Convergence Results for Neutrosophic Amplitude Amplification

1. Introduction

Quantum amplitude amplification is one of the fundamental techniques underlying quantum search algorithms. In its standard form, an algorithm $\mathcal{A}$ prepares a state in which a certain subspace is interpreted as the set of “good” states. Repeated applications of a suitable Grover-type iterate rotate the state toward this good subspace. Brassard, Høyer, Mosca, and Tapp established the general amplitude-amplification framework and showed that a success probability a can be amplified using a number of applications proportional to $1/\sqrt{a}$ rather than the classical $1/a$ scaling [1]. Grover’s original search algorithm is recovered as a special case of this framework [2,3].

The standard theory assumes that the distinction between good and bad states is exact. In many applications, however, information about the success event may involve uncertainty, incomplete information, or contradictory information. Neutrosophic theory was introduced by Smarandache as a framework in which truth, indeterminacy, and falsity can be represented separately [6]. This makes it natural to consider a generalized amplitude-amplification model in which the outcome is not described by a single scalar probability.

Neutrosophic approaches to quantum computation have subsequently been considered in the literature. In particular, neutrosophic quantum computing has been formulated by extending the three-component neutrosophic representation to quantum states and quantum operations [7,8].

The purpose of this paper is to establish a rigorous convergence framework for such a neutrosophic amplitude-amplification process. Our main objectives are:

• to define a neutrosophic representation of the good-state amplitude;
• to construct an abstract neutrosophic amplification map;
• to derive explicit convergence conditions;
• to estimate the limiting truth, indeterminacy, and falsity components;
• to recover ordinary amplitude amplification as a special case.

The central point is that the three neutrosophic components should not be forced into a classical probability interpretation. Instead, they are treated as independent descriptors of the epistemic status of the amplified event.

2. Preliminaries

2.1. Classical Amplitude Amplification

Let $\mathcal{H}$ be a Hilbert space and let $\mathcal{A}$ be a unitary state-preparation operator. We write

$$\mathcal{A}\left|0\right.〉=\sqrt{a}\left|{\psi }_G\right.〉+\sqrt{1-a}\left|{\psi }_B\right.〉,0\le a\le 1,$$

(1)

where $\left|{\psi }_G\right.〉$ belongs to the good subspace and $\left|{\psi }_B\right.〉$ belongs to its orthogonal complement.

It is convenient to introduce an angle $\theta$ satisfying

$${sin}^2\theta =a,0\le \theta \le {\displaystyle \frac{\pi }{2}}.$$

(2)

Then

$$\mathcal{A}\left|0\right.〉=sin\theta \left|{\psi }_G\right.〉+cos\theta \left|{\psi }_B\right.〉.$$

(3)

The amplitude-amplification operator is constructed from two reflections. Its action remains inside the two-dimensional space

$$\mathcal{K}=span\{\left|{\psi }_G\right.〉,\left|{\psi }_B\right.〉\}.$$

(4)

After k amplification steps, the ideal state has the form

$$\left|{\Psi }_k\right.〉=sin\left((2k+1)\theta \right)\left|{\psi }_G\right.〉+cos\left((2k+1)\theta \right)\left|{\psi }_B\right.〉.$$

(5)

Consequently, the probability of obtaining a good state is

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

(6)

These formulas are the basic geometric mechanism behind quantum amplitude amplification [1,4].

2.2. Neutrosophic Triples

We recall the basic neutrosophic representation.

Definition 2.1. 

A neutrosophic triple is an ordered triple

$$\nu =(T,I,F),$$

(7)

where $T,I,F\in [0,1]$ represent, respectively, the degrees of truth, indeterminacy, and falsity.

Unlike an ordinary probability vector, a neutrosophic triple need not satisfy

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

(8)

This separation of truth, indeterminacy, and falsity is one of the basic features of neutrosophic theory [6].

Definition 2.2. 

A neutrosophic amplitude datum is a triple

$$\mathcal{N}=(a_T,a_I,a_F),$$

(9)

where each component belongs to $[0,1]$. The components are interpreted as the truth-success, indeterminate- success, and false-success amplitudes, respectively.

For mathematical convenience, we distinguish between an amplitude and its associated intensity.

Definition 2.3. 

Given a neutrosophic amplitude

$$\mathbf{\alpha }=({\alpha }_T,{\alpha }_I,{\alpha }_F),$$

(10)

its associated neutrosophic intensity is

$$\mathcal{P}\left(\mathbf{\alpha }\right)=\left(|{\alpha }_T{|}^2,|{\alpha }_I{|}^2,{\left|{\alpha }_F\right|}^2\right).$$

(11)

Remark 2.4. 

The three components should not automatically be interpreted as mutually exclusive probabilities. They describe three aspects of the epistemic status of the amplified event.

3. Neutrosophic Quantum States

Let

$$\left|\Psi \right.〉={\alpha }_G\left|{\psi }_G\right.〉+{\alpha }_B\left|{\psi }_B\right.〉,|{\alpha }_G{|}^2+{\left|{\alpha }_B\right|}^2=1.$$

(12)

In the neutrosophic setting, the good amplitude is represented by

$${\mathbf{\alpha }}_G=({\alpha }_T,{\alpha }_I,{\alpha }_F).$$

(13)

We associate with it the neutrosophic success intensity

$${\mathcal{N}}_G=(T_G,I_G,F_G),$$

(14)

where

$$T_G=|{\alpha }_T{|}^2,I_G=|{\alpha }_I{|}^2,F_G={\left|{\alpha }_F\right|}^2.$$

(15)

The three quantities can be regarded as a vector

$${\mathbf{n}}_G=\left(\begin{array}{c}{T}_G\\ I_G\\ F_G\end{array}\right).$$

(16)

This representation allows us to study the evolution of the three components separately.

4. Neutrosophic Amplitude Amplification

Definition 4.1. 

Let $\mathcal{A}$ be a quantum state-preparation operator and let $\mathcal{G}$ denote a good-state oracle. A neutrosophic amplitude amplification process is an iterative sequence

$${\mathcal{N}}_{k+1}={\Phi }_k\left({\mathcal{N}}_k\right),k\ge 0,$$

(17)

where

$${\mathcal{N}}_k=(T_k,I_k,F_k)$$

(18)

and ${\Phi }_k$ is a map acting on the neutrosophic state space.

We first consider the autonomous case.

Definition 4.2. 

A neutrosophic amplification map is a map

$$\Phi :{[0,1]}^3⟶{[0,1]}^3$$

(19)

of the form

$$\Phi (T,I,F)=\left({\Phi }_T(T,I,F),{\Phi }_I(T,I,F),{\Phi }_F(T,I,F)\right).$$

(20)

The ideal objective is to increase the truth component while reducing the indeterminacy and falsity components.

Definition 4.3. 

A fixed point

$${\mathcal{N}}_{*}=(T_{*},I_{*},F_{*})$$

(21)

of Φ satisfies

$$\Phi \left({\mathcal{N}}_{*}\right)={\mathcal{N}}_{*}.$$

(22)

5. The Ideal Neutrosophic Amplification Model

Suppose the initial good-state probability is

$$a={sin}^2\theta .$$

(23)

We define the ideal truth component after k amplification iterations by

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

(24)

The remaining epistemic components are modeled by

$$I_k=(1-T_k){\eta }_I,$$

(25)

and

$$F_k=(1-T_k){\eta }_F,$$

(26)

where

$${\eta }_I,{\eta }_F\in [0,1]$$

(27)

are fixed uncertainty parameters.

Definition 5.1. 

The ideal neutrosophic amplitude amplification sequence associated with $(\theta ,{\eta }_I,{\eta }_F)$ is

$${\mathcal{N}}_k=\left({sin}^2\left((2k+1)\theta \right),{\eta }_I{cos}^2\left((2k+1)\theta \right),{\eta }_F{cos}^2\left((2k+1)\theta \right)\right).$$

(28)

This model is particularly useful because it preserves the ordinary Grover geometry in the truth component while explicitly tracking indeterminacy and falsity.

6. Convergence in the Ideal Case

In general, the sequence

$${sin}^2\left((2k+1)\theta \right)$$

(29)

does not converge as $k\to \infty$. Instead, amplitude amplification is normally stopped at an iteration number for which the success probability is close to one [1].

Thus, convergence must be formulated carefully.

Definition 6.1. 

A neutrosophic amplification sequence $\left\{{\mathcal{N}}_k\right\}$ is said to be ε-convergent to

$${\mathcal{N}}_{*}=(T_{*},I_{*},F_{*})$$

(30)

if there exists $k_0$ such that $\parallel {\mathcal{N}}_k-{\mathcal{N}}_{*}{\parallel }_2<\epsilon$ for the relevant stopping iterations $k\ge k_0$.

For the ideal amplification process, we instead consider a sequence of stopping times.

Let $k_{*}$ be an integer satisfying

$$\left|(2k_{*}+1)\theta -{\displaystyle \frac{\pi }{2}}\right|\le \theta .$$

(31)

Then the following estimate holds.

Proposition 6.2. 

For the ideal amplification sequence,

$$T_{k_{*}}={sin}^2\left((2k_{*}+1)\theta \right),$$

(32)

and therefore

$$1-T_{k_{*}}={cos}^2\left((2k_{*}+1)\theta \right).$$

(33)

Moreover,

$$I_{k_{*}}={\eta }_I(1-T_{k_{*}}),F_{k_{*}}={\eta }_F(1-T_{k_{*}}).$$

(34)

Proof. 

The first equality follows directly from the definition of the ideal amplification sequence. Since

$${sin}^{2}x+{cos}^{2}x=1,$$

we have

$$1-T_{k_{*}}=1-{sin}^2\left((2k_{*}+1)\theta \right)={cos}^2\left((2k_{*}+1)\theta \right).$$

The formulas for $I_{k_{*}}$ and $F_{k_{*}}$ then follow from their definitions. □

7. Perturbed Neutrosophic Amplification

Real implementations may contain phase errors, oracle errors, decoherence, approximation errors, and imperfect knowledge of the initial success amplitude. We therefore consider a perturbed model.

Let

$${\mathcal{N}}_{k+1}=\Phi \left({\mathcal{N}}_k\right)+{\mathbf{e}}_k,$$

(35)

where

$${\mathbf{e}}_k=(e_k^T,e_k^I,e_k^F)$$

(36)

is an error vector.

Assume that $\Phi$ is Lipschitz continuous:

$${∥\Phi \left(\mathcal{N}\right)-\Phi \left(\mathcal{M}\right)∥}_2\le q{∥\mathcal{N}-\mathcal{M}∥}_2,0\le q<1.$$

(37)

Theorem 7.1 

(Convergence under contractive amplification). Let

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

(38)

be a contraction with constant $0\le q<1$. Suppose that

$${\mathcal{N}}_{k+1}=\Phi \left({\mathcal{N}}_k\right)+{\mathbf{e}}_k$$

(39)

and that

$${∥{\mathbf{e}}_k∥}_2\le \epsilon$$

(40)

for all k. Then Φ has a unique fixed point ${\mathcal{N}}_{*}$ and

$${∥{\mathcal{N}}_k-{\mathcal{N}}_{*}∥}_2\le q^k{∥{\mathcal{N}}_0-{\mathcal{N}}_{*}∥}_2+{\displaystyle \frac{\epsilon }{1-q}}.$$

(41)

Proof. 

Since $\Phi$ is a contraction on the complete metric space ${[0,1]}^3$, the Banach fixed-point theorem implies that there exists a unique ${\mathcal{N}}_{*}$ such that

$$\Phi \left({\mathcal{N}}_{*}\right)={\mathcal{N}}_{*}.$$

Therefore

$$\begin{array}{cc}\hfill {∥{\mathcal{N}}_{k+1}-{\mathcal{N}}_{*}∥}_2& ={∥\Phi \left({\mathcal{N}}_k\right)+{\mathbf{e}}_k-\Phi \left({\mathcal{N}}_{*}\right)∥}_2\hfill \\ & \le {∥\Phi \left({\mathcal{N}}_k\right)-\Phi \left({\mathcal{N}}_{*}\right)∥}_2+{∥{\mathbf{e}}_k∥}_2\hfill \\ & \le q{∥{\mathcal{N}}_k-{\mathcal{N}}_{*}∥}_2+\epsilon .\hfill \end{array}$$

Iterating this inequality gives

$${∥{\mathcal{N}}_k-{\mathcal{N}}_{*}∥}_2\le q^k{∥{\mathcal{N}}_0-{\mathcal{N}}_{*}∥}_2+\epsilon {\displaystyle \sum _{j=0}^{k-1}}{q}^j.$$

Since

$${\displaystyle \sum _{j=0}^{k-1}}{q}^j={\displaystyle \frac{1-q^k}{1-q}}\le {\displaystyle \frac{1}{1-q}},$$

we obtain

$${∥{\mathcal{N}}_k-{\mathcal{N}}_{*}∥}_2\le q^k{∥{\mathcal{N}}_0-{\mathcal{N}}_{*}∥}_2+{\displaystyle \frac{\epsilon }{1-q}}.$$

□

Corollary 7.2. 

If

$${\epsilon }_k⟶0$$

(42)

and

$${\mathcal{N}}_{k+1}=\Phi \left({\mathcal{N}}_k\right)+{\mathbf{e}}_k,$$

(43)

where Φ is a contraction, then

$${\mathcal{N}}_k⟶{\mathcal{N}}_{*}.$$

(44)

8. Componentwise Convergence

The previous theorem gives convergence in the Euclidean norm. We now derive componentwise estimates.

Suppose

$$\Phi (T,I,F)=({\Phi }_T(T,I,F),{\Phi }_I(T,I,F),{\Phi }_F(T,I,F)).$$

(45)

Let

$${\mathcal{N}}_{*}=(T_{*},I_{*},F_{*}).$$

(46)

Proposition 8.1. 

Under the assumptions of the convergence theorem,

$$|T_k-T_{*}|\le q^k{∥{\mathcal{N}}_0-{\mathcal{N}}_{*}∥}_2+{\displaystyle \frac{\epsilon }{1-q}},$$

(47)

$$|I_k-I_{*}|\le q^k{∥{\mathcal{N}}_0-{\mathcal{N}}_{*}∥}_2+{\displaystyle \frac{\epsilon }{1-q}},$$

(48)

and

$$|F_k-F_{*}|\le q^k{∥{\mathcal{N}}_0-{\mathcal{N}}_{*}∥}_2+{\displaystyle \frac{\epsilon }{1-q}}.$$

(49)

Proof. 

For every component of a vector $\mathbf{x}\in {\mathbb{R}}^3$,

$$|x_j|\le {∥\mathbf{x}∥}_2.$$

Applying this observation to

$${\mathcal{N}}_k-{\mathcal{N}}_{*}$$

and using the previous theorem gives all three estimates. □

The estimates show that a persistent perturbation produces an asymptotic error floor of order

$${\displaystyle \frac{\epsilon }{1-q}}.$$

(50)

9. Truth Amplification and Uncertainty Suppression

A natural requirement for a neutrosophic amplification algorithm is that the truth component increase while the uncertainty components decrease.

Definition 9.1. 

A neutrosophic amplification map Φ is called truth-enhancing if

$${\Phi }_T(T,I,F)\ge T$$

(51)

whenever

$$T\le T_{*}.$$

(52)

It is called uncertainty-suppressing if

$${\Phi }_I(T,I,F)\le I$$

(53)

and

$${\Phi }_F(T,I,F)\le F.$$

(54)

Theorem 9.2. 

Suppose that Φ is truth-enhancing and uncertainty-suppressing. If

$$T_k\le T_{*}$$

(55)

for all k, then $\left\{T_k\right\}$ is nondecreasing. If, in addition,

$$I_{k+1}\le I_k,F_{k+1}\le F_k,$$

(56)

then the three sequences

$$\left\{T_k\right\},\left\{I_k\right\},\left\{F_k\right\}$$

(57)

are convergent.

Proof. 

Since

$$T_{k+1}\ge T_k,$$

the sequence $\left\{T_k\right\}$ is nondecreasing. Because

$$0\le T_k\le 1,$$

it is bounded above and therefore convergent.

Similarly,

$$I_{k+1}\le I_k$$

and

$$F_{k+1}\le F_k.$$

Since all components belong to $[0,1]$, both sequences are bounded below by zero and hence converge. □

10. A Linear Neutrosophic Model

Consider the affine model

$$\left(\begin{array}{c}{T}_{k+1}\\ I_{k+1}\\ F_{k+1}\end{array}\right)=A\left(\begin{array}{c}{T}_k\\ I_k\\ F_k\end{array}\right)+\mathbf{b},$$

(58)

where

$$A=\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right).$$

(59)

Assume

$$\rho \left(A\right)<1,$$

(60)

where $\rho \left(A\right)$ denotes the spectral radius.

Theorem 10.1. 

If $\rho \left(A\right)<1$, then the linear neutrosophic amplification system has a unique fixed point

$${\mathcal{N}}_{*}={(I-A)}^{-1}\mathbf{b}$$

(61)

and

$${\mathcal{N}}_k-{\mathcal{N}}_{*}=A^k({\mathcal{N}}_0-{\mathcal{N}}_{*}).$$

(62)

Consequently,

$${\mathcal{N}}_k⟶{\mathcal{N}}_{*}.$$

(63)

Proof. 

A fixed point satisfies

$${\mathcal{N}}_{*}=A{\mathcal{N}}_{*}+\mathbf{b},$$

so

$$(I-A){\mathcal{N}}_{*}=\mathbf{b}.$$

Since $\rho \left(A\right)<1$, the matrix $I-A$ is invertible and

$${\mathcal{N}}_{*}={(I-A)}^{-1}\mathbf{b}.$$

Subtracting the fixed-point equation from the iteration gives

$${\mathcal{N}}_{k+1}-{\mathcal{N}}_{*}=A({\mathcal{N}}_k-{\mathcal{N}}_{*}).$$

Induction yields

$${\mathcal{N}}_k-{\mathcal{N}}_{*}=A^k({\mathcal{N}}_0-{\mathcal{N}}_{*}).$$

The condition $\rho \left(A\right)<1$ implies

$$A^k⟶0,$$

and hence

$${\mathcal{N}}_k⟶{\mathcal{N}}_{*}.$$

□

11. Connection with Standard Amplitude Amplification

The neutrosophic framework should contain the classical theory as a special case.

Suppose that

$$I_k=F_k=0$$

(64)

for all k, and let

$$T_k=p_k.$$

(65)

Then

$${\mathcal{N}}_k=(p_k,0,0).$$

(66)

For the standard amplitude-amplification iteration,

$$p_k={sin}^2\left((2k+1)\theta \right),$$

(67)

where

$${sin}^2\theta =a.$$

(68)

Proposition 11.1. 

The neutrosophic amplification model reduces to ordinary quantum amplitude amplification when

$$I_k=F_k=0$$

(69)

and

$$T_k=p_k.$$

(70)

Proof. 

Under these assumptions,

$${\mathcal{N}}_k=(p_k,0,0).$$

Thus the only nonzero component is the ordinary success probability

$$p_k={sin}^2\left((2k+1)\theta \right),$$

which is precisely the standard amplitude-amplification formula [1]. □

12. Convergence Rate

Suppose that the target success probability is

$$1-\delta ,0<\delta <1.$$

(71)

We require

$${sin}^2\left((2k+1)\theta \right)\ge 1-\delta .$$

(72)

Equivalently,

$$\left|(2k+1)\theta -{\displaystyle \frac{\pi }{2}}\right|\le arcsin\sqrt{\delta }.$$

(73)

Hence the desired iteration number is approximately

$$k\approx {\displaystyle \frac{\pi }{4\theta }}-{\displaystyle \frac{1}{2}}.$$

(74)

For small a,

$$\theta =arcsin\sqrt{a}\sim \sqrt{a},$$

(75)

and consequently

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

(76)

This recovers the quadratic improvement of amplitude amplification over classical repetition [1].

13. Neutrosophic Error Bounds

We now combine the classical amplification error with neutrosophic uncertainty.

Let

$${\mathcal{N}}_k=(T_k,I_k,F_k)$$

(77)

and suppose that the ideal truth component is

$$T_k^{\left(0\right)}={sin}^2\left((2k+1)\theta \right).$$

(78)

Assume

$$|T_k-T_k^{\left(0\right)}|\le {\epsilon }_T,$$

(79)

$$|I_k-I_k^{\left(0\right)}|\le {\epsilon }_I,$$

(80)

and

$$|F_k-F_k^{\left(0\right)}|\le {\epsilon }_F.$$

(81)

Then

Proposition 13.1. 

The total neutrosophic error satisfies

$${∥{\mathcal{N}}_k-{\mathcal{N}}_k^{\left(0\right)}∥}_2\le \sqrt{{\epsilon }_T^2+{\epsilon }_I^2+{\epsilon }_F^2}.$$

(82)

Proof. 

By definition,

$${∥{\mathcal{N}}_k-{\mathcal{N}}_k^{\left(0\right)}∥}_2^2=|T_k-T_k^{\left(0\right)}{|}^2+|I_k-I_k^{\left(0\right)}{|}^2+{|F_k-F_k^{\left(0\right)}|}^2.$$

Using the three assumed bounds gives

$${∥{\mathcal{N}}_k-{\mathcal{N}}_k^{\left(0\right)}∥}_2^2\le {\epsilon }_T^2+{\epsilon }_I^2+{\epsilon }_F^2.$$

Taking square roots proves the result. □

14. Example

Consider an initial success probability

$$a=0.01.$$

(83)

Then

$$\theta =arcsin\left(0.1\right).$$

(84)

The ideal truth component after k iterations is

$$T_k={sin}^2((2k+1)arcsin\left(0.1\right)).$$

(85)

Suppose that

$${\eta }_I=0.20,{\eta }_F=0.10.$$

(86)

The neutrosophic state is therefore

$${\mathcal{N}}_k=\left(T_k,0.20(1-T_k),0.10(1-T_k)\right).$$

(87)

For example, at an iteration for which

$$T_k=0.95,$$

(88)

we obtain

$${\mathcal{N}}_k=(0.95,0.01,0.005).$$

(89)

Thus the amplification has produced a high truth component while leaving comparatively small indeterminacy and falsity components.

15. A Fixed-Point Interpretation

The previous sections show that two notions of convergence occur naturally.

First, the ideal Grover-type iteration is an oscillatory rotation. Its useful convergence is therefore convergence along selected stopping iterations toward a desired success threshold.

Second, a noisy or regularized neutrosophic amplification map may be contractive. In that case ordinary fixed-point convergence applies.

This distinction is important.

Remark 15.1. 

One should not claim that the ordinary Grover iteration converges monotonically as $k\to \infty$. The quantity

$${sin}^2\left((2k+1)\theta \right)$$

is generally oscillatory. The appropriate statement is that there exist stopping iterations at which the success probability approaches the desired value. Contractive neutrosophic models, on the other hand, can possess genuine asymptotic convergence.

16. Robustness with Respect to Initial Uncertainty

Let two initial neutrosophic states be

$${\mathcal{N}}_0=(T_0,I_0,F_0)$$

(90)

and

$${\tilde{\mathcal{N}}}_0=({\tilde{T}}_0,{\tilde{I}}_0,{\tilde{F}}_0).$$

(91)

Suppose $\Phi$ is a contraction with constant $q<1$. Then

$${∥{\Phi }^k\left({\mathcal{N}}_0\right)-{\Phi }^k\left({\tilde{\mathcal{N}}}_0\right)∥}_2\le q^k{∥{\mathcal{N}}_0-{\tilde{\mathcal{N}}}_0∥}_2.$$

(92)

Thus initial uncertainty is exponentially suppressed in the contractive model.

Theorem 16.1. 

Under a contractive neutrosophic amplification map, two initial neutrosophic states become asymptotically indistinguishable:

$$\underset{k\to \infty }{lim}{∥{\Phi }^k\left({\mathcal{N}}_0\right)-{\Phi }^k\left({\tilde{\mathcal{N}}}_0\right)∥}_2=0.$$

(93)

Proof. 

Since $0\le q<1$,

$$q^k⟶0.$$

The stated inequality therefore implies the result. □

17. Discussion

The theory developed here separates three effects that are usually combined in a single success probability.

The first component,

$$T_k,$$

measures the degree to which the amplification procedure supports the good outcome.

The second component,

$$I_k,$$

measures unresolved or indeterminate information.

The third component,

$$F_k,$$

measures information supporting the opposite outcome.

This decomposition permits amplification to be studied even when the classification of a state as good or bad is not completely reliable.

The principal mathematical result is the contractive convergence estimate

$${∥{\mathcal{N}}_k-{\mathcal{N}}_{*}∥}_2\le q^k{∥{\mathcal{N}}_0-{\mathcal{N}}_{*}∥}_2+{\displaystyle \frac{\epsilon }{1-q}}.$$

(94)

The first term represents the transient error and decays geometrically. The second term represents the asymptotic error generated by persistent perturbations.

The framework is also compatible with standard quantum amplitude amplification. When the indeterminacy and falsity components vanish, the ordinary quadratic amplification law is recovered.

18. Conclusions

We have introduced a mathematical framework for neutrosophic amplitude amplification and established several convergence results.

The principal conclusions are:

1.

A quantum success amplitude can be extended to a three-component neutrosophic quantity.

2.

The classical amplitude-amplification process is recovered when the indeterminacy and falsity components vanish.

3.

A contractive neutrosophic amplification map possesses a unique fixed point.

4.

Under bounded perturbations, the amplification sequence remains within an explicit error neighborhood of the fixed point.

5.

The truth, indeterminacy, and falsity components can be analyzed separately through componentwise convergence estimates.

6.

The ordinary quadratic quantum speedup

$$O(1/\sqrt{a})$$

remains the natural complexity scale for the ideal amplitude amplification component.

The framework suggests several directions for future research, including neutrosophic amplitude estimation, fixed-point neutrosophic search, noisy neutrosophic quantum channels, variational neutrosophic algorithms, and the relationship between neutrosophic quantum states and generalized quantum measurements.