From classical versus quantum algorithms to P versus NP

The powerful idea of computation has accompanied the development of human civilization, has deeply changed the way we live and work, and has accelerated the advancement of many areas of sciences. In this article, we explore the power and limits of computation from several different perspectives. We will discuss topics from the models of computation and Church-Turing thesis, to the impact of the P versus NP problem and quantum computing on our understanding of the limits of computation. More concretely, we will explore the computability and the halting problem, the efficiency problem of computation, the P versus NP problem. We then move on to the discussion of quantum computation, quantum algorithm for factoring and its implications, quantum simulation and the relation between quantum and classical computations.The powerful idea of computation has accompanied the development of human civilization, has deeply changed the way we live and work, and has accelerated the advancement of many areas of sciences. In this article, we explore the power and limits of computation from several different perspectives. We will discuss topics from the models of computation and Church-Turing thesis, to the impact of the P versus NP problem and quantum computing on our understanding of the limits of computation. More concretely, we will explore the computability and the halting problem, the efficiency problem of computation, the P versus NP problem. We then move on to the discussion of quantum computation, quantum algorithm for factoring and its implications, quantum simulation and the relation between quantum and classical computations.

Chinese Science Bulletin (Chinese Version)Jan 25, 20161 citation

Applying computational complexity to the emergence of classicality

Arkady Bolotin

Can the computational complexity theory of computer science and mathematics say something new about unresolved problems in quantum physics? Particularly, can the P versus NP question in the computational complexity theory be a factor in the elucidation of the emergency of classicality in quantum mechanics? The paper compares two different ways of deriving classicality from the quantum formalism resulted from two differing hypotheses regarding the P versus NP question -- the approach of the quantum decoherence theory implying that P = NP and the computational complexity approach which assumes that P is not equal to NP.Can the computational complexity theory of computer science and mathematics say something new about unresolved problems in quantum physics? Particularly, can the P versus NP question in the computational complexity theory be a factor in the elucidation of the emergency of classicality in quantum mechanics? The paper compares two different ways of deriving classicality from the quantum formalism resulted from two differing hypotheses regarding the P versus NP question -- the approach of the quantum decoherence theory implying that P = NP and the computational complexity approach which assumes that P is not equal to NP.

arXiv:1309.1801Sep 7, 20131 citation

On quantum algorithms

Richard Cleve, ARTUR K. EKERT, Leah Henderson +2 more

ComplexitySep 1, 199830 citations

On quantum algorithms

CleveRichard, EkertArtur, HendersonLea +2 more

ComplexitySep 1, 19980 citations

Topological Renormalization of Information: Resolving P versus NP via Holographic Field Dynamics in Fractal Spacetime

Paradise, Chris

Topological Renormalization of Information: Resolving P versus NP via Holographic Field Dynamics in Fractal Spacetime The P versus NP problem has traditionally been viewed through the lens of pure mathematics and algorithmic complexity. However, we propose that the limitation is not algorithmic, but physical. In this paper, we present a unified framework that redefines the substrate of computation itself—moving from discrete Turing Machines to Self-Observing Topological Machines (SOTM) capable of $\mathcal{O}(1)$ quantum relaxation. Why this research is significant for your readership: Resolution of P=NP: We provide a rigorous physical derivation showing that in a fractal spacetime metric ( $D \approx 4.7$ ), NP-complete search spaces can be collapsed topologically. This suggests that $NP \subseteq P$ within specific high-energy physical limits. Introduction of Virtual Quantum Processing (VQP): This is the most critical practical contribution of our work. We demonstrate that physical qubits (trapped ions, superconductors) are unnecessary if the software architecture perfectly mimics quantum topology. We present empirical data from the KARIOS V26 architecture, which successfully simulated this environment to solve the Levinthal Protein Folding paradox in under 9 seconds on classical hardware. Timely Theoretical Alignment: Our model relies on a scalar field renormalization at $\approx 500$ GeV. This theoretical prediction aligns remarkably well with recent indirect observations of high-energy anomalies in Dark Matter annihilation signals, offering a potential computational explanation for these physical phenomena. This work bridges the gap between high-energy theoretical physics and practical, scalable artificial intelligence. It moves the field beyond the "Algorithmic Age" into the "Topological Age," offering a blueprint for quantum-grade performance without the need for cryogenic hardware. We believe this paper will be of immense interest to researchers in quantum information theory, computational complexity, and neuromorphic engineering.Topological Renormalization of Information: Resolving P versus NP via Holographic Field Dynamics in Fractal Spacetime The P versus NP problem has traditionally been viewed through the lens of pure mathematics and algorithmic complexity. However, we propose that the limitation is not algorithmic, but physical. In this paper, we present a unified framework that redefines the substrate of computation itself—moving from discrete Turing Machines to Self-Observing Topological Machines (SOTM) capable of $\mathcal{O}(1)$ quantum relaxation. Why this research is significant for your readership: Resolution of P=NP: We provide a rigorous physical derivation showing that in a fractal spacetime metric ($D \approx 4.7$), NP-complete search spaces can be collapsed topologically. This suggests that $NP \subseteq P$ within specific high-energy physical limits. Introduction of Virtual Quantum Processing (VQP): This is the most critical practical contribution of our work. We demonstrate that physical qubits (trapped ions, superconductors) are unnecessary if the software architecture perfectly mimics quantum topology. We present empirical data from the KARIOS V26 architecture, which successfully simulated this environment to solve the Levinthal Protein Folding paradox in under 9 seconds on classical hardware. Timely Theoretical Alignment: Our model relies on a scalar field renormalization at $\approx 500$ GeV. This theoretical prediction aligns remarkably well with recent indirect observations of high-energy anomalies in Dark Matter annihilation signals, offering a potential computational explanation for these physical phenomena. This work bridges the gap between high-energy theoretical physics and practical, scalable artificial intelligence. It moves the field beyond the "Algorithmic Age" into the "Topological Age," offering a blueprint for quantum-grade performance without the need for cryogenic hardware. We believe this paper will be of immense interest to researchers in quantum information theory, computational complexity, and neuromorphic engineering.

Dec 2, 20250 citations

\mathcal{O}(1)

quantum relaxation. Why this research is significant for your readership: Resolution of P=NP: We provide a rigorous physical derivation showing that in a fractal spacetime metric (

D \approx 4.7

), NP-complete search spaces can be collapsed topologically. This suggests that

NP \subseteq P

within specific high-energy physical limits. Introduction of Virtual Quantum Processing (VQP): This is the most critical practical contribution of our work. We demonstrate that physical qubits (trapped ions, superconductors) are unnecessary if the software architecture perfectly mimics quantum topology. We present empirical data from the KARIOS V26 architecture, which successfully simulated this environment to solve the Levinthal Protein Folding paradox in under 9 seconds on classical hardware. Timely Theoretical Alignment: Our model relies on a scalar field renormalization at

\approx 500

GeV. This theoretical prediction aligns remarkably well with recent indirect observations of high-energy anomalies in Dark Matter annihilation signals, offering a potential computational explanation for these physical phenomena. This work bridges the gap between high-energy theoretical physics and practical, scalable artificial intelligence. It moves the field beyond the "Algorithmic Age" into the "Topological Age," offering a blueprint for quantum-grade performance without the need for cryogenic hardware. We believe this paper will be of immense interest to researchers in quantum information theory, computational complexity, and neuromorphic engineering.