How to Compute Using Quantum Walks

Abstract

Quantum walks are widely and successfully used to model diverse physical processes. This leads to computation of the models, to explore their properties. Quantum walks have also been shown to be universal for quantum computing. This is a more subtle result than is often appreciated, since it applies to computations run on qubit-based quantum computers in the single walker case, and physical quantum walks in the multi-walker case (quantum cellular automata). Nonetheless, quantum walks are powerful tools for quantum computing when correctly applied. In this paper, I explain the relationship between quantum walks as models and quantum walks as computational tools, and give some examples of their application in both contexts.

Quantum communication protocols by quantum walks with two coins

“…Kendon et al studied state transfer in quantum walks on various graphs, and they mainly considered continuous time quantum walks. They found that very few graphs can realize perfect state transfer [22,23]. By discrete-time quantum walk search algorithm, M. Stefaňàk et al discussed state transfer on star graphs and complete bipartite graphs [24].…”

Quantum Neurobiology

“…Quantum neurobiological approaches could seek to interrelate three domains: scientific theories of time, temporal modes of cognition, and the underlying time morphology of biological processes. Physics findings related to time include time entanglement (temporal correlations have a different structure than spatial correlations [161]), the Floquet model [134], time symmetry breaking [162], time evolution [141], and quantum walks [136]. In the cognitive domain, Kantian neuroscience shows empirically how the spontaneity of cognition is demonstrated by the constitutive role of the brain in processing incoming sensory input [163] and correlates neurobiology and philosophy [164].…”

“…The second requirement the AdS/Brain theory addresses is the issue that different neural dynamics paradigms define the system evolution at each scale tier of the neural signaling operation [38]. Floquet periodicity [134,135] propelled with continuous-time quantum walks [136] is selected as the basis for a multiscalar model of brain network, neuron, synapse, and ion channel dynamics, as these formalisms flexibly accommodate varying dynamical regimes within a system.…”

Quantum Information Science

“…Quantum walks are the quantum version of the random walk used in finance to characterize market behavior on the model of Brownian motion [12]. Classical random walks are based on Markov (stochastic) processes and a coin flip, and quantum walks too proceed on the basis of a coin flip, using a quantum coin-flip operator such as a Hadamard coin (which flips a qubit into a one or zero).…”