Computational neuroscience models have been used for understanding neural dynamics in the brain and how they may be altered when physiological or other conditions change. We review and develop a data-driven approach to neuroimaging data called the energy landscape analysis. The methods are rooted in statistical physics theory, in particular the Ising model, also known as the (pairwise) maximum entropy model and Boltzmann machine. The methods have been applied to fitting electrophysiological data in neuroscience for a decade, but their use in neuroimaging data is still in its infancy. We first review the methods and discuss some algorithms and technical aspects. Then, we apply the methods to functional magnetic resonance imaging data recorded from healthy individuals to inspect the relationship between the accuracy of fitting, the size of the brain system to be analysed and the data length.This article is part of the themed issue ‘Mathematical methods in medicine: neuroscience, cardiology and pathology’.

The inevitable presence of decoherence effects in systems suitable for quantum computation necessitates effective error-correction schemes to protect information from noise. We compute the stability of the toric code to depolarization by mapping the quantum problem onto a classical disordered eight-vertex Ising model. By studying the stability of the related ferromagnetic phase both via large-scale Monte Carlo simulations and via the duality method, we are able to demonstrate an increased error threshold of 18.9(3)% when noise correlations are taken into account. Remarkably, this agrees within error bars with the result for a different class of codes-topological color codes-where the mapping yields interesting new types of interacting eight-vertex models.Comment: 10 pages, 6 figures, 1 table - see Physics Viewpoint by D. Gottesman [http://physics.aps.org/articles/v5/50

We present an analysis leading to precise locations of the multicritical points for spin glasses on regular lattices. The conventional technique for determination of the location of the multicritical point was previously derived using a hypothesis emerging from duality and the replica method. In the present study, we propose a systematic technique, by an improved technique, giving more precise locations of the multicritical points on the square, triangular, and hexagonal lattices by carefully examining the relationship between two partition functions related with each other by the duality. We can find that the multicritical points of the +/-J Ising model are located at p{c}=0.890813 on the square lattice, where p{c} means the probability of J{ij}=J(>0) , at p{c}=0.835985 on the triangular lattice, and at p{c}=0.932593 on the hexagonal lattice. These results are in excellent agreement with recent numerical estimations.

New model-independent compact representations of imaginary-time data are presented in terms of the intermediate representation (IR) of analytical continuation. We demonstrate the efficiency of the IR through continuous-time quantum Monte Carlo calculations of an Anderson impurity model. We find that the IR yields a significantly compact form of various types of correlation functions. This allows the direct quantum Monte Carlo measurement of Green's functions in a compressed form, which considerably reduces the computational cost and memory usage. Furthermore, the present framework will provide general ways to boost the power of cutting-edge diagrammatic/quantum Monte Carlo treatments of many-body systems.

The quantum speed limit (QSL), or the energy-time uncertainty relation, describes the fundamental maximum rate for quantum time evolution and has been regarded as being unique in quantum mechanics. In this study, we obtain a classical speed limit corresponding to the QSL using the Hilbert space for the classical Liouville equation. Thus, classical mechanics has a fundamental speed limit, and QSL is not a purely quantum phenomenon but a universal dynamical property of the Hilbert space. Furthermore, we obtain similar speed limits for the imaginary-time Schrödinger equations such as the master equation.Introduction.-Noncommutativity is one of the most important components of quantum mechanics. The Heisenberg uncertainty principle [1] stems from the canonical commutation relations [2]. Because this consequence cannot appear in classical systems, Heisenbergs uncertainty principle is a purely quantum phenomenon. The product of energy and time has the same dimensions as the product of position and momentum, which naively implies the existence of a similar relation between energy and time. However, the time operator, which satisfies the canonical commutation relations for the Hamiltonian, does not exist in a realistic model [3], and thus, there is no energy-time uncertainty principle that strictly corresponds to Heisenberg's uncertainty principle. Properly formulating the uncertainty relation for energy and time is a delicate issue that is still being discussed [4,5].The first rigorous derivation of an analogous uncertainty principle for energy and time was given by Mandelstam and Tamm [6] in which they determined that the product of the energy variance and time required for a state to be orthogonal to its initial state was greater than Planck's constant. This result implied that quantum mechanics has a fundamental speed limit characterized by Planck's constant, and thus, this inequality is called the energy-time uncertainty relation or quantum speed limit (QSL). The quantum speed limit can be also regarded as a trade-off between energy and time in the variance of a state. Investigating the restrictions on the time evolution of quantum dynamics is an interesting and important problem, and there are many related works: an alternative quantum speed limit [7], the shortest time for quantum computation [8], cases on mixed states [9], time-dependent systems [10][11][12], and open systems [13][14][15], geometric derivations of the QSL [16][17][18][19][20], and various applications [21][22][23][24][25][26][27][28].Note that the QSL is a strictly different concept than Heisenberg's uncertainty principle. Nevertheless, since QSL appears in a similar context to Heisenberg's uncertainty principle, QSL has been considered a purely quantum phenomenon with no corresponding concept in classical mechanics. Recent studies [12,[29][30][31] have im-