scholarly journals Cancellation of vacuum diagrams and the long-time limit in out-of-equilibrium diagrammatic quantum Monte Carlo

2019 ◽  
Vol 100 (8) ◽  
Author(s):  
Alice Moutenet ◽  
Priyanka Seth ◽  
Michel Ferrero ◽  
Olivier Parcollet
Keyword(s):  
Monte Carlo ◽  
Quantum Monte Carlo ◽  
Time Limit ◽  
Long Time

QUANTUM MONTE CARLO SIMULATIONS OF FERMIONS: A MATHEMATICAL ANALYSIS OF THE FIXED-NODE APPROXIMATION

2006 ◽  
Vol 16 (09) ◽  
pp. 1403-1440 ◽  
Author(s):  
ERIC CANCÈS ◽  
BENJAMIN JOURDAIN ◽  
TONY LELIÈVRE
Keyword(s):  
Monte Carlo ◽  
Importance Sampling ◽  
Mathematical Analysis ◽  
Ground State ◽  
Quantum Monte Carlo ◽  
Stochastic Methods ◽  
Sampling Function ◽  
Long Time ◽  
Nodal Surfaces ◽  
Fixed Node

The Diffusion Monte Carlo (DMC) method is a powerful strategy to estimate the ground state energy E0 of an N-body Schrödinger Hamiltonian H = -½Δ + V with high accuracy. It consists of writing E0 as the long-time limit of an expectation value of a drift-diffusion process with a source term, and numerically simulating this process by means of a collection of random walkers. As for a number of stochastic methods, a DMC calculation makes use of an importance sampling function ψI which hopefully approximates some ground state ψ0 of H. In the fermionic case, it has been observed that the DMC method is biased, except in the special case when the nodal surfaces of ψI coincide with those of a ground state of H. The approximation due to the fact that, in practice, the nodal surfaces of ψI differ from those of the ground states of H, is referred to as the Fixed Node Approximation (FNA). Our purpose in this paper is to provide a mathematical analysis of the FNA. We prove that, under convenient hypotheses, a DMC calculation performed with the importance sampling function ψI, provides an estimation of the infimum of the energy 〈ψ, Hψ〉 on the set of the fermionic test functions ψ that exactly vanish on the nodal surfaces of ψI.

scholarly journals A Discontinuity of the Energy of Quantum Walk in Impurities

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1134
Author(s):  
Kenta Higuchi ◽  
Takashi Komatsu ◽  
Norio Konno ◽  
Hisashi Morioka ◽  
Etsuo Segawa
Keyword(s):  
Unit Circle ◽  
Stationary State ◽  
Discrete Time ◽  
Quantum Walk ◽  
Time Limit ◽  
The Other ◽  
Unitary Matrix ◽  
Initial State ◽  
Time Quantum ◽  
Long Time

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

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