Quantum simulations of one dimensional quantum systems

@article{Somma2016QuantumSO,
  title={Quantum simulations of one dimensional quantum systems},
  author={Rolando D. Somma},
  journal={Quantum Inf. Comput.},
  year={2016},
  volume={16},
  pages={1125-1168}
}
  • R. Somma
  • Published 21 March 2015
  • Computer Science
  • Quantum Inf. Comput.
We present quantum algorithms for the simulation of quantum systems in one spatial dimension, which result in quantum speedups that range from superpolynomial to polynomial. We first describe a method to simulate the evolution of the quantum harmonic oscillator (QHO) based on a refined analysis of the Trotter-Suzuki formula that exploits the Lie algebra structure. For total evolution time $t$ and precision $\epsilon>0$, the complexity of our method is $ O(\exp(\gamma \sqrt{\log(N/\epsilon… 

Figures and Tables from this paper

Quantum Algorithms for Simulating the Lattice Schwinger Model
TLDR
This work performs a tight analysis of low-order Trotter formula simulations of the Schwinger model, using recently derived commutator bounds, and gives upper bounds on the resources needed for simulations in both scenarios.
Corrected quantum walk for optimal Hamiltonian simulation
TLDR
A method to simulate Hamiltonian evolution on a quantum computer by repeatedly using a superposition of steps of a quantum walk, then applying a correction to the weightings for the numbers of steps, to obtain complexity which is the same as the lower bound up to double-logarithmic factors for all parameter regimes.
Quantum algorithms for Gibbs sampling and hitting-time estimation
TLDR
This work presents quantum algorithms for solving two problems regarding stochastic processes, one of which estimates the hitting time of a Markov chain and the other quadratically improves the dependence on 1/\epsilon and $1/\Delta of the analog classical algorithm for hitting-time estimation.
Quantum Simulation of Hawking Radiation Using VQE Algorithm on IBM Quantum Computer
Quantum computers have an exponential speed-up advantage over classical computers. One of the most prominent utilities of quantum computers is their ability to study complex quantum systems in
Quantum simulation of scattering in the quantum Ising model
We discuss real time evolution for the quantum Ising model in one spatial dimension with $N_s$ sites. In the limit where the nearest neighbor interactions $J$ in the spatial directions are small,
Nearly tight Trotterization of interacting electrons
TLDR
It suffices to use O gates to simulate electronic structure in the plane-wave basis with $n$ spin orbitals and $\eta$ electrons up to a negligible factor, improving the best previous result in second quantization while outperforming the first-quantized simulation when $n=\mathcal{O}\left(\eta^2\right)$.
Quantum phase estimation for a class of generalized eigenvalue problems
Quantum phase estimation provides a path to quantum computation of solutions to Hermitian eigenvalue problems $Hv = \lambda v$, such as those occurring in quantum chemistry. It is natural to ask
Quantum simulation of discretized harmonic oscillator
In this work, we conduct a quantum simulation of a particle in a harmonic oscillator potential on a quantum chip provided by IBM quantum experience platform. The simulation is carried out in two
Quantum Algorithm Implementations for Beginners
TLDR
This review aims to explain the principles of quantum programming, which are quite different from classical programming, with straightforward algebra that makes understanding of the underlying fascinating quantum mechanical principles optional.
Study of the effect of quantum noise on the accuracy of the Schrödinger equation simulation on a quantum computer using the Zalka-Wiesner method
The study of the effect of quantum noise on the accuracy of modeling quantum systems on a quantum computer using the Zalka-Wiesner method is carried out. The efficiency of the developed methods and
...
...

References

SHOWING 1-10 OF 61 REFERENCES
Quantum Algorithms for Quantum Field Theories
TLDR
A quantum algorithm to compute relativistic scattering probabilities in a massive quantum field theory with quartic self-interactions in spacetime of four and fewer dimensions is developed and achieves exponential speedup over the fastest known classical algorithm.
Using Quantum Computers for Quantum Simulation
TLDR
The theoretical and experimental development of quantum simulation using quantum computers is surveyed, from the first ideas to the intense research efforts currently underway.
Simulating a quantum magnet with trapped ions
To gain deeper insight into the dynamics of complex quantum systems we need a quantum leap in computer simulations. We cannot translate quantum behaviour arising from superposition states or
Quantum algorithms for fermionic simulations
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,
Quantum simulations of physics problems
If a large Quantum Computer (QC) existed today, what type of physical problems could we efficiently simulate on it that we could not simulate on a classical Turing machine? In this paper we argue
Simulating Quantum Dynamics On A Quantum Computer
TLDR
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.
Quantum computation and quantum information
  • T. Paul
  • Physics
    Mathematical Structures in Computer Science
  • 2007
This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers deal
Simulating chemistry using quantum computers.
TLDR
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
TLDR
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.
Preparing ground States of quantum many-body systems on a quantum computer.
TLDR
A quantum computer can prepare the ground state of a quantum system as efficiently as it does for classical systems, and this result is presented to the case of interacting quantum particles.
...
...