Recent findings of Markov violations challenge Markov random walk processes for decision making. On the other hand, quantum walk processes explain these Markov violations in a natural way, but they have only been applied to binary alternative decision making. In this work, we propose a general framework for extending quantum walk processes to multi-alternative decision making. The multi-alternative quantum walk model operates in a direct sum space of the alternatives, with Hamiltonian built for each pair of alternatives to model context effects. Order effects come naturally from the matrix non-commutativity of the Hamiltonians. Future works built on this framework can connect parameters of the models with the expected utilities.
Recent findings of Markov violations challenge Markov random walk processes for decision making. On the other hand, quantum walk processes explain these Markov violations in a natural way, but they have only been applied to binary alternative decision making. In this work, we propose a general framework for extending quantum walk processes to multi-alternative decision making. The multi-alternative quantum walk model operates in a direct sum space of the alternatives, with Hamiltonian built for each pair of alternatives to model context effects. Order effects come naturally from the matrix non-commutativity of the Hamiltonians. Future works built on this framework can connect parameters of the models with the expected utilities.
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.
Classical models of decision-making under uncertainty, grounded in expected utility theory and Bayesian probability, often fail to account for systematic cognitive anomalies such as preference reversals, order effects, ambiguity aversion, and context-dependent choices. Quantum cognition models offer a novel theoretical framework that applies the mathematical principles of quantum probability—such as superposition, interference, and non-commutativity—to cognitive processes without assuming physical quantum mechanisms in the brain. By representing cognitive states as probabilistic superpositions that collapse through decision “measurements,” these models provide parsimonious explanations for violations of classical rationality. This paper examines how quantum cognition frameworks capture contextual influence, belief updating, and interference effects in uncertain decision environments. Drawing on findings from cognitive psychology and behavioral economics, the study highlights the relevance of quantum cognition for applications in economic choice modeling, legal decision analysis, and artificial intelligence. The analysis suggests that quantum probability offers a robust alternative to classical approaches when modeling human decision-making in complex, ambiguous, and context-sensitive situations.
Signal processing is important for modern technologies such as digital communication systems and sensor networks, and is used to enhance and optimize the quality of transmitted signals. However, traditional methods and design frameworks are insufficient for the demands of future communication networks, which are expected to be highly complex and constantly changing. Our research focuses on a mathematical model known as a Markov model, which represents a system as a series of states linked by transitions. This model can be applied to various signal processing challenges in telecommunications, such as the modeling of communication channels, sources of information, and the correction of distortion and noise in transmitted signals. We examine the random walk algorithm, a simple and efficient way to simulate the behavior of a Markov chain, as it can provide valuable insights into the performance of communication systems and help to optimize their design. However, it may not be able to accurately model systems with complex transitions or systems that exhibit long-term correlations. In order to address the challenges and limitations of classical random walk, we propose the use of quantum random walk, which is a quantum mechanical version of classical random walk. It describes the motion of particles in a probabilistic way, similar to classical random walk, but with the added complexity of quantum mechanics. We also provide quantum walk algorithm with Hadamard gate implementation for coin operator.
A framework for finite-dimensional quantum Markov chains on Hilbert spaces is introduced. Quantum Markov chains generalize both classical Markov chains with possibly hidden states and existing models of quantum walks on finite graphs. Quantum Markov chains are based on Markov operations that may be applied to quantum systems and include quantum measurements, for example. It is proved that quantum Markov chains are asymptotically stationary and hence possess ergodic and entropic properties. With a quantum Markov chain one may associate a quantum Markov process, which is a stochastic process in the classical sense. Generalized Markov chains allow a representation with respect to a generalized Markov source model with definite (but possibly hidden) states relative to which observables give rise to classical stochastic processes. It is demonstrated that this model allows for observables to violate Bell’s inequality.
In the world of computational probing, random walks and quantum walks are two separate but tightly linked paradigms. Classical stochastic processes like random walks are based on the random movement of particles in mathematical space and have been extensively used in data mining, network analysis, computer vision, etc. Quantum walks, by contrast generalize random walks into the quantum realm, taking the benefit of the quantum superposition and interference in order to investigate multiple paths at the same time. Quantum walks provide significant potential for algorithmic development: in quantum computing and communication, and in Scientific modelling and simulation. This paper provides an in-depth survey of random walks and quantum walks, outlining historical context, fundamental principles, applications, challenges, and future prospects. In this paper, motivated by the abstractness of both random walks and quantum walks, we survey the field of computational exploration at large scales, coalescing the myriad notions and difficult challenges addressed by existing research efforts on random and quantum walks, inferring a nuanced and multifaceted view on how both random walks and quantum walks are involved in paving the way to future development in many fields.
Spatial search and walk processes. Spatial search by quantum walk is database search with the additional constraint that one is required to move through the search space that obeys some locality structure. For example, the data items may be stored at the vertices of a two-dimensional grid. The requirement of moves along the edges of the grid captures the cost of accessing different items starting from some fixed position in the database. One of possible ways of carrying out spatial search is by performing a random walk on the search space or its quantum analog, a quantum walk. The complexity of spatial search by quantum walk is strongly tied to the quantum hitting time [19] of the walk. Let S , with jS j D n, be a finite set of states. Assume that a subset M S of states are marked. We are given a procedure C that, on input x 2 S and an associated data structure d.x/, checks whether the state x is marked. The goal is either to find a marked state when promised that M ¤ ; (search version) or to determine whether M is nonempty (decision version). The algorithm progresses in stages. In the setup stage, we access some state of S (usually a random state). In the walk stage we move from state to state, performing a spatial walk as described below. The moves are called updates. In addition, in the walk stage we perform checks to see if the current state is marked at steps selected by the algorithm. In the classical setting, the transition probabilities of the spatial walk are described by a stochastic matrix P D .px;y/x;y2S . This makes the walk a Markov chain. In every move the algorithm must perform a random transition according to P . The possible x ! y moves, i.e., those with px;y ¤ 0, form the edges of a (directed) graph G, and we say that the Markov chain P has locality structure G. We define the search problem in the classical setting, which carries over to the quantum case with little modification: