantum method outperforms classical counterparts would strengthen the paper's contributions.\n* Reliance on a Trusted Third Party: The architecture relies on a trusted third party (helper) to ensure the integrity of the Bell pairs used in quantum teleportation.  Exploring alternatives that reduce or eliminate the need for a trusted third party would enhance the framework's practicality and security.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces a novel method for secure and distributed quantum kernel-based machine learning.  The authors introduce new quantum feature maps for polynomial, radial basis function (RBF), and Laplacian kernels, enabling secure computation of these kernels when data is distributed across multiple parties.  The proposed framework leverages quantum teleportation to ensure data security during transmission and is validated using a simulator from IBM (Qiskit Aer Simulator) on multiple public datasets.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Critical changes: \n* Expand Kernel Evaluation: Implement and evaluate the proposed quantum feature maps for polynomial, RBF, and Laplacian kernels on the Qiskit Aer Simulator.  This would involve optimizing the quantum circuits for these kernels to fit within the simulator's qubit limitations or exploring alternative simulation platforms with higher qubit capacities. \n* Explore Multi-Party Configurations: Extend the evaluation beyond the two-party configuration to include scenarios with three or more parties.\n\nProposed change for strengthening the work:\n* Consider Malicious Entities: Extend the security analysis to consider actively malicious participants who may deviate from the protocol or attempt to compromise data privacy.  This could involve employing techniques from quantum cryptography and secure multi-party computation to enhance the framework's robustness against malicious attacks.", "broader_impact_concerns": "N/A.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The contribution seems minor -- connecting two domains, but worth writing down.  It is not clear if it will ultimately have applications, and the most interesting part was not able to be simulated.  But the authors provided some code to eventually enable it.  

See more discussions above on the Claims of Evidence and Audience.
Decision: Accept (poster)