We introduce refined concepts for neutrosophic quantum computing such as neutrosophic quantum states and transformation gates, neutrosophic Hadamard matrix, coherent and decoherent superposition states, entanglement and measurement notions based on neutrosophic quantum states. We also give some observations using these principles. We present a number of quantum computational matrix transformations based on neutrosophic logic and clarify quantum mechanical notions relying on neutrosophic states. The paper is intended to extend the work of Smarandache by introducing a mathematical framework for neutrosophic quantum computing and presenting some results.
We introduce refined concepts for neutrosophic quantum computing such as neutrosophic quantum states and transformation gates, neutrosophic Hadamard matrix, coherent and decoherent superposition states, entanglement and measurement notions based on neutrosophic quantum states. We also give some observations using these principles. We present a number of quantum computational matrix transformations based on neutrosophic logic and clarify quantum mechanical notions relying on neutrosophic states. The paper is intended to extend the work of Smarandache by introducing a mathematical framework for neutrosophic quantum computing and presenting some results.
Quantum computing exploits superposition and entanglement to perform computations beyond the capabilities of classical machines. However, practical quantum systems operate under substantial environmental noise, measurement uncertainty, and decoherence, leading to ambiguous or partially determined quantum states. Conventional probabilistic descriptions capture stochastic behaviour but do not explicitly represent indeterminacy arising from incomplete information or measurement ambiguity. In this work, we introduce a neutrosophic representation of qubit states that integrates principles of Neutrosophic Logic with the mathematical framework of Quantum Computing. In the proposed model, a qubit is characterised by three independent parameters representing truth, indeterminacy, and falsity. This formulation enables explicit representation of ambiguous quantum states and uncertain measurement outcomes. We extend the neutrosophic formalism to quantum gate operations and analyse how the model captures uncertainty propagation in quantum circuits. Through theoretical examples and graphical analysis, we demonstrate that neutrosophic representation provides a richer description of uncertainty compared with conventional probabilistic approaches. The proposed framework offers a conceptual pathway for improved modelling of noisy intermediate-scale quantum devices and may contribute to the development of uncertainty-aware quantum computing architectures.
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 mechanics reveals a reality where uncertainty is not merely epistemic but ontological, challenging classical logics that rely on determinacy. We introduce neutrosophic logic as a novel framework to formalise this intrinsic indeterminacy. Unlike probability or fuzzy systems, neutrosophy treats truth (T), falsity (F), and indeterminacy (I) as independent dimensions, enabling a richer representation of quantum superposition, entanglement, and measurement collapse. We show that neutrosophic triplets capture the undefined yet real state of qubits before observation and provide a coherent description of entangled correlations beyond probabilistic models. This approach offers a unifying formalism that accommodates both structural indeterminacy and outcome probabilities, bridging foundational debates in quantum theory with practical advances in quantum computation and information science. By explicitly integrating indeterminacy into mathematical modelling, neutrosophy advances our understanding of quantum reality and suggests new avenues for experimental validation and quantum technology design.
Neutrosophic information processing extends conventional information processing by explicitly representing three logically independent components: truth, indeterminacy, and falsity. While neutrosophic models have been extensively developed at the mathematical level, their physical implementation remains comparatively underdeveloped. This paper investigates possible physical architectures for neutrosophic information processing, with particular emphasis on quantum systems. We distinguish three levels of implementation. At the first level, ordinary physical devices are used to represent neutrosophic information as a classical data structure. At the second level, physical systems such as photonic, spin, superconducting, atomic, or solid-state platforms encode the truth, indeterminacy, and falsity components in different physical degrees of freedom. At the third level, a genuinely quantum-neutrosophic processor is considered, in which quantum superposition, measurement uncertainty, noise, and explicitly represented indeterminacy are integrated into a single computational model. A mathematical framework based on neutrosophic triples (T, I, F) is introduced together with neutrosophic bits, density operators, measurement maps, and elementary transformation operators. We then discuss possible implementations using multilevel quantum systems, photonic polarization, superconducting circuits, trapped ions, and hybrid quantum-classical architectures. The paper emphasizes the distinction between physical uncertainty already present in quantum mechanics and an explicit logical indeterminacy variable. This distinction is essential for designing experimental systems capable of processing neutrosophic information. The paper concludes with a research program for experimental neutrosophic processors and identifies several open problems concerning noise, calibration, readout, error correction, and complexity.
This presentation provides a comprehensive overview of the foundational principles of quantum computation. It begins by contrasting classical and quantum information, highlighting how quantum bits (qubits) leverage superposition, entanglement, and interference to process information in ways that surpass classical limits. The document explains the four postulates of quantum mechanics: state space (qubits as unit vectors in a Hilbert space), time evolution (via unitary operators), measurement (probabilistic collapse of the state), and composite systems (tensor products of subsystems). Key concepts such as the Bloch sphere representation, quantum gates (e.g., Hadamard, CNOT, Pauli gates), and quantum circuits are discussed, illustrating how quantum operations manipulate qubit states. The presentation also covers quantum evolution, the gate model for computation, and the universality of certain gate sets, emphasizing the potential for quantum parallelism and the transformative impact of quantum computing on fields like cryptography and optimization.
A remarkable application of quantum mechanical concepts of coherent superposition and quantum entanglement is a quantum computer which can solve certain problems at speeds unbelievably faster than the conventional computer. In this chapter, the basic principles and the conditions for the implementation of the quantum computer are introduced and the limitations imposed by the probabilistic nature of quantum mechanics and the inevitable decoherence phenomenon are discussed. Next the basic building blocks, the quantum logic gates, are introduced. These include the Hadamard, the CNOT, and the quantum phase gates. After these preliminaries, the implementation of the Deutsch algorithm, quantum teleportation, and quantum dense coding in terms of the quantum logic gates are discussed. It is also shown how the Bell states can be produced and measured using a sequence of quantum logic gates.