Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/9801041 · Physical Review Letters

Nonlinear Quantum Mechanics Implies Polynomial-Time Solution for NP -Complete and # P Problems

Daniel S. Abrams, Seth Lloyd

Published Nov 2, 1998• 249 citations
arXivHTMLPDF

Abstract

If quantum states exhibit small nonlinearities during time evolution, then quantum computers can be used to solve $\mathrm{NP}$-complete and # $P$ problems in polynomial time. We provide algorithms that solve $\mathrm{NP}$-complete and # $P$ oracle problems by exploiting nonlinear quantum logic gates. Using the Weinberg model as a simple example, the explicit construction of these gates is derived from the underlying physics. Nonlinear quantum algorithms are also presented using Polchinski type nonlinearities which do not allow for superluminal communication.

DOI
10.1103/physrevlett.81.3992

Related papers

NP in BQP with Nonlinearity

Phil Gossett

If one modifies the laws of Quantum Mechanics to allow nonlinear evolution of quantum states, this paper shows that NP-complete problems would be efficiently solvable in polynomial time with bounded probability (NP in BQP). With that (admittedly very unlikely) assumption, this is demonstrated by describing a polynomially large network of quantum gates that solves the 3SAT problem with bounded probability in polynomial time. As in a previous paper by Abrams and Lloyd (but by a somewhat simpler argument), allowing nonlinearity in the laws of Quantum Mechanics would prove the "weak Church-Turing thesis" to be false. General Relativity is suggested as a possible mechanism to supply the necessary nonlinearity.

arXiv:quant-ph/9804025•Apr 9, 1998•1 citation

Quantum Computing, NP-complete Problems and Chaotic Dynamics

Masanori Ohya, Igor Vasil'evich Volovich

An approach to the solution of NP-complete problems based on quantum computing and chaotic dynamics is proposed. We consider the satisfiability problem and argue that the problem, in principle, can be solved in polynomial time if we combine the quantum computer with the chaotic dynamics amplifier based on the logistic map. We discuss a possible implementation of such a chaotic quantum computation by using the atomic quantum computer with quantum gates described by the Hartree-Fock equations. In this case, in principle, one can build not only standard linear quantum gates but also nonlinear gates and moreover they obey to Fermi statistics. This new type of entaglement related with Fermi statistics can be interesting also for quantum communication theory.

arXiv:quant-ph/9912100•Dec 21, 1999•16 citations

Quantum computers and intractable (NP-complete) computing problems

V. Černý

In this paper we discuss physical aspects of intractable (NP-complete) computing problems. We show, using a specific model, that a quantum-mechanical computer can in principle solve an NP-complete problem in polynomial time; however, it would use an exponentially large energy for that computation. We conjecture that our model reflects a complementarity principle concerning the time and the energy needed to perform an NP-complete computation.

Physical Review A•Jul 1, 1993•38 citations

A Quantum Computing Based Solution Approach for NP-Complete Problems

Seda Nur Gungor, Mehmet Karaköse

Although NP-complete problems have solutions that can be verified in polynomial time, their exponential computational complexity presents significant challenges for classical computing methods. This study focuses on transferring various NP-complete problems into the quantum computing framework. The application of specific quantum computing algorithms is proposed for solving these problems. Within the scope of this study, a reusable quantum component with problem-specific constraints is designed, and a quantum circuit model is introduced to adapt the Quantum Feasibility Labeling algorithm for solving the Maximum Clique problem. Additionally, the Quick Quantum Search algorithm, which performs a search with a single oracle call, is integrated into the solution of the Maximum Independent Set problem. The proposed quantum circuits provide illustrative examples of how the components of an NP-complete problem can be represented in a quantum computing environment and how problem-specific constraints can be implemented within a quantum circuit. The utilization of modular circuit designs effectively demonstrates the applicability of larger and more complex problems in quantum environments. These approaches not only enable more efficient use of computational resources compared to traditional methods but also lead to significant reductions in solution times. This study provides a comprehensive framework that not only demonstrates the practicality of quantum circuits for small-scale NP-complete problems but also lays the groundwork for addressing larger and more complex challenges in future quantum computing applications.

•Feb 19, 2025•4 citations

A New Quantum Algorithm for NP-complete Problems

Makoto Ohya, Masanori Ohya, Igor V. Volovich

An approach to the solution of NP-complete problems based on quantum computing and chaotic dynamics is proposed. We consider the satisfiability problem and argue that the problem, in principle, can be solved in polynomial time if we combine the quantum computer with the chaotic dynamics amplifier based on the logistic map.

•Jan 1, 2002•1 citation

Linear and nonlinear quantum algorithms made explicit

Mitchell Chiew

Quantum computing literature is unstandardized and often-times unwieldy. We design and formalise explicit quantum algorithms with rubrics, pseudocode, quantum circuit diagrams, and exact order-of-magnitude expressions for their complexity. We devise new linear quantum algorithms for representing permutations of objects that outperform the previous world standard. Nonlinear quantum computing is a speculative field, and has the potential to solve NP-complete and harder problems. We find shortcomings in existing nonlinear quantum algorithms, and devise our own explicit algorithm that leaves no doubt as to the power of nonlinear quantum computing. Our work shows the advantage of making quantum algorithms explicit.

•Jan 1, 2019•0 citations
If quantum states exhibit small nonlinearities during time evolution, then quantum computers can be used to solve NP\mathrm{NP}NP-complete and # PPP problems in polynomial time. We provide algorithms that solve NP\mathrm{NP}NP-complete and # PPP oracle problems by exploiting nonlinear quantum logic gates. Using the Weinberg model as a simple example, the explicit construction of these gates is derived from the underlying physics. Nonlinear quantum algorithms are also presented using Polchinski type nonlinearities which do not allow for superluminal communication.
If one modifies the laws of Quantum Mechanics to allow nonlinear evolution of quantum states, this paper shows that NP-complete problems would be efficiently solvable in polynomial time with bounded probability (NP in BQP). With that (admittedly very unlikely) assumption, this is demonstrated by describing a polynomially large network of quantum gates that solves the 3SAT problem with bounded probability in polynomial time. As in a previous paper by Abrams and Lloyd (but by a somewhat simpler argument), allowing nonlinearity in the laws of Quantum Mechanics would prove the "weak Church-Turing thesis" to be false. General Relativity is suggested as a possible mechanism to supply the necessary nonlinearity.
An approach to the solution of NP-complete problems based on quantum computing and chaotic dynamics is proposed. We consider the satisfiability problem and argue that the problem, in principle, can be solved in polynomial time if we combine the quantum computer with the chaotic dynamics amplifier based on the logistic map. We discuss a possible implementation of such a chaotic quantum computation by using the atomic quantum computer with quantum gates described by the Hartree-Fock equations. In this case, in principle, one can build not only standard linear quantum gates but also nonlinear gates and moreover they obey to Fermi statistics. This new type of entaglement related with Fermi statistics can be interesting also for quantum communication theory.
Although NP-complete problems have solutions that can be verified in polynomial time, their exponential computational complexity presents significant challenges for classical computing methods. This study focuses on transferring various NP-complete problems into the quantum computing framework. The application of specific quantum computing algorithms is proposed for solving these problems. Within the scope of this study, a reusable quantum component with problem-specific constraints is designed, and a quantum circuit model is introduced to adapt the Quantum Feasibility Labeling algorithm for solving the Maximum Clique problem. Additionally, the Quick Quantum Search algorithm, which performs a search with a single oracle call, is integrated into the solution of the Maximum Independent Set problem. The proposed quantum circuits provide illustrative examples of how the components of an NP-complete problem can be represented in a quantum computing environment and how problem-specific constraints can be implemented within a quantum circuit. The utilization of modular circuit designs effectively demonstrates the applicability of larger and more complex problems in quantum environments. These approaches not only enable more efficient use of computational resources compared to traditional methods but also lead to significant reductions in solution times. This study provides a comprehensive framework that not only demonstrates the practicality of quantum circuits for small-scale NP-complete problems but also lays the groundwork for addressing larger and more complex challenges in future quantum computing applications.
Quantum computing literature is unstandardized and often-times unwieldy. We design and formalise explicit quantum algorithms with rubrics, pseudocode, quantum circuit diagrams, and exact order-of-magnitude expressions for their complexity. We devise new linear quantum algorithms for representing permutations of objects that outperform the previous world standard. Nonlinear quantum computing is a speculative field, and has the potential to solve NP-complete and harder problems. We find shortcomings in existing nonlinear quantum algorithms, and devise our own explicit algorithm that leaves no doubt as to the power of nonlinear quantum computing. Our work shows the advantage of making quantum algorithms explicit.