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Neutrosophic Quantum Algorithms: Neutrosophic Shor’s and Grover’s Algorithms

Saeid Jafari

Published Sep 29, 2026• 0 citations
Publisher

Abstract

Quantum computation provides algorithms that exploit superposition, interference, and entanglement to solve certain computational problems more efficiently than known classical algorithms. Two fundamental examples are Shor’s factoring algorithm and Grover’s search algorithm. In practical computational settings, however, information associated with quantum states, oracle responses, and measurements may be uncertain or indeterminate. In this paper, we introduce a mathematical framework for neutrosophic quantum algorithms in which quantum states are supplemented by three independent information components representing truth, indeterminacy, and falsity. We define neutrosophic quantum states, neutrosophic quantum measurements, neutrosophic quantum oracles, and scalarization functions. Using these concepts, we formulate neutrosophic versions of Grover’s search algorithm and Shor’s factoring algorithm. For Grover’s algorithm, the classical binary oracle is generalized to an oracle whose response may carry truth, indeterminacy, and falsity degrees. For Shor’s algorithm, neutrosophic information is incorporated into modular exponentiation, quantum Fourier transformation, measurement, and period estimation. We prove that the proposed construction contains the ordinary quantum algorithms as a special case. We also discuss complexity and indeterminacy propagation. The proposed theory is an information-theoretic mathematical extension of quantum algorithms. It is not intended to modify the physical postulates of quantum mechanics.

DOI
10.20944/preprints202609.2494.v1

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Quantum computation provides algorithms that exploit superposition, interference, and entanglement to solve certain computational problems more efficiently than known classical algorithms. Two fundamental examples are Shor’s factoring algorithm and Grover’s search algorithm. In practical computational settings, however, information associated with quantum states, oracle responses, and measurements may be uncertain or indeterminate. In this paper, we introduce a mathematical framework for neutrosophic quantum algorithms in which quantum states are supplemented by three independent information components representing truth, indeterminacy, and falsity. We define neutrosophic quantum states, neutrosophic quantum measurements, neutrosophic quantum oracles, and scalarization functions. Using these concepts, we formulate neutrosophic versions of Grover’s search algorithm and Shor’s factoring algorithm. For Grover’s algorithm, the classical binary oracle is generalized to an oracle whose response may carry truth, indeterminacy, and falsity degrees. For Shor’s algorithm, neutrosophic information is incorporated into modular exponentiation, quantum Fourier transformation, measurement, and period estimation. We prove that the proposed construction contains the ordinary quantum algorithms as a special case. We also discuss complexity and indeterminacy propagation. The proposed theory is an information-theoretic mathematical extension of quantum algorithms. It is not intended to modify the physical postulates of quantum mechanics.
Quantum computing is a modern new paradigm of computation similarly exploiting the standards of quantum mechanics like superposition, entanglement and quantum interference to finish computations a good way to be unfathomable for classical computers. This paper follows the trails to the theoretical foundations of quantum computing and offers with a few essential algorithms for quantum computing like Shor's set of rules for factorization, Grover's set of rules for search, Quantum Fourier Transform and Quantum Approximate Optimization Algorithm (QAOA). This paper specializes in how complicated the computation is, mathematical ideas underlying such algorithms and their feasible makes use of in cryptography, optimization and simulation in quantum. By sifting thru the interaction among quantum mechanics and algorithmic design, this paintings offers a few perception into the mechanisms that permit for quantum speedup, the cutting-edge obstacles due to hardware obstacles and mistakess charges and what may be carried out withinside the destiny each theoretically and empirically.
Abstract This guide is based on the IBM Quantum course Phase Estimation and Factoring and focuses on Grover’s algorithm, a quantum algorithm for unstructured search problems that provides a quadratic speedup over classical approaches. In an unstructured search problem, we are given a function (often represented as an oracle) that marks one or more “correct” solutions among NNN possible candidates. A classical algorithm requires on the order of O(N)O(N)O(N) evaluations in the worst case. In contrast, Grover’s algorithm finds a solution using only O(N)O(\sqrt{N})O(N) oracle queries. This quadratic improvement implies: The quantum algorithm requires roughly the square root of the number of operations needed classically. Any classical algorithm for unstructured search must have cost at least quadratic relative to Grover’s quantum procedure. Core Idea of the Algorithm Grover’s algorithm relies on: Preparation of a uniform superposition (typically using Hadamard gates). An oracle that marks valid solutions via phase inversion. The Grover diffusion operator, which amplifies the probability amplitude of the marked states through interference. By repeatedly applying the Grover operator, the amplitude of the desired solution grows while others diminish, allowing measurement to reveal the correct answer with high probability. Practical Considerations Despite its theoretical elegance and wide applicability, the quadratic speedup of Grover’s algorithm is unlikely to yield near-term practical advantage. Modern classical hardware operates at extremely high clock rates and is vastly more mature than current quantum devices. For problem sizes accessible to today’s quantum computers, classical systems remain dominant. However, history shows that even sub-quadratic classical improvements — such as: Fast Fourier Transform (FFT) Efficient sorting algorithms like quicksort and mergesort have had transformative practical impact. The crucial distinction is that Grover’s algorithm depends on fundamentally different hardware — quantum processors — which are still in early development. Long-Term Perspective As quantum hardware advances in qubit count, coherence time, and error correction, Grover’s quadratic advantage may become practically significant for: Cryptographic key search Optimization heuristics Combinatorial search problems Amplitude amplification subroutines in broader quantum algorithms Through hands-on experimentation on IBM Q, this lesson demonstrates how theoretical amplitude amplification translates into executable quantum circuits, reinforcing the relationship between interference, probability amplitudes, and computational complexity.
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Quantum computation has attracted much attention since it was shown by Shor and Grover the possibility to implement quantum algorithms able to realize, respectively, factoring and searching in a faster way than any other known classical algorithm. It is possible to use Grover algorithm, taking profit of its ability to find a specific value in a unordered database, to find, for example, the zero of a logical function; the minimal or maximal value in a database or to recognize if an odd number is prime or not. Here we show quantum algorithms to solve those cited mathematical problems. The solution requires the use of a quantum bit string comparator being used as oracle. This quantum circuit compares two quantum states and identifies if they are equal or, otherwise, which of them is the largest. Moreover, we also show the quantum bit string comparator allow us to implement conditional statements in quantum computation, a fundamental structure for designing of algorithms.
This chapter provides a comprehensive introduction to the foundational concepts and advanced developments of mathematical perspectives on quantum information theory. Beginning with the distinctions between classical and quantum information, we explore key measures of quantum correlations such as entanglement entropy, mutual information, and entanglement of purification. Various entanglement quantifiers – including concurrence and negativity – are discussed in the context of bipartite and mixed states. The chapter also introduces quantum channels, noise models, and their physical realizations in systems ranging from superconducting qubits to black holes. We present key quantum algorithms, including Shor’s factoring algorithm and Grover’s search, along with their underlying principles such as the quantum Fourier transform. Bridging theory and practice, this chapter sets the stage for further exploration into quantum error correction, quantum thermodynamics, and the role of information in fundamental physics. The material is designed to be accessible to graduate students, researchers, and professionals across the disciplines of physics, computer science, and engineering.
Quantum computing, with its potential to revolutionize computation, relies fundamentally on the development of efficient algorithms to leverage its unparalleled processing capabilities. This research delves into the creation, exploration, and refinement of quantum computing algorithms, focusing on their applications in optimization, cryptography, and machine learning. Quantum algorithms such as Shor's and Grover's have demonstrated remarkable advantages in factorization and search problems, while more recent innovations are tackling complex challenges in global optimization and data analysis. By integrating principles of quantum mechanics—such as superposition, entanglement, and interference—these algorithms promise exponential speed-ups over classical counterparts in specific domains. This study critically analyzes existing quantum algorithms, proposes advancements in hybrid quantum-classical frameworks, and explores their implications for practical problem-solving. Through simulations and theoretical evaluations, it aims to bridge the gap between quantum theory and real-world applications, contributing to the evolution of quantum computing as a transformative technology.