scholarly journals Quasi-adiabatic quantum Monte Carlo algorithm for quantum evolution in imaginary time

2013 ◽  
Vol 87 (17) ◽  
Author(s):  
Cheng-Wei Liu ◽  
Anatoli Polkovnikov ◽  
Anders W. Sandvik
Keyword(s):  
Monte Carlo ◽  
Quantum Monte Carlo ◽  
Monte Carlo Algorithm ◽  
Quantum Evolution ◽  
Imaginary Time

scholarly journals Quantum adiabatic algorithms, small gaps, and different paths

2011 ◽  
Vol 11 (3&4) ◽  
pp. 181-214
Author(s):  
Edward Farhi ◽  
Jeffrey Goldstone ◽  
David Gosset ◽  
Sam Gutmann ◽  
Harvey B. Meyer ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Excited State ◽  
Ground State ◽  
Quantum Monte Carlo ◽  
Monte Carlo Algorithm ◽  
Imaginary Time ◽  
Monte Carlo Data ◽  
First Excited State ◽  
Time Quantum

We construct a set of instances of 3SAT which are not solved efficiently using the simplest quantum adiabatic algorithm. These instances are obtained by picking random clauses all consistent with two disparate planted solutions and then penalizing one of them with a single additional clause. We argue that by randomly modifying the beginning Hamiltonian, one obtains (with substantial probability) an adiabatic path that removes this difficulty. This suggests that the quantum adiabatic algorithm should in general be run on each instance with many different random paths leading to the problem Hamiltonian. We do not know whether this trick will help for a random instance of 3SAT (as opposed to an instance from the particular set we consider), especially if the instance has an exponential number of disparate assignments that violate few clauses. We use a continuous imaginary time Quantum Monte Carlo algorithm in a novel way to numerically investigate the ground state as well as the first excited state of our system. Our arguments are supplemented by Quantum Monte Carlo data from simulations with up to 150 spins.

A survey on pure sampling in quantum Monte Carlo methods

2013 ◽  
Vol 91 (7) ◽  
pp. 505-510 ◽  
Author(s):  
Stuart M. Rothstein
Keyword(s):  
Monte Carlo ◽  
Ground State ◽  
Quantum Monte Carlo ◽  
Sampling Methods ◽  
Electrical Response ◽  
Distribution Functions ◽  
Monte Carlo Algorithm ◽  
Position Operator ◽  
Trial Solution ◽  
Mixed Distribution

The most commonly employed diffusion Monte Carlo algorithm and some of its variants afford a way to sample configuration space from a so-called “mixed distribution”, the product of an input trial solution to the Schrödinger equation for the ground state and its unknown exact solution. This mixed distribution is sufficient to compute the ground state energy and other properties represented by operators that commute with the Hamiltonian. These energy-related properties are exact, save for a small bias introduced by the input trial function’s incorrect exchange nodes, the so-called “fixed-node error”. However, properties represented by operators that commute with the position operator are also of interest. When calculated by sampling from the mixed distribution, these properties are much more strongly biased by the input trial function. Our objective is to review methods that allow sampling from the desired “pure” distribution, one that is unbiased except for the exchange node error. Thereby, one accurately calculates physical properties such as the dipole and other electrical moments, electrical response properties of molecules, and particle distribution functions for clusters. We survey the results of calculations that employ pure-sampling methods through what has been published in year 2012. Our review also touches on truly exact sampling methods.

A SINGLE HOLE IN THE TWO-DIMENSIONAL INFINITE-U HUBBARD MODEL IN AN ORBITAL MAGNETIC FIELD

1994 ◽  
Vol 08 (06) ◽  
pp. 707-725
Author(s):  
S. V. MESHKOV ◽  
J. C. ANGLÈS D'AURIAC
Keyword(s):  
Magnetic Field ◽  
Monte Carlo ◽  
Hubbard Model ◽  
Ground State ◽  
Quantum Monte Carlo ◽  
Applied Field ◽  
Monte Carlo Algorithm ◽  
Two Dimensional ◽  
Thermodynamical Properties ◽  
Single Hole

Using an original Quantum Monte Carlo algorithm, we study the thermodynamical properties of a single hole in the two-dimensional infinite-U Hubbard model at finite temperature. We investigate the energy and the spin correlators as a function of an external orbital magnetic field. This field is found to destroy the Nagaoka ferromagnetism and to induce chirality in the spin background. The applied field is partially screened by a fictitious magnetic field coming from the chirality. Our algorithm allows us to reach a temperature low enough to discuss the ground state properties of the model.

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