Koujin Takeda

Analysis of CDMA systems that are characterized by eigenvalue spectrum

An approach to analyze the performance of the code division multiple access (CDMA) scheme, which is a core technology used in modern wireless communication systems, is provided. The approach characterizes the objective system by the eigenvalue spectrum of a cross-correlation matrix composed of signature sequences used in CDMA communication, which enable us to handle a wider class of CDMA systems beyond the basic model reported by Tanaka in Europhys. Lett., 54 (2001) 540. The utility of the scheme is shown by analyzing a system in which the generation of signature sequences is designed for enhancing the orthogonality.

Cavity approach to the spectral density of sparse symmetric random matrices.

The spectral density of various ensembles of sparse symmetric random matrices is analyzed using the cavity method. We consider two cases: matrices whose associated graphs are locally treelike, and sparse covariance matrices. We derive a closed set of equations from which the density of eigenvalues can be efficiently calculated. Within this approach, the Wigner semicircle law for Gaussian matrices and the Marcenko-Pastur law for covariance matrices are recovered easily. Our results are compared with numerical diagonalization, showing excellent agreement.

Self-dual random-plaquette gauge model and the quantum toric code

Abstract We study the four-dimensional Z 2 random-plaquette lattice gauge theory as a model of topological quantum memory, the toric code in particular. In this model, the procedure of quantum error correction works properly in the ordered (Higgs) phase, and phase boundary between the ordered (Higgs) and disordered (confinement) phases gives the accuracy threshold of error correction. Using self-duality of the model in conjunction with the replica method, we show that this model has exactly the same mathematical structure as that of the two-dimensional random-bond Ising model, which has been studied very extensively. This observation enables us to derive a conjecture on the exact location of the multicritical point (accuracy threshold) of the model, p c =0.889972…, and leads to several nontrivial results including bounds on the accuracy threshold in three dimensions.

Nonexponential decay of an unstable quantum system: Small-Q-value s-wave decay

We study the decay process of an unstable quantum system, especially the deviation from the exponential decay law. We show that the exponential period no longer exists in the case of the s-wave decay with small Q value, where the Q value is the difference between the energy of the initially prepared state and the minimum energy of the continuous eigenstates in the system. We also derive the quantitative condition that this kind of decay process takes place and discuss what kind of system is suitable to observe the decay.

Spectral density of random graphs with topological constraints

The spectral density of random graphs with topological constraints is analysed using the replica method. We consider graph ensembles featuring generalised degree-degree correlations, as well as those with a community structure. In each case an exact solution is found for the spectral density in the form of consistency equations depending on the statistical properties of the graph ensemble in question. We highlight the effect of these topological constraints on the resulting spectral density.

Statistical mechanical analysis of the Kronecker channel model for multiple-input multiple-output wireless communication.

The Kronecker channel model of wireless communication is analyzed using statistical mechanics methods. In the model, spatial proximities among transmission/reception antennas are taken into account as certain correlation matrices, which generally yield nontrivial dependence among symbols to be estimated. This prevents accurate assessment of the communication performance by naively using a previously developed analytical scheme based on a matrix integration formula. In order to resolve this difficulty, we develop a formalism that can formally handle the correlations in Kronecker models based on the known scheme. Unfortunately, direct application of the developed scheme is, in general, practically difficult. However, the formalism is still useful, indicating that the effect of the correlations generally increase after the fourth order with respect to correlation strength. Therefore, the known analytical scheme offers a good approximation in performance evaluation when the correlation strength is sufficiently small. For a class of specific correlation, we show that the performance analysis can be mapped to the problem of one-dimensional spin systems in random fields, which can be investigated without approximation by the belief propagation algorithm.

Statistical mechanical analysis of the linear vector channel in digital communication

A statistical mechanical framework to analyze linear vector channel models in digital wireless communication is proposed for a large system. The framework is a generalization of that proposed for code-division multiple-access systems in Takeda et al (2006 Europhys. Lett. 76 1193) and enables the analysis of the system in which the elements of the channel transfer matrix are statistically correlated with each other. The significance of the proposed scheme is demonstrated by assessing the performance of an existing model of multi-input multi-output communication systems.

Statistical mechanical analysis of compressed sensing utilizing correlated compression matrix

We investigate a reconstruction limit of compressed sensing for a reconstruction scheme based on the L1-norm minimization utilizing a correlated compression matrix with a statistical mechanics method. We focus on the compression matrix modeled as the Kronecker-type random matrix studied in research on multiple-input multiple-output wireless communication systems. We found that strong one-dimensional correlations between expansion bases of original information slightly degrade reconstruction performance.

Exact location of the multicritical point for finite-dimensional spin glasses: a conjecture

We present a conjecture on the exact location of the multicritical point in the phase diagram of spin glass models in finite dimensions. By generalizing our previous work, we combine duality and gauge symmetry for replicated random systems to derive formulae which make it possible to understand all the relevant available numerical results in a unified way. The method applies to non-self-dual lattices as well as to self-dual cases, in the former case of which we derive a relation for a pair of values of multicritical points for mutually-dual lattices. The examples include the ±J and Gaussian Ising spin glasses on the square, hexagonal and triangular lattices, the Potts and Zq models with chiral randomness on these lattices, and the three-dimensional ±J Ising spin glass and the random plaquette gauge model.

Statistical mechanical assessment of a reconstruction limit of compressed sensing: Toward theoretical analysis of correlated signals

We provide a scheme for exploring the reconstruction limits of compressed sensing by minimizing the general cost function under the random measurement constraints for generic correlated signal sources. Our scheme is based on the statistical mechanical replica method for dealing with random systems. As a simple but non-trivial example, we apply the scheme to a sparse autoregressive model, where the first differences in the input signals of the correlated time series are sparse, and evaluate the critical compression rate for a perfect reconstruction. The results are in good agreement with a numerical experiment for a signal reconstruction.