Stochastic Methods

Kuo-Lun Huang, Vincent Gaudet, Masoud Salehi

Stochastic decoding is a decoding algorithm for Low-Density Parity-Check (LDPC) codes that leads to lowered routing congestion. Randomization is crucial for generating stochastic bits in stochastic decoders. In this paper, we propose a scaling method for stochastic LDPC decoding over the binary symmetric channel model that is independent of the channel error probability, and thus reduces the complexity in the conversion of received messages into stochastic streams. The scaling method not only improves the error rate performance, but lowers the error floor for stochastic decoding.

Daniel F. Walls, Gerard J Milburn

Tomo Munehisa, Yasuko Munehisa

We describe a further development of the stochastic state selection method, which is a kind of Monte Carlo method we have proposed in order to numerically study large quantum spin systems. In the stochastic state selection method we make a sampling which is simultaneous for many states. This feature enables us to modify the method so that a number of given constraints are satisfied in each sampling. In this paper we discuss this modified stochastic state selection method that will be called the constrained stochastic state selection method in distinction from the previously proposed one (the conventional stochastic state selection method) in this paper. We argue that, by virtue of the constrained sampling, some quantities obtained in each sampling become more reliable, i.e. their statistical fluctuations are less than those from the conventional stochastic state selection method. In numerical calculations of the spin- 1/2 quantum Heisenberg antiferromagnet on a 36-site triangular lattice we explicitly show that data errors in our estimation of the ground state energy are reduced. Then we successfully evaluate several low-lying energy eigenvalues of the model on a 48-site lattice. Our results support that this system can be described by the theory based on the spontaneous symmetry breaking in the semiclassical Néel ordered antiferromagnet.

Khaldoon Ghanem

The stochastic sampling method (StochS) is used for the analytic continuation of quantum Monte Carlo data from the imaginary axis to the real axis. Compared to the maximum entropy method, StochS does not have explicit parameters, and one would expect the results to be unbiased. We present a very efficient algorithm for performing StochS and use it to study the effect of the discretization grid. Surprisingly, we find that the grid affects the results of StochS acting as an implicit default model. We provide a recipe for choosing a reliable StochS grid.To reduce the effect of the grid, we extend StochS into a gridless method (gStochS) by sampling the grid points from a default model instead of having them fixed. The effect of the default model is much reduced in gStochS compared to StochS and depends mainly on its width rather than its shape. The proper width can then be chosen using a simple recipe like we did in StochS.Finally, to avoid fixing the width, we go one step further and extend gStochS to sample over a whole class of default models with different widths. The extended method (eStochS) is then able to automatically relocate the grid points and concentrate them in the important region. Test cases show that eStochS gives good results resolving sharp features in the spectrum without the need for fine tuning a default model.

A. O. Bolivar