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

@article{Poulin2015TheTS,
title={The Trotter step size required for accurate quantum simulation of quantum chemistry},
author={David Poulin and Matthew B. Hastings and Dave Wecker and Nathan Wiebe and Andrew C. Doberty and Matthias Troyer},
journal={Quantum Inf. Comput.},
year={2015},
volume={15},
pages={361-384}
}
• Published 19 June 2014
• Computer Science
• Quantum Inf. Comput.
The simulation of molecules is a widely anticipated application of quantum computers. However, recent studies [1, 2] have cast a shadow on this hope by revealing that the complexity in gate count of such simulations increases with the number of spin orbitals N as N8, which becomes prohibitive even for molecules of modest size N ∼ 100. This study was partly based on a scaling analysis of the Trotter step required for an ensemble of random artificial molecules. Here, we revisit this analysis and…
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## References

SHOWING 1-10 OF 48 REFERENCES
Quantum Algorithms for Quantum Chemistry based on the sparsity of the CI-matrix
• Physics
• 2013
Quantum chemistry provides a target for quantum simulation of considerable scientific interest and industrial importance. The majority of algorithms to date have been based on a second-quantized
Improving quantum algorithms for quantum chemistry
• Computer Science
Quantum Inf. Comput.
• 2015
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.
Simulation of electronic structure Hamiltonians using quantum computers
• Physics
• 2011
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,
Simulating chemistry using quantum computers.
• Physics
Annual review of physical chemistry
• 2011
This review discusses to what extent the ideas in quantum computation, now a well-established field, have been applied to chemical problems and describes algorithms that achieve significant advantages for the electronic-structure problem, the simulation of chemical dynamics, protein folding, and other tasks.
Simulated Quantum Computation of Molecular Energies
• Physics
Science
• 2005
Calculations of the water and lithium hydride molecular ground-state energies have been carried out on a quantum computer simulator using a recursive phase-estimation algorithm and mapping of the molecular wave function to the quantum bits are described.
Simulating Quantum Dynamics On A Quantum Computer
• Computer Science
• 2010
A range of techniques to simulate Hamiltonians with badly behaved derivatives are proposed, including using adaptive time steps, adapting the order of the integrators, and omitting regions about discontinuities.
Black-box hamiltonian simulation and unitary implementation
• Computer Science
Quantum Inf. Comput.
• 2012
This work gives the best known simulation of sparse Hamiltonians with constant precision, and shows that a black-box unitary can be performed with bounded error using O(N2/3(log logN)4/3) queries to its matrix elements.
Quantum algorithms for fermionic simulations
• Physics, Computer Science
• 2001
We investigate the simulation of fermionic systems on a quantum computer. We show in detail how quantum computers avoid the dynamical sign problem present in classical simulations of these systems,
Entangled quantum electronic wavefunctions of the Mn₄CaO₅ cluster in photosystem II.
• Physics, Chemistry
Nature chemistry
• 2013
Through entanglement maps, this work reveals rich information contained in the wavefunctions on bonding changes in the cycle, and suggests that current candidates that have been recently distinguished using parameterized spin models should be reassessed.
Efficient Simulation of Quantum Systems by Quantum Computers
The time evolution of the wave function of a quantum mechanical many particle system can be implemented very efficiently on a quantum computer and the computational cost is comparable to a conventional simulation of the corresponding classical system.