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Variational quantum evolution equation solver.

Fong Yew Leong1, Wei-Bin Ewe2, Dax Enshan Koh2

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This study introduces a variational quantum algorithm for solving evolution equations using implicit time-stepping. Encoded source states accelerate convergence, demonstrating efficient scaling for quantum computation of differential equations.

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Area of Science:

  • Quantum computing
  • Computational physics
  • Numerical analysis

Background:

  • Variational quantum algorithms (VQAs) are emerging for solving differential equations.
  • Near-term quantum computers present opportunities for novel computational approaches.
  • Efficient numerical methods are crucial for simulating complex physical systems.

Purpose of the Study:

  • To propose a novel variational quantum algorithm for solving general evolution equations.
  • To investigate implicit time-stepping methods for the Laplacian operator.
  • To enhance convergence speed using encoded source states.

Main Methods:

  • Developed a variational quantum algorithm employing implicit time-stepping.
  • Utilized encoded source states derived from previous solutions for initialization.
  • Performed statevector simulations for the heat equation to analyze scaling.
  • Extended the algorithm to higher-order schemes like Crank-Nicolson.
  • Proposed a semi-implicit scheme for non-linear systems.

Main Results:

  • Encoded source states significantly improve convergence over random re-initialization.
  • Demonstrated the algorithm's time complexity scaling with Ansatz volume for gradient estimation.
  • Showcased time-to-solution scaling with the diffusion parameter for the heat equation.
  • Validated the algorithm's applicability to reaction-diffusion and Navier-Stokes equations through proof-of-concept.

Conclusions:

  • The proposed VQA offers an efficient method for solving evolution equations on quantum computers.
  • The use of encoded states is key to faster convergence in quantum simulations.
  • The algorithm is adaptable to various time-stepping schemes and complex non-linear equations.