Stochastic Gradient Approximation: An Efficient Method to Optimize Many-Body Wave Functions

Harju, Ari; Barbiellini, B.; Siljamäki, S.; Nieminen, Risto M.; Ortíz, Gerardo

doi:10.1103/physrevlett.79.1173

Cited by 75 publications

(87 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Firstly, unlike the local optimization methods, the GO method update all tensors simultaneously and also the noise in the gradient may help avoid the local minima, which has some similarity to the simulated annealing technique. 25 Secondly, the MC sweeps can be easily and massively parallelized. Thirdly, it is easy to deal with the systems beyond nearest neighbor interactions, such as J 1 -J 2 square Heisenberg model.…”

Section: Methodsmentioning

confidence: 99%

Gradient optimization of finite projected entangled pair states

et al. 2017

View full text Add to dashboard Cite

The projected entangled pair states (PEPS) methods have been proved to be powerful tools to solve the strongly correlated quantum many-body problems in two-dimension. However, due to the high computational scaling with the virtual bond dimension D, in a practical application PEPS are often limited to rather small bond dimensions, which may not be large enough for some highly entangled systems, for instance, the frustrated systems. The optimization of the ground state using imaginary time evolution method with simple update scheme may go to a larger bond dimension. However, the accuracy of the rough approximation to the environment of the local tensors is questionable. Here, we demonstrate that combining the imaginary time evolution method with simple update, Monte Carlo sampling techniques and gradient optimization will offer an efficient method to calculate the PEPS ground state. By taking the advantages of massive parallel computing, we can study the quantum systems with larger bond dimensions up to D=10 without resorting to any symmetry. Benchmark tests of the method on the J1-J2 model give impressive accuracy compared with exact results.

show abstract

Section: Methodsmentioning

confidence: 99%

Gradient optimization of finite projected entangled pair states

et al. 2017

View full text Add to dashboard Cite

show abstract

“…For example, by choosing the step size to decrease as μ t ∝ 1/t α , with N fixed and 0 < α 1, we are guaranteed convergence to some local minimum. 18 Equivalently, the noise could be reduced at each step by increasing N ∝ t β , or some combination of both.…”

Section: B Optimizationmentioning

confidence: 99%

Variational Monte Carlo with the multiscale entanglement renormalization ansatz

Ferris

Vidal

2012

Phys. Rev. B

View full text Add to dashboard Cite

Monte Carlo sampling techniques have been proposed as a strategy to reduce the computational cost of contractions in tensor network approaches to solving many-body systems. Here, we put forward a variational Monte Carlo approach for the multiscale entanglement renormalization ansatz (MERA), which is a unitary tensor network. Two major adjustments are required compared to previous proposals with nonunitary tensor networks. First, instead of sampling over configurations of the original lattice, made of L sites, we sample over configurations of an effective lattice, which is made of just ln(L) sites. Second, the optimization of unitary tensors must account for their unitary character while being robust to statistical noise, which we accomplish with a modified steepest descent method within the set of unitary tensors. We demonstrate the performance of the variational Monte Carlo MERA approach in the relatively simple context of a finite quantum spin chain at criticality, and discuss future, more challenging applications, including two-dimensional systems.

show abstract

“…Reintroducing the plain (not importance-sampled) projection operator F = 1−τ (H −E 0 ), we can rewrite the reweighting factor for a simple Gutzwiller wavefunction into (21) with the ratioF (R ′ , R)/F (R ′ , R) = 1 for R ′ = R, and many such polynomials, therefore the order of the polynomial representing (20) increases with the number n of Monte Carlo steps. It is therefore not practical to directly estimate the ever increasing number of coefficients for the reweighting factor.…”

Section: B Correlated Samplingmentioning

confidence: 99%

“…General methods for achieving this are correlated sampling 15,20 or the recently developed stochastic gradient approximation. 21 Exploiting the particular form of Gutzwiller wavefunctions, we find a different approach, which is equivalent to correlated sampling: We observe that expectation values can be rewritten as the quotient of two polynomials (i.e. a rational function) in the Gutzwiller parameters.…”

Section: Introductionmentioning

confidence: 99%

Optimization of Gutzwiller wave functions in quantum Monte Carlo

1999

View full text Add to dashboard Cite

Gutzwiller functions are popular variational wavefunctions for correlated electrons in Hubbard models. Following the variational principle, we are interested in the Gutzwiller parameters that minimize e.g. the expectation value of the energy. Rewriting the expectation value as a rational function in the Gutzwiller parameters, we find a very efficient way for performing that minimization. The method can be used to optimize general Gutzwiller-type wavefunctions both, in variational and in fixed-node diffusion Monte Carlo. 71.10.Fd, 71.27.+a, 02.70.Lq

show abstract

Stochastic Gradient Approximation: An Efficient Method to Optimize Many-Body Wave Functions

Cited by 75 publications

References 13 publications

Gradient optimization of finite projected entangled pair states

Gradient optimization of finite projected entangled pair states

Variational Monte Carlo with the multiscale entanglement renormalization ansatz

Optimization of Gutzwiller wave functions in quantum Monte Carlo

Contact Info

Product

Resources

About