scholarly journals Application of Quantum—Markov Open System Models to Human Cognition and Decision

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 990
Author(s):  
Jerome Busemeyer ◽  
Qizi Zhang ◽  
S. N. Balakrishnan ◽  
Zheng Wang
Keyword(s):  
Random Walk ◽  
Open System ◽  
Choice Behavior ◽  
Quantum Walk ◽  
Decision Time ◽  
Human Cognition ◽  
System Models ◽  
Human Choice ◽  
Single Process ◽  
Good Potential

Markov processes, such as random walk models, have been successfully used by cognitive and neural scientists to model human choice behavior and decision time for over 50 years. Recently, quantum walk models have been introduced as an alternative way to model the dynamics of human choice and confidence across time. Empirical evidence points to the need for both types of processes, and open system models provide a way to incorporate them both into a single process. However, some of the constraints required by open system models present challenges for achieving this goal. The purpose of this article is to address these challenges and formulate open system models that have good potential to make important advancements in cognitive science.

scholarly journals Quantum walks and elliptic integrals

2010 ◽  
Vol 20 (6) ◽  
pp. 1091-1098 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Random Walk ◽  
Generating Function ◽  
Elliptic Integral ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Elliptic Integrals ◽  
Two Dimensional ◽  
One Dimensional ◽  
Similar Expression ◽  
Return Probability

Pólya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.

scholarly journals NON-UNIFORM MIXING OF QUANTUM WALK ON CYCLES

2007 ◽  
Vol 05 (06) ◽  
pp. 781-793 ◽  
Author(s):  
WILLIAM ADAMCZAK ◽  
KEVIN ANDREW ◽  
LEON BERGEN ◽  
DILLON ETHIER ◽  
PETER HERNBERG ◽  
...  
Keyword(s):  
Random Walk ◽  
Uniform Distribution ◽  
Continuous Time ◽  
Quantum Walk ◽  
Complete Graphs ◽  
Circulant Graph ◽  
Mixing Properties ◽  
Time Quantum ◽  
Average Distribution ◽  
Instantaneous Distribution

A classical lazy random walk on cycles is known to mix with the uniform distribution. In contrast, we show that a continuous-time quantum walk on cycles exhibits strong non-uniform mixing properties. First, we prove that the instantaneous distribution of a quantum walk on most even-length cycles is never uniform. More specifically, we prove that a quantum walk on a cycle Cnis not instantaneous uniform mixing, whenever n satisfies either: (a) n = 2u, for u ≥ 3; or (b) n = 2uq, for u ≥ 1 and q ≡ 3 (mod 4). Second, we prove that the average distribution of a quantum walk on any Abelian circulant graph is never uniform. As a corollary, the average distribution of a quantum walk on any standard circulant graph, such as the cycles, complete graphs, and even hypercubes, is never uniform. Nevertheless, we show that the average distribution of a quantum walk on the cycle Cnis O(1/n)-uniform.

scholarly journals A random walk approach to quantum algorithms

2006 ◽  
Vol 364 (1849) ◽  
pp. 3407-3422 ◽  
Author(s):  
Vivien M Kendon
Keyword(s):  
Random Walk ◽  
Quantum Dynamics ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Walk ◽  
Quantum Random Walk ◽  
Small Quantum ◽  
Starting Point ◽  
Quantum Version ◽  
Speed Up

The development of quantum algorithms based on quantum versions of random walks is placed in the context of the emerging field of quantum computing. Constructing a suitable quantum version of a random walk is not trivial; pure quantum dynamics is deterministic, so randomness only enters during the measurement phase, i.e. when converting the quantum information into classical information. The outcome of a quantum random walk is very different from the corresponding classical random walk owing to the interference between the different possible paths. The upshot is that quantum walkers find themselves further from their starting point than a classical walker on average, and this forms the basis of a quantum speed up, which can be exploited to solve problems faster. Surprisingly, the effect of making the walk slightly less than perfectly quantum can optimize the properties of the quantum walk for algorithmic applications. Looking to the future, even with a small quantum computer available, the development of quantum walk algorithms might proceed more rapidly than it has, especially for solving real problems.

scholarly journals Efficient circuits for quantum walks

2010 ◽  
Vol 10 (5&6) ◽  
pp. 420-434
Author(s):  
C.-F. Chiang ◽  
D. Nagaj ◽  
P. Wocjan
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Probability Distributions ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Problem Size ◽  
Integrable Probability ◽  
The Difference ◽  
General Method

We present an efficient general method for realizing a quantum walk operator corresponding to an arbitrary sparse classical random walk. Our approach is based on Grover and Rudolph's method for preparing coherent versions of efficiently integrable probability distributions \cite{GroverRudolph}. This method is intended for use in quantum walk algorithms with polynomial speedups, whose complexity is usually measured in terms of how many times we have to apply a step of a quantum walk \cite{Szegedy}, compared to the number of necessary classical Markov chain steps. We consider a finer notion of complexity including the number of elementary gates it takes to implement each step of the quantum walk with some desired accuracy. The difference in complexity for various implementation approaches is that our method scales linearly in the sparsity parameter and poly-logarithmically with the inverse of the desired precision. The best previously known general methods either scale quadratically in the sparsity parameter, or polynomially in the inverse precision. Our approach is especially relevant for implementing quantum walks corresponding to classical random walks like those used in the classical algorithms for approximating permanents \cite{Vigoda, Vazirani} and sampling from binary contingency tables \cite{Stefankovi}. In those algorithms, the sparsity parameter grows with the problem size, while maintaining high precision is required.

scholarly journals An Analysis of Decision Under Risk in Rats

2018 ◽  
Author(s):  
Christine M. Constantinople ◽  
Alex T. Piet ◽  
Carlos D. Brody
Keyword(s):  
Choice Behavior ◽  
Neural Substrates ◽  
Neural Mechanisms ◽  
Reference Dependence ◽  
Decision Making Under Risk ◽  
Internal Reference ◽  
Decision Under Risk ◽  
Psychological Phenomena ◽  
Human Choice ◽  
Behavioral Economic Theory

AbstractProspect Theory is the predominant behavioral economic theory describing decision-making under risk. It accounts for near universal aspects of human choice behavior whose prevalence may reflect fundamental neural mechanisms. We now apply Prospect Theory’s framework to rodents, using a task in which rats chose between guaranteed and probabilistic rewards. Like humans, rats distorted probabilities and showed diminishing marginal sensitivity, in which they were less sensitive to differences in larger rewards. They exhibited reference dependence, in which the valence of outcomes (gain or loss) was determined by an internal reference point reflecting reward history. The similarities between rats and humans suggest conserved neural substrates, and enable application of powerful molecular/circuit tools to study mechanisms of psychological phenomena from behavioral economics.

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