scholarly journals Path-sum solution of the Weyl quantum walk in 3 + 1 dimensions

2017 ◽  
Vol 375 (2106) ◽  
pp. 20160394 ◽  
Author(s):  
G. M. D'Ariano ◽  
N. Mosco ◽  
P. Perinotti ◽  
A. Tosini
Keyword(s):  
Cayley Graph ◽  
Degrees Of Freedom ◽  
Transition Amplitude ◽  
Quantum Walk ◽  
Transition Matrices ◽  
Unitary Evolution ◽  
Position Representation ◽  
Quantum Revolution ◽  
The Given ◽  
Internal Degrees Of Freedom

We consider the Weyl quantum walk in 3+1 dimensions, that is a discrete-time walk describing a particle with two internal degrees of freedom moving on a Cayley graph of the group , which in an appropriate regime evolves according to Weyl's equation. The Weyl quantum walk was recently derived as the unique unitary evolution on a Cayley graph of that is homogeneous and isotropic. The general solution of the quantum walk evolution is provided here in the position representation, by the analytical expression of the propagator, i.e. transition amplitude from a node of the graph to another node in a finite number of steps. The quantum nature of the walk manifests itself in the interference of the paths on the graph joining the given nodes. The solution is based on the binary encoding of the admissible paths on the graph and on the semigroup structure of the walk transition matrices. This article is part of the themed issue ‘Second quantum revolution: foundational questions’.

scholarly journals Quantum Walk in Terms of Quantum Bernoulli Noise and Quantum Central Limit Theorem for Quantum Bernoulli Noise

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Caishi Wang ◽  
Ce Wang ◽  
Yuling Tang ◽  
Suling Ren
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Degrees Of Freedom ◽  
Probability Distributions ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Quantum Central Limit Theorem ◽  
Creation Operators ◽  
Internal Degrees Of Freedom

As a unitary quantum walk with infinitely many internal degrees of freedom, the quantum walk in terms of quantum Bernoulli noise (recently introduced by Wang and Ye) shows a rather classical asymptotic behavior, which is quite different from the case of the usual quantum walks with a finite number of internal degrees of freedom. In this paper, we further examine the structure of the walk. By using the Fourier transform on the state space of the walk, we obtain a formula that links the moments of the walk’s probability distributions directly with annihilation and creation operators on Bernoulli functionals. We also prove some other results on the structure of the walk. Finally, as an application of these results, we establish a quantum central limit theorem for the annihilation and creation operators themselves.

Reversibly Sampling Conformations and Binding Modes Using Molecular Darting

2020 ◽  
Author(s):  
Samuel C. Gill ◽  
David Mobley
Keyword(s):  
Monte Carlo ◽  
Ligand Binding ◽  
Binding Sites ◽  
Degrees Of Freedom ◽  
Dynamics Simulation ◽  
Hiv Integrase ◽  
Binding Modes ◽  
System A ◽  
Multiple Binding ◽  
Internal Degrees Of Freedom

<div>Sampling multiple binding modes of a ligand in a single molecular dynamics simulation is difficult. A given ligand may have many internal degrees of freedom, along with many different ways it might orient itself a binding site or across several binding sites, all of which might be separated by large energy barriers. We have developed a novel Monte Carlo move called Molecular Darting (MolDarting) to reversibly sample between predefined binding modes of a ligand. Here, we couple this with nonequilibrium candidate Monte Carlo (NCMC) to improve acceptance of moves.</div><div>We apply this technique to a simple dipeptide system, a ligand binding to T4 Lysozyme L99A, and ligand binding to HIV integrase in order to test this new method. We observe significant increases in acceptance compared to uniformly sampling the internal, and rotational/translational degrees of freedom in these systems.</div>

scholarly journals A novel class of translationally invariant spin chains with long-range interactions

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
B. Basu-Mallick ◽  
F. Finkel ◽  
A. González-López
Keyword(s):  
Long Range ◽  
Degrees Of Freedom ◽  
Spin Chains ◽  
Open Chain ◽  
New Class ◽  
Spin Interactions ◽  
Long Range Interactions ◽  
Spin Reversal ◽  
Twisted Yangian ◽  
Internal Degrees Of Freedom

Abstract We introduce a new class of open, translationally invariant spin chains with long-range interactions depending on both spin permutation and (polarized) spin reversal operators, which includes the Haldane-Shastry chain as a particular degenerate case. The new class is characterized by the fact that the Hamiltonian is invariant under “twisted” translations, combining an ordinary translation with a spin flip at one end of the chain. It includes a remarkable model with elliptic spin-spin interactions, smoothly interpolating between the XXX Heisenberg model with anti-periodic boundary conditions and a new open chain with sites uniformly spaced on a half-circle and interactions inversely proportional to the square of the distance between the spins. We are able to compute in closed form the partition function of the latter chain, thereby obtaining a complete description of its spectrum in terms of a pair of independent su(1|1) and su(m/2) motifs when the number m of internal degrees of freedom is even. This implies that the even m model is invariant under the direct sum of the Yangians Y (gl(1|1)) and Y (gl(0|m/2)). We also analyze several statistical properties of the new chain’s spectrum. In particular, we show that it is highly degenerate, which strongly suggests the existence of an underlying (twisted) Yangian symmetry also for odd m.

scholarly journals Three-state quantum walk on the Cayley Graph of the Dihedral Group

2021 ◽  
Vol 20 (3) ◽  
Author(s):  
Ying Liu ◽  
Jia-bin Yuan ◽  
Wen-jing Dai ◽  
Dan Li
Keyword(s):  
Cayley Graph ◽  
Dihedral Group ◽  
Quantum Walk

scholarly journals Dissipative dynamics of a particle coupled to a field via internal degrees of freedom

2021 ◽  
Vol 103 (5) ◽  
Author(s):  
Kanupriya Sinha ◽  
Adrián Ezequiel Rubio López ◽  
Yiğit Subaşı
Keyword(s):  
Degrees Of Freedom ◽  
Dissipative Dynamics ◽  
Internal Degrees Of Freedom

scholarly journals Optimal Relabeling of Water Molecules and Single-Molecule Entropy Estimation

Biophysica ◽  
2021 ◽  
Vol 1 (3) ◽  
pp. 279-296
Author(s):  
Federico Fogolari ◽  
Gennaro Esposito
Keyword(s):  
Single Molecule ◽  
Degrees Of Freedom ◽  
Water Molecules ◽  
Maximum Information ◽  
Nearest Neighbours ◽  
Biomolecular Simulations ◽  
Solvent Molecules ◽  
Solvation Entropy ◽  
Dynamics Simulations ◽  
Internal Degrees Of Freedom

Estimation of solvent entropy from equilibrium molecular dynamics simulations is a long-standing problem in statistical mechanics. In recent years, methods that estimate entropy using k-th nearest neighbours (kNN) have been applied to internal degrees of freedom in biomolecular simulations, and for the rigorous computation of positional-orientational entropy of one and two molecules. The mutual information expansion (MIE) and the maximum information spanning tree (MIST) methods were proposed and used to deal with a large number of non-independent degrees of freedom, providing estimates or bounds on the global entropy, thus complementing the kNN method. The application of the combination of such methods to solvent molecules appears problematic because of the indistinguishability of molecules and of their symmetric parts. All indistiguishable molecules span the same global conformational volume, making application of MIE and MIST methods difficult. Here, we address the problem of indistinguishability by relabeling water molecules in such a way that each water molecule spans only a local region throughout the simulation. Then, we work out approximations and show how to compute the single-molecule entropy for the system of relabeled molecules. The results suggest that relabeling water molecules is promising for computation of solvation entropy.

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