Optimal Scalarization Functions for Neutrosophic Quantum Search

1. Introduction

Quantum computation provides computational procedures that exploit superposition, interference, and measurement. One of the most important examples is quantum search. Grover’s algorithm demonstrates that an unstructured search problem can be solved with a quadratic improvement in query complexity compared with classical exhaustive search [2]. The general amplitude-amplification framework extends this idea to arbitrary quantum procedures that generate “good” states with nonzero probability [1].

The conventional formulation of quantum search assumes that a search oracle gives a definite binary answer. For a computational basis state $|x\rangle$, one normally has a predicate

$$\chi \left(x\right)\in \{0,1\}.$$

The state is therefore classified either as marked or unmarked.

Many decision problems do not possess such a sharp classification. Information about a candidate may be incomplete, inconsistent, or partially indeterminate. Neutrosophic set theory was introduced to represent truth, indeterminacy, and falsity as separate components [4]. In a single-valued neutrosophic setting, a candidate may be represented by

$$N=(T,I,F),T,I,F\in [0,1].$$

These three components need not satisfy the classical probabilistic constraint $T+I+F=1$.

Score and accuracy functions have subsequently been used to transform neutrosophic information into quantities suitable for comparison and decision making [5]. A substantial literature has studied the construction and optimization of such score functions [6,7,8].

The purpose of this paper is to investigate the corresponding problem in quantum search. More precisely, suppose that every candidate x has a neutrosophic evaluation

$$N_x=(T_x,I_x,F_x).$$

A quantum search procedure needs a mechanism for converting this three-component information into a scalar quantity that determines how strongly the state $|x\rangle$ should be treated as a desirable candidate.

This motivates the following question:

Which scalarization function is mathematically appropriate for a neutrosophic quantum search problem?

We do not assume that there is a universally optimal scalarization. Instead, we formulate “optimality” relative to explicit criteria: discrimination between candidates, sensitivity to indeterminacy, robustness with respect to perturbations, and compatibility with amplitude amplification.

The main contributions are:

(i)

a mathematical model of neutrosophic search states;

(ii)

a family of normalized scalarization functions;

(iii)

sufficient conditions for monotonicity and boundedness;

(iv)

an indeterminacy-sensitive nonlinear scalarization;

(v)

an optimization criterion for selecting scalarization parameters;

(vi)

a connection between scalarized utility and quantum amplitude amplification;

(vii)

theoretical examples illustrating how different scalarizations produce different search priorities.

The paper is intended as a mathematical framework rather than as a claim that neutrosophic information changes the standard unitary laws of quantum mechanics.

2. Preliminaries

2.1. Neutrosophic Evaluations

We begin with the standard single-valued neutrosophic representation.

Definition 1.

Asingle-valued neutrosophic numberis a triple

$$N=(T,I,F)\in {[0,1]}^3,$$

where T, I, and F denote respectively the degrees of truth, indeterminacy, and falsity.

The independence of the three components is one of the principal features of neutrosophic modeling [4].

Remark 1.

Unlike a probability distribution, a neutrosophic number is not required to satisfy

$$T+I+F=1.$$

Consequently, the scalarization of a neutrosophic number should not automatically be interpreted as a probability.

For two neutrosophic numbers

$$N_1=(T_1,I_1,F_1),N_2=(T_2,I_2,F_2),$$

one natural partial order is

$$N_1⪯N_2$$

whenever

$$T_1\le T_2,I_1\ge I_2,F_1\ge F_2.$$

Thus larger truth and smaller indeterminacy and falsity correspond to a more desirable evaluation.

2.2. Score Functions

A score function is a mapping

$$S:{[0,1]}^3⟶\mathbb{R}$$

that assigns a scalar value to a neutrosophic number.

Several score functions have been proposed for ranking neutrosophic numbers. In particular, score–accuracy approaches use scalar quantities to distinguish alternatives whose truth, indeterminacy, and falsity components differ [5]. More recent work has investigated optimized and alternative score functions because different scalarizations can produce different rankings [6,8].

For the quantum-search setting, however, the scalarization has an additional role: it determines the strength with which a candidate is treated by the search procedure.

3. Neutrosophic Quantum Search

3.1. Search Space

Let

$$\mathcal{H}={\mathbb{C}}^N$$

be an N-dimensional Hilbert space with computational basis

$$\{|0\rangle ,|1\rangle ,\dots ,|N-1\rangle \}.$$

A general normalized search state is

$$|\psi \rangle ={\displaystyle \sum _{x=0}^{N-1}}{\alpha }_x|x\rangle ,{\displaystyle \sum _{x=0}^{N-1}}{\left|{\alpha }_x\right|}^2=1.$$

In the standard Grover problem, a Boolean oracle determines whether x is marked. The original quantum-search construction and its quadratic query improvement are described by Grover [2].

In a neutrosophic search problem, we instead associate with every candidate x a neutrosophic evaluation

$$N_x=(T_x,I_x,F_x).$$

Thus the search problem is represented by the collection

$$\mathcal{N}=\{N_x:x\in \{0,\dots ,N-1\}\}.$$

3.2. Scalarized Utility

Let

$$S:{[0,1]}^3\to [0,1]$$

be a scalarization function.

We define the scalarized utility of candidate x by

$$u_x=S(T_x,I_x,F_x).$$

The quantity $u_x$ should be interpreted as a decision utility or search preference, rather than automatically as a physical quantum probability.

Definition 2.

A scalarization function

$$S:{[0,1]}^3\to [0,1]$$

is calledquantum-search admissibleif it satisfies:

1.

$S(1,0,0)=1$;

2.

$S(0,1,1)=0$;

3.

S is continuous;

4.

S is nondecreasing in T;

5.

S is nonincreasing in I;

6.

S is nonincreasing in F.

These conditions guarantee that a candidate with more truth and less indeterminacy and falsity cannot receive a lower scalarized utility.

4. A Family of Scalarization Functions

4.1. Linear Scalarization

The simplest family is

$$S_{\mathrm{L}}(T,I,F)={\lambda }_{T}T-{\lambda }_{I}I-{\lambda }_{F}F,$$

where

$${\lambda }_T,{\lambda }_I,{\lambda }_F\ge 0.$$

To normalize the function to $[0,1]$, we define

$$\begin{array}{|c|}\hline S_{\mathrm{N}}(T,I,F)={\displaystyle \frac{{\lambda }_{T}T+{\lambda }_I(1-I)+{\lambda }_F(1-F)}{{\lambda }_T+{\lambda }_I+{\lambda }_F}}\\ \hline\end{array}$$

provided

$${\lambda }_T+{\lambda }_I+{\lambda }_F>0.$$

This construction is related to the general philosophy of neutrosophic score functions, while allowing the relative importance of truth, indeterminacy, and falsity to be explicitly controlled [5,6].

Proposition 1.

The normalized linear scalarization $S_{\mathrm{N}}$ maps ${[0,1]}^3$ into $[0,1]$.

Proof. 

Since

$$0\le T\le 1,0\le 1-I\le 1,0\le 1-F\le 1,$$

we have

$$0\le {\lambda }_{T}T+{\lambda }_I(1-I)+{\lambda }_F(1-F)\le {\lambda }_T+{\lambda }_I+{\lambda }_F.$$

Division by the positive denominator gives

$$0\le S_{\mathrm{N}}(T,I,F)\le 1.$$

   □

4.2. Indeterminacy-Penalized Scalarization

In many search applications, indeterminacy should not simply enter linearly. We therefore consider

$$S_{\mathrm{IP}}(T,I,F)={\displaystyle \frac{{\lambda }_{T}T+{\lambda }_I{(1-I)}^{\gamma }+{\lambda }_F{(1-F)}^{\delta }}{{\lambda }_T+{\lambda }_I+{\lambda }_F}}$$

where

$${\lambda }_T,{\lambda }_I,{\lambda }_F\ge 0,\gamma ,\delta >0.$$

The parameters $\gamma$ and $\delta$ control the nonlinear response to indeterminacy and falsity.

For example:

$$\gamma >1$$

produces a stronger penalty as indeterminacy approaches one, while

$$0<\gamma <1$$

makes the scalarization more sensitive to small amounts of indeterminacy.

Theorem 1.

For $\gamma ,\delta >0$, the function $S_{\mathrm{IP}}$ is continuous on ${[0,1]}^3$ and satisfies

$${\displaystyle \frac{\partial S_{\mathrm{IP}}}{\partial T}}\ge 0,{\displaystyle \frac{\partial S_{\mathrm{IP}}}{\partial I}}\le 0,{\displaystyle \frac{\partial S_{\mathrm{IP}}}{\partial F}}\le 0.$$

Proof. 

Continuity follows from continuity of the functions

$$T,{(1-I)}^{\gamma },{(1-F)}^{\delta }.$$

Furthermore,

$${\displaystyle \frac{\partial S_{\mathrm{IP}}}{\partial T}}={\displaystyle \frac{{\lambda }_T}{{\lambda }_T+{\lambda }_I+{\lambda }_F}}\ge 0.$$

Also,

$${\displaystyle \frac{\partial S_{\mathrm{IP}}}{\partial I}}=-{\displaystyle \frac{{\lambda }_I\gamma {(1-I)}^{\gamma -1}}{{\lambda }_T+{\lambda }_I+{\lambda }_F}}\le 0,$$

and similarly

$${\displaystyle \frac{\partial S_{\mathrm{IP}}}{\partial F}}=-{\displaystyle \frac{{\lambda }_F\delta {(1-F)}^{\delta -1}}{{\lambda }_T+{\lambda }_I+{\lambda }_F}}\le 0.$$

   □

5. Entropy-Sensitive Scalarization

The previous functions use the three neutrosophic components directly. We now introduce a measure of uncertainty based on the binary entropy function

$$h\left(p\right)=-plogp-(1-p)log(1-p),$$

with the convention $0log0=0$.

For a neutrosophic evaluation, define

$$H(T,I,F)={\displaystyle \frac{h\left(T\right)+h\left(I\right)+h\left(F\right)}{3log2}}.$$

Then

$$0\le H(T,I,F)\le 1.$$

We define the entropy-sensitive scalarization

$$\begin{array}{|c|}\hline S_{\mathrm{E}}(T,I,F)=S_{\mathrm{N}}(T,I,F)\left(1-\rho H(T,I,F)\right),\\ \hline\end{array}$$

where

$$0\le \rho \le 1.$$

The parameter $\rho$ controls the degree to which uncertainty complexity influences the scalarized utility.

Remark 2.

The entropy term should be interpreted as a modeling device rather than as the von Neumann entropy of a quantum state. It measures the complexity of the neutrosophic evaluation itself.

6. Robust Scalarization

Suppose that the neutrosophic values are uncertain within intervals

$$T\in [T^{-},T^{+}],I\in [I^{-},I^{+}],F\in [F^{-},F^{+}].$$

For a scalarization function S, define its worst-case utility by

$$S_{min}=\underset{\begin{array}{c}T\in [T^{-},T^{+}]\\ I\in [I^{-},I^{+}]\\ F\in [F^{-},F^{+}]\end{array}}{min}S(T,I,F).$$

Similarly, define

$$S_{max}=\underset{\begin{array}{c}T\in [T^{-},T^{+}]\\ I\in [I^{-},I^{+}]\\ F\in [F^{-},F^{+}]\end{array}}{max}S(T,I,F).$$

The robustness interval is therefore

$$\mathcal{R}\left(N\right)=[S_{min},S_{max}].$$

For monotone scalarizations, the extrema occur at the endpoints.

Proposition 2.

If S is nondecreasing in T and nonincreasing in I and F, then

$$S_{min}=S(T^{-},I^{+},F^{+})$$

and

$$S_{max}=S(T^{+},I^{-},F^{-}).$$

Proof. 

The minimum is obtained by choosing the smallest truth value and the largest indeterminacy and falsity values. The maximum is obtained by the opposite choices.    □

This observation gives a computationally simple robust scalarization criterion.

7. Optimality Criteria

The word “optimal” must be defined relative to a specified objective. We therefore introduce three quantitative criteria.

7.1. Discrimination

Let

$$N_a=(T_a,I_a,F_a)$$

and

$$N_b=(T_b,I_b,F_b)$$

be two candidates.

The discrimination gap is

$$D_S(N_a,N_b)=|S\left(N_a\right)-S\left(N_b\right)|.$$

For a collection of candidates

$$\mathcal{X}=\{N_1,\dots ,N_N\},$$

define the average discrimination by

$$D\left(S\right)={\displaystyle \frac{2}{N(N-1)}}\sum _{i<j}|S\left(N_i\right)-S\left(N_j\right)|.$$

Large values of $D\left(S\right)$ correspond to stronger separation of candidates.

7.2. Indeterminacy Sensitivity

Define

$$P_I\left(S\right)=-{\displaystyle \frac{\partial S}{\partial I}}.$$

The average indeterminacy penalty is

$${\overline{P}}_I\left(S\right)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{x=1}^N}\left(-{\displaystyle \frac{\partial S}{\partial I}}\left(N_x\right)\right).$$

A decision maker may impose a target value

$$\eta >0$$

and require

$${\overline{P}}_I\left(S\right)\ge \eta .$$

7.3. Robustness

Let

$${\Delta }_T,{\Delta }_I,{\Delta }_F$$

denote perturbation bounds. The local robustness of S can be measured by the Lipschitz quantity

$$L_S=\underset{(T,I,F)}{sup}\left(\left|{\displaystyle \frac{\partial S}{\partial T}}\right|+\left|{\displaystyle \frac{\partial S}{\partial I}}\right|+\left|{\displaystyle \frac{\partial S}{\partial F}}\right|\right).$$

A smaller value of $L_S$ means that small perturbations of the neutrosophic information have a smaller effect on the scalarized utility.

8. An Optimization Problem for Scalarization

Let

$$\theta =({\lambda }_T,{\lambda }_I,{\lambda }_F,\gamma ,\delta )$$

denote the scalarization parameters.

We write

$$S_{\theta }$$

for the corresponding scalarization function.

A general scalarization-design problem can be written as

$$\begin{array}{|c|}\hline {\displaystyle \underset{\theta \in \Theta }{max}}\left\{\alpha D\left(S_{\theta }\right)+\beta {\overline{P}}_I\left(S_{\theta }\right)-{\gamma }_0{L}_{S_{\theta }}\right\},\\ \hline\end{array}$$

where

$$\alpha ,\beta ,{\gamma }_0\ge 0$$

are application-dependent weights.

The feasible set $\Theta$ may contain conditions such as

$${\lambda }_T+{\lambda }_I+{\lambda }_F=1,$$

$${\lambda }_T,{\lambda }_I,{\lambda }_F\ge 0,$$

and

$$\gamma ,\delta \in [{\gamma }_{min},{\gamma }_{max}].$$

This formulation makes explicit that an “optimal” scalarization is dependent on the intended use of the quantum-search algorithm.

9. Connection with Quantum Amplitude Amplification

Amplitude amplification generalizes Grover’s search algorithm. If a quantum procedure produces a good state with probability a, then amplitude amplification can increase the success probability using a number of applications proportional to

$$O\left({\displaystyle \frac{1}{\sqrt{a}}}\right)$$

under the standard assumptions [1].

We now connect this result to neutrosophic scalarization.

Suppose that the initial state is

$$|\psi \rangle ={\displaystyle \sum _{x=0}^{N-1}}{\alpha }_x|x\rangle .$$

Let

$$u_x=S\left(N_x\right)$$

be the scalarized utility of candidate x.

We define a threshold

$$\tau \in [0,1]$$

and declare

$$x\in G_{\tau }$$

if and only if

$$u_x\ge \tau .$$

The set

$$G_{\tau }=\{x:S\left(N_x\right)\ge \tau \}$$

is the scalarized good-state set.

The corresponding initial good-state probability is

$$a_{\tau }=\sum _{x\in G_{\tau }}{\left|{\alpha }_x\right|}^2.$$

Amplitude amplification then acts on the decomposition

$$|\psi \rangle =\sqrt{a_{\tau }}|G\rangle +\sqrt{1-a_{\tau }}|B\rangle ,$$

where $|G\rangle$ and $|B\rangle$ are normalized good and bad components.

The angle

$${\theta }_{\tau }=arcsin\sqrt{a_{\tau }}$$

determines the amplification dynamics.

After k Grover-type iterations, the idealized good-state probability is

$$P_k={sin}^2\left((2k+1){\theta }_{\tau }\right).$$

Thus the scalarization function influences the quantum search indirectly through the set $G_{\tau }$ and therefore through $a_{\tau }$.

10. Optimality for Quantum Search

The previous section leads to a natural quantum-search objective.

For a scalarization S, define

$$a(S,\tau )=\sum _{x:S\left(N_x\right)\ge \tau }{\left|{\alpha }_x\right|}^2.$$

The ideal amplitude-amplification iteration count is approximately

$$K(S,\tau )\approx {\displaystyle \frac{\pi }{4\sqrt{a(S,\tau )}}}.$$

Therefore, for a fixed threshold $\tau$, scalarizations that produce different good-state sets may produce different search costs.

However, maximizing $a(S,\tau )$ is not by itself sufficient. A scalarization that classifies nearly every state as good may yield a large initial success probability but provide little discrimination.

We therefore introduce the quantum-search utility functional

$$\begin{array}{|c|}\hline Q(S,\tau )={\displaystyle \frac{\Delta (S,\tau )}{1+\kappa K(S,\tau )}},\\ \hline\end{array}$$

where $\kappa >0$ and

$$\Delta (S,\tau )$$

is a discrimination measure between the accepted and rejected states.

One possible definition is

$$\Delta (S,\tau )=\left|{\displaystyle \frac{1}{|G_{\tau }|}}\sum _{x\in G_{\tau }}S\left(N_x\right)-{\displaystyle \frac{1}{|B_{\tau }|}}\sum _{x\in B_{\tau }}S\left(N_x\right)\right|.$$

This quantity measures the separation between accepted and rejected candidates.

11. A Main Theorem

We now state a basic result concerning threshold-based neutrosophic quantum search.

Theorem 2.

Let $S:{[0,1]}^3\to [0,1]$ be a continuous quantum-search admissible scalarization. Let

$$G_{\tau }=\{x:S\left(N_x\right)\ge \tau \}.$$

Assume that

$$a_{\tau }=\sum _{x\in G_{\tau }}{\left|{\alpha }_x\right|}^2>0.$$

Then the corresponding amplitude-amplification procedure requires

$$O\left(a_{\tau }^{-1/2}\right)$$

oracle applications to obtain a good state with constant success probability, subject to the standard amplitude-amplification assumptions.

Proof. 

By construction, the normalized initial state can be decomposed as

$$|\psi \rangle =\sqrt{a_{\tau }}|G\rangle +\sqrt{1-a_{\tau }}|B\rangle .$$

Amplitude amplification rotates the state in the two-dimensional subspace spanned by $|G\rangle$ and $|B\rangle$. Writing

$${\theta }_{\tau }=arcsin\sqrt{a_{\tau }},$$

after k iterations the success probability is

$$P_k={sin}^2\left((2k+1){\theta }_{\tau }\right).$$

Choosing

$$k=O\left({\displaystyle \frac{1}{{\theta }_{\tau }}}\right)$$

produces constant success probability. Since

$${\theta }_{\tau }=arcsin\sqrt{a_{\tau }}=\Theta \left(\sqrt{a_{\tau }}\right)$$

for small $a_{\tau }$, we obtain

$$k=O\left(a_{\tau }^{-1/2}\right).$$

This is the standard amplitude-amplification scaling [1].    □

12. Comparison of Scalarization Functions

We compare four scalarization families.

1.

Linear normalized scalarization:

$$S_{\mathrm{N}}.$$

2.

Nonlinear indeterminacy-penalized scalarization:

$$S_{\mathrm{IP}}.$$

3.

Entropy-sensitive scalarization:

$$S_{\mathrm{E}}.$$

4.

A simple truth-dominant scalarization:

$$S_{\mathrm{T}}(T,I,F)={\displaystyle \frac{{\lambda }_{T}T+{\lambda }_I(1-I)+{\lambda }_F(1-F)}{{\lambda }_T+{\lambda }_I+{\lambda }_F}}.$$

Although $S_{\mathrm{T}}$ has the same algebraic form as $S_{\mathrm{N}}$, it is useful to emphasize that different parameter choices represent different decision policies.

The literature on neutrosophic decision making shows that alternative score functions can lead to different rankings, which motivates application-specific scalarization rather than assuming a single universal score function [6,8].

13. Illustrative Example

Consider three search candidates

$$N_1=(0.90,0.05,0.10),$$

$$N_2=(0.80,0.05,0.05),$$

and

$$N_3=(0.75,0.20,0.05).$$

Consider first the equal-weight linear scalarization

$$S(T,I,F)={\displaystyle \frac{T+(1-I)+(1-F)}{3}}.$$

Then

$$S\left(N_1\right)={\displaystyle \frac{0.90+0.95+0.90}{3}}=0.9167,$$

approximately, while

$$S\left(N_2\right)={\displaystyle \frac{0.80+0.95+0.95}{3}}=0.9000,$$

and

$$S\left(N_3\right)={\displaystyle \frac{0.75+0.80+0.95}{3}}=0.8333.$$

Hence

$$S\left(N_1\right)>S\left(N_2\right)>S\left(N_3\right).$$

Now suppose that indeterminacy is strongly penalized through

$$S_{\mathrm{IP}}(T,I,F)={\displaystyle \frac{T+{(1-I)}^2+(1-F)}{3}}.$$

Then

$$S_{\mathrm{IP}}\left(N_1\right)={\displaystyle \frac{0.90+{\left(0.95\right)}^2+0.90}{3}},$$

while

$$S_{\mathrm{IP}}\left(N_2\right)={\displaystyle \frac{0.80+{\left(0.95\right)}^2+0.95}{3}}.$$

The nonlinear scalarization changes the relative contribution of indeterminacy and can therefore change the separation between candidates.

This example illustrates an important principle: the scalarization function is not merely a technical preprocessing step. It is part of the mathematical specification of the search problem.

14. Threshold Selection

The threshold $\tau$ plays a central role.

For a given scalarization S, define

$$G_{\tau }=\{x:S\left(N_x\right)\ge \tau \}.$$

If $\tau$ is too small, then

$$|G_{\tau }|$$

may become large, reducing the discriminating power of the search.

If $\tau$ is too large, then

$$|G_{\tau }|$$

may become very small, possibly reducing the initial success probability.

We therefore define the threshold-selection problem

$$\begin{array}{|c|}\hline {\tau }^{*}\in \underset{\tau \in [0,1]}{arg\; max}Q(S,\tau ).\\ \hline\end{array}$$

In a practical implementation, the threshold may be chosen from the finite set

$$\{S\left(N_0\right),S\left(N_1\right),\dots ,S\left(N_{N-1}\right)\}.$$

This reduces the continuous optimization problem to a finite search.

15. Robust Neutrosophic Quantum Search

Suppose that each candidate has interval-valued neutrosophic information:

$$N_x=([T_x^{-},T_x^{+}],[I_x^{-},I_x^{+}],[F_x^{-},F_x^{+}]).$$

A conservative scalarization is

$$\underline{S}\left(N_x\right)=S(T_x^{-},I_x^{+},F_x^{+}).$$

A candidate is accepted only when

$$\underline{S}\left(N_x\right)\ge \tau .$$

This produces a robust good-state set

$$G_{\tau }^{\mathrm{rob}}=\{x:\underline{S}\left(N_x\right)\ge \tau \}.$$

The corresponding success probability is

$$a_{\tau }^{\mathrm{rob}}=\sum _{x\in G_{\tau }^{\mathrm{rob}}}{\left|{\alpha }_x\right|}^2.$$

The robust quantum-search complexity therefore becomes

$$O\left({\displaystyle \frac{1}{\sqrt{a_{\tau }^{\mathrm{rob}}}}}\right).$$

This formulation is particularly useful when the neutrosophic parameters are estimated from noisy or incomplete information.

16. Algorithmic Framework

The proposed method can be summarized as follows.

1.

Construct the search space

$$\{|x\rangle :x=0,\dots ,N-1\}.$$

2.

Assign a neutrosophic evaluation

$$N_x=(T_x,I_x,F_x)$$

to each candidate.

3.

Select a parameterized scalarization

$$S_{\theta }.$$

4.

Compute

$$u_x=S_{\theta }\left(N_x\right).$$

5.

Select a threshold $\tau$.

6.

Define

$$G_{\tau }=\{x:u_x\ge \tau \}.$$

7.

Construct the corresponding good-state oracle.

8.

Apply amplitude amplification.

9.

Measure the final state.

10.

Validate the measured candidate using the original neutrosophic information.

The final validation step is important because scalarization is a decision mapping and does not itself constitute physical measurement.

17. Pseudocode

Input:

    Search space X

    Neutrosophic evaluations N[x] = (T[x], I[x], F[x])

    Scalarization S_theta

    Threshold tau

    Initial quantum state |psi>

For each x in X:

    u[x] = S_theta(T[x], I[x], F[x])

Define:

    Good = {x : u[x] >= tau}

Prepare the quantum state |psi>.

Construct an oracle that identifies states in Good.

Estimate the initial good-state probability a.

Apply amplitude amplification approximately

    pi/(4 sqrt(a))

times.

Measure the final quantum state.

Return the measured candidate and

its neutrosophic evaluation.

18. Discussion

The proposed framework highlights three different layers of uncertainty.

First, the search space may contain many possible candidates.

Second, the evaluation of each candidate may itself be uncertain or indeterminate, which is represented by the neutrosophic triple

$$(T,I,F).$$

Third, the quantum algorithm introduces probabilistic measurement outcomes.

These three forms of uncertainty should not be conflated.

In particular, the indeterminacy component I is not the same thing as quantum measurement probability. It belongs to the information model used to evaluate candidates. The scalarization function provides the bridge between this information model and the binary good/bad oracle required by standard amplitude amplification.

This distinction is important for maintaining mathematical consistency.

The framework also shows why there cannot be a universally optimal scalarization independent of the application. If falsity is considered more serious than indeterminacy, then

$${\lambda }_F>{\lambda }_I$$

may be appropriate. If indeterminacy is particularly undesirable, one may choose

$${\lambda }_I\gg {\lambda }_T$$

or use a nonlinear exponent $\gamma >1$.

Conversely, an application that tolerates incomplete information may use a smaller indeterminacy penalty.

Thus optimality is properly formulated as a constrained optimization problem over a family of scalarization functions.

19. Theoretical Properties of the Framework

We record several consequences.

Proposition 3.

Suppose that

$$S\left(N_1\right)\ge S\left(N_2\right)$$

whenever

$$N_1⪰N_2.$$

Then the threshold rule

$$N\mapsto {\mathbf{1}}_{\left\{S\right(N)\ge \tau \}}$$

is monotone with respect to the neutrosophic order.

Proof. 

If $N_1⪰N_2$, then

$$S\left(N_1\right)\ge S\left(N_2\right).$$

Therefore, whenever

$$S\left(N_2\right)\ge \tau ,$$

we also have

$$S\left(N_1\right)\ge \tau .$$

Hence the threshold classifier is monotone. □

Corollary 1.

For an admissible scalarization, increasing truth or decreasing indeterminacy and falsity cannot cause a candidate to move from the good set to the bad set.

Proof. 

This follows immediately from the monotonicity of the scalarization. □

20. Relation to Existing Neutrosophic Score Functions

The classical neutrosophic literature contains several score, accuracy, certainty, and ranking mechanisms. For example, score and accuracy functions have been developed for single-valued and interval-valued neutrosophic information [5]. Other studies have proposed optimized score functions for multi-attribute decision problems [6]. More recent work has continued to investigate limitations of existing score functions and construct improved ranking mechanisms [8].

The present paper differs in its target application. The scalarization is not used merely to rank alternatives. Instead, it determines the membership of candidates in a quantum-search acceptance set:

$$G_{\tau }=\{x:S\left(N_x\right)\ge \tau \}.$$

Consequently, the scalarization affects both the decision ordering and the amplitude-amplification dynamics.

This provides a natural mathematical interface between neutrosophic decision theory and quantum search.

21. Conclusion

We have developed a mathematical framework for optimal scalarization in neutrosophic quantum search.

The central object is a scalarization

$$S:{[0,1]}^3\to [0,1]$$

that maps the truth, indeterminacy, and falsity components of a neutrosophic evaluation into a scalar search utility.

We introduced linear, nonlinear, entropy-sensitive, and robust scalarization mechanisms. We proved basic boundedness and monotonicity properties and formulated an optimization problem incorporating discrimination, indeterminacy sensitivity, and robustness.

The main connection with quantum search is obtained through the threshold set

$$G_{\tau }=\{x:S\left(N_x\right)\ge \tau \}.$$

Once this set is defined, standard amplitude amplification gives the query scaling

$$O\left({\displaystyle \frac{1}{\sqrt{a_{\tau }}}}\right),$$

where

$$a_{\tau }=\sum _{x\in G_{\tau }}{\left|{\alpha }_x\right|}^2.$$

Thus the scalarization affects quantum-search performance through the definition of the good-state set.

An important conclusion is that “optimal” cannot be understood without specifying the objective. A scalarization that maximizes discrimination may not minimize sensitivity to perturbations, and a scalarization that strongly penalizes indeterminacy may classify fewer states as acceptable. Therefore, future work should investigate application-specific optimization of scalarization parameters.

Several directions remain open. These include adaptive scalarization during amplitude amplification, variational optimization of scalarization parameters, interval-valued and dynamic neutrosophic oracles, noisy quantum implementations, and rigorous query-complexity bounds for neutrosophic search models.