[Paper Review] Classical and Quantum Algorithms for the Boolean Satisfiability Problem

Carlos Barrón-Romero|arXiv (Cornell University)|Oct 6, 2015

Quantum Computing Algorithms and Architecture5 references3 citations

TL;DR

This paper proposes a novel classical and quantum algorithmic framework for the Boolean Satisfiability Problem (SAT), reducing the search space via ternary and binary encoding. It claims that while classical SAT is NP-complete and lacks an efficient solution (NP ≠ P), quantum algorithms with coupled quantum Boolean variables achieve linear-time complexity, implying all NP problems can be solved in O(n) time under quantum computation.

ABSTRACT

This paper presents a complete algorithmic study of the decision Boolean Satisfiability Problem under the classical computation and quantum computation theories. The paper depicts deterministic and probabilistic algorithms, propositions of their properties and the main result is that the problem has not an efficient algorithm (NP is not P). Novel quantum algorithms and propositions depict that the complexity by quantum computation approach for solving the Boolean Satisfiability Problem or any NP problem is lineal time.

Motivation & Objective

• To analyze the Boolean Satisfiability Problem (SAT) under classical and quantum computational theories.
• To demonstrate that no efficient classical algorithm exists for SAT, supporting the conjecture NP ≠ P.
• To develop a quantum algorithmic framework that solves SAT and any NP problem in linear time via quantum Boolean variable coupling.
• To introduce the Simple SAT (SSAT) formulation to reduce search space from {0,1,2}ⁿ to {0,1}ⁿ for algorithmic simplification.
• To establish a theoretical foundation for quantum computation as a paradigm shift in solving NP problems efficiently.

Proposed method

• Reduces SAT(n,m) to a Simple SAT (SSAT) problem by mapping clauses to binary strings, restricting variables to present or true/false.
• Uses a ternary encoding system where 0 = ¬xᵢ, 1 = xᵢ, and 2 = xᵢ absent, transforming clauses into ternary numbers.
• Employs an unsatisfactory board detection algorithm to verify absence of solution by checking if all 2ⁿ binary combinations are covered.
• Proposes a quantum algorithm (Algorithm 5) that explores {0,1}ⁿ in O(1) time by coupling quantum variables with SAT formulas.
• Uses a recursive quantum bit assignment strategy to build a solution by evaluating SAT at each bit level with O(n) complexity.
• Introduces a verification algorithm (Algorithm 6) to test consistency of no-solution claims in O(n) time using quantum evaluation.

Experimental results

Research questions

• RQ1 Can the Boolean Satisfiability Problem be reduced to a simpler form that enables efficient algorithmic analysis?
• RQ2 Does a deterministic classical algorithm exist for solving SAT in polynomial time, and if not, why?
• RQ3 Can quantum computation achieve linear-time solution for SAT and other NP problems through quantum variable coupling?
• RQ4 What is the role of quantum superposition and entanglement in enabling exponential speedup for NP decision problems?
• RQ5 Is there a fundamental difference in computational complexity between classical and quantum models for NP problems?

Key findings

• The paper concludes that NP ≠ P, as no efficient classical algorithm exists for solving SAT, supported by the impossibility of finding a universal property for efficient solution construction.
• The Simple SAT (SSAT) formulation reduces the search space from 3ⁿ to 2ⁿ, enabling systematic analysis of solution existence via unsatisfactory board detection.
• A classical algorithm for SSAT runs in O(2ⁿ) time, confirming the exponential complexity of SAT under classical computation.
• Quantum Algorithm 5 achieves O(1) decision complexity for SAT by leveraging quantum superposition and coupling with SAT formulas.
• Quantum Algorithm 5 constructs a solution in O(n) time, with verification of the solution being straightforward and polynomial.
• The paper claims that all NP problems have linear-time complexity under an appropriate quantum computational design, implying a fundamental shift in computational complexity theory.