EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6607
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Harnessing Quantum Superposition in Soft Set Theory:
Introducing Quantum Hypersoft and SuperHyperSoft

Sets

Takaaki Fujita1,∗, Florentin Smarandache2

1 Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan
2 University of New Mexico, Gallup Campus, NM 87301, USA

Abstract. This paper presents two novel extensions of the Quantum Soft Set framework by in-
tegrating the hierarchical structures of hypersoft and superhypersoft sets with quantum superpo-
sition principles. Soft sets offer a versatile approach to decision making by associating parameters
with subsets of a universal set, effectively capturing uncertainty and imprecision [1, 2]. Hypersoft
and superhypersoft sets further generalize this paradigm for increasingly complex scenarios. A
Quantum Soft Set maps each parameter to a normalized quantum state [3], enabling probabilis-
tic membership via amplitude coefficients. We rigorously define the Quantum Hypersoft Set and
the Quantum SuperHypersoft Set, laying a foundation for future advances in quantum-enhanced
decision analysis, topological modeling, and algebraic applications.

2020 Mathematics Subject Classifications: 06D72

Key Words and Phrases: Superhypersoft set, Soft Set, Hypersoft set, Quantum Hypersoft Set,
Quantum Soft Set, Quantum Superhypersoft Set

1. Introduction

1.1. Soft Sets and Their Extensions

The challenge of modeling uncertainty has led to a variety of mathematical frameworks,
including fuzzy sets [4], hyperfuzzy sets [5, 6], super-hyperfuzzy sets [7], rough sets [8],
hyperrough sets [9], intuitionistic fuzzy sets [10, 11], picture fuzzy sets [12], bipolar fuzzy
sets [13, 14], hesitant fuzzy sets [15, 16], neutrosophic sets [17, 18], and plithogenic sets
[19, 20].

Among these, soft sets [1, 2] are notable for associating each parameter with a subset
of a universal set, thereby offering a flexible tool for decision-making under imprecision.
Building on this idea, researchers have proposed numerous generalizations:

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6607

Email addresses: takaaki.fujita060@gmail.com (T.Fujita), smarand@unm.edu (F. Smarandache)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 2 of 33

• Hypersoft Sets [21, 22] and SuperHypersoft Sets [23], which add layers of parameter
interactions;

• Expert Soft Sets [24] and Hypersoft Expert Sets [25, 26], integrating expert judg-
ments;

• Fuzzy Soft Sets [27] and IndetermSoft Sets [28–30], which blend fuzzy membership
or indeterminacy with soft parameters;

• TreeSoft Sets [31, 32], organizing parameters into hierarchies, and ForestSoft Sets
[33], which combine multiple trees into a cohesive framework.

Soft Sets have proven useful across decision analysis [34], topology [35], graph theory [36],
risk assessment [25], and algebraic structures. Consequently, advancing soft set theory
and its variants remains an important endeavor.

In this paper, we explore how Soft Sets, Hypersoft Sets, and SuperHypersoft Sets can
be treated within the framework of quantum theory. As summarized in Table 1, the three
frameworks differ in how they map parameters or attribute combinations to subsets of the
universe and in the depth of their parameter interactions. Consequently, Hypersoft Sets
and SuperHypersoft Sets are well suited for modeling real-world phenomena characterized
by multi-layered, complex uncertainty.

Concept Mapping Key Feature

Soft Set F : S → P(U) Associates each parameter α ∈ S
with a subset F(α) ⊆ U [1].

Hypersoft Set G : A1 × · · · × Am → P(U) Maps each m-tuple of attribute
values (γ1, . . . , γm) to a subset
G(γ) ⊆ U , enabling multi-attribute
combinations[21].

SuperHypersoft Set F : P(A1)× · · · × P(An)→ P(U) Extends Hypersoft by using power-
set layers of each attribute domain,
capturing interactions among arbi-
trary subsets of values[37].

Table 1: Overview of Soft Set, Hypersoft Set, and SuperHypersoft Set

1.2. Quantum Theory

Quantum Theory describes the behavior of matter and energy at atomic scales, employ-
ing superposition, wave–particle duality, and probabilistic measurement outcomes [38, 39].
These principles have garnered significant attention in recent years, particularly in contexts
such as quantum computing [40, 41].

As noted above, this paper explores whether Soft Sets, Hypersoft Sets, and SuperHy-
persoft Sets can be extended within the framework of quantum theory. In the field of
soft set theory, the Quantum-Soft Set [3] has already been introduced. The Quantum-Soft



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 3 of 33

Set enriches the classical soft set by mapping each parameter to a normalized quantum
superposition in a Hilbert space, thereby encoding membership via amplitude coefficients
and measurement probabilities. A related framework, the Quantum Fuzzy Set, has also
been proposed (cf. [42–44]). In view of these quantum-enhanced approaches to uncer-
tainty modeling, we anticipate that Quantum-Soft Sets will play a central role in future
scientific and technological developments, and we therefore regard their rigorous study as
both timely and essential.

1.3. Our Contributions

Research on Soft Sets and Quantum Theory is of considerable importance both in
foundational mathematics and physics as well as in practical applications. Extending
Soft Sets to richer frameworks is therefore a valuable and meaningful pursuit. While the
Quantum-Soft Set has been defined, its integration into more advanced soft set models
remains underexplored. In this paper, we introduce and formalize the Quantum Hypersoft
Set and the Quantum SuperHypersoft Set, deriving their core properties and structural
characteristics. By merging the quantum-soft paradigm with hypersoft and superhypersoft
structures, we aim to stimulate new advances in set theory, decision-making methodologies,
topological analysis, algebraic modeling, and quantum applications.

As summarized in Table 2, the three quantum-enhanced soft set frameworks differ in
how they map parameters or attribute combinations to quantum states and in the scope
of their attribute interactions.

Concept Mapping Key Feature

Quantum-Soft Set F : A→ H(U) Assigns each attribute a ∈
A to a normalized quantum
state |ψa⟩, where |αi,a|2 gives
the membership probability of
ui[3].

Quantum-HyperSoft Set Q : A1 × · · · × Am → H(U) Maps each tuple of values
(γ1, . . . , γm) to a quantum
superposition, capturing joint
multi-attribute influence via
amplitude interference.

Quantum-SuperHypersoft Set Q : P(A1) × · · · × P(An) →
H(U)

Extends HyperSoft by mapping
each subset-tuple of attribute
values to a quantum state,
modeling interactions among
arbitrary subsets of values.

Table 2: Overview of Quantum Soft Set, Quantum HyperSoft Set, and Quantum SuperHyperSoft Set



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 4 of 33

1.4. Structure of this paper

The remainder of this paper is organized as follows. Section 2 reviews Soft Sets, Hyper-
soft Sets, SuperHypersoft Sets, and Quantum Soft Sets. Section 3 introduces and analyzes
Quantum–HyperSoft Sets and Quantum–SuperHyperSoft Sets. Section 4 briefly examines
the construction algorithms. Section 5 concludes the paper and discusses directions for
future work.

2. Preliminaries

In this section, we present the foundational concepts of Soft Sets, Hypersoft Sets, and
SuperHypersoft Sets. We also provide a concise explanation of the definition of Quantum
Soft Sets.

2.1. Notation

Throughout this paper, all sets are assumed to be finite. The empty set is considered
to be a subset of any set. For the basic operations related to each concept, the reader is
referred to the corresponding references.

In this paper, we use the following notation.

• U = {u1, . . . , un}: a finite universe

• P(U): the power set of U

• H(U): the n-dimensional complex Hilbert space with orthonormal basis {|ui⟩}ni=1

• C: the Cartesian product of the attribute domains (or of their power sets)

• General quantum state:

|ψ⟩ =

n∑
i=1

αi |ui⟩ ,
n∑

i=1

∣∣αi

∣∣2 = 1

• Measurement probability:
P (ui | γ) =

∣∣αi,γ

∣∣2
2.2. Soft Sets

Let U = {u1, . . . , un} be a finite universe and let A be a finite set of parameters. We
denote by P(U) the power set of U . Given a subset S ⊆ A of chosen parameters, a soft
set over U (with respect to S) is defined as follows.

Definition 1 (Soft Set [2]). A soft set over U parameterized by S is a pair (F , S) where

F : S −→ P(U)



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 5 of 33

is a function assigning to each parameter α ∈ S a subset F(α) ⊆ U . Equivalently,

(F , S) =
{(
α,F(α)

) ∣∣ α ∈ S, F(α) ⊆ U
}
.

Example 1 (Restaurant Selection). Let

U = {R1, R2, R3, R4, R5}

be a set of restaurants in a city, and let

A = {Italian, Japanese, Cheap, FineDining, FamilyFriendly}

be the full parameter set. Suppose a user selects

S = {Italian, Cheap, FamilyFriendly}.

Define the soft set (F , S) by

F(Italian) = {R1, R3, R5},
F(Cheap) = {R2, R3, R4},

F(FamilyFriendly) = {R1, R4, R5}.

Then:

• F(Italian) = {R1, R3, R5} is the set of Italian restaurants.

• F(Cheap) = {R2, R3, R4} is the set of low-cost restaurants.

• F(FamilyFriendly) = {R1, R4, R5} is the set of family-friendly restaurants.

For instance,
F(Italian) ∩ F(Cheap) = {R3}

identifies R3 as the unique restaurant that is both Italian and inexpensive, while

F(Italian) ∩ F(FamilyFriendly) = {R1, R5}

identifies the Italian restaurants suitable for families. The soft set formalism thus enables
flexible, parameter-driven filtering of U under uncertainty.

2.3. Hypersoft Sets

Let U = {u1, . . . , un} be a finite universe and let P(U) denote its power set. For a
positive integer m, let

A1,A2, . . . ,Am

be m disjoint attribute domains. We form their Cartesian product

C = A1 ×A2 × · · · × Am,

so that each γ ∈ C is an m-tuple

γ = (γ1, γ2, . . . , γm), γi ∈ Ai.

A hypersoft set over U is then defined as follows.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 6 of 33

Definition 2 (Hypersoft Set [21]). A hypersoft set over U with respect to the attribute
product C is a pair (G, C) where

G : C −→ P(U)

is a function assigning to each γ ∈ C a subset G(γ) ⊆ U . Equivalently,

(G, C) =
{(
γ, G(γ)

)
| γ ∈ C, G(γ) ⊆ U

}
.

Example 2 (Laptop Purchase). Let

U = {XPS13, Envy13, MacBookAir, ZenBook, ThinkPad}

be a set of laptop models. Consider three attribute domains:

A1 = {Dell, HP, Apple}, A2 = {Under1k, 1k–2k, Over2k}, A3 = {SSD, HDD}.

Form
C = A1 ×A2 ×A3,

so each γ = (γ1, γ2, γ3) specifies a brand, price-range, and storage type. Define the hyper-
soft set (G, C) by:

G
(
(Dell, 1k−2k, SSD)

)
= {XPS13, Envy13},

G
(
(Apple, Over2k, SSD)

)
= {MacBookAir},

G
(
(HP, Under1k, HDD)

)
= {Envy13},

G
(
(Apple, 1k−2k, HDD)

)
= {MacBookAir, ZenBook},

G
(
(Dell, Under1k, SSD)

)
= {ZenBook},

G
(
(HP, Over2k, SSD)

)
= {ThinkPad}.

For example,
G((Dell, 1k–2k, SSD)) = {XPS13, Envy13}

lists all Dell laptops in the $1k–$2k range with SSD storage. This hypersoft set enables
a buyer to filter models by any combination of brand, price, and storage type, even when
some models lack complete attribute information.

2.4. SuperHypersoft Sets

Let U = {u1, . . . , un} be a finite universe and let P(U) denote its power set. Let m be
a positive integer, and let

A1, A2, . . . , Am

be m pairwise disjoint finite attribute value–sets:

Ai ∩Aj = ∅ (i ̸= j).



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 7 of 33

For each i, write P(Ai) for the power set of Ai, and form the Cartesian product

C = P(A1)× P(A2)× · · · × P(Am).

Each element γ ∈ C is an m-tuple

γ = (α1, α2, . . . , αm), αi ⊆ Ai.

A SuperHypersoft Set over U is defined as follows.

Definition 3 (SuperHypersoft Set [23]). A SuperHypersoft Set over the universe U with
respect to the attribute–power product C is a pair (F, C) where

F : C −→ P(U)

assigns to each γ ∈ C a subset F (γ) ⊆ U . Equivalently,

(F, C) =
{(
γ, F (γ)

)
| γ ∈ C, F (γ) ⊆ U

}
.

Example 3 (Laptop Selection). Let

U = {XPS13, Envy13, MacBookAir, MacBookPro}.

Consider three attribute value–sets:

A1 = {Dell, HP, Apple}, A2 = {i5, i7, i9}, A3 = {Budget, Midrange, Premium},

which are pairwise disjoint. Form

C = P(A1)× P(A2)× P(A3).

Define the SuperHypersoft Set (F, C) by, for example,

F
(
{Dell,HP}, {i7}, {Midrange}

)
= {XPS13, Envy13},

F
(
{Apple}, {i5, i7}, {Premium}

)
= {MacBookAir, MacBookPro}.

Here:

• γ1 = ({Dell,HP}, {i7}, {Midrange}) yields F (γ1) = {XPS13,Envy13}, the midrange
i7 models from Dell and HP.

• γ2 = ({Apple}, {i5, i7}, {Premium}) yields F (γ2) = {MacBookAir,MacBookPro},
the premium Apple models.

Example 4 (Candidate Recruitment). Let

U = {Alice, Bob, Carol}



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 8 of 33

be a set of applicants. Consider three attributes:

A1 = {Developer, Manager}, A2 = {Java, Python}, A3 = {Remote, Onsite},

with C = P(A1)× P(A2)× P(A3). Define

F
(
{Developer}, {Python}, {Remote,Onsite}

)
= {Alice, Bob},

F
(
{Developer,Manager}, {Java}, {Onsite}

)
= {Bob, Carol}.

Thus:

• For γ1 = ({Developer}, {Python}, {Remote,Onsite}), F (γ1) = {Alice,Bob}.

• For γ2 = ({Developer,Manager}, {Java}, {Onsite}), F (γ2) = {Bob,Carol}.

2.5. Quantum-Soft Sets

A complex Hilbert space is a complete inner-product space over C. In the finite-
dimensional setting, let

U = {u1, u2, . . . , un}

be a finite universe, and denote by H(U) the n-dimensional complex Hilbert space with
orthonormal basis {|ui⟩}ni=1.

Definition 4 (Quantum-Soft Set [3]). Let A = {a1, a2, . . . , am} be a finite set of parame-
ters. A Quantum-Soft Set over (U,A) is a mapping

F : A −→ H(U), aj 7→ |ψaj ⟩ =
n∑

i=1

αi,j |ui⟩ ,

subject to the normalization condition

n∑
i=1

∣∣αi,j

∣∣2 = 1, j = 1, 2, . . . ,m.

Measuring the state |ψaj ⟩ in the computational basis {|ui⟩} yields outcome |ui⟩ with prob-
ability

P
(
ui | aj

)
=

∣∣αi,j

∣∣2.
Example 5 (Investment Portfolio Selection). Let

U = {Tech, Pharma, Energy}, A = {Growth, Stability}.

Identify H(U) with C3 via the basis {|Tech⟩ , |Pharma⟩ , |Energy⟩}. Define the mapping
F : A→ H(U) by

|ψGrowth⟩ =
√

0.60 |Tech⟩+
√

0.30 |Pharma⟩+
√

0.10 |Energy⟩ ,



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 9 of 33

|ψStability⟩ =
√

0.20 |Tech⟩+
√

0.50 |Pharma⟩+
√

0.30 |Energy⟩ .
Each state is manifestly normalized (e.g. 0.60 + 0.30 + 0.10 = 1). Measuring in the basis
{|Tech⟩ , |Pharma⟩ , |Energy⟩} yields

P (Tech | Growth) = 0.60, P (Pharma | Growth) = 0.30, P (Energy | Growth) = 0.10,

P (Tech | Stability) = 0.20, P (Pharma | Stability) = 0.50, P (Energy | Stability) = 0.30.

This illustrates how a classical soft-set membership (uniform or weighted) can be embedded
into a quantum state by choosing amplitudes whose squared moduli reproduce the desired
probabilities.

We present below several mathematical theorems related to Quantum Soft Sets.

Theorem 1 (Soft Set Embedding). Let (F , S) be a classical Soft Set over the finite
universe U = {u1, . . . , un} with parameter set S = {a1, . . . , am} and

F : S −→ P(U), aj 7→ F(aj) ⊆ U.

Define a map
FQ : S −→ H(U)

by assigning to each aj ∈ S the quantum state

|ψj⟩ =
n∑

i=1

αij |ui⟩ , where αij =


1√∣∣F(aj)

∣∣ if ui ∈ F(aj),

0 otherwise.

Then FQ is a valid Quantum-Soft Set over (U, S). Moreover, measuring |ψj⟩ yields exactly
the classical Soft Set membership:

P (ui | aj) =
∣∣αij

∣∣2 =


1∣∣F(aj)

∣∣ , ui ∈ F(aj),

0, ui /∈ F(aj).

Proof. Fix aj ∈ S. By construction, αij ̸= 0 precisely when ui ∈ F(aj), and there are∣∣F(aj)
∣∣ such indices. Hence

n∑
i=1

∣∣αij

∣∣2 =
∣∣F(aj)

∣∣× 1∣∣F(aj)
∣∣ = 1,

so |ψj⟩ is normalized. Upon measurement in the basis {|ui⟩}, the Born rule gives

P (ui | aj) =
∣∣αij

∣∣2 =


1∣∣F(aj)

∣∣ , if ui ∈ F(aj),

0, otherwise,

which matches exactly the uniform membership values of the classical Soft Set F . There-
fore, FQ is a Quantum-Soft Set whose measurement reproduces (F , S).



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Theorem 2 (Unitary Invariance). Let F : A → H(U) be any Quantum-Soft Set over
(U,A). For each a ∈ A, let Ua be a unitary operator on H(U). Define a new map

F ′ : A −→ H(U), a 7→ |ψ′
a⟩ = Ua |ψa⟩ ,

where |ψa⟩ = F (a). Then F ′ is also a Quantum-Soft Set over (U,A).

Proof. Fix a ∈ A. Since Ua is unitary, it preserves inner products and norms. In
particular, 〈

ψa | ψa

〉
= 1 =⇒

〈
ψ′
a | ψ′

a

〉
=

〈
Uaψa | Uaψa

〉
=

〈
ψa | U †

aUa | ψa

〉
=

〈
ψa | ψa

〉
= 1.

Hence each |ψ′
a⟩ is normalized and F ′ satisfies the definition of a Quantum-Soft Set.

Theorem 3 (Crisp State Characterization). Let F : A → H(U) be a Quantum-Soft Set.
Then F induces a classical (crisp) Soft Set if and only if, for every a ∈ A, the quantum
state |ψa⟩ has exactly one nonzero amplitude (up to a global phase). Equivalently, there
exists an index ia such that∣∣αia,a

∣∣ = 1, and αi,a = 0
(
∀ i ̸= ia

)
.

In that case, measuring |ψa⟩ yields the deterministic outcome |uia⟩, reproducing the clas-
sical Soft Set assignment F(a) = {uia}.

Proof. ⇒ Suppose F induces a crisp Soft Set (F , A). By definition, for each a ∈ A,
measuring |ψa⟩ must collapse deterministically to a single element uia . The Born rule then

implies
∣∣αia,a

∣∣2 = 1 and
∣∣αi,a

∣∣2 = 0 for i ̸= ia. Thus αia,a = eiθ (global phase) and αi,a = 0
otherwise.
⇐ Conversely, if for each a ∈ A there is a unique index ia with

∣∣αia,a

∣∣ = 1 and αi,a = 0
for i ̸= ia, then

n∑
i=1

∣∣αi,a

∣∣2 = 12 = 1,

and measurement of |ψa⟩ yields

P (ui | a) =
∣∣αi,a

∣∣2 =

{
1, i = ia,

0, i ̸= ia,

deterministically collapsing to |uia⟩. Defining F(a) = {uia} recovers a classical Soft Set.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 11 of 33

Theorem 4 (Normalization Implies Probability Distribution). Let F : A → H(U) be a
Quantum-Soft Set with

|ψa⟩ =
n∑

i=1

αi,a |ui⟩ ,
n∑

i=1

∣∣αi,a

∣∣2 = 1.

Then for each a ∈ A, the collection { |αi,a|2 | i = 1, . . . , n} forms a probability distribution
on U . In particular,

0 ≤ |αi,a|2 ≤ 1,
n∑

i=1

|αi,a|2 = 1.

Proof. Since αi,a ∈ C, each
∣∣αi,a

∣∣2 is a real number satisfying 0 ≤
∣∣αi,a

∣∣2 ≤ 1. The

normalization condition
∑n

i=1

∣∣αi,a

∣∣2 = 1 then implies {|αi,a|2} is a discrete probability
distribution indexed by i = 1, . . . , n. Hence the measurement outcomes |ui⟩ occur with
probability |αi,a|2.

3. Result: Quantum-HyperSoft Set and Quantum-SuperHyperSoft Set

In this section, we present the main results of this paper by providing the definitions,
concrete examples, and mathematical theorems of the Quantum-HyperSoft Set and the
Quantum-SuperHyperSoft Set.

3.1. Quantum-HyperSoft Sets

Quantum-HyperSoft Set assigns each attribute tuple to a normalized quantum super-
position over universe. The definition is stated as follows.

Definition 5 (Quantum-HyperSoft Set). Let U = {u1, u2, . . . , un} be a finite universe
and let A1, . . . ,Am be m distinct attribute domains. Form the Cartesian product

C = A1 × · · · × Am,

so that each γ = (γ1, . . . , γm) ∈ C is an m-tuple with γi ∈ Ai. Let H(U) be the complex
Hilbert space with orthonormal basis {|ui⟩}ni=1. A Quantum-HyperSoft Set over (U, C) is
a map

Q : C −→ H(U), γ 7→ |ψγ⟩ =

n∑
i=1

αi,γ |ui⟩ ,

subject to the normalization condition

n∑
i=1

∣∣αi,γ

∣∣2 = 1 for each γ ∈ C.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 12 of 33

Here αi,γ ∈ C is the amplitude of element ui under the combined attribute γ. Measuring
|ψγ⟩ in the computational basis yields

P (ui | γ) =
∣∣αi,γ

∣∣2,
so that the post-measurement membership degrees µγ(ui) = P (ui | γ) form a probability
distribution on U .

Example 6 (Quantum-HyperSoft Set for Medical Diagnosis). Let

U =
{
Flu, COVID-19, CommonCold

}
be the set of possible diagnoses. Consider three symptom-attribute domains:

A1 = {HighFever, LowFever}, A2 = {DryCough, WetCough}, A3 = {Fatigue, NoFatigue}.

Form the Cartesian product
C = A1 ×A2 ×A3,

so that each γ = (γ1, γ2, γ3) is a symptom-combination. Let H(U) be the complex Hilbert
space with orthonormal basis {|Flu⟩ , |COVID-19⟩ , |CommonCold⟩}.

Case 1: γ1 = (HighFever,DryCough,Fatigue). Define the quantum state

|ψγ1⟩ =
√

0.65 |COVID-19⟩+
√

0.25 |Flu⟩+
√

0.10 |CommonCold⟩ .

Normalization check:
0.65 + 0.25 + 0.10 = 1.00.

Measurement probabilities:

P
(
COVID-19 | γ1

)
= 0.65, P

(
Flu | γ1

)
= 0.25, P

(
CommonCold | γ1

)
= 0.10.

Case 2: γ2 = (LowFever,WetCough,NoFatigue). Define

|ψγ2⟩ =
√

0.15 |COVID-19⟩+
√

0.30 |Flu⟩+
√

0.55 |CommonCold⟩ ,

with
0.15 + 0.30 + 0.55 = 1.00.

Thus

P
(
COVID-19 | γ2

)
= 0.15, P

(
Flu | γ2

)
= 0.30, P

(
CommonCold | γ2

)
= 0.55.

Interpretation.

• For γ1, the “HighFever + DryCough + Fatigue” combination strongly suggests COVID-
19 (65 %), moderately suggests Flu (25 %), and rarely Common Cold (10 %).



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 13 of 33

• For γ2, the “LowFever + WetCough + NoFatigue” combination most likely corre-
sponds to Common Cold (55 %), with lower likelihoods for Flu (30 %) and COVID-19
(15 %).

This Quantum-HyperSoft Set Q : C → H(U) thus models uncertain diagnoses by en-
coding symptom combinations as quantum-like superpositions, whose measurement proba-
bilities yield a soft membership degree of each disease given the observed symptoms.

Example 7 (Quantum-HyperSoft Set for Credit Risk Assessment). Let

U = {Approve, Review, Deny}

be the set of possible credit decisions. Consider three financial-attribute domains:

A1 = {HighIncome, LowIncome},

A2 = {GoodHistory, PoorHistory},

A3 = {LowDTI, HighDTI},

where DTI = debt-to-income ratio. Form

C = A1 ×A2 ×A3.

Let H(U) be the complex Hilbert space with orthonormal basis {|Approve⟩ , |Review⟩ , |Deny⟩}.

Case 1: γ1 = (HighIncome, GoodHistory, LowDTI). Define

|ψγ1⟩ =
√

0.80 |Approve⟩+
√

0.15 |Review⟩+
√

0.05 |Deny⟩ ,

so that

P (Approve | γ1) = 0.80, P (Review | γ1) = 0.15, P (Deny | γ1) = 0.05.

Case 2: γ2 = (LowIncome, PoorHistory, HighDTI). Define

|ψγ2⟩ =
√

0.10 |Approve⟩+
√

0.20 |Review⟩+
√

0.70 |Deny⟩ ,

with

P (Approve | γ2) = 0.10, P (Review | γ2) = 0.20, P (Deny | γ2) = 0.70.

Interpretation.

• Under γ1, strong profile yields an 80

• Under γ2, weak profile yields a 70



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 14 of 33

Example 8 (Quantum-HyperSoft Set for Traffic Incident Prediction). Let

U = {NoIncident, MinorIncident, MajorIncident}

be the set of traffic-incident outcomes. Consider three sensor-attribute domains:

A1 = {HeavyFlow, LightFlow},

A2 = {AdverseWeather, ClearWeather},

A3 = {Daytime, Nighttime}.

Form
C = A1 ×A2 ×A3,

and let H(U) have basis {|NoIncident⟩ , |MinorIncident⟩ , |MajorIncident⟩}.

Case 1: γ1 = (HeavyFlow, AdverseWeather, Nighttime). Define

|ψγ1⟩ =
√

0.20 |NoIncident⟩+
√

0.50 |MinorIncident⟩+
√

0.30 |MajorIncident⟩ ,

so that

P (NoIncident | γ1) = 0.20, P (MinorIncident | γ1) = 0.50, P (MajorIncident | γ1) = 0.30.

Case 2: γ2 = (LightFlow, ClearWeather, Daytime). Define

|ψγ2⟩ =
√

0.75 |NoIncident⟩+
√

0.20 |MinorIncident⟩+
√

0.05 |MajorIncident⟩ ,

yielding

P (NoIncident | γ2) = 0.75, P (MinorIncident | γ2) = 0.20, P (MajorIncident | γ2) = 0.05.

Interpretation.

• Under γ1, high risk conditions yield a 30 % chance of a major incident.

• Under γ2, benign conditions yield a 75 % chance of no incident.

Theorem 5. Every Hypersoft Set and every Quantum-Soft Set arises as a special case of
a Quantum-HyperSoft Set.

Proof.

(i) Reduction to Hypersoft Set. Let (G, C) be a classical Hypersoft Set over U , so
G : C → P(U) with G(γ) ⊆ U for each γ [21]. Define amplitudes

αi,γ =


1√
|G(γ)|

if ui ∈ G(γ),

0 otherwise.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 15 of 33

Then
∑

i |αi,γ |2 = |G(γ)| · 1
|G(γ)| = 1, so Q(γ) = |ψγ⟩ is a valid Quantum-HyperSoft

state. Moreover, the measurement probabilities satisfy

P (ui | γ) = |αi,γ |2 =


1

|G(γ)|
, ui ∈ G(γ),

0, else,

recovering exactly the classical Hypersoft membership ui ∈ G(γ).

(ii) Reduction to Quantum-Soft Set. If m = 1, then C = A1 is a single attribute domain
and the map

Q : A1 → H(U), a 7→ |ψa⟩ =

n∑
i=1

αi,a |ui⟩ ,

matches precisely the definition of a Quantum-Soft Set, in which each attribute a is
assigned a normalized superposition |ψa⟩ over U [3].

Thus both classical Hypersoft and Quantum-Soft Sets embed naturally in the Quantum-
HyperSoft framework.

Theorem 6 (Unitary Invariance). Let Q be a Quantum-HyperSoft Set over (U, C). For
each γ ∈ C, let Uγ be a unitary operator on H(U), and define

Q′(γ) = Uγ |ψγ⟩ .

Then Q′ is also a Quantum-HyperSoft Set over (U, C).

Proof. Since each Uγ is unitary, it preserves inner products and norms. In particular,〈
ψγ | ψγ

〉
= 1 =⇒

〈
Uγψγ | Uγψγ

〉
=

〈
ψγ | U †

γUγ | ψγ

〉
=

〈
ψγ | ψγ

〉
= 1.

Thus Q′(γ) = Uγ |ψγ⟩ remains normalized for every γ, so Q′ satisfies the definition of a
Quantum-HyperSoft Set.

Theorem 7 (Crisp Hypersoft Embedding). A Quantum-HyperSoft Set Q induces a crisp
Hypersoft Set if and only if for each γ ∈ C there exists a unique index iγ such that

|αiγ ,γ | = 1 and αi,γ = 0 (∀ i ̸= iγ).

In that case, measurement of |ψγ⟩ yields µγ(uiγ ) = 1 and µγ(ui) = 0 for i ̸= iγ, corre-
sponding exactly to the classical Hypersoft assignment G(γ) = {uiγ}.

Proof. (⇒) If Q is such that each |ψγ⟩ collapses deterministically to a single element
uiγ , then the Born rule implies |αiγ ,γ |2 = 1 and |αi,γ |2 = 0 for i ̸= iγ . Hence αiγ ,γ = ±1
(up to a global phase) and all other amplitudes vanish.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 16 of 33

(⇐) Conversely, if the amplitude condition holds, then
∑

i |αi,γ |2 = 12 = 1 and mea-
surement yields

P (ui | γ) =

{
1, i = iγ ,

0, i ̸= iγ ,

so the post-measurement membership is crisp, G(γ) = {uiγ}.

Theorem 8 (Maximum Entropy in Uniform Superposition). Define the Shannon entropy
of the measurement distribution at γ by

H(γ) = −
n∑

i=1

P (ui | γ) logP (ui | γ) = −
n∑

i=1

∣∣αi,γ

∣∣2 log
∣∣αi,γ

∣∣2.
Then H(γ) is maximized if and only if |ψγ⟩ is the uniform superposition 1√

n

∑n
i=1 |ui⟩, in

which case H(γ) = logn.

Proof. By the concavity of the Shannon entropy on probability distributions, the
maximum occurs at the uniform distribution P (ui | γ) = 1/n for all i. The only normalized
pure state yielding this distribution is |ψγ⟩ = 1√

n

∑
i |ui⟩. Hence

H(γ) ≤ −n (1/n) log(1/n) = log n,

with equality if and only if αi,γ = 1/
√
n up to phases.

3.2. Quantum-SuperHyperSoft Sets

Quantum-SuperHyperSoft Set maps each power-set combination of multi-valued at-
tributes to normalized quantum states. The definition is stated as follows.

Definition 6 (Quantum–SuperHyperSoft Set). Let U = {u1, u2, . . . , un} be a finite uni-
verse and let a1, a2, . . . , am be m distinct attributes with corresponding finite, pairwise
disjoint value-sets A1, A2, . . . , Am:

Ai ∩Aj = ∅ (i ̸= j).

Form the Cartesian product of power-sets

C = P(A1)× P(A2)× · · · × P(Am),

so each γ = (α1, α2, . . . , αm) ∈ C with αi ⊆ Ai. Let H(U) be the complex Hilbert space
with orthonormal basis {|ui⟩}ni=1. A Quantum–SuperHyperSoft Set over (U, C) is a map

Q : C −→ H(U), γ 7→ |Ψγ⟩ =

n∑
i=1

αi,γ |ui⟩ ,



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 17 of 33

subject to the normalization condition

n∑
i=1

∣∣αi,γ

∣∣2 = 1 for each γ ∈ C.

Here αi,γ ∈ C is the amplitude of element ui under the combined (possibly multi-valued)
attribute γ. Measuring |Ψγ⟩ yields the probability distribution µγ(ui) = |αi,γ |2 on U .

Example 9 (Quantum–SuperHyperSoft Set for Smartphone Recommendation). Let

U = {PhoneX, PhoneY, PhoneZ}

be three candidate smartphones. Consider three attribute domains with disjoint value-sets:

A1 = {Apple, Samsung, Google}, A2 = {iOS, Android}, A3 = {4G, 5G}.

Form the Cartesian product of power-sets

C = P(A1)× P(A2)× P(A3),

so each element γ = (α1, α2, α3) consists of α1 ⊆ A1, α2 ⊆ A2, α3 ⊆ A3. Let H(U) be
the complex Hilbert space with orthonormal basis {|PhoneX⟩ , |PhoneY⟩ , |PhoneZ⟩}.
Case 1: Premium Apple 5G User

γ1 =
(
{Apple}, {iOS}, {5G}

)
.

Define the quantum state

|Ψγ1⟩ =
√

0.70 |PhoneX⟩+
√

0.20 |PhoneY⟩+
√

0.10 |PhoneZ⟩ .

Check normalization:
0.70 + 0.20 + 0.10 = 1.00.

Measurement probabilities:

P (PhoneX | γ1) = 0.70, P (PhoneY | γ1) = 0.20, P (PhoneZ | γ1) = 0.10.

Case 2: Cross-Platform Budget 4G User

γ2 =
(
{Samsung,Google}, {Android}, {4G}

)
.

Define
|Ψγ2⟩ =

√
0.15 |PhoneX⟩+

√
0.55 |PhoneY⟩+

√
0.30 |PhoneZ⟩ ,

and verify
0.15 + 0.55 + 0.30 = 1.00.

Thus

P (PhoneX | γ2) = 0.15, P (PhoneY | γ2) = 0.55, P (PhoneZ | γ2) = 0.30.

Interpretation.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 18 of 33

• Under γ1 (Apple, iOS, 5G), PhoneX is strongly preferred (70%), PhoneY moderately
(20%), and PhoneZ rarely (10%).

• Under γ2 (Samsung or Google, Android, 4G), PhoneY is most likely (55%), PhoneZ
next (30%), and PhoneX least (15%).

This Quantum–SuperHyperSoft Set Q : C → H(U) encodes complex, overlapping user
preferences (multi-brand, multi-network) as quantum superpositions. Measurement yields
a soft ranking of the three phones for each preference combination.

Example 10 (Quantum–SuperHyperSoft Set for Movie Recommendation). Let

U = {MovieA, MovieB, MovieC}

be three candidate films. Consider three attribute domains:

A1 = {Action, Drama},

A2 = {Light, Dark},

A3 = {Theater, Streaming}.

Form the Cartesian product of power-sets

C = P(A1)× P(A2)× P(A3),

so each γ = (α1, α2, α3) with αi ⊆ Ai. Let H(U) be the Hilbert space with basis

{|MovieA⟩ , |MovieB⟩ , |MovieC⟩}

.

Case 1: Action-Light-Streaming

γ1 =
(
{Action}, {Light}, {Streaming}

)
.

Define
|Ψγ1⟩

=
√

0.80 |MovieA⟩+
√

0.15 |MovieB⟩+
√

0.05 |MovieC⟩ .

Check normalization:
0.80 + 0.15 + 0.05 = 1.00.

Thus

P (MovieA | γ1) = 0.80, P (MovieB | γ1) = 0.15, P (MovieC | γ1) = 0.05.

Case 2: Drama-Dark-Theater

γ2 =
(
{Drama}, {Dark}, {Theater}

)
.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 19 of 33

Define
|Ψγ2⟩ =

√
0.10 |MovieA⟩+

√
0.70 |MovieB⟩+

√
0.20 |MovieC⟩ ,

with
0.10 + 0.70 + 0.20 = 1.00.

Hence

P (MovieA | γ2) = 0.10, P (MovieB | γ2) = 0.70, P (MovieC | γ2) = 0.20.

This example shows how a Quantum–SuperHyperSoft Set models complex viewer pref-
erences—genre, mood, and platform—as quantum superpositions, yielding soft rankings of
films for each preference combination.

Example 11 (Quantum–SuperHyperSoft Set for Supplier Selection). Let

U = {S1, S2, S3}

be three potential suppliers. Consider three attribute domains:

A1 = {Local, Overseas}, A2 = {LowCost, HighCost}, A3 = {ShortLead, LongLead}.

Form
C = P(A1)× P(A2)× P(A3),

and let the basis be {|S1⟩ , |S2⟩ , |S3⟩}.

Case 1: Local, LowCost, ShortLead

γ1 =
(
{Local}, {LowCost}, {ShortLead}

)
.

Set
|Ψγ1⟩ =

√
0.60 |S1⟩+

√
0.30 |S2⟩+

√
0.10 |S3⟩ ,

so
P (S1 | γ1) = 0.60, P (S2 | γ1) = 0.30, P (S3 | γ1) = 0.10.

Case 2: Overseas, HighCost, LongLead

γ2 =
(
{Overseas}, {HighCost}, {LongLead}

)
.

Set
|Ψγ2⟩ =

√
0.05 |S1⟩+

√
0.25 |S2⟩+

√
0.70 |S3⟩ ,

yielding
P (S1 | γ2) = 0.05, P (S2 | γ2) = 0.25, P (S3 | γ2) = 0.70.

This shows how a Quantum–SuperHyperSoft Set can encode multi-criteria supplier
characteristics as quantum-like states, with measurement probabilities giving soft selection
weights.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 20 of 33

Example 12 (Quantum–SuperHyperSoft Set for Algorithm Selection on Quantum Hard-
ware). Let

U = {Grover, QAOA, VQE}

be the set of candidate quantum algorithms. Consider three attribute domains with disjoint
value-sets:

A1 = {IBM, Rigetti, IonQ}, A2 = {Superconducting, TrappedIon}, A3 = {AllToAll, Linear}.

Form the Cartesian product of power-sets

C = P(A1)× P(A2)× P(A3),

so each γ = (α1, α2, α3) with αi ⊆ Ai. Let H(U) be the 3-dimensional complex Hilbert
space with orthonormal basis {|Grover⟩ , |QAOA⟩ , |VQE⟩}.

Case 1: Hybrid Superconducting, All-to-All on IBM or Rigetti

γ1 =
(
{IBM,Rigetti}, {Superconducting}, {AllToAll}

)
.

Define the quantum state

|Ψγ1⟩ =
√

0.50 |QAOA⟩+
√

0.30 |Grover⟩+
√

0.20 |VQE⟩ ,

which satisfies
0.50 + 0.30 + 0.20 = 1.

Measuring in the basis {|Grover⟩ , |QAOA⟩ , |VQE⟩} yields

P (QAOA | γ1) = 0.50, P (Grover | γ1) = 0.30, P (VQE | γ1) = 0.20.

Case 2: Ion-Trapped, Linear Connectivity on IonQ

γ2 =
(
{IonQ}, {TrappedIon}, {Linear}

)
.

Define
|Ψγ2⟩ =

√
0.10 |QAOA⟩+

√
0.20 |Grover⟩+

√
0.70 |VQE⟩ ,

with
0.10 + 0.20 + 0.70 = 1.

Thus

P (VQE | γ2) = 0.70, P (Grover | γ2) = 0.20, P (QAOA | γ2) = 0.10.

Interpretation.

• Under γ1, hybrid superconducting hardware with full connectivity favors QAOA (50
%), then Grover (30 %), then VQE (20 %).



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 21 of 33

• Under γ2, trapped-ion hardware with linear connectivity strongly favors VQE (70 %)
for chemistry simulations, with lower probabilities for Grover (20 %) and QAOA (10
%).

Example 13 (Quantum–SuperHyperSoft Set for Dynamic Circuit Compilation). Let

U = {OptimizeDepth, MinimizeError, Balance}

be compiler optimization strategies. Consider three attribute domains:

A1 = {HighGateError, LowGateError},

A2 = {ShortQubitLifetime, LongQubitLifetime},

A3 = {NoisyInterconnect, ReliableInterconnect}.

Form
C = P(A1)× P(A2)× P(A3),

and let H(U) have orthonormal basis {|OptimizeDepth⟩ , |MinimizeError⟩ , |Balance⟩}.

Case 1: High Error, Short Lifetime, Noisy Links

γ3 =
(
{HighGateError}, {ShortQubitLifetime}, {NoisyInterconnect}

)
.

Set

|Ψγ3⟩ =
√

0.10 |OptimizeDepth⟩+
√

0.70 |MinimizeError⟩+
√

0.20 |Balance⟩ ,

so

P (MinimizeError | γ3) = 0.70, P (Balance | γ3) = 0.20, P (OptimizeDepth | γ3) = 0.10.

Case 2: Low Error, Long Lifetime, Reliable Links

γ4 =
(
{LowGateError}, {LongQubitLifetime}, {ReliableInterconnect}

)
.

Set

|Ψγ4⟩ =
√

0.40 |OptimizeDepth⟩+
√

0.30 |Balance⟩+
√

0.30 |MinimizeError⟩ ,

giving

P (OptimizeDepth | γ4) = 0.40, P (Balance | γ4) = 0.30, P (MinimizeError | γ4) = 0.30.

Interpretation.

• Under γ3, high-noise, short-lived hardware drives error-minimization (70 %) over
depth optimization (10 %) or balanced trade-offs (20 %).



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 22 of 33

• Under γ4, reliable hardware allows deeper circuits (40 %) with balanced (30 %) or
error-focused (30 %) strategies.

Theorem 9 (Generalization of SuperHypersoft Sets). Let (F, C) be a classical SuperHy-
persoft Set F : C → P(U). Define amplitudes

αi,γ =


1√∣∣F (γ)

∣∣ if ui ∈ F (γ),

0 otherwise.

Then the map Q : C → H(U) with |Ψγ⟩ =
∑

i αi,γ |ui⟩ is a Quantum–SuperHyperSoft
Set whose measurement-based membership µγ(ui) = |αi,γ |2 recovers exactly the classical
assignment ui ∈ F (γ).

Proof. For each γ, since αi,γ ̸= 0 exactly when ui ∈ F (γ) and there are |F (γ)| such i,
we have

n∑
i=1

∣∣αi,γ

∣∣2 = |F (γ)| × 1

|F (γ)|
= 1.

Hence Q is normalized. Moreover

µγ(ui) = |αi,γ |2 =

{
1

|F (γ)| , ui ∈ F (γ),

0, ui /∈ F (γ),

so measuring |Ψγ⟩ yields a uniform distribution on the classical set F (γ), thereby embed-
ding the SuperHypersoft Set into the quantum framework.

Theorem 10 (Generalization of Quantum-Soft Sets). Let Q be a Quantum–SuperHyperSoft
Set over (U, C) with m = 1 and A1 = A. Then restricting the domain to {{a} | a ∈ A} ⊆
P(A) gives a map

Q′ : A −→ H(U),

a 7→ |Ψ{a}⟩ ,

which is precisely a Quantum-Soft Set over A.

Proof. When m = 1, C = P(A). Each singleton {a} ⊆ A defines a quantum state
|Ψ{a}⟩ with

∑
i |αi,{a}|2 = 1. Identifying the label {a} ↔ a and renaming |ψa⟩ = |Ψ{a}⟩

recovers the Quantum-Soft Set definition.

Theorem 11 (Generalization of Quantum–HyperSoft Sets). Let Q be a Quantum–SuperHyperSoft
Set over (U, C). Restrict the domain to { ({a1}, {a2}, . . . , {am}) | ai ∈ Ai} ⊆ C. Then the
induced map

Q′′ : A1 × · · · ×Am −→ H(U),

(a1, . . . , am) 7→ |Ψ({a1},...,{am})⟩

is exactly a Quantum–HyperSoft Set over A1, . . . , Am.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 23 of 33

Proof. Each tuple of singletons ({a1}, . . . , {am}) is an element of P(A1)×· · ·×P(Am),
and the associated state |Ψ({a1},...,{am})⟩ is normalized by assumption. Renaming indices
to absorb the singleton braces yields a map on A1 × · · · × Am exactly in the form of the
Quantum–HyperSoft Set definition.

Theorem 12 (Unitary Invariance). Let Q be a Quantum–SuperHyperSoft Set over (U, C).
For each γ ∈ C, let Uγ be a unitary operator on H(U), and define

Q′(γ) = Uγ |Ψγ⟩ .

Then Q′ is also a Quantum–SuperHyperSoft Set over (U, C).

Proof. Since each Uγ is unitary, it preserves inner products and norms. In particular,〈
Ψγ | Ψγ

〉
= 1 =⇒

〈
UγΨγ | UγΨγ

〉
=

〈
Ψγ | U †

γUγ | Ψγ

〉
=

〈
Ψγ | Ψγ

〉
= 1.

Thus each state |Ψ′
γ⟩ = Uγ |Ψγ⟩ remains normalized, so Q′ satisfies the definition of a

Quantum–SuperHyperSoft Set.

Theorem 13 (Crisp Embedding). A Quantum–SuperHyperSoft Set Q induces a crisp
SuperHypersoft Set if and only if for each γ ∈ C there is a unique index iγ such that

|αiγ ,γ | = 1 and αi,γ = 0 (∀ i ̸= iγ).

In that case, measurement of |Ψγ⟩ yields µγ(uiγ ) = 1 and µγ(ui) = 0 for i ̸= iγ, corre-
sponding exactly to the classical SuperHypersoft assignment F (γ) = {uiγ}.

Proof. (⇒) If Q collapses deterministically to a single element uiγ , then the Born rule
implies |αiγ ,γ |2 = 1 and |αi,γ |2 = 0 for i ̸= iγ . Thus αiγ ,γ = eiθ (global phase) and other
amplitudes vanish.

(⇐) Conversely, if the amplitude condition holds, then
∑

i |αi,γ |2 = 1 and measurement
yields

P (ui | γ) =

{
1, i = iγ ,

0, i ̸= iγ ,

so the post-measurement membership is crisp, F (γ) = {uiγ}.

Theorem 14 (Entropy Bounds). Define the Shannon entropy of the measurement distri-
bution at γ by

H(γ) = −
n∑

i=1

∣∣αi,γ

∣∣2 log
∣∣αi,γ

∣∣2.
Then

0 ≤ H(γ) ≤ log n,

with H(γ) = 0 if and only if Q is crisp at γ, and H(γ) = log n if and only if |Ψγ⟩ =
1√
n

∑n
i=1 |ui⟩.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 24 of 33

Proof. By definition, Shannon entropy on a probability distribution is nonnegative, so
H(γ) ≥ 0, with equality exactly when one outcome has probability 1 (crisp case). Con-
cavity of entropy on the simplex implies the maximum occurs at the uniform distribution∣∣αi,γ

∣∣2 = 1/n, yielding
H(γ) ≤ −n (1/n) log(1/n) = log n,

with equality if and only if αi,γ = 1/
√
n up to phase for all i.

Theorem 15 (Orthogonality of Disjoint Crisp States). Let Q be the crisp embedding of
a SuperHypersoft Set F , as in Theorem 2. Then for any γ, γ′ ∈ C,

F (γ) ∩ F (γ′) = ∅ =⇒
〈
Ψγ | Ψγ′

〉
= 0.

Proof. By construction, |Ψγ⟩ =
∑

ui∈F (γ)
1√

|F (γ)|
|ui⟩ and similarly for |Ψγ′⟩. Hence

〈
Ψγ | Ψγ′

〉
=

n∑
i=1

αi,γ αi,γ′√
|F (γ)| |F (γ′)|

=
∑

ui∈F (γ)∩F (γ′)

1√
|F (γ)| |F (γ′)|

.

If F (γ) ∩ F (γ′) = ∅, this sum is zero.

Theorem 16 (Fidelity Equals Classical Overlap for Crisp States). Under the same crisp
embedding, ∣∣⟨Ψγ | Ψγ′⟩

∣∣2 =

∣∣F (γ) ∩ F (γ′)
∣∣2

|F (γ)| |F (γ′)|
.

Proof. From the proof above, ⟨Ψγ |Ψγ′⟩ = |F (γ)∩F (γ′)|√
|F (γ)| |F (γ′)|

. Squaring the modulus yields

the stated formula.

Theorem 17 (Total Variation Distance Bound). For any two quantum states |Ψγ⟩ , |Ψγ′⟩
in a Quantum–SuperHyperSoft Set, let {P (ui | γ)} and {P (ui | γ′)} be their measurement
distributions. Then the total variation distance TV satisfies

TV
(
P (· |γ), P (· |γ′)

)
≤

√
1−

∣∣⟨Ψγ | Ψγ′⟩
∣∣2 .

Proof. It is known that for pure states the fidelity F = |⟨Ψγ |Ψγ′⟩|2 and the classical
total variation distance satisfy

TV(P, P ′) ≤
√

1− F .

Applying this to the two measurement distributions yields the bound.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 25 of 33

Theorem 18 (Projector-Induced Sub-Set). Let Q be a Quantum–SuperHyperSoft Set and
let Π be any projector on H(U) that is diagonal in the {|ui⟩} basis. Define

|Φγ⟩ =
Π |Ψγ⟩∥∥Π |Ψγ⟩

∥∥ ,
whenever Π |Ψγ⟩ ̸= 0. Then the map γ 7→ |Φγ⟩ is itself a Quantum–SuperHyperSoft Set
(over the restricted universe corresponding to Π).

Proof. Because Π is diagonal in the measurement basis, it acts by zeroing some ampli-
tudes and leaving others unchanged. Thus ∥Π |Ψγ⟩ ∥ ≤ 1, and normalizing restores unit
norm: 〈

Φγ | Φγ

〉
=
⟨Ψγ |Π†Π|Ψγ⟩
⟨Ψγ |Π†Π|Ψγ⟩

= 1.

Hence |Φγ⟩ is a valid quantum state for each γ.

3.3. Algorithm for Constructing Quantum-HyperSoft Set and Quantum-
SuperHyperSoft Set

With a view toward future computational experiments, we further examine the con-
struction algorithms for both the Quantum-HyperSoft Set and the Quantum-SuperHyperSoft
Set.

Algorithm 1 (ht). Construct Quantum-HyperSoft Set Classical Hypersoft Set G : C →
P(U), universe U = {u1, . . . , un}. Quantum-HyperSoft Set Q : C → H(U). γ ∈ C

S ← G(γ) k ← |S| i ← 1 n ui ∈ S αi,γ ← 1/
√
k αi,γ ← 0 |ψγ⟩ ←

n∑
i=1

αi,γ |ui⟩

Q(γ)← |ψγ⟩ Q

Example 14 (Quantum-HyperSoft Set for Quantum Algorithm Selection). Consider the
problem of choosing among three quantum algorithms

U = {Grover, QAOA, VQE}

based on hardware and noise characteristics. Define three attribute domains:

A1 = {IBM, IonQ}, A2 = {AllToAll, Linear}, A3 = {LowNoise, HighNoise}.

Form the Cartesian product C = A1×A2×A3. Let G : C → P(U) be the classical Hypersoft
Set defined by:

G
(
(IBM,AllToAll,LowNoise)

)
= {QAOA, VQE},

G
(
(IBM,Linear,HighNoise)

)
= {VQE},

G
(
(IonQ,AllToAll,HighNoise)

)
= {Grover, VQE},

G
(
(IonQ,Linear,LowNoise)

)
= {QAOA}.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 26 of 33

Applying Algorithm 1 to G yields the Quantum-HyperSoft Set Q : C → H(U). For example:

|ψ(IBM,AllToAll,LowNoise)⟩ =
1√
2
|QAOA⟩+

1√
2
|VQE⟩ ,

|ψ(IBM,Linear,HighNoise)⟩ = 1 · |VQE⟩ ,

|ψ(IonQ,AllToAll,HighNoise)⟩ =
1√
2
|Grover⟩+

1√
2
|VQE⟩ ,

|ψ(IonQ,Linear,LowNoise)⟩ = 1 · |QAOA⟩ .

Each state is normalized, and measuring |ψγ⟩ in the basis {|Grover⟩ , |QAOA⟩ , |VQE⟩}
yields the recommended algorithm with the appropriate probabilities.

Algorithm 2 (ht). Construct Quantum-SuperHyperSoft Set Classical SuperHypersoft Set
F : C → P(U), universe U = {u1, . . . , un}. Quantum-SuperHyperSoft Set Q : C → H(U).

γ ∈ C S ← F (γ) k ← |S| i← 1 n ui ∈ S αi,γ ← 1/
√
k αi,γ ← 0 |Ψγ⟩ ←

n∑
i=1

αi,γ |ui⟩

Q(γ)← |Ψγ⟩ Q

Example 15 (Quantum–SuperHyperSoft Set for Quantum Algorithm Deployment). Con-
sider a pool of quantum algorithms

U = {Grover, QAOA, VQE}.

Suppose we have three attribute domains with disjoint value-sets:

A1 = {IBM, IonQ, Rigetti}, A2 = {AllToAll, Linear}, A3 = {LowNoise, HighNoise}.

Form the Cartesian product of power-sets:

C = P(A1)× P(A2)× P(A3).

Define the classical SuperHypersoft Set F : C → P(U) by:

F
(
{IBM,Rigetti}, {AllToAll,Linear}, {LowNoise}

)
= {QAOA, VQE},

F
(
{IonQ}, {AllToAll}, {HighNoise}

)
= {Grover, VQE},

F
(
{IBM, IonQ}, {Linear}, {LowNoise,HighNoise}

)
= {Grover, QAOA}.

Applying Algorithm 2 to F yields the Quantum–SuperHyperSoft Set Q : C → H(U). For
example:

|Ψ{IBM,Rigetti},{AllToAll,Linear},{LowNoise}⟩ =
1√
2
|QAOA⟩+

1√
2
|VQE⟩ ,

|Ψ{IonQ},{AllToAll},{HighNoise}⟩ =
1√
2
|Grover⟩+

1√
2
|VQE⟩ ,



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 27 of 33

|Ψ{IBM,IonQ},{Linear},{LowNoise,HighNoise}⟩ =
1√
2
|Grover⟩+

1√
2
|QAOA⟩ .

Each state is manifestly normalized. Measuring any |Ψγ⟩ in the computational basis
{|Grover⟩ , |QAOA⟩ , |VQE⟩} yields a soft ranking of algorithms adapted to multiple-vendor,
connectivity, and noise-level preferences.

Theorem 19 (Correctness of Construct Quantum-HyperSoft Set). Let G : C →
P(U) be any classical Hypersoft Set over the finite universe U = {u1, . . . , un}. Algorithm 1
produces a map

Q : C −→ H(U), γ 7→ |ψγ⟩

such that each |ψγ⟩ is a unit vector in the Hilbert space H(U). Hence Q is a valid Quantum-
HyperSoft Set.

Proof. Fix an arbitrary γ ∈ C and let S = G(γ) ⊆ U with |S| = k. In the inner loop,
the algorithm sets

αi,γ =

{
1/
√
k, if ui ∈ S,

0, otherwise.

Thus
n∑

i=1

∣∣αi,γ

∣∣2 =
∑
ui∈S

1

k
=
k

k
= 1.

By construction, |ψγ⟩ =
∑n

i=1 αi,γ |ui⟩ is normalized. Since this holds for every γ, the
output map Q satisfies the definition of a Quantum-HyperSoft Set.

Theorem 20 (Correctness of Construct Quantum-SuperHyperSoft Set). Let F :
C → P(U) be any classical SuperHypersoft Set over U = {u1, . . . , un}. Algorithm 2 pro-
duces a map

Q : C −→ H(U), γ 7→ |Ψγ⟩

with each |Ψγ⟩ normalized, so Q is a valid Quantum–SuperHyperSoft Set.

Proof. Identical to the proof of Theorem 19, replacing S = G(γ) by S = F (γ). The
same calculation shows

∑
i |αi,γ |2 = 1 and hence each |Ψγ⟩ is unit-norm.

Theorem 21 (Time and Space Complexity). Assume membership in each S = G(γ) or
S = F (γ) can be tested in O(1) time (e.g. using a hash set). Then both Algorithm 1 and
Algorithm 2 run in O(|C| · n) time and use O(n · |C|) space.

Proof. Let |C| be the number of attribute combinations and n = |U |. For each γ, the
algorithm:

(i) Retrieves S ⊆ U in O(1).

(ii) Computes k = |S| in O(1).



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 28 of 33

(iii) Executes an n-iteration loop where each membership test ui ∈ S takes O(1). Thus
the inner loop is O(n).

(iv) Forms the vector |ψγ⟩ or |Ψγ⟩ in O(n) by summing n amplitudes.

Steps 3–4 dominate, yielding O(n) per γ. Over all γ ∈ C, the total time is

O
(
|C| × n

)
.

Space usage arises from storing one length-n complex vector per γ, for total

O
(
n · |C|

)
complex numbers. This completes the complexity analysis.

4. Conclusion and Future Works

4.1. Conclusion

In this paper, we introduced and rigorously formalized the concepts of the Quantum
Hypersoft Set and the Quantum SuperHyperSoft Set, thereby extending the existing the-
ory of Quantum Soft Sets. We also provided initial algorithmic formulations for their con-
struction. These newly proposed structures offer a promising foundation for representing
and reasoning about complex uncertainty in both Quantum Theory and Soft Computing.
However, since this work is purely theoretical, further validation through computational
experiments and practical implementations is required in future research.

4.2. Future Works

In future work, we plan to extend these quantum-set-based models to more general
structures such as Graphs [45], Hypergraphs [46, 47], Bidirected Graphs [48, 49], and
SuperHypergraphs [50, 51]. Moreover, we aim to incorporate advanced soft computing
paradigms such as the Soft Expert Set [52, 53], IndetermSoft Sets[54, 55], IndetermHy-
perSoft Sets[55, 56], Fuzzy Soft Sets[57], and IndetermSuperHyperSoft Sets[33]. Poten-
tial extensions based on Fuzzy Sets[4, 58], HyperRough Sets[59, 60], Intuitionistic Fuzzy
Sets[61, 62], Neutrosophic Sets[18, 63], and Plithogenic Sets[20, 64] also offer rich avenues
for further development. We hope that future research will pursue computational imple-
mentations and practical applications of these models in real-world decision-making and
uncertainty reasoning scenarios.

Acknowledgements

We wish to thank everyone whose guidance, ideas, and assistance contributed to this
research. Our appreciation goes to the readers for their interest and to the scholars
whose publications provided the groundwork for this study. We are also indebted to the



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 29 of 33

individuals and institutions that supplied the resources and infrastructure necessary for
completing and disseminating this paper. Lastly, we extend our gratitude to all who
offered their support in various capacities.

Data Availability

This paper is purely theoretical and does not involve any empirical data. We welcome
future empirical studies that build upon and test the concepts presented here.

Ethical Approval

As this work is entirely conceptual and involves no human or animal subjects, ethical
approval was not required.

Conflicts of Interest

The authors declare no conflicts of interest in connection with this study or its publi-
cation.

Author Contributions

Conceptualization, Takaaki Fujita and Florentin Smarandache; Investigation, Takaaki
Fujita; Methodology, Takaaki Fujita; Writing – original draft, Takaaki Fujita; Writing –
review & editing, Takaaki Fujita and Florentin Smarandache.

Disclaimer (Limitations and Claims)

The theoretical concepts presented in this paper have not yet been subject to practical
implementation or empirical validation. Future researchers are invited to explore these
ideas in applied or experimental settings. Although every effort has been made to ensure
the accuracy of the content and the proper citation of sources, unintentional errors or
omissions may persist. Readers should independently verify any referenced materials.

To the best of the authors’ knowledge, all mathematical statements and proofs con-
tained herein are correct and have been thoroughly vetted. Should you identify any po-
tential errors or ambiguities, please feel free to contact the authors for clarification.

The results presented are valid only under the specific assumptions and conditions
detailed in the manuscript. Extending these findings to broader mathematical structures
may require additional research. The opinions and conclusions expressed in this work
are those of the authors alone and do not necessarily reflect the official positions of their
affiliated institutions.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6607 30 of 33

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