• Corpus ID: 237532296

A Partially Random Trotter Algorithm for Quantum Hamiltonian Simulations

@article{Jin2021APR,
  title={A Partially Random Trotter Algorithm for Quantum Hamiltonian Simulations},
  author={Shi Jin and Xiantao Li},
  journal={ArXiv},
  year={2021},
  volume={abs/2109.07987}
}
Given the Hamiltonian, the evaluation of unitary operators has been at the heart of many quantum algorithms. Motivated by existing deterministic and random methods, we present a hybrid approach, where Hamiltonians with large amplitude are evaluated at each time step, while the remaining terms are evaluated at random. The bound for the mean square error is obtained, together with a concentration bound. The mean square error consists of a variance term and a bias term, arising respectively from… 

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References

SHOWING 1-10 OF 55 REFERENCES

Random Compiler for Fast Hamiltonian Simulation.

A randomized compiler for Hamiltonian simulation where gate probabilities are proportional to the strength of a corresponding term in the Hamiltonian, especially suited to electronic structure Hamiltonians relevant to quantum chemistry.

Improving quantum algorithms for quantum chemistry

Improvements to the standard Trotter-Suzuki based algorithms used in the simulation of quantum chemistry on a quantum computer by modifying how Jordan-Wigner transformations are implemented to reduce their cost from linear or logarithmic in the number of orbitals to a constant.

Faster quantum simulation by randomization

By simply randomizing how the summands are ordered, one can prove stronger bounds on the quality of approximation for product formulas of any given order, and thereby give more efficient simulations of Hamiltonian dynamics.

On the Chemical Basis of Trotter-Suzuki Errors in Quantum Chemistry Simulation

It is argued that chemical properties, such as the maximum nuclear charge in a molecule and the filling fraction of orbitals, can be decisive for determining the cost of a quantum simulation.

Quantum simulation via randomized product formulas: Low gate complexity with accuracy guarantees

This work provides a comprehensive analysis of a single realization of the random product formula produced by qDRIFT, and proves that a typical realizing of the randomized product formula approximates the ideal unitary evolution up to a small diamond-norm error.

Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure

It is found that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol, and three ordering strategies are proposed, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian.

Simulation of electronic structure Hamiltonians using quantum computers

Over the last century, a large number of physical and mathematical developments paired with rapidly advancing technology have allowed the field of quantum chemistry to advance dramatically. However,

Time-dependent unbounded Hamiltonian simulation with vector norm scaling

It is demonstrated that under suitable assumptions of the Hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of theHamiltonian increases using Trotter type methods.

The Trotter step size required for accurate quantum simulation of quantum chemistry

This study presents an alternative simulation scheme and shows that it can sometimes outperform existing schemes, but that this possibility depends crucially on the details of the simulated molecule.

Quantum speedup of Monte Carlo methods

  • A. Montanaro
  • Computer Science
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2015
A quantum algorithm which can accelerate Monte Carlo methods in a very general setting, achieving a near-quadratic speedup over the best possible classical algorithm.
...