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Quantum Simulation of the Factorization Problem.

Jose Luis Rosales1, Vicente Martin1

  • 1Center for Computational Simulation, ETS Ingenieros Informáticos, Universidad Politécnica de Madrid, Campus Montegancedo, E28660 Madrid, Spain.

Physical Review Letters
|November 26, 2016
PubMed
Summary

This study introduces a quantum simulator approach for number factorization, deriving a Hamiltonian for a new arithmetic function. The quantum system

Area of Science:

  • Quantum Computing
  • Number Theory
  • Computational Physics

Background:

  • Feynman proposed quantum simulators as computers by finding suitable Hamiltonians.
  • The Pólya-Hilbert conjecture suggested proving the Riemann hypothesis via spectral decomposition of Hermitian operators.
  • Number factorization is a computationally intensive problem with implications for cryptography.

Purpose of the Study:

  • To investigate the use of a quantum simulator for the problem of integer factorization (N=xy).
  • To formulate a new arithmetic function for factorization and derive its corresponding Hamiltonian.
  • To analyze the quantum system's spectrum and its relation to number theoretic functions.

Main Methods:

  • Derivation of the Hamiltonian for a physical system simulating a novel arithmetic function for factorization.

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  • Exact spectral solution of the quantum system without external conditions.
  • Analysis of the quantum system's predictions for the prime counting function.

    • Main Results:

    • A Hamiltonian was derived for a quantum system simulating an arithmetic function based on primes below sqrt[N].
    • The quantum system's spectrum was solved exactly.
    • For x≪sqrt[N], the system predicts a prime counting function nearly identical to Riemann's R(x) function.

    Conclusions:

    • The study demonstrates a quantum simulation approach to number factorization.
    • The derived quantum system's behavior offers a novel prediction for the prime counting function, distinct from analytic number theory.
    • The results stem purely from the quantum theory of the simulator, offering a new perspective on number theoretic problems.