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A Perfect Path to Computational Biology and Quantum Computing

Celina M.H. de Figueiredo

Published May 31, 2025• 0 citations
OpenAlex

Abstract

I revisit my contributions to the P versus NP millennium problem and the computational complexity of combinatorial problems, especially those arising in Computational Biology and Quantum Computing, through 20 PhD theses, mine and of my students. I explain how the dichotomy NP-complete versus polynomial-time of long-standing problems together with their multivariate analysis is settled. Yet, intriguing questions remain.

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I revisit my contributions to the P versus NP millennium problem and the computational complexity of combinatorial problems, especially those arising in Computational Biology and Quantum Computing, through 20 PhD theses, mine and of my students. I explain how the dichotomy NP-complete versus polynomial-time of long-standing problems together with their multivariate analysis is settled. Yet, intriguing questions remain.
The P vs. NP problem is a central open question in theoretical computer science, asking whether every problem whose solution can be verified in polynomial time can also be solved in polynomial time. This paper presents an interdisciplinary exploration of this problem by integrating mathematical analysis, quantum computing, and machine learning.We analyze the limitations of polynomial-time reductions among NP-complete problems under the Exponential Time Hypothesis (ETH), providing detailed proofs that offer new insights into the structural complexities within NP. Additionally, we develop a novel quantum-classical hybrid algorithm that combines Grover's algorithm with classical heuristics to solve specific instances of NP-complete problems more efficiently. This includes a comprehensive complexity analysis, scalability discussions, and detailed practical implementation aspects such as oracle constructions.Furthermore, we propose a machine learning framework based on Graph Neural Networks (GNNs) to capture patterns within NP-complete problems. This approach is theoretically justified, enhances heuristic methods, and demonstrates superior performance through extensive experimental comparisons with classical heuristics and exact solvers.By integrating these methods, this work offers fresh perspectives on the P vs. NP problem and contributes to the broader understanding of computational complexity.
We address two sets of long-standing open questions in linear algebra and probability theory, from a computational complexity perspective: stochastic matrix divisibility, and divisibility and decomposability of probability distributions. We prove that finite divisibility of stochastic matrices is an NP-complete problem, and extend this result to nonnegative matrices, and completely-positive trace-preserving maps, i.e. the quantum analogue of stochastic matrices. We further prove a complexity hierarchy for the divisibility and decomposability of probability distributions, showing that finite distribution divisibility is in P, but decomposability is NP-hard. For the former, we give an explicit polynomial-time algorithm. All results on distributions extend to weak-membership formulations, proving that the complexity of these problems is robust to perturbations.
The P versus NP problem stands as the foundational challenge of theoretical computer science and computational complexity. Traditional approaches attempt to resolve the question by treating computation purely as discrete Boolean logic operations over finite symbolic strings. In this paper, we establish that NP-hardness is not an arbitrary logical obstacle, but the exact computational manifestation of the physical and geometric phase space of the universe. Utilizing High-Resolution Quantum Field Theory (HR-QCFT), Rough Operator Algebra (ROA), and the Seonggil Theory of Composite Torsion (STCT), we prove that the non-satisfiable constraints of NP-complete problems are isomorphic to the macroscopic topological scar Rdefect generated by the zero-divisor collapse from 16-dimensional sedenions (S) tooctonions (O). By mapping deterministic algorithms onto rough differential equations driven by paths of vanishing H¨older continuity α_n → 0, we demonstrate that any polynomial-timeheuristic algorithm must truncate the infinite-dimensional High-Tensor signature algebra. Crucially, due to the non-associative G_2-triality of STCT, the non-vanishing Associator anomaly exponentially amplifies truncation errors via non-commutative arithmetic friction η. This forces the algorithmic transition operator ˆMTuring into spectral instability, resulting in a total determinant collapse det( ˆM_Turing) = 0. Consequently, we prove P ̸ = NP as an absolute geometric law governing the computational manifold of the physical universe.