Alternative sampling for variational quantum Monte Carlo

Expectation values of physical quantities may accurately be obtained by the evaluation of integrals within Many-Body Quantum mechanics, and these multi-dimensional integrals may be estimated using Monte Carlo methods. In a previous publication it has been shown that for the simplest, most commonly applied strategy in continuum Quantum Monte Carlo, the random error in the resulting estimates is not well controlled. At best the Central Limit theorem is valid in its weakest form, and at worst it is invalid and replaced by an alternative Generalised Central Limit theorem and non-Normal random error. In both cases the random error is not controlled. Here we consider a new `residual sampling strategy' that reintroduces the Central Limit Theorem in its strongest form, and provides full control of the random error in estimates. Estimates of the total energy and the variance of the local energy within Variational Monte Carlo are considered in detail, and the approach presented may be generalised to expectation values of other operators, and to other variants of the Quantum Monte Carlo method.Comment: 14 pages, 9 figure

Error Performance of Channel Coding in Random Access Communication

A new channel coding approach was proposed in [1] for random multiple access communication over the discrete-time memoryless channel. The coding approach allows users to choose their communication rates independently without sharing the rate information among each other or with the receiver. The receiver will either decode the message or report a collision depending on whether reliable message recovery is possible. It was shown that, asymptotically as the codeword length goes to infinity, the set of communication rates supporting reliable message recovery can be characterized by an achievable region which equals Shannon's information rate region possibly without a convex hull operation. In this paper, we derive achievable bounds on error probabilities, including the decoding error probability and the collision miss detection probability, of random multiple access systems with a finite codeword length. Achievable error exponents are obtained by taking the codeword length to infinity.Comment: submitted to IEEE Transactions on Information Theor

Simulation of infinitely divisible random fields

Two methods to approximate infinitely divisible random fields are presented. The methods are based on approximating the kernel function in the spectral representation of such fields, leading to numerical integration of the respective integrals. Error bounds for the approximation error are derived and the approximations are used to simulate certain classes of infinitely divisible random fields.Comment: 41 pages, 3 figure

Error exponents of typical random codes

We define the error exponent of the typical random code as the long-block limit of the negative normalized expectation of the logarithm of the error probability of the random code, as opposed to the traditional random coding error exponent, which is the limit of the negative normalized logarithm of the expectation of the error probability. For the ensemble of uniformly randomly drawn fixed composition codes, we provide exact error exponents of typical random codes for a general discrete memoryless channel (DMC) and a wide class of (stochastic) decoders, collectively referred to as the generalized likelihood decoder (GLD). This ensemble of fixed composition codes is shown to be no worse than any other ensemble of independent codewords that are drawn under a permutation--invariant distribution (e.g., i.i.d. codewords). We also present relationships between the error exponent of the typical random code and the ordinary random coding error exponent, as well as the expurgated exponent for the GLD. Finally, we demonstrate that our analysis technique is applicable also to more general communication scenarios, such as list decoding (for fixed-size lists) as well as decoding with an erasure/list option in Forney's sense.Comment: 26 pages, submitted for publicatio