Quantum Speedup and Mathematical Solutions of Implementing Bio-Molecular Solutions for the Independent Set Problem on...

Area of Science:

• Computational Biology
• Quantum Computing
• Algorithm Design

Background:

• The independent-set problem is a fundamental challenge in graph theory with significant computational complexity.
• Existing bio-molecular and quantum algorithms have limitations in terms of speed and scalability.

Purpose of the Study:

• To propose a novel bio-molecular algorithm for solving the independent-set problem.
• To demonstrate how this bio-molecular approach, when combined with quantum computing, can achieve a quadratic speed-up.
• To explore the potential of quantum-molecular algorithms for other computational problems like element distinctness.

Main Methods:

• Development of a bio-molecular algorithm with specific operational complexities (O(n^2+m) operations, O(2^n) DNA strands).
• Construction of a Boolean circuit derived from bio-molecular solutions.
• Application of a quantum-molecular algorithm to analyze query complexity bounds.

Main Results:

• The bio-molecular algorithm achieves O(n^2+m) biological operations and O(2^n) DNA strands.
• A derived Boolean circuit efficiently finds maximal independent sets.
• The quantum-molecular algorithm reduces the query complexity for the element distinctness problem.
• Experimental validation on IBM's quantum computer for a small independent set problem.

Conclusions:

• Bio-molecular computers can generate Boolean circuits enabling quantum computers to achieve quadratic speed-ups for the independent-set problem.
• Quantum computing can enhance the speed and scalability of bio-molecular computers.
• The proposed quantum-molecular algorithm offers improved query complexity for the element distinctness problem.