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Geometric Algebra Jordan-Wigner Transformation for Quantum Simulation.

Grégoire Veyrac1, Zeno Toffano1

  • 1Laboratoire Signaux et Systèmes (L2S), UMR 8506, CentraleSupélec, Université Paris-Saclay, CNRS, 91190 Gif-sur-Yvette, France.

Entropy (Basel, Switzerland)
|May 24, 2024
PubMed
Summary

Geometric Algebra (GA) offers a new method for quantum simulation of fermionic systems, simplifying quantum circuits. This approach reformulates transformations like the Jordan-Wigner transformation for more efficient quantum computations.

Keywords:
Geometric algebraquantum computingquantum simulation

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Area of Science:

  • Quantum Information Science
  • Computational Chemistry
  • Quantum Computing Algorithms

Background:

  • Quantum simulation of electronic Hamiltonians requires transformations (e.g., Jordan-Wigner, Bravyi-Kitaev) to handle fermionic properties.
  • These transformations often necessitate additional circuit levels in quantum computations, increasing complexity.
  • A more direct method for incorporating fermionic properties into quantum circuits is needed.

Purpose of the Study:

  • To propose and investigate the use of Geometric Algebra (GA) methods for quantum simulation of fermionic systems.
  • To reformulate existing quantum transformations and Hamiltonians within the GA framework.
  • To demonstrate the application of GA for building quantum simulation circuits, specifically for the Hydrogen molecule.

Main Methods:

  • Application of the Witt basis method within Geometric Algebra (GA) to reformulate the Jordan-Wigner transformation (JWT).
  • Expression of various quantum gates using the GA-based JWT formulation.
  • Rewriting general one- and two-electron Hamiltonians and constructing a quantum simulation circuit for the Hydrogen molecule using GA.
  • Reformulation of the quantum Ising Hamiltonian within the GA framework.

Main Results:

  • A novel GA-based reformulation of the Jordan-Wigner transformation is presented.
  • The GA framework is shown to be well-suited for expressing quantum gates and Hamiltonians relevant to fermionic systems.
  • A quantum simulation circuit for the Hydrogen molecule is successfully constructed using the proposed GA methods.
  • The quantum Ising Hamiltonian is also reformulated in this framework.

Conclusions:

  • Geometric Algebra provides a more straightforward and potentially efficient method for incorporating fermionic properties in quantum simulations.
  • The proposed GA approach simplifies the representation of quantum gates and Hamiltonians, potentially reducing quantum circuit complexity.
  • This work opens new avenues for applying GA in quantum computing for molecular simulations and other fermionic systems.