An Adaptive Quantum Circuit of Dempster's Rule of Combination for Uncertain Pattern Classification
Authors: Fuyuan Xiao
Yu Zhou
Witold Pedrycz
Venue: NeurIPS 2025 poster ()
Keywords: Uncertain pattern classification; Quantum evidence theory; Adaptive quantum circuit; Dempster's rule of combination; Quantum amplitude encoding mechanism; Quantum measurement outcome representation
Primary area: theory

Abstract: In pattern classification, efficient uncertainty reasoning plays a critical role, particularly in real-time applications involving noisy data, ambiguous class boundaries, or overlapping categories. Leveraging the advanced computational power of quantum computing, an Adaptive Quantum Circuit for Dempster’s Rule of Combination (AQC-DRC) is proposed to address efficient classification under uncertain environments. The AQC-DRC is developed within the framework of quantum evidence theory (QET) and facilitates decision-making based on quantum basic probability and plausibility levels, which is a generalized Bayesian inference method. The AQC-DRC provides a deterministic computation of DRC, ensuring that quantum fusion outcomes in uncertain pattern classification are exactly aligned with those of the classical method, while simultaneously achieving exponential reductions in the computational complexity of evidence combination and significantly improving fusion efficiency. It is founded that the quantum basic probability amplitude function in QET, as a generalized quantum probability amplitude, can be naturally utilized to express the quantum amplitude encoding. In addition, the quantum basic probability in QET, as a generalized quantum probability, naturally forms a quantum basic probability distribution and can be used to represent quantum measurement outcomes for quantum basic probability level decision-making. Furthermore, the quantum plausibility function in QET also can be naturally used to express the quantum measurement outcomes for quantum plausibility level decision-making. These findings enrich the physical understanding of quantum amplitude encoding and quantum measurement outcomes, offering broad application prospects for representing and processing uncertain knowledge in pattern classification.

An Adaptive Quantum Circuit of Dempster's Rule of Combination for Uncertain Pattern Classification
Fuyuan Xiao"
Yu Zhou
School of Big Data and Software Engineering Chongqing University Chongqing, China 401331
School of Big Data and Software Engineering Chongqing University Chongqing, China 401331
xiaofuyuan&equ. ede.e
Witold Pedryez
Department of Electrical and Computer Engineering
University of Alberta
Edmonton, AB T6G 2R3, Canada
Systems Research Institute, Polish Academy of Sciences
00-901 Warsaw, Poland,
National Information Processing Institute
00-608 Warsaw, Poland
upedrycz&ualbertaa.
Abstract
In pattern classification, efficient uncertainty reasoning plays critical role, partic- ularly in real-time applications involving noisy data, ambiguous class boundaries, overlapping categories. Leveraging the advanced computational power of quan- tum computing, an Adaptive Quantum Circuit for Dempster's Rule of Combina- tion (AQC-DRC) is proposed address efficient classification under uncertain environments. The AQC-DRC is developed within the framework of quantum evi- dence theory (QET) and facilitates decision-makee making based on quantum basic prob- ability and plausibility levels, which is generalized Bayesian inference method. The AQC-DRC provides deterministic computation DRC, ensuring that quan- tum fusion outcomes in uncertain pattern classification are exactly aligned with those of the classical method, while simultaneously achieving exponential redue- tions in the computational complexity of evidence combination and significantly improving fusion efficiency. plitude function in QET, as generalized quantum probability amplitude, can be naturally utilized to express the quantum amplitude encoding. In addition, the quantum basic probability in QET, a generalized quantum probability, naturally forms a quantum basic probability distribution and can be used to represent quan- measurement outcomes for quantum basic probability level decision-making Furthermore, the quantum plausibility function QET also can be naturally used express the quantum measurement outcomes for quantum plausibility level decision-making. These findings enrich the physical understanding of quantum amplitude encoding and quantum measurement outcomes, offering broad appli- cation prospects for representing and processing uncertain knowledge in pattern classification.
founded that the quantum basic probability am-
"Corresponinng author: Fuyuan Xiao
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Introduction
Uncertainty and its dynamics is inherent in many real-world scenarios such as medical diagnosis, intelligent transportation, financial risk assessment, fault detection, and multi-sensor data fusion [II- G]. In the field of pattern classification, uncertainty reasoning is also essential, particularly when data noisy, class boundaries are ambiguous, there overlap between categories. Properly modeling uncertainty in these situations contributes to enhanced robustness and reliability of classification systems [Z-I0]. To address uncertainty reasoning in such complex environments, numerous theories and methods have been proposed, including Dempster-Shie -hapfer evidence theory [ [2], evidential reasoning IT3
D number theory [[S]], belief rule base [[6]]. complex evidence theory [[АА]
random permutation sets [[T8-22], and quantum evidence theory [23]] [21], among others [[5]]. These approaches provide solid theoretical foundations and practical tools for modeling and managing
uncertainty in various areas, such as evolutionary games [26 27], network intrusion detection [28].
classification [29, [0], social dilemma experiments [3]] 53), and reliability evaluation [3=]]
As an effective approach for uncertainty reasoning, DempsterShafer evidence theory (DSET) [[]] [2] offers powerful framework for representing and managing uncertainty through the basic prob- ability assignment (BPA) function [651-77] The Dempster's rule of combination (DRC) [III], a
core component of DSET, possesses several desirable properties that make particularly suitable for multisource information fusion -E.- image segmentation [22]] time series [93]], decision making [44 A5], engincering management [96], and others [Р7]] (1) Commutativity: ensures that the fusion result remains invariant regardless of the order in which evidence is combined; (2) Asso- ciativity: provides the system with flexible capabilities for structured and sequential fusion; and (3) Consistency: guarantees that, in the absence of new valid information, the outcome of the evidence combination remains unchanged. These advantages support flexible integration of multisource in- formation, enable recursive and incremental computation, and facilitate the scalability of reasoning systems [HS]. However, the computational complexity of DRC increases exponentially with the number of elements in the frame of discernment [49 550].
The rapid development of quantum computing offers new research perspective for addressing the computational complexity challenges in Dempster's 's rule of combination of DempsterShafer evi- dence theory [5][-56]. Leveraging the principles of quantum parallelism and quantum entanglement, quantum computing provides the potential to significantly accelerate the processing of large-scale uncertain information with extensive applications [57 6ll]. In particular, it opens up new possibil- ities for overcoming the exponential computation complexity issues inherent in classical evidence reasoning frameworks based on DRC, thus providing an innovative approach to efficient information fusion and decision-making [62]]
Leveraging the advanced computational power of quantum computing, an Adaptive Quantum Cir- cuit for Dempster's Rule of Combination (AQC-DRC) is proposed to address efficient classification under uncertain environments. The AQC-DRC is developed within the framework of quantum evi- dence theory (QET) and facilitates decision-making based on quantum basic probability and plausi- bility levels, which is generalized quantum Bayesian inference method. The AQC-DRC provides deterministic computation of DRC, ensuring that quantum fusion outcomes in uncertain pattern classification are exactly aligned with those of the classical method, while simultaneously achieving exponential reductions in the computational complexity of evidence combination and significantly improving fusion efficiency.
In this study, it is founded that the quantum basic probability amplitude (QBPA) function in QET [2 [4]], as a generalized quantum probability amplitude, can be naturally used express the quantum amplitude encoding. In addition, the quantum basic probability (QBP) in QET, as a generalized quantum probability, naturally forms quantum basic probability distribution (QBPD) and can be used to represent quantum measurement outcomes for basic probability level decision- making. Furthermore, the quantum plausibility (QPI) function QET also can be naturally used to express the quantum measurement outcomes for quantum plausibility level decision-makinn making. These findings open new perspectives and enrich the physical understanding of quantum amplitude encod- ing and quantum measurement outcomes, offering broad application prospects for representing and processing uncertain knowledge in areas such as pattern classification, recognition, and decision- making.

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2 Preliminaries
2.1 DSET: DempsterShafer evidence theory [0
Definition (Frame of discernment). Let Q be a frame of discernment (FOD), consisting of a set of
mutually exclusive and collectively nonempty events:
Ω (hs, h2,................... he..... bn.-
(1)
Let 2 be the power set of 22, denoted as:
2 = 10, (4a) (ha). (hn Jo (hi, 123,h....... {hi, he...., hat...,,hhh.... hih...., Ot where an empty set 03).
(2)
Definition (Hypothesis or proposition). H2 defined as a hypothesis or proposition when H, C Ω
Definition 3 (Basic probability assignment). In FOD Ω basic probability assignment (BPA) or basic belief assignment (BBA) m, also called a mass function, is defined as a mapping:
m:
2 + (0, 1).
(3)
satisfying
(0) 0 and Σ m(H,) 1.
(4)
HyCo
Definition 4 (Focal element). Let m be BPA. VH, S if (H,) > Hj is called a focal element in DSET.
Definition 5 (Plausibility function). Let H, and He be two propositions such that Hj, H, S Ω. plausibility function Pl, mapping from 2 to [0, 1), is defined by
PI(H))
m 1- - Bel(H,), H₃ = Hj, Σ
(5)
H
which the belief function Bel(H, En.gn, m(H,), measuring the strength of the evidence
in favor of proposition Hz.
Definition 6 (Dempster's rule combination). Let my and m2 be two independent BPAs FOD Ω with propositions H,, H C Ω, respectively. Dempster's 's rule of combination (DRC), represented
in the form my m2, is defined by
mi (HH)) HenHa=M,
Σ
H, * 0,
1 1- K
mi mH,
(6)
0.
H, 0,
with the conflict coefficient K between my and m2:
K
Σ HenH =
(HHH)))((H)).
(7)
2.2 QET: Quantum evidence theory
123
24
Definition 7 (Quantum frame of discernment). Let & be quantum frame of discernment (QFOD) consisting of a set of mutually exclusive and collectively nonempty events, each of which is expressed as an orthonormal basis Øy in a Hilbert space:
(oi...., Sg...,,
(8)
Let nd be the power set of db denoted as:
2 = (0, (61), {62). ..., { J. (6102)..... (0102... .ohi2....h......... osh...,,4) which can be simply represented (1) is empty set):
(9)
2 (0, Φ1, 02, ................................ On, $12..... ... 012 ...... ... d.
(10)
3

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Definition 8 (Quantum hypothesis or proposition), 1 is defined as quantum hypothesis or propo- sition when v Ø.
Definition 9 (Quantum basic probability amplitude function). A quantum basic probability ampli- nude (QBPA) function Qua in QFOD Q, also referred quantum mass function, is defined mapping:
Qui 2 +C
(11)
satisfying
Qni(0) and Qu((vy) ((v)) ((05))e
Vs 5
Σ Vy
Qu(()))
(12)
which V-I: p((;)) ) E [0. 1] represents the modulus of Qp((v))) ((byy) denotes phase term of QI (1b);: and QQ (11,) (1by) denotes the modulus squared of Qm((1)))
Note that, QBPA is a generalized quantum probability amplitude. When the QBPA are only assigned singleton states, the QBPA is referred to quantum probability amplitude.
Definition 10 (Quantum focal element). Let Qn be QBPA. Véy S Φ if Qu((,)) or (pbbi 0. 2 is called a quantum focal element in QET.
Definition 11 (Quantum basic probability function). The quantum basic probability function of Qa. also referred as quantum basic probubility distribution (QBPD), defined definedas
M: [0,1]],
(13)
and satisfies:
M(0) and M(4;)) = 1Qm Qm;)))
1y S
Σ M((), 1,
(14)
where (w,))2 = (v, (w) = (2(())) = =3 + v3. in which (m(())) the complex conjugate of Q1)(1))). e.g., Qm((y) = Ij yjū; and M(b,) (1b S ②) is called quantum basic probability ( (QBP), which represents the observed degree of belief or support to v5
Note that, QBP is a generalized quantum probability. When the quantum basic probability are only assigned to singleton states, the QBP is referred to as quantum probability.
Definition 12 (Quantum plausibility function). Let Qui be QBPA with proposition dy S 4.
quantum plausibility (QP1) function QET, mapping from 2 [0, 11, is defined by:
(P((
Σ #200570
Qa((1)))
1// C
(15)
According to Eq. (14) Eq. (L5) can also be represented as:
QPI(
Σ
(w5))=
Σ M(Hp)) My S
(16)
Therefore, when M m, Eq.
(L)) becomes:
QPI(v)) Σ m((p)),
4 S 4,
(17)
which is consistent with the classical Pl in DSET.
3 AQC-DRC: Adaptive Quantum Circuit for Dempster's Rule of Combination
The AQC-DRC consists of the following three components: 1) quantum amplitude encoding for BPA, 2) construction of the adaptive quantum circuit for DRC, and 3) measurement in the adaptive quantum circuit for decision decision-making. The flowchart is displayed in Fig. M.

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0. Raw data
Quantum amplitude encoding!
3. Measurement the adaptive quantum circuit
3.1. Quantum basic probability level declslon-making
q1
42
Q2
10y On
+
Un
On
sepal length
sepal width
:
petal length
petal width
nn
PQx k..1
QGBP: QM,
BBAS my m2
3.2. Quantum plausibility level decision-making
1QxR
2. Construction of adaptive quantum circuit
9
92
[1 [1Qxe))
Uc
4n
10)@n
PQ
12ma)
Uc
Pux k
10)8n
3Q4
Setosa
QPI(Se)
[10mx)
Uc
QPI
Versicolour
QPI(Ve)
Classification
80)8n
PQuk k...
Virginica
QPI(Vi)
D
Figure 1: The flowchart of the adaptive quantum circuit for Dempster's rule of combination.
3.1 Quantum amplitude encoding for BPA
In this section, the QBPA in QET expressed for quantum amplitude encoding mechanism. this context, BPA is encoded superposition over n-qubit quantum state.
Definition 13 (QBPA-based quantum amplitude encoding mechanism). Let Qua be a QBPA on the with quantum proposition ti S. Φ. The QBPA-based quantum amplitude encoding mechanism, also referred to the quantum superposition state of the QBPA, is defined as:
QFOD & = { .i..... Φn)
(Qu) = Σ Qu((s))(us)
Σ
vy
(18)
Qu((e))) Σ
(et,) =1,
c
vy c
where
(1by)
185z)
185m) ... ..-18))) 1..1)6))66511 18pn 6zz 852551).
(19)
1=1
and
1, of E Vy.
65s
(20)
10
of $ vy.
When ((bb)) 0, the QBPA expression for quantum amplitude encoding be represented as:
(Qu) = Σ p(())) CY
and
(+6,)) 11
(21)
Mi C
Definition 14 (Qaantum amplitude encoding of BPA). Let m& be BPA the FOD {di...., di.... .0. 2n with proposition v S 4. Considering QBPA-based quantum amplitude en- coding mechanism, a BPA my is encoded into the amplitudes of an n-qubit state as:
Q ) Us (m) (0)
Σ $
Qua, (155)195)
Σ 4.50
pP (a(1))) 18 1b3))
(22)
5

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satisfying
(aa(ts)e -/m = -V mm/ss) and Σ
Vma(( =1,
(23)
where (4))) is defined Definition
1. and Ug denotes state preparation oracle or operator.
Each BPA my is encoded as a superposition over an n-qubit quantum state with regards the QBPA-based quantum amplitude encoding mechanism. Therefore, collection of k BPAs (m1.... m....... m is encoded to set of & distinct n-qubit quantum states (Qu,))..,,IQu..)) (Q.. (Q..)..... (QM,)} with (Qu,) = (m)))) @n.
3.2 Construction of an adaptive quantum circuit for DRC
The encoded quantum states (0Q,)))...,,I 10m.. corresponding to the BPAs (m1,.... : ] will be combined by series of specific quantum operators, which is cate- gorized as one types of Uc designed through Toffoli gates.
[Q,,)}
To be specific, two encoded quantum states (QM,) and (Qms) are considered corresponding to the BPAs my and m2, the inputs of Uc are [Qu,). ). (Qm,). and (0) 8n -uubit. After applying Uc based on the Toffoli gate, the output is mapped (0) 8m qubits, which serve auxiliary qubits for storing the processed information. Subsequently, partial trace over the subsystem [Qı,, Qu, is performed which characterizes the state of the
using Tr(Qu, Qm,) Qm,) to obtain the reduced density matrix PQwes subsystem PQuin1
Triqu, Rw,) (Uc10)
(QM,) (QMa)(Qua (Qaaal (Qu,) (QM,, (0t)
PQ
Triuu, Sac,
Uc|0)
Tn @ PQN; & pam, Ut)
Σ
Σ
II
(1by)) 10 (o5))
(24)
ne Ve 15h52
Σ
II
mm mm)()))
# n 15452
where (vxx) is defined in Definition IT. is concluded that for arbitrary encoded quantum states corresponding to the BPAs, a Uc operator of the Toffoli gate implemented.
Hence, h (2 < l < k) encoded quantum states {[QM,). the BPAs (m1.... m& ). h Uc operations based on Toffoli gates are recursively implemented the (h- Uc
Qm,)) are considered corresponding
accordance with the DRC to fuse h BPAs. Following the recursive application
Toffoli gate, we perform a partial trace over the subsystem (QM,,... Qns, } using TriQus ...Qu,,
to obtain the reduced density matrix PQm, which characterizes the state the corresponding subsystem:
paaei....
TriCu, Ouick
1)...
Uc 0) Tt (0)
Th & petna @ palm U1)
Σ
Σ
II
pa (ubs)) (e (abe)
(25)
ne may 15kck
II W mogy 1ShSk
Σ
Σ
ma. (m,(w,)) [wr)(eill
indicates all the encoded quantum states corresponding to the BPAs are fused through the adaptive quantum circuit of DRC by - Uc Tof- foli gates. A total of [0) 8(2--1n qubits are required to complete the DRC, where (0) (k) qubits are used for encoding the quantum states of the BPAs, and (0) [k-(k)) qubits are allocated as auxiliary qubits for storing the processed information.
where (1.)) is defined in Definition I3.
When h
3.3 Measurement in the adaptive quantum circuit for decision-making
We define two types of measurement operators in terms of the quantum basic probability level and the plausibility level decision decision-mmkkigg for different application requirements.

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3.3.1 Quantum measurement for quantum basic probability level fecision-making
We first define measurement operator for quantum basic probability level decision-making Af- ter that, define a QBP expression for quantum measurement outcomes. On this basis, we can measure the results from the adaptive quantum circuit of DRC.
Definition 15 (Measurement operator for quantum basic probability level decision-makinga The measurement operator Monp is defined for the quantum basic probability level decision-making
MQBP = [Miosilue S Φ), Mjax) (16))
(26)
where (abe) is defined in Definition
Definition 16 (QBP expression for quantum measurement outcomes). Let poua be a density operator with regards to the trace of the output of Uc. Let MQBr {Miess = (ex S D be set of measurement operators. The quantum basic probability (QBP) expression for quantum measurement outcomes is defined as:
(Mioy PROC....)
Σ mag
II
Ph (1)) (1 (e
the IShSR
M(()) Σ Tr
II (
(27)
2 mej vu ISASA 70
which forms quantum basic probability distribution (QBPD) M.
After implementing the measurement operator MQBP, for v/. S D * 0, because
(0 (vs)) mn (19)) and m(th)) M(1))), the combined BPA can be generated:
Σ
II
V maa()s))
m9 Ve 1ShSk
m(x)) = M(1)
(28)
Σ
Σ
πI
seConey=sd = 1shgk
41.200
For vhr 0. we have
K= = Tr (M M POme...)
= Σ II
(a (00
Σ
II
ma (v:;)
(29)
noy 8 ishSk
Dee 8 Bighs
Then, for 2/e S 4, 2/e # 0. we also have
Σ m
II 1SASk
mm(())))
Tr (M/jsr) pome..a
m(x)) 1 1- (Mjjy pam.....
(30)
When implementing AQC-DRC based on the quantum measurement for quantum basic probability level decision decision-making, denoted as AQC-DRCOBP. for [/)) 1, decision can be made follow:
arg (M))) arg max (m and D = the
(31)
3.3.2 Quantum measurement for quantum plausibility level decision-making
Starting from the density matrix PQu,,... we perform partial trace over the auxiliary (n 1) qubits (gn tw -w-1 ... .11 using Tr. au+10u-1....)) This yields the reduced density matrix
7

---

PQ which describes the state associated with the w-th qubit:
La 1 (pecis...)
PQ
II
Pa (a((s)e 11]11 |1)(1+
1-
Σ
Σ
II
po (on(0s)e
10)(0)
One EWt n u. IShih
ou Ea m IShSA
(1
II
Vmn((s))
[1)
E
Σ Σ F ISASK
II
(v;n))))))
10)/0|.
- IShSk
(32)
surement operator MQPi is defined for quantum plausibility level decision decision-making The mea-
Definition 17 (Measurement operator for quantum plausibility level decision-making),
MQp = [Mailu E (0,1)), Miu = =lu)(u).
(33)
Definition 18 (QPI expression for quantum measurement outcomes). Let Pūre.... be the density
operator of the with qubit in terms of the output of UC, Let MQr = (Mis) = (u)/ E (0. 13}
and MQBP = {Mie)) = ((x)) (txllvé C 23 be a set of measurement operators. The quantum plausibility (QPI) expression for quantum measurement outcomes is defined as:
ΣE BW ne, 1ShSt
II
Pa (e((t))e
Tr (Min PoOA.....
QPI( 1- (Mjo porn.....)
(34)
After implementing the measurement operator MOPL, because |yx (1))))) (et)) = mm (
and PI(W..) = (P)( for each element &ie (1 S w S n) in FOD, the PI for dur can be generated:
Σ Σ
Vmn(
Pl(d.) = QPI().) tW neyy me IShSk
(35)
1-K
Since 1/(1 K) is a constant normalization factor, it does not affect the decision-making outcome, and we can simply Eq. ((54) for decision- makiig to improve computational efficiency witout loss of generality:
QPI() = Tr (M PaN.....
(36)
When implementing AQC-DRC based on the quantum plausibility level decision-making, denoted as AQC-DRCQri decision can be made follow:
w = arg max (QPI((w)) arg max(()))) and Pw-
(37)
Therefore, based on the definitions and rigorous mathematical derivations in Sections 5. can observe that the proposed AQC-DRCouP yields results consistent with DRC at the quantum basic probability level, while AQC-DRCQri aligns with DRC+QPI at the quantum plausibility level. These results demonstrate deterministic computation and guarantee that no information lost.
4 Application in pattern classification
In the experimental setup, three distinct datasets namely Iris (Ir), Abalone (Ab), and Knowledge (Kn) are chosen from the UCI repository https://archive.issucii .dd.ee/de....i. For the Iris dataset, all at- tributes and classes are selected. Whereas, due the limitation of available qubits, for the Abalone dataset, we randomly select five attributes (Length, Diameter, Height, Whole_weigh cera_weight) and two classes (Rings and Rings 15); for the Knowledge dataset, all classes are selected, but three attributes (STR, LPR, and PEG) are randomly select. The details of each dataset, including the selected attributes and classes, are summarized in Table II The approach from [66]] is adopted to generate the BPA for each attribute. The individual BPAs of each dataset
and Vis-

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subsequently aggregated to perform classification using different methods. To evaluate the effective- ness of the proposed methods, we compare the classical DRC [I DRC+QPI, QC-DRRonP [62], QC-DRConp+ (the QC-DRCQBP is used firstly to obtain the QBP, and then to combine the QBP into the QPI using the classical process), and the proposed AQC-DRCOBP: AQC-DRCoup+QQ AQC-DRCQBP is used firstly to obtain the QBP, and then to combine the QBP into the QPI using the classical process) and AQC-DRCopI based on the average accuracy (Acc) obtained from k-fold cross- voss-validation (K 5). To analyze the robustness of different methods, we further evaluate the average accuracy performance of each method under varying numbers of measurement shots (s).
(the
Table I: Dataset information
Table 2: Computational complexity analysis
Ir
Ab
Kn
Method
Complexity
Qubits
# Class
3
2
4
DRC
O(KN22
# Attribute
4
5
3
DRC + QPI
O(kN22m)
# Instance
150
218
403
QC-DRC
O + kN)
3n
QC-DRC + QPI
Okn + kN + +nN)
3n
AQC-DRCQuP QUP
O(kn + N)
(2k 1 )y
AQC-DRCQ + QPI
Okn + nN) O(kn²)
(2k 1 y
AQC-DRCag
(2k: 1)mz
AQC QO
AOC-DRC.u--Q9
BUC QPI
AQC DRCqx
ERC QP1)
AQC DRC.
BBC QM QCORCCR
AQC-DRCus
QU-DXGGgo
QC-DRtiqur
XC-DDCCuu~Q0P
DIr
AgC-UBCC
AQC-DXCCg
AQC-DRSQUP
0.55
03
3,10
1110
6.8-
0.90
0.99
2
0.925
-0.05
6.7-
9.1
00,00
6.6
12
14
10
10
12
Anpt0
(ngle)
Eegi0
(a) Iris.
(b) Abalone.
(c) Knowledge.
Figure 2: Comparison of the classical and quantum methods in terms of Acc and Dif metrics.
The experimental results are displayed in Fig. D and Tables AIAd It is observed that as the number of shots s increases from 2 to 25, 2T, 25, 29, 210, 221, 222 213 and finally 214 for Iris dataset, the average accuracies of QC-DRCOBP change as follows: 0.9, 0.9333, 0.9467, 0.9333, 0.9533, 0.9333, 0.9467, 0.9333, 0.9533, and 0.9467; the average accuracies of QC-DRCoRP+QPP change as follows: 0.9333, 0.9067, 0.9467, 0.95, 0.9467, 0.94, 0.9467, 0.94, 0.9333, and 0.9533; the aver- age accuracies of
change as follows: 0,9067, 0.9133, 0.9333, 0,9467, 0.94, 0,9467, 0.94, 0.9333, 0.9467, and 0.9533; the average accuracies of AQC-DRCoBF+QP) change as follows: 0.92. 0.94, 0.9533, 0.9467, 0.9467, 0.94, 0.94, 0.96. 0.9467, and 0.94; the average accuracics of AQC-DRCqR change as follows: 0.84, 0.8733, 0.9533, 0.9533, 0.92, 0.9333, 0.94, 0.94, 0.9467, and 0.9467. While the classical DRC and DRC+QPI maintain an average accuracy of 0.9467, the ac- the proposed AQC-DRCoBP, AQC-DRCqur+QU AQC-DRCoR gradually approach those of the classical models. Similarly, on the Abalone dataset, classical DRC and DRC+QPI achieve accuracies of 0.9823; on the Knowledge dataset, they reach 0.8539 and 0,8591, respectively. The corresponding quantum methods, including QC-DRCOBP. QC- DRCOBP+QPP QPI, AQC-DRCQsP+OPF and AQC-DRCOp, also demonstrate consistent trend of con- verging toward the classical models' performance.
curacies of QC-DRCouP.
and
On the other hand, we define evaluation metric, Dif, to quantify the difference between the classi- fication results of the classical DRC, and those produced by the QC-DRC QC-DRCoBP+@UP the proposed AQC-DRCOEP, AQC-DRCour+0U and AQC-DRCop. The metrie Dif, under different numbers of shots s for k-fold cross-validation is defined as:
Method
5 Mc, (Method) U Mc (DRC) Mc." (Method) n Mc* (DRC)] (38)
Dif,
DRC
|Tc

---

where Mc. (Method) denotes the misclassified instances by other methods, including QC- ORC OC- the proposed AQC-DRCOBP- AQC-DRCoap+ and AQC-DRCop: Mc" (cR(D represents the misclassified instances by the classical DRC; Te* denotes the test in- stances in the x-th fold of cross-validation; and |- represents the number of instances. StD denotes the standard deviation of Dif, with respect to k-fold cross-validation. The experimental results are displayed in Fig. @ and Tables A7A12
It is observed that as the number of shots & increases from 2 to 2 the Dif values of the QC-DRCOBP. QC-DRCoBr+QPI the proposed AQC-DRCoar AQC- DRCoBp+ QPI, and AQC-DRCqPI relative to the classical DRC gradually converge to zero in terms of three distinct datasets.
5 Computational complexity analysis
Assume that there are n elements in the frame of discernment (FOD) and k pieces of evidence, with total of N focal elements. As shown in Table 2 the time complexities of the classical DRC and DRC+QPI are O( In contrast, with sufficient auxiliary qubits, the time complexities of QC-DRCQBP. QC-DRCogr+QPI. AQC-DRCOBP- AQC-DRCoBP+QP and AQC-DRCQPL, in terms of both the circuit depth and the normalization process, are O + kN), O(kn + AN + nN), O(kn + N). O(kn nN)) and O(kn²), ) respectively. Comparative analysis reveals that QC- AQC-DRCOBP. AQC-DRCqup+OP and AQC-DRCqp all achieve exponential reductions in time complexity relative to the classical DRC and DRC+QPI. Notably, AQC-DRCoup improves upon QC-DRCous by reducing the time complexity by (k 1)N; AQC- DRCo improves upon QC-DRCaRp+QUP by reducing the time complexity by kN. Further- more, since 1 < N < 2n, when N = 2", AQC-DRCqpI achieves an exponential improvement over QC-DRCoRP. QC-DRCoRp+QUU
DRCOBP. QC-DRCQBP+
AQC-DRCom and AQC-DRCoap+QP1 Regarding space com- plexity, QC-DRCOBP. QC-DRConP+QP" require 3n qubits. In comparison, AQC-DRCqnP, AQC- DPC and AQC-DRCqPI require (2k 1)n qubits, growing linearly with both the number of elements n in the FOD and the number of evidence sources k. The results indicate that the proposed AQC-DRCQBP. compared to existing approaches. Specifically, AQC-DRCQUI demonstrates optimal performance as N tends toward 2n. A detailed analysis can be found in Appendix AI
OQC-DRCogp+QP) and AQC-DRCoPI exhibit superior time complexity
Limitation and conclusion
In this paper, address efficient classification under uncertain environments, we propose an adaptive quantum circuit for Dempster's 's rule of combination (AQC-DRC) to support quantum basic probabil- ity and plausibility level doecision-making within the framework of quantum evidence theory (QET). The AQC-DRC, as a generalized quantum Bayesian inference method, enables deterministic com- putation of DRC, ensuring that quantum fusion outcomes in uncertain pattern classification are fully consistent with those of the classical method. Furthermore, it achieves an exponential reduction in computational complexity, positioning as a promising approach for real-time quantum multisource information fusion. The architecture of the proposed AQC-DRC is conceptually straightforward and highly scalable, which facilitates its practical implementation. It is observed that the quantum ba- sic probability amplitude function in QET can naturally express the quantum amplitude encoding. The quantum basic probability in QET, forming the quantum basic probability distribution, can di- rectly express quantum measurement outcomes for basic probability level decision-making while the quantum plausibility function in QET can also naturally represent the quantum measurement outcomes for plausibility-Ieve decision-maie making. These insights not only broaden the understanding of QET. but also provide a more intuitive physical interpretation of quantum amplitude encoding and quantum measurement outcomes.
However, this study revealed that the current version of AQC-DRC is unable to process complex- valued input data, thereby constraining its applicability. Future research will focus on extending AQC-DRC's capabilities through further developments in quantum computing techniques. Notably, Xu et al. [b proposed a multimodal optimization strategy, which is excellent strategy to enhance multimodal data and obtain preferable classification results. Consequently, this work holds the potential to incorporate such strategies to establish trustworthy fusion and decision-makinn framework for handling heterogencous data in future applications.
the complementarity
10

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Acknowledgments
The authors greatly appreciate the reviewers' suggestions and the area chair's encouragement. This research is supported by the National Natural Science Foundation of China (No. 62473067), Xiaomi Young Talents Program, Chongqing Talents: Exceptional Young Talents Project (No. cstc2022ycjh- bgzxm0070), and Chongqing Overseas Scholars Innovation Program (No. cx2022024).
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Appendix
A1 Computational complexity analysis
All algorithms in Table D can be divided into two parts: the combination process, which involves obtaining the complete combined evidence or QPI: and the arciision-making process, which involves using the obtained evidence or QPI to reach a decision. Assume that there are elements in the frame of discernment (FOD) and i pieces of evidence, with total of N focal elements.
1) DRC: When combining two pieces of evidence, the computational complexity of computing any of the focal elements is O(22)). Since there are N focal elements in total, the computational com- plexity of getting the complete combination result is O(N22n). The total number of combinations is k
the total complexity of combination process is O(kN22m). After the combination process, a decision is made from n elements in the FOD. The computational complexity of the decision-making process is (n)). Thus, the total computational complexity of DRC OkN222n). (O(kN2 + n) = )(kN22")
2) DRC+QPI: The computational complexity required to compute the QPI of a focal element from QBP in the classical process is O(N). In the experimental process we made the decision to use the QPI of elements in FOD, so the computational complexity to compute QPI is O(nN). Since the computational complexity of the DRC is OlkN22"), the total computational complexity of combi- + nN) Okk222)) n. kN when 2 2). The computational complexity of decision-making process is Onn). Thus, the total computational complexity of DRC+QPI is Ok::22n).
nation process is O ). (O(kN222
3) QC-DRCQBP: The computational complexity of the quantum line part we use the depth of the quantum circuit to denote. The QC-DRCQBP can only combine two pieces of evidence at time. For the combination of two evidences the number of Toffoli gates used n, so the computational complexity of the quantum circuit part is (n)). There are N measurements obtained from quantum circuit, each of which requires a normalization process of complexity O(1), so the computational complexity of the normalization part is ( The computational complexity of two evidence combinations is (n + N). In total, k - combinations have to be performed, so the computational complexity of combination process is O(kn + k:N). The computational complexity of decision- making process is (n). Thus, the total computational complexity of QC-DRCqBP is O(kn + kNN..
4) QC-DRCQBP+ The computational complexity of QC-DRCQBP is Okkn + kN), and the computational complexity of the classical process of generating QPI from QBP is (nnN). There- fore, the total computational complexity of combination process is O(kn + KN + nN).. The compu- tational complexity of decision-making process is O(n). Thus, the total computational complexity of QC-DRCquP+QP1 is O(kn + kN + n.N..
can combine more than one piece of evidence at a time and the number required Toffoli (CCNOT) gate is (k- - 11). Thus, the complexity of the quantum circuit Okn)) The complexity of the normalization part is consistent with QC-DRCqEP and is ONN). Thus the total computational complexity of combination process is O(kn + N). The computational complexity of decision- makinn process is On)). Thus, the total computational complexity of AQC- DRCqBP is O(k++N),
5) AQC-DRCqBp: The
6) AQC-DRCQEP The computational complexity of AQC-DRCQEP is O(kn + N), and the computational complexity of the classical process of generating QPI from QBP is OnnN). There- fore, the total computational complexity of combination process is Okn + n.V.. (O(kn N nN) = (kn + (n + I)N) O(kn + n.))). The computational complexity of decision-makinn process is (n). Thus, the total computational complexity of AQC-DRCqBP+QP) is O(kn + nN).
7) AQC-DRCQrI The algorithm involves measuring n qubits individually, so a total of n quantum circuits are needed to complete it. The computational complexity of each of these quantum circuits is consistent with that of the quantum circuit of AQC-DRCOBP, which is O(k)), and thus its total computational complexity of all quantum circuits O(kn²). The computational complexity of decision-making process is On)). Thus, the total computational complexity of AQC-DRCQFl is Okn?).
15

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A2 Experimental environment
The classical DRC and DRC+QPI methods were executed using Python 3.8.20 on a Windows 10 system equipped with a 13th Gen Intelő Core 19-13900K CPU (3.00 GHz), 128 GB RAM, and an NVIDIA GeForee RTX 4090 GPU. The quantum methods, including QC-DRCQBP, QC- DRCQBP+QP1 the proposed AQC-DRCQBP, AQC-DRCQBP+QP. and AQC-DRCopI, were mainly executed on the lonQ quantum simulator (https://iogqccom))
A3 Experimental results
Table Al: Comparison of classification accuracies under w-th fold of in terms of classical/quantum basic probability level decision-making.
for Iris dataset
K-th fold of cross validvaali
Method
Shots
Acc
2
4
DRC
1.0000
0.9000
0.9000
0.9667
0.9667
0.9467
QC-DRCoBP
32
0.9000
0.9333
0.8667
0.9000
0.9000
0.9000
64
0.9333
0.9333
0.9333
0,9333
0.9333
0.9333
128
1.0000
0.9333
0,9000
0,9333
0,9667
0.9467
256
1. 1.0000
0.9000
0,9333
0,9000
0,9333
0,9333
512
1.0000
0.8667
0.9333
0.9667
1.0000
0.9533
1024
1.0000
0.9000
0.9000
0.9333
0.9333
0.9333
2048
1.0000
0.9333
0.8667
0.9667
0.9667
0.9467
4096
1.0000
0.8667
0,9000
0,9333
0.9667
0,9333
8192
1.0000
0.9000
0.9000
0,9667
1,0000
0.9533
16384
1.0000
0.9000
0,9000
0.9667
0.9667
0.9467
AQC-DRCqHP
32
0.9000
0.8667
0.9000
0.9000
0.9667
0.9067
64
0.9000
0.9333
0.8667
0,9000
0.9667
0.9133
128
0.9667
0,9000
0.9333
0,9333
0.9333
0.9333
256
1.0000
0.9000
0.9000
0,9667
0.9667
0.9467
512
1.0000
0,9000
0,9000
0,9333
0.9667
0,9400
1024
1.0000
0.9000
0,9333
0,9333
0,9667
0.9467
2048
1.0000
0.8667
0.9333
0.9667
0.9333
0.9400
4096
1.0000
0.8667
0.9000
0.9667
0.9333
0.9333
8192
1.0000
0.8667
0.9000
0.9667
1.0000
0.9467
16384
1.0000
0.8667
0.9333
0.9667
1.0000
0.9533
16

---

Table A2: Comparison of classification accuracies under w-th fold of cross cross-validation for Iris dataset in terms of classical/quantum plausibility level decision-making.
W-th fold of cross-validation
Method
Shots
Acc
3
4
DRC+QPI
1.0000
0.9000
0,9000
0.9667
0.9667
0.9467
QC-DRCqur+QPI
32
0.9000
0.7333
0.8667
0.8000
0.9000
0.8400
64
0.8667
0,8000
0,9000
0.8667
0,9333
0.8733
128
1.0000
0.9333
0.9333
0,9667
0.9333
0.9533
256
1.0000
0.9333
0,9000
0,9667
0,9667
0,9533
512
1.0000
0.8667
0.8667
0.9333
0,9333
0.9200
1024
1.0000
0,9000
0.9000
0.9333
0,9333
0.9333
2048
1.0000
0.8667
0.9000
0.9333
1.0000
0.9400
4096
1,0000
0.8667
0.9000
0.9667
0.9667
0.9400
8192
1.0000
0.9000
0.8667
0.9667
1.0000
0.9467
16384
1.0000
0.9000
0.9000
0.9667
0.9667
0.9467
QPI
0.9667
0.9667
0.9000
0,9000
0.9333
0.9333
64
0.9000
0.8333
0.9000
0.9333
0.9667
0.9067
128
1.0000
0.9667
0.8667
0.9000
1 1.0000
0.9467
256
1.0000
0.9000
0,9333
0,9600
0.9667
0,9500
512
1.0000
0.9000
0,9000
0.9667
0.9667
0.9467
1024
1.0000
0.8667
0,9000
0,9667
0. 0,9667
0.9400
2048
1.0000
0,9000
0.9000
0.9667
0.9667
0.9467
4096
1.0000
0.8667
0.8667
0.9667
1.0000
0,9400
8192
1.0000
0.8667
0.8667
0.9667
0.9667
0.9333
16384
1.0000
0.9000
0.9333
0.9667
0.9667
0.9533
AQC-DRCOPI
32
0.9667
0.9333
0.8667
0,9000
0,9333
0,9200
1.0000
0,9000
0.9000
0.9667
0.9333
0.9400
128
1.0000
0,9333
0.9000
0.9667
0.9667
0.9533
256
1.0000
0.9333
0.9000
0.9333
0.9667
0.9467
512
1.0000
0.9333
0.9000
0.9333
0.9667
0.9467
1024
1.0000
0.8667
0,9000
0.9667
0.9667
0,9400
2048
1.0000
0.9000
0.8667
0.9667
0.9667
0.9400
4096
1.0000
0.9333
0.9000
0.9667
1.0000
0.9600
8192
1.0000
0.8667
0.9333
0,9667
0.9667
0.9467
16384
1.0000
0.8667
0.9000
0.9667
0.9667
0.9400
17

---

Table A3: Comparison of classification accuracies under w-th fold of cross validation for Abalone dataset in terms of classical/quantum basic probability level decision-main mmaiing.
K-th fold cross- validation
Method
Shots
Acc
2
DRC
1.0000
1.0000
0.9767
1,0000
0.9348
0.9823
QC-DRCoaP
32
0.9535
1,0000
0.9535
0.9767
0.8261
0.9420
64
0.9535
0.9767
0.9302
1,0000
0.8913
0.9504
128
1,0000
1,0000
0,9535
0.9767
0.8696
0.9600
256
1,0000
0.9767
0.9535
1,0000
0.9130
0.9687
512
1.0000
1.0000
0.9767
1,0000
0.9348
0,9823
1024
1.0000
1,0000
0.9767
1,0000
0,9565
0.9867
1.0000
1,0000
0,9767
1,0000
0,9348
0.9823
4096
1.0000
1_0000
1.0000
1.0000
0.9348
0.9870
8192
1.0000
1,0000
0.9767
1,0000
0.9348
0.9823
16384
1.0000
1.0000
0.9767
1,0000
0.9348
0.9823
AQC-DRCQBP
32
1.0000
1.0000
0.9767
1,0000
0.9565
0.9867
64
1.0000
0.9767
0.9767
1,0000
0.9348
0.9777
128
1.0000
1.0000
0.9767
1,0000
0.9348
0.9823
256
1.0000
1,0000
0.9767
1,0000
0.9565
0.9867
512
1.0000
1,0000
0.9767
1,0000
0.9348
0.9823
1024
1,0000
1,0000
0.9767
1,0000
0.9348
0,9823
2048
1.0000
1.0000
0,9767
1,0000
0.9348
0,9823
4096
1.0000
1,0000
0.9767
1,0000
0.9348
0.9823
8192
1.0000
1,0000
0.9767
1,0000
0,9348
0.9823
16384
1.0000
1.0000
0.9767
1.0000
0.9348
0.9823
18

---

Table A4: Comparison of classification accuracies under x-th fold of cross cross- validation for Abalone dataset in terms of classical/quantumm plausibility level decision-making.
W-th fold of cross-validation
Method
Shots
Acc
3
DRC+QPI
1.0000
1.0000
0.9767
1.0000
0.9348
0.9823
QC-DRCqur+QPI
32
1.0000
0.9767
1.0000
1.0000
0.9348
0.9823
64
1.0000
1.0000
1,0000
1,0000
0,9348
0,9870
128
1.0000
1.0000
0.9767
1.0000
0.9348
0.9823
256
1.0000
1.0000
0.9767
1.0000
0,9565
0.9867
512
1.0000
10000
0.9767
1.0000
0.9565
0.9867
1024
1.0000
1. 1.0000
0.9767
1,0000
0,9348
0.9823
2048
1.0000
1.0000
0.9767
1.0000
0.9348
0.9823
4096
1.0000
1.0000
0.9767
1.0000
0.9348
0.9823
8192
1.0000
1.0000
0.9767
1,0000
0.9348
0.9823
16384
1.0000
1.0000
0.9767
1,0000
0.9348
0.9823
QPI
0.9070
1.0000
0.9535
0.9070
0.7609
0.9057
64
0.9535
0.9767
0.9535
1.0000
0.8478
0.9463
128
1.0000
0.9767
0,9535
0.9767
0.8913
0.9597
256
1.0000
1.0000
0.9767
1.0000
0.9348
0,9823
512
1.0000
0.9767
0.9767
1.0000
0.9348
0.9777
1024
1.0000
1.0000
1,0000
1.0000
0.9565
0.9913
2048
1.0000
10000
0.9767
1.0000
0.9348
0.9823
4096
1.0000
1.0000
0.9767
1,0000
0,9348
0.9823
8192
1.0000
1.0000
0.9767
1.0000
0.9565
0.9867
16384
1.0000
1.0000
0.9767
1.0000
0.9348
0.9823
AQC-DRCOPI
32
0,9767
0.9767
0,9302
0,9070
0.8261
0.9234
1.0000
0.9767
0.9767
1.0000
0.8261
0.9559
128
1.0000
0.9767
0.9767
1.0000
0,9565
0.9820
256
1.0000
0.9767
0.9767
1,0000
0.9348
0.9777
512
1.0000
0.9767
0.9767
1.0000
0.9348
0.9777
1024
1.0000
0.9767
1,0000
1.0000
0.9348
0,9823
2048
1.0000
0.9767
0.9767
1,0000
0.9348
0.9777
4096
1.0000
1.0000
0.9767
1.0000
0.9348
0.9823
8192
1.0000
1.0000
0.9767
1.0000
0.9348
0.9823
16384
1.0000
1.0000
0.9767
1.0000
0,9348
0.9823
19

---

Table A5: Comparison of classification accuracies under w-th fold of cross cross- validation for Knowledge dataset terms of classical/quantum basic probability level decision decision-making.
K-th fold cross- validation
Method
Shots
Acc
2
DRC
0.8667
0.8312
0.8267
0,8592
0.8861
0.8539
QC-DRCoaP
32
0.6267
0.5455
0.6133
0.5775
0.6582
0.6042
64
0.8000
0.6883
0.6667
0.7324
0,6456
0.7066
128
0.7467
0.8052
0,6800
0.7887
0.7595
0.7560
256
0.8533
0.7792
0.7733
0.8028
0.8481
0.8114
512
0.8667
0.7922
0.8400
0,8169
0.8354
0,8302
1024
0.8667
0.7922
0.8667
0,8451
0.8734
0.8488
0.8667
0.8052
0,8400
0.8451
0.8861
0.8486
4096
0.8667
0.8312
0.8000
0.8592
0.8987
0.8511
8192
0.8800
0.8182
0.8400
0.8873
0.8861
0.8623
16384
0.8667
0.8442
0.8267
0,8310
0.8861
0.8509
AQC-DRCQBP
32
0.6400
0.5844
0.6800
0.5634
0.6329
0.6201
64
0.6800
0.6883
0.6000
0.7183
0.6962
0.6766
128
0.6933
0.7532
0.7333
0.7183
0.7468
0.7290
256
0.7467
0.7532
0.7600
0.7606
0.8861
0.7813
512
0.8667
0.8052
0.8133
0.8169
0.8101
0.8224
1024
0,8667
0,7532
0.7733
0,8592
0.8987
0,8302
2048
0,8800
0,8182
0,8400
0,8169
0.8734
0.8457
4096
0.8800
0.8182
0.8267
0.8592
0.8734
0.8515
8192
0.8667
0.8052
0.8267
0.8592
0.8734
0.8462
16384
0.8800
0.8312
0.8267
0.8451
0.8861
0.8538
20

---

Table A6: Comparison classification accuracies under w-th fold of cross crass-validation for Knowledge dataset in terms of classical/quantum) plausibility level decision-making.
W-th fold of cross-validation
Method
Shots
Acc
3
DRC+QPI
0.8667
0.8571
0.8267
0.8592
0.8861
0.8591
QC-DRCqur+QPI
32
0.6267
0.5844
0.6267
0.6479
0.7089
0.6389
64
0.7333
0.7013
0,6133
0.6901
0.7342
0,6945
128
0.7600
0.7403
0.7067
0.7887
0.7215
0.7434
256
0.7467
0.7403
0.7867
0,8028
0,8481
0.7849
512
0.8267
0.8442
0.8133
0.8592
0,8354
0.8358
1024
0.8533
0.8442
0.8400
0.8451
0,8481
0.8461
2048
0.8667
0.8182
0.8400
0.8451
0.8734
0.8487
4096
0.8667
0.8442
0.8533
0.8732
0.8861
0.8647
8192
0.8533
0.8312
0.8533
0.8732
0.8861
0.8594
16384
0.8667
0.8571
0.8267
0.8592
0.8861
0.8591
QPI
0.6000
0.6494
0.5600
0.5915
0.6203
0.6042
64
0.7200
0.7403
0.6533
0.7183
0.7468
0.7157
128
0.7467
0.7792
0.7333
0.8310
0.8354
0.7851
256
0.8133
0.7532
0.7733
0.7746
0.8481
0.7925
512
0.8267
0.8182
0.8400
0.8169
0.8734
0.8350
1024
0.8667
0.8442
0.8400
0.8873
0.8861
0.8648
2048
0.8400
0.8571
0.8400
0.8592
0.8861
0.8565
4096
0.8533
0.8442
0.8267
0.8451
0.8861
0.8511
8192
0.8667
0.8442
0.8400
0.8732
0.8861
0.8620
16384
0.8667
0.8571
0.8267
0.8732
0.8861
0.8620
AQC-DRCOPI
32
0.6133
0.6753
0.6533
0,7042
0.6203
0.6533
0.7067
0.7403
0.6933
0.7324
0.7595
0.7264
128
0.8133
0.7403
0.8000
0.7324
0.7722
0.7716
256
0.8133
0.7662
0.8133
0.8169
0.8734
0.8166
512
0.8667
0.7662
0.8000
0.8310
0.8608
0,8249
1024
0.8533
0.8182
0.8267
0.8732
0.8861
0,8515
2048
0.8533
0.8312
0.8133
0.8732
0.8987
0.8540
4096
0.8533
0.8442
0.8267
0.8592
0.8734
0.8513
8192
0.8667
0.8312
0.8133
0.8592
0.8861
0.8513
16384
0.8667
0.8571
0.8133
0.8592
0.8861
0.8565
21

---

Table A7: Comparison of Dif and StD metrics under x-fold cross-validation for Iris dataset in terms of quantum basic probability level decision maaiinn..m.in.....
x--th fold of cross-validation
Method
Shots
Dif
StD
2
4
5
QC-DRCoap
32
0.1000
0.1000
0.1000
0.0667
0.1333
0.1000
0.0211
64
0.0667
0.1000
0.0333
0.0333
0.1000
0.0667
0.0298
128
0,0000
0.1000
0.0667
0.0333
0.0667
0.0533
0.0340
256
0.0000
0,0000
0.0333
0.0667
0.0333
0.0267
0.0249
512
0,0000
0,0333
0.0333
0.0000
0,0333
0.0200
0.0163
1024
0.0000
0.0667
0.0000
0.0333
0.0333
0.0267
0.0249
2048
0,0000
0,0333
0.0333
0.0000
0.0000
0,0133
0.0163
4096
0.0000
0.0333
0.0000
0.0333
0.0000
0.0133
0.0163
8192
0.0000
0.0000
0.0000
0.0000
0.0333
0.0067
0.0133
16384
0,0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
AQC-DRCOBP
32
0.1000
061667
0.0000
0.0667
0.0667
0.0800
0.0542
64
0.1000
0,1000
0.0333
0.0667
0.0667
0.0733
0.0249
0.0333
0.0667
0.0333
0.0333
0.1000
0.0533
0.0267
256
0.0000
0.0000
0.0667
0.0000
0.0000
0.0133
0.0267
512
0.0000
0.0000
0.0000
0.0333
0.0000
0.0067
0.0133
1024
0.0000
0.0667
0.0333
0.0333
0.0000
0.0267
0.0249
2048
0,0000
0,0333
0.0333
0.0000
0,0333
0,0200
0.0163
4096
0.0000
0,0333
0.0667
0.0000
0.0333
0.0267
0.0249
8192
0.0000
0,0333
0.0000
0.0000
0.0333
0,0133
0.0163
16384
0.0000
0.0333
0.0333
0.0000
0.0333
0.0200
0.0163
22

---

Table A8: Comparison of Dif and St1> metrics under s-fold validatiin for Iris dataset in terms of quantum plausibility level decision-making.
x-th fold of cross-validation
Method
Shots
Dif
StD
2
3
QC-DRCoar+QPI
32
0.1000
0.1667
0.1000
0.1667
0.1333
0.1333
0.0298
64
0.1333
0.1667
0.0667
0.1000
0.1000
0.1133
0.0340
128
0.0000
0.0333
0.0333
0.0000
0.1000
0.0333
0.0365
256
0,0000
0,0333
0.0667
0.0000
0.0667
0.0333
0.0298
512
0.0000
0.0333
0.0333
0.0333
0.0333
0.0267
0.0133
1024
0.0000
0.0000
0.0667
0.0333
0.0333
0.0267
0.0249
2048
0.0000
0.0333
0.0000
0,0333
0.0333
0.0200
0.0163
4096
0.0000
0.0333
0.0000
0.0000
0.0000
0.0067
0.0133
8192
0.0000
0.0000
0.0333
0.0000
0.0333
0.0133
0.0163
16384
0,0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
OPI
32
0.0333
0.0667
0.0667
0.0667
0.1000
0.0667
0.0211
0,1000
0.0667
0.0667
0.0333
0.0000
0.0533
0.0340
128
0.0000
0.0667
0.0333
0.0667
0.0333
0.0400
0.0249
256
0.0000
0.0667
0.0333
0.0000
0.0000
0.0200
0.0267
512
0.0000
0.0667
0.0667
0.0000
0.0000
0.0267
0.0327
1024
0,0000
0,0333
0.0667
0.0000
0.0000
0.0200
0.0267
2048
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
4096
0.0000
0.0333
0.0333
0.0000
0.0333
0.0200
0.0163
8192
0.0000
0.0333
0.0333
0.0000
0.0000
0.0133
0.0163
16384
0.0000
0.0000
0.0333
0.0000
0.0000
0.0067
0.0133
AQC-DRCQPI
32
0.0333
0,1000
0.0333
0.0667
0.0333
0.0533
0.0267
64
0.0000
0.1333
0.0667
0.0000
0. 10000
0.0600
0.0533
128
0.0000
0.1000
0.0667
0.0000
0.0000
0.0333
0.0422
256
0.0000
0.0333
0.0000
0.0333
0.0000
0.0133
0.0163
512
0.0000
0.0333
0.0000
0.0333
0.0000
0.0133
0.0163
1024
0.0000
0.0333
0.0000
0.0000
0.0000
0.0067
0.0133
2048
0.0000
0.0000
0.0333
0.0000
0.0000
0.0067
0.0133
4096
0,0000
0.0333
0.0000
0.0000
0.0333
0.0133
0.0163
8192
0.0000
0.0333
0.0333
0.0000
0.0000
0.0133
0.0163
16384
0.0000
0.0333
0.0000
0.0000
0.0000
0.0067
0.0133
23

---

Table A9: Comparison of Dif and StD metrics under x-fold cross-validation for Abalone dataset in terms of quantum basic probability level pecision-making.
x-th fold of cross-validation
Method
Shots
Dif
StD
2
5
QC-DRCoap
32
0.0465
0.0000
0.0233
0.0233
0.1522
0.0490
0.0536
64
0.0465
0.0233
0.0465
0.0000
0.0870
0.0406
0.0289
128
0,0000
0.0000
0.0233
0.0233
0.0652
0.0223
0.0238
256
0.0000
0.0233
0.0233
0.0000
0.0217
0.0137
0.0112
512
0,0000
0,0000
0.0000
0.0000
0,0000
0.0000
0,0000
1024
0.0000
0,0000
0.0000
0.0000
0.0217
0.0043
0.0087
2048
0,0000
0,0000
0.0000
0.0000
0.0000
0.0000
0.0000
4096
0.0000
0.0000
0.0233
0.0000
0.0000
0.0047
0.0093
8192
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
16384
0,0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
AQC-DRCOBP
32
0.0000
0,0000
0.0000
0.0000
0.0217
0.0043
0.0087
64
0.0000
0.0 0.0233
0.0000
0.0000
0.0000
0.0047
0.0093
0.0000
0,0000
0.0000
0.0000
0.0000
0.0000
0.0000
256
0.0000
0.0000
0.0000
0.0000
0.0217
0.0043
0.0087
512
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
1024
0.0000
0,0000
0.0000
0.0000
0.0000
0.0000
0.0000
2048
0,0000
0,0000
0.0000
0.0000
0,0000
0.0000
0.0000
4096
0.0000
0,0000
0.0000
0.0000
0.0000
0.0000
0.0000
8192
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
16384
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
24

---

Table A00: Comparison Dif and StD metrics under k-fold cross- cross-validation for Abalone dataset in terms of quantum plausibility level decision-making
x-th fold of cross-validation
Method
Shots
Dif
StD
2
3
QC-DRCoar+QPI
32
0.0000
0.0233
0.0233
0.0000
0.0000
0.0093
0.0114
64
0.0000
0.0000
0.0233
0.0000
0.0000
0.0047
0.0093
128
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
256
0,0000
0,0000
0.0000
0.0000
0.0217
0.0043
0.0087
512
0.0000
0.0000
0.0000
0.0000
0.0217
0.0043
0.0087
1024
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
2048
0.0000
0.0000
0.0000
0,0000
0.0000
0.0000
0.0000
4096
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
8192
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
16384
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
OPI
32
0.0930
0.0000
0.0233
0.0930
1739
0.0766
0.0612
0.0465
0.0233
0.0233
0.0000
0.1304
0.0447
0.0453
128
0.0000
0.0233
0.0233
0.0233
0.0435
0.0226
0.0138
256
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
512
0.0000
0.0233
0.0000
0.0000
0.0000
0.0047
0.0093
1024
0,0000
0,0000
0.0233
0.0000
0.0217
0.0090
0.01
2048
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
4096
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
8192
0.0000
0.0000
0.0000
0.0000
0.0217
0.0043
0.0087
16384
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
AQC-DRCQPI
32
0.0233
0,0233
0.0465
0.09 0.0930
0.1522
0.0676
0.0493
64
0.0000
0.0233
0.0465
0. 0.0000
O. 1087
0.0357
0.0404
128
0.0000
0.0233
0.0000
0.0000
0.0217
0.0090
0.0110
256
0.0000
0.0233
0.0465
0.0000
0.0000
0.0140
0.0186
512
0.0000
0.0233
0.0000
0.0000
0.0000
0.0047
0.0093
1024
0.0000
0.0233
0.0233
0.0000
0.0000
0.0093
0.0114
2048
0.0000
0.0233
0.0000
0.0000
0.0000
0.0047
0.0093
4096
0,0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
8192
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
16384
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
25

---

Table All: Comparison of Dif and StD metrics under k-fold cross-validation for Knowledge dataset in terms of quantum basic probability level decision-making
x-th fold of cross-validation
Method
Shots
Dif
StD
2
5
QC-DRCoap
32
0.3200
0.3117
0,2933
0.3380
0.2785
0.3083
0.0207
64
0.1200
0.2727
0.2133
0.1831
0.2911
0.2161
0.0619
128
0.1467
0.1039
0.2267
0.1549
0.1772
0.1619
0.0402
256
0.0400
0.0779
0.1067
0.1408
0.0633
0.0857
0.0350
512
0,0000
0,0649
0.0400
0.0704
0,0506
0.0452
0,0250
1024
0.0000
0.0649
0.0667
0.0423
0.0380
0.0424
0.0241
2048
0.0000
0,0779
0.0667
0.0141
0.0000
0,0317
0.0337
0.0000
0.0519
0.0267
0.0563
0.0127
0.0295
0.0218
8192
0.0133
0.0130
0.0400
0.0563
0.0000
0.0245
0.0205
16384
0,0000
0.0130
0.0000
0.0563
0.0000
0.0139
0.0218
AQC-DRCOBP
32
0.3333
0,3766
0.2800
0.3803
0.3038
0.3348
0.0395
64
0.2667
0,2208
0.3067
0.1690
0.2658
0.2458
0.0470
0.1733
0.1818
0.1467
0,1690
0.1646
0.1671
0.0117
256
0.1200
0.2078
0.1200
0.1268
0.0253
0.1200
0.0578
512
0.0267
0.0779
0.0400
0.1268
0.0759
0.0695
0.0349
1024
0.0267
0.0779
0.1067
0.0563
0.0380
0.0611
0.0286
2048
0,0133
0,0130
0.0133
0.0423
0.0127
0,0189
0.0117
4096
0.0133
0,0390
0.0267
0.0282
0.0127
0.0240
0.0099
8192
0.0000
0.0779
0.0267
0.0000
0.0127
0.0234
0.0290
16384
0.0133
0.0260
0.0000
0.0141
0.0000
0.0107
0.0098
26

---

Table A12: Comparison of Dif and StD metries under k-fold cross- volsdataaiitioon for Knowledge dataset in terms of quantum plausibility level decision-making.
x-th fold of cross-validation
Method
Shots
Dif
StD
2
3
QC-DRCoar+QPI
32
0.2933
0.3247
0.3600
0.2958
0.3038
0.3155
0.0248
64
0.2133
0.2078
0.2933
0.2254
0,2025
0.2285
0.0333
128
0.1333
0.1429
0.1733
0.1268
0.1899
0.1532
0.0243
256
0.1200
0.1688
0.1200
0.1408
0,0886
0.1277
0.0265
512
0,0400
0.0649
0.0933
0.0845
0.0759
0.0717
0.0184
1024
0.0133
0.0390
0.0933
0.0423
0.0380
0.0452
0.0262
2048
0.0267
0.0390
0.0400
0,0423
0.0127
0.0321
0.0111
4096
0.0000
0.0130
0.0267
0.0141
0.0000
0.0107
0.0100
8192
0.0133
0.0260
0.0533
0.0141
0.0000
0.0213
0.0180
16384
0.0000
0.0000
0.0533
0.0282
0.0000
0.0163
0.0215
OPI
32
0,2667
0.2857
0.3467
0.2958
0,3165
0.3023
0.0274
0.1733
0.1688
0,2800
0.1972
0.1899
0,2018
0.0404
128
0.1200
0.1299
0.1467
0.0563
0.
0.1159
0.0310
256
0.0533
0.1558
0.1067
0.0845
0.0886
0.0978
0.0337
512
0.0400
0.1169
0.0400
0.0986
0.0127
0.0616
0.0394
1024
0,0000
0,0130
0.0400
0.0282
0,0253
0.0213
0.0137
2048
0.0267
0.0000
0.0400
0.0563
0.0253
0.0297
0.0186
4096
0.0133
0.0130
0.0800
0.0423
0.0000
0.0297
0.0287
8192
0.0000
0.0390
0.0400
0.0141
0.0000
0.0186
0.0178
16384
0.0000
0.0000
0.0533
0.0141
0.0000
0.0135
0.0207
AQC-DRCQPI
32
0.3067
0.3117
0.3867
0,2113
0.3418
0,3116
0.0577
64
0.1600
0.1948
0.1867
0.2113
0.2278
0.1961
0.0229
128
0.0800
0.1429
0.0800
0.181
0.1139
0,1200
0.0393
256
0.0533
0.0909
0.0400
0.0704
0.0380
0.0585
0.0199
512
0.0533
0.1169
0.0533
0.0563
0.0506
0.0661
0.0255
1024
0.0400
0.0390
0.0800
0.0141
0.0000
0.0346
0.0273
2048
0.0133
0.0260
0.0667
0.0141
0.0127
0.0265
0.0207
4096
0.0133
0.0130
0.0000
0.0282
0.0380
0.0185
0.0132
8192
0.0000
0.0260
0.0133
0.0000
0.0000
0.0079
0.0104
16384
0.0000
0.0000
0.0400
0.0282
0.0000
0.0136
0.0171
27

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34


Reviews:
[{"review_id": "RPkvyuGK9L", "reviewer": "Reviewer_9Y4M", "summary": "To address efficient classification under uncertain environments, the authors proposed an adaptive quantum circuit for Dempster’s rule of combination (AQC-DRC) to support quantum basic probability and plausibility level decision-making within the framework of quantum evidence theory (QET).", "questions": "1. What are the QC-DRC_QBP + QPl and AQC-DRC_QBP + QPl? The methods of QC-DRC_QBP + QPl and AQC-DRC_QBP + QPl should be explained in the text.\n2. What are the purpose of two decision-making levels of quantum basic probability and quantum plausibility?\n3. In what situations are quantum basic probability and quantum plausibility used?", "limitations": "Yes", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths：-- The AQC-DRC, as a generalized quantum Bayesian inference method, enabled deterministic computation of DRC, ensuring that quantum fusion outcomes in uncertain pattern classification are fully consistent with those of the classical method.\n-- Furthermore, the proposed AQC-DRC achieved an exponential reduction in computational complexity, positioning it as a promising approach for real-time quantum multisource information fusion.\n-- The architecture of the proposed AQC-DRC is conceptually straightforward and highly scalable, which facilitates its practical implementation.\n-- The quantum basic probability amplitude function in QET can naturally express the quantum amplitude encoding. The quantum basic probability in QET, forming the quantum basic probability distribution, can directly express quantum measurement outcomes for basic probability level decision-making, while the quantum plausibility function in QET can also naturally represent the quantum measurement outcomes for plausibility-level decision-making.\n-- These insights not only broaden the understanding of QET, but also provide a more intuitive physical interpretation of quantum amplitude encoding and quantum measurement outcomes.\nWeaknesses：\nDue to the limitations of quantum hardware, the proposed AQC-DRC is unable to handle high-dimensional data.", "was_revised": false, "final_justification": "", "rebuttal": "Questions:\n\nQ1: What are the QC-DRC_QBP + QPl and AQC-DRC_QBP + QPl? The methods of QC-DRC_QBP + QPl and AQC-DRC_QBP + QPl should be explained in the text.\n\nA1: We sincerely appreciate your constructive feedback.\nThe QC-DRC_QBP is a non-adaptive quantum circuit for Dempster's rule of combination, which is used to compute quantum basic probability (QBP) for quantum basic probability level decision-making. The QC-DRC_QBP + QPl is a method that first uses QC-DRC_QBP to obtain QBP and then uses the classical process to combine QBP into QPl. Similarly, AQC-DRC_QBP + QPl is a method that first uses AQC-DRC_QBP to obtain QBP and then uses the classical process to combine QBP into QPl.\nYour suggestion made us realize that our explanation of this part was not clear enough, and we have made changes to this part that will be presented in the final draft.\n\nQ2: What are the purpose of two decision-making levels of quantum basic probability and quantum plausibility?\n\nA2: We sincerely appreciate your constructive feedback.\n1. The decision-making level of quantum basic probability can describe uncertain, partially unknown, and completely unknown quantum probabilities. For a three-class frame of discernment {$\\phi_1, \\phi_2, \\phi_3$}, using 3 qubits as an example: \n(1) The measurement result $\\ket{001}$ represents the quantum probability of \"supporting class $\\phi_1$\"; \n(2) The measurement result $\\ket{011}$ represents the quantum probability of an \"uncertain state between class $\\phi_1$ and class $\\phi_2$\"; \n(3) The measurement result $\\ket{111}$ represents the quantum probability of being \"completely unknown (uncertain state among class $\\phi_1$, class $\\phi_2$ and class $\\phi_3$)\". The decision-making level of quantum plausibility can provide an upper bound on quantum probability, representing an optimistic estimate. This level can be used mainly in scenarios with high security requirements, such as medical diagnostics or military intelligence analysis. For example, event $\\phi_1$ represents an infectious disease. The quantum basic probability (QBP) of a person contracting the disease is M$(\\phi_1) = 0.3$ that is lower than 0.5, whereas the quantum plausibility (QPl) is QPl$(\\phi_1) = 0.95$. Although the person cannot be directly identified as having the disease, the possibility that the person has the disease should still not be ignored for safety reasons.\n2. In addition, from the computational complexity point of view, the computational complexity of AQC-DRC_QPl is $O(kn^2)$, while the computational complexity of AQC-DRC_QBP is $O(kn+N)$. Therefore, if the usage scenario of the algorithm does not require significant bias towards one of the two levels, one option is to select an algorithm with lower computational complexity, based on the relationship between the size of FOD $n$ and the number of the focal elements $N$.\nYour suggestion made us realize that our explanation of this part was not clear enough, and we have made changes to this part that will be presented in the final draft.\n\nQ3: In what situations are quantum basic probability and quantum plausibility used?\n\nA3: We sincerely appreciate your constructive feedback.\nQuantum basic probability and quantum plausibility are used in decision-making and classification tasks under quantum frameworks, particularly when dealing with uncertainty, partial ignorance, or complete ignorance.\n1. Quantum basic probability is applied when one needs to model and quantify uncertain, partially unknown, or completely unknown events. It provides a quantum basic probability, allowing more expressive representations than classical probabilities.\n2. Quantum plausibility is used when an optimistic estimation or an upper bound of the quantum probability is required, such as in quantum decision fusion or quantum belief updating. It helps assess the maximum possible support for a quantum hypothesis or event, particularly useful in ambiguous scenarios.\nTogether, quantum basic probability and quantum plausibility are essential in quantum information processing, and quantum decision theory."}, {"review_id": "USYvcxtFCs", "reviewer": "Reviewer_XsYs", "summary": "The authors propose an Adaptive Quantum Circuit for Dempsters Rule of Combination (AQC-DRC) calculations in Dempster-Shafer (DS) theory of evidence which combines two basic probability assignments (BPA).  After defining basic DS concepts such as evidence, plausibility, BBA, etc. in quantum terms, the lay out their Adaptive Quantum Circuit which consists of a quantum amplitude encoding for BPA, a construction of the adaptive quantum circuit for DRC, and a measurement in the adaptive quantum circuit for subsequent decision-making.  They implement methods for basic probability and plausibility with the AQC-DRC.  They evaluate their quantum system on three classification datasets comparing their approach with the classical DS classification and show that the quantum methods gradually approach the classical results as the number of measurement shots increases while achieving an exponential speedup over classical methods.", "questions": "Have you thought about mapping conditional beliefs and plausibilities into the quantum framework, i.e., Bl(A|B) and Pl(A|B)?  These are particularly useful as lower and upper bounds for Bayesian conditionals P(A|B).  Also, can you please clarify how you arrived at the various quantum complexity results?", "limitations": "I would move the limitations section to the regular text", "rating": 5, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\nThe mapping of classical DRC into a quantum framework and then showing the viability and exponential speedup without significant loss of accuracy is important for the future when quantum computers will have evolved to a point where the are available for performing the types of classifications tasks the authors discuss.\n\nWeakness:\n\nThe DRC is know to be problematic and I wonder if the authors could also implement the Fagin-Halpern conditional which provides a better method for information fusion without the problems of the DCR.  It would have been also good to discuss any potential near-term applications of their AQC-DRC.", "was_revised": true, "final_justification": "I have no more updates based on the authors' feedback.", "rebuttal": "Weakness:\n\nW1: The (Dempster rule of combination) DRC is know to be problematic and I wonder if the authors could also implement the Fagin-Halpern conditional which provides a better method for information fusion without the problems of the DCR.\n\nA1: We sincerely appreciate your constructive feedback.\nDue to space constraints, the implementation of the quantum Fagin-Halpern conditional method will be considered in our near future work.\n\nW2: It would have been also good to discuss any potential near-term applications of their AQC-DRC.\n\nA2: We sincerely appreciate your constructive feedback.\nWe agree that discussing near-term applications of the AQC-DRC would add value. In the short term, AQC-DRC could be applied to areas such as:\n1) quantum-enhanced sensor fusion, where uncertain and conflicting measurements from multiple sources must be integrated efficiently;\n2) decision-making under uncertainty in domains like autonomous systems, medical diagnosis, and threat assessment.\nWe plan to discuss them in the future work and explore them in future work.\n\nQuestions:\n\nQ1: Have you thought about mapping conditional beliefs and plausibilities into the quantum framework, i.e., $Bl(A|B)$ and $Pl(A|B)$?\n\nA1: Thank you for the insightful question.\nYes, the idea of mapping conditional beliefs and plausibilities $Bl(A|B)$ and $Pl(A|B)$ into the quantum framework is both important and promising. In future work, we plan to explore quantum analogues of conditional belief and plausibility to reflect updated knowledge given new evidence. This would further enhance the expressive power and interpretability of quantum-based reasoning systems.\n\nQ2: Also, can you please clarify how you arrived at the various quantum complexity results?\n\nA2: We sincerely appreciate your constructive feedback.\nAll the algorithms in Table 2 of the manuscript can be divided into two parts: the combination process, which involves obtaining the complete combined evidence or QPl; and the decision-making process, which involves using the obtained evidence or QPl to reach a decision. Assume that there are $n$ elements in the frame of discernment (FOD) and $k$ pieces of evidence, with a total of $N$ focal elements.\n1) DRC: When combining two pieces of evidence, the computational complexity of computing any of the focal elements is $O(2^{2n})$. Since there are $N$ focal elements in total, the computational complexity of getting the complete combination result is $O(N2^{2n})$. The total number of combinations is $k-1$, so the total complexity of combination process is $O(kN2^{2n})$. After the combination process, a decision is made from n elements in the FOD. The computational complexity of the decision-making process is $O(n)$. Thus, the total computational complexity of DRC is $O(kN2^{2n})$. ($O(kN2^{2n}+n)=O(kN2^{2n})$)\n2) DRC+QPl: The computational complexity required to compute the QPl of a focal element from QBP in the classical process is $O(N)$. In the experimental process we made the decision to use the QPl of elements in FOD, so the computational complexity to compute QPl is $O(nN)$. Since the computational complexity of the DRC is $O(kN2^{2n})$, the total computational complexity of combination process is $O(kN2^{2n})$. ($O(kN2^{2n}+nN)=O(kN2^{2n})$, as $nN<kN2^{2n}$ when $n≥2$). The computational complexity of decision-making process is $O(n)$. Thus, the total computational complexity of DRC+QPl is $O(kN2^{2n})$.\n3) QC-DRC_QBP: The computational complexity of the quantum line part we use the depth of the quantum circuit to denote. The QC-DRC_QBP can only combine two pieces of evidence at a time. For the combination of two evidences the number of Toffoli gates used is $n$, so the computational complexity of the quantum circuit part is $O(n)$. There are $N$ measurements obtained from quantum circuit, each of which requires a normalization process of complexity $O(1)$, so the computational complexity of the normalization part is $O(N)$. The computational complexity of two evidence combinations is $O(n+N)$. In total, $k-1$ combinations have to be performed, so the computational complexity of combination process is $O(kn+kN)$. The computational complexity of decision-making process is $O(n)$. Thus, the total computational complexity of QC-DRC_QBP is $O(kn+kN)$.\n4) QC-DRC_QBP+QPl: The computational complexity of QC-DRC_QBP is $O(kn+kN)$, and the computational complexity of the classical process of generating QPl from QBP is $O(nN)$. Therefore, the total computational complexity of combination process is $O(kn+kN+nN)$. The computational complexity of decision-making process is $O(n)$. Thus, the total computational complexity of QC-DRC_QBP+QPl is $O(kn+kN+nN)$.\n5) AQC-DRC_QBP: The AQC-DRC_QBP can combine more than one piece of evidence at a time and the number required CNOT gate is $(k-1)n$. Thus, the complexity of the quantum circuit is $O(kn)$. The complexity of the normalization part is consistent with QC-DRC_QBP and is $O(N)$. Thus the total computational complexity of combination process is $O(kn+N)$. The computational complexity of decision-making process is $O(n)$. Thus, the total computational complexity of AQC-DRC_QBP is $O(kn+N)$.\n6) AQC-DRC_QBP+QPl: The computational complexity of AQC-DRC_QBP is $O(kn+N)$, and the computational complexity of the classical process of generating QPl from QBP is $O(nN)$. Therefore, the total computational complexity of combination process is $O(kn+nN)$. ($O(kn+N+nN)=O(kn+(n+1)N)=O(kn+nN)$). The computational complexity of decision-making process is $O(n)$. Thus, the total computational complexity of AQC-DRC_QBP+QPl is $O(kn+nN)$.\n7) AQC-DRC_QPl: The algorithm involves measuring $n$ qubits individually, so a total of $n$ quantum circuits are needed to complete it. The computational complexity of each of these quantum circuits is consistent with that of the quantum circuit of AQC-DRC_QBP, which is $O(kn)$, and thus its total computational complexity of all quantum circuits is $O(kn^2)$. The computational complexity of decision-making process is $O(n)$. Thus, the total computational complexity of AQC-DRC_QPl is $O(kn^2)$.\n\nLimitations:\n\nL1: I would move the limitations section to the regular text.\n\nA1: Thank you for your suggestion. We agree that limitations should be included in the main text. We have made changes that will be presented in the final draft."}, {"review_id": "BdSSIwgehq", "reviewer": "Reviewer_FwDi", "summary": "The manuscript proposes a so-called quantum version of the DS evidence framework, and suggests implementing the model's computation using adaptive quantum circuits. However, there some fundamental misuses in the basic formalization of the manuscript.", "questions": "Essentially, the probabilistic structure corresponding to quantum mechanics is a non-classical probabilistic structure that relaxes the closure property of sigma algebra with respect to countable unions in Kolmogorov's probability axioms. Does the authors of the paper know that this structure is not isomorphic to the mathematical structure underlying the DS theory, and therefore cannot be directly expressed in a quantum formalism?", "limitations": "Essentially, the probabilistic structure corresponding to quantum mechanics is a non-classical probabilistic structure that relaxes the closure property of sigma algebra with respect to countable unions in Kolmogorov's probability axioms. Does the authors of the paper know that this structure is not isomorphic to the mathematical structure underlying the DS theory, and therefore cannot be directly expressed in a quantum formalism?", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "In the main mathematical formalization of the manuscript, there exist misuses (or at least a misleading uses). For example, under the specification of Dirac's bra-ket notation, |ϕ1ϕ2> refers to the simultaneous occurrence of events ϕ1 and ϕ2 on the (two) subsystems of the composite system, respectively. However, the |ϕ1ϕ2> defined in the manuscript (in accordance with the convention of Dempster-Shafer theory) refers to the occurrence of either ϕ1 or ϕ2 on the same system. Overall, although the formalization in the manuscript employs the fundamental formalization of quantum mechanics (Dirac's bra-ket notation), it is contrary to the intrinsic interpretation of the formalization of quantum mechanics in terms of substantive meaning.\n\nBecause this misuse was fundamental, the subsequent formalizations (Definitions 8, 9 and 10, etc.) in the manuscript also deviated from the essential meaning of quantum mechanics. Therefore, the entire formalization of the manuscript and its algorithm model should not be regarded as being quantum in nature. It is merely a heuristic method that shares some formal characteristics with quantum mechanics, but its substantive meaning contradicts the fundamental interpretation of quantum mechanics.", "was_revised": true, "final_justification": "The research motivation and significance are not particularly impressive.", "rebuttal": "Weakness:\n\nW1: In the main mathematical formalization of the manuscript, there exist misuses (or at least a misleading uses). For example, under the specification of Dirac's bra-ket notation, $\\ket{\\phi_1\\phi_2}$ refers to the simultaneous occurrence of events $\\phi_1$ and $\\phi_2$ on the (two) subsystems of the composite system, respectively. However, the $\\ket{\\phi_1\\phi_2}$ defined in the manuscript (in accordance with the convention of Dempster-Shafer theory) refers to the occurrence of either $\\phi_1$ or $\\phi_2$ on the same system. Overall, although the formalization in the manuscript employs the fundamental formalization of quantum mechanics (Dirac's bra-ket notation), it is contrary to the intrinsic interpretation of the formalization of quantum mechanics in terms of substantive meaning.\nBecause this misuse was fundamental, the subsequent formalizations (Definitions 8, 9 and 10, etc.) in the manuscript also deviated from the essential meaning of quantum mechanics. Therefore, the entire formalization of the manuscript and its algorithm model should not be regarded as being quantum in nature. It is merely a heuristic method that shares some formal characteristics with quantum mechanics, but its substantive meaning contradicts the fundamental interpretation of quantum mechanics.\n\nA1: Thank you very much for your constructive and rigorous comments. We would like to respond as follows:\n1) Inspired by quantum probability theory, which generalizes classical probability theory within the mathematical framework of quantum mechanics, we proposed a quantum evidence theory that integrates Dempster-Shafer (DS) evidence theory with quantum probability. The aim is to address uncertainty and ambiguity in complex decision tasks.\n2) In doing so, we discussed the mathematical equivalence of fundamental concepts in quantum evidence theory within Hilbert space, to further explore the intrinsic relationship between quantum evidence theory and the mathematical framework of quantum mechanics. We sincerely apologize for any confusion this may have led to.\n3) Just as classical probability evolved into quantum probability, and Euclidean geometry extended into Riemannian geometry, we are inspired to propose quantum evidence theory as a natural extension for more complex application scenarios. On one hand, it has successfully modeled interference effects observed in classification decision task. On the other hand, the notions of quantum basic probability amplitude and quantum basic probability give new physical meaning to quantum amplitude encoding and quantum measurement from an information-theoretic perspective.\n4) In the revised manuscript, we will clearly state that the expression $\\phi_1\\phi_2$ denotes a composite quantum proposition, and that the corresponding ket notation $\\ket{\\phi_1\\phi_2}$ is a mathematical representation in Hilbert space, equivalent to how composite quantum proposition are encoded.\nIn summary, we appreciate your critical insight and will make sure to clarify these distinctions in the revision to avoid any misinterpretation of the model’s intent and theoretical nature.\n\nQuestion:\n\nQ1: Essentially, the probabilistic structure corresponding to quantum mechanics is a non-classical probabilistic structure that relaxes the closure property of sigma algebra with respect to countable unions in Kolmogorov's probability axioms. Does the authors of the paper know that this structure is not isomorphic to the mathematical structure underlying the DS theory, and therefore cannot be directly expressed in a quantum formalism?\n\nA1: Thank you for your deep and insightful question.\n1) As we answered for W1, inspired by quantum probability theory, which generalizes classical probability theory within the mathematical framework of quantum mechanics, we proposed a quantum evidence theory that integrates Dempster-Shafer (DS) evidence theory with quantum probability. The aim is to address uncertainty and ambiguity in complex decision tasks. In addition, we discussed the mathematical equivalence of fundamental concepts in quantum evidence theory within Hilbert space, to further explore the intrinsic relationship between quantum evidence theory and the mathematical framework of quantum mechanics.\n2) For example, $\\phi_1\\phi_2$ denotes a composite quantum proposition, and that the corresponding ket notation $\\ket{\\phi_1\\phi_2}$ is a mathematical representation in Hilbert space, equivalent to how composite quantum proposition are encoded. Take a two-class frame of discernment {$\\phi_1, \\phi_2$}, using 2 qubits as an example: The measurement result $\\ket{11}$, corresponding to composite quantum proposition $\\phi_1\\phi_2$ mathematical representation $\\ket{\\phi_1\\phi_2}$ in Hilbert space, represents the quantum probability for composite quantum proposition $\\phi_1\\phi_2$. The measurement result $\\ket{11}$ represents the quantum probability of being \"completely unknown (uncertain state between class $\\phi_1$ and class $\\phi_2$)\".\nWe appreciate your critical insight. And, we will modify and clarify this positioning more explicitly in the revised manuscript to prevent any misunderstanding regarding the nature of our formalism."}, {"review_id": "DKlBzC8Hk4", "reviewer": "Reviewer_rA57", "summary": "This paper proposes an Adaptive Quantum Circuit for Dempster's Rule of Combination (AQC-DRC) to address uncertainty in pattern classification. The method leverages quantum evidence theory (QET) to provide efficient evidence fusion while maintaining consistency with classical Dempster-Shafer theory results. The AQC-DRC achieves exponential reductions in computational complexity and supports decision-making based on both quantum basic probability and plausibility levels. The authors present rigorous theoretical derivations, experimental evaluations on datasets such as Iris and Abalone, and a detailed comparison with classical and quantum methods.", "questions": "1. Why the quantum basic probability amplitude function in QET can be naturally utilized to express the quantum amplitude encoding?\n2. What’s different between the quantum basic probability level decision-making and quantum plausibility level decision-making?\n3. Figure A1 is a main flowchart of the adaptive quantum circuit for Dempster’s rule of combination. It should be placed in the main text.", "limitations": "yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths\nIn this paper, the authors proposed an Adaptive Quantum Circuit for Dempster’s Rule of Combination (AQC-DRC) to address efficient classification under uncertain environments by leveraging the advanced computational power of quantum computing. The AQC-DRC is developed within the framework of quantum evidence theory (QET) and facilitates decision-making based on quantum basic probability and plausibility levels, which is a generalized Bayesian inference method. The main strengths of this paper are summarized as follows:\n1. The AQC-DRC provides a deterministic computation of DRC, ensuring that quantum fusion outcomes in uncertain pattern classification are exactly aligned with those of the classical method, while simultaneously achieving exponential reductions in the computational complexity of evidence combination and significantly improving fusion efficiency.\n2. It is founded that the quantum basic probability amplitude function in QET, as a generalized quantum probability amplitude, can be naturally utilized to express the quantum amplitude encoding. In addition, the quantum basic probability in QET, as a generalized quantum probability, naturally forms a quantum basic probability distribution and can be used to represent quantum measurement outcomes for quantum basic probability level decision-making. Furthermore, the quantum plausibility function in QET also can be naturally used to express the quantum measurement outcomes for quantum plausibility level decision-making.\n3. These findings enrich the physical understanding of quantum amplitude encoding and quantum measurement outcomes.\nOverall, this work is a generalized Bayesian inference method and is very meaningful to offer broad application prospects for representing and processing uncertain knowledge in pattern classification.\n\nWeaknesses\n1. As the authors mentioned in the paper. The current version of AQC-DRC is unable to process complex-valued input data, thereby constraining its applicability.\n2. Why the quantum plausibility function in QET can be used to represent quantum measurement outcomes for quantum basic probability level decision-making, and can be naturally used to express the quantum measurement outcomes for quantum plausibility level decision-making?\n3. The section of Preliminaries is too long. The content should be concise.", "was_revised": true, "final_justification": "The paper presents AQC-DRC within QET, delivering deterministic fusion exactly equivalent to classical DRC while achieving exponential reductions in evidence-combination complexity and supporting both QBP- and QPl-level decisions. The rebuttal convincingly clarifies that QBPA naturally fits amplitude encoding (normalization-consistent) and rigorously maps measurement outcomes to QBP and QPl, with clear guidance on use cases and complexity trade-offs. Experiments on Iris and Abalone substantiate correctness and efficiency; the authors will streamline the Preliminaries and move the main flowchart into the text to improve clarity. Therefore, I would like to keep my positive score.", "rebuttal": "Weakness:\n\nW1: Why the quantum plausibility function in QET can be used to represent quantum measurement outcomes for quantum basic probability level decision-making, and can be naturally used to express the quantum measurement outcomes for quantum plausibility level decision-making?\n\nA1: We sincerely appreciate your constructive feedback.\nIn our concluding section, we mentioned “The quantum basic probability in QET, forming the quantum basic probability distribution, can directly express quantum measurement outcomes for basic probability level decision-making, while the quantum plausibility function in QET can also naturally represent the quantum measurement outcomes for plausibility-level decision-making.” This section presents two conclusions:\n1)\tThe quantum basic probability in QET, forming the quantum basic probability distribution, can directly express quantum measurement outcomes for basic probability level decision-making.\nAccording to Definition 16, we obtain the QBP expression for quantum measurement outcomes after a rigorous mathematical proof. In other words, quantum measurement outcomes generated by $U_M^{QBP}$ can be simply computed to obtain QBP for basic probability level decision-making.\n2)\tThe quantum plausibility function in QET can also naturally represent the quantum measurement outcomes for plausibility-level decision-making.\nSimilarly, according to Definition 18, we obtain the QPl expression for quantum measurement outcomes after rigorous mathematical proof. In other words, quantum measurement outcomes generated by $U_M^{QPl}$ can be simply calculated to obtain QPl. Based on these two reasons, we arrive at these two conclusions mentioned in the conclusion section.\n\nW2: The section of Preliminaries is too long. The content should be concise.\n\nA2: Thank you for your suggestion. We agree that Preliminaries should be more concise. We have made changes to this section that will be presented in the final draft.\n\nQuestion: \n\nQ1: Why the quantum basic probability amplitude function in QET can be naturally utilized to express the quantum amplitude encoding?\n\nA1: We sincerely appreciate your constructive feedback.\nThe quantum basic probability amplitude (QBPA) function is defined in Definition 9. According to this definition, the QBPA must satisfy the conditions outlined in Eq. (12), one of which is: $\\sum_{\\ket{\\psi_j}\\in \\ket{\\Phi}}|\\mathbf{Q_M}(\\ket{\\psi_j})|^2=1$. In quantum computing, a superposition state must satisfy the condition that the sum of the squares of each mods of amplitude equals one, consistent with the conditions in Eq. (12). Therefore, the quantum basic probability amplitude function in QET can be naturally utilized to express the quantum amplitude encoding.\n\nQ2: What’s different between the quantum basic probability level decision-making and quantum plausibility level decision-making?\n\nA2: Thank you for your questions and suggestions.\n1) The decision-making level of quantum basic probability can describe uncertain, partially unknown, and completely unknown quantum probabilities. For a three-class frame of discernment {$\\phi_1, \\phi_2, \\phi_3$}, using 3 qubits as an example: The measurement result $\\ket{001}$ represents the quantum probability of \"supporting class $\\phi_1$\"; The measurement result $\\ket{011}$ represents the quantum probability of an \"uncertain state between class $\\phi_1$ and class $\\phi_2$\"; The measurement result $\\ket{111}$ represents the quantum probability of being \"completely unknown (uncertain state among class $\\phi_1$, class $\\phi_2$ and class $\\phi_3$)\". The decision-making level of quantum plausibility can provide an upper bound on quantum probability, representing an optimistic estimate. This level can be used mainly in scenarios with high security requirements, such as medical diagnostics or military intelligence analysis. For example, event $\\phi_1$ represents an infectious disease. The quantum basic probability (QBP) of a person contracting the disease is M$(\\phi_1) = 0.3$ that is lower than 0.5, whereas the quantum plausibility (QPl) is QPl$(\\phi_1) = 0.95$. Although the person cannot be directly identified as having the disease, the possibility that the person has the disease should still not be ignored for safety reasons.\n2) In addition, from the computational complexity point of view, the computational complexity of AQC-DRC_QPl is $O(kn^2)$, while the computational complexity of AQC-DRC_QBP is $O(kn+N)$. Therefore, if the usage scenario of the algorithm does not require significant bias towards one of the two levels, one option is to select an algorithm with lower computational complexity, based on the relationship between the size of FOD $n$ and the number of the focal elements $N$.\nYour suggestion made us realize that our explanation of this part was not clear enough, and we have made changes to this part that will be presented in the final draft.\n\nQ3: Figure A1 is a main flowchart of the adaptive quantum circuit for Dempster’s rule of combination. It should be placed in the main text.\n\nA3: We sincerely appreciate your constructive feedback.\nWe agree that Figure A1, which illustrates the main flowchart of the adaptive quantum circuit for Dempster’s rule of combination, plays an important role. In the revised manuscript, if space permits, we will move this figure from the appendix to the main text to improve clarity and help readers better understand the overall workflow and structure of the algorithm."}]

Author rebuttal:

Meta-review: This paper proposes an Adaptive Quantum Circuit for Dempster's Rule of Combination (AQC-DRC), a quantum algorithm designed to accelerate evidence fusion in pattern classification under uncertainty. The authors claim their method achieves an exponential reduction in computational complexity compared to classical Dempster-Shafer Theory (DST) while guaranteeing that the quantum fusion outcomes are deterministically identical to the classical ones. The work is framed within a proposed "Quantum Evidence Theory" (QET), which the authors argue provides new physical insights into quantum amplitude encoding and measurement. Three of the four reviewers ultimately supported acceptance, with two high-confidence reviewers rating it as "Strong Accept" and "Accept".

The paper's primary strengths, as highlighted by the reviewers, are its novelty and potential impact. It presents a concrete quantum algorithm for a well-established, computationally expensive classical method. Reviewer 9Y4M noted that the AQC-DRC "achieved an exponential reduction in computational complexity, positioning it as a promising approach for real-time quantum multisource information fusion." [9Y4M] Reviewer rA57 praised the fact that it "provides a deterministic computation of DRC, ensuring that quantum fusion outcomes in uncertain pattern classification are exactly aligned with those of the classical method." [rA57] The main weakness stems from a fundamental critique by Reviewer FwDi, who argued that the paper's formalism misuses quantum mechanical principles. This reviewer contended that the model is "fundamentally still a classical probability model that is jointly measurable, rather than a quantum probability model... Calling this kind of definition 'quantum' would cause considerable confusion in thinking." [FwDi] Other minor weaknesses include the near-term infeasibility on current quantum hardware and the use of DRC, which is known to be problematic in some scenarios.

The author rebuttal was comprehensive and effectively addressed most concerns. For reviewers 9Y4M, XsYs, and rA57, the authors provided detailed clarifications on terminology, the motivation for different decision-making levels, and the derivation of complexity results, leading all three to affirm their positive ratings. The most significant discussion was with Reviewer FwDi, which involved a multi-turn debate. The authors successfully refuted the reviewer's initial misunderstanding that they were modeling a linear problem with an exponential state space by clarifying that DST itself operates on an exponential power set. While the reviewer conceded this, they maintained a more philosophical objection regarding the formalism's "quantumness." In my assessment, while Reviewer FwDi's critique is technically nuanced and important for the quantum foundations community, the authors' primary claim—the design of a quantum circuit that correctly and efficiently implements DRC—stands. The strong support from two high-confidence experts in the area outweighs this more philosophical disagreement.

Given the clear technical contribution, the demonstrated theoretical speedup, and the strong support from a majority of the reviewers, I recommend acceptance. The paper is technically solid and presents a novel application of quantum computing to a known ML-related problem. The debate surrounding the foundational interpretation of the formalism, while valid, does not invalidate the paper's core result concerning the quantum circuit implementation. The work is a good fit for the conference, contributing to the growing intersection of quantum computing and machine learning.
Decision: Accept (poster)