Bounds of Quantum Mixing Processes via Controlled Walks
Date
2020-04-29
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Abstract
Quantum mixing is a category of processes of reaching a stationary state when starting from a single initial vertex via quantum walks. In this thesis, we introduce the quantum mixing processes based on the framework of controlled quantum walks in [DH17a]. We first discuss why Szegedy's walks are not well-suited for the quantum mixing processes. We then show that the controlled quantum walk is a suitable framework for achieving a quadratic speed-up in the quantum mixing processes, because of its relationship with interpolated walks [KMOR16] and the phase estimation algorithm [CEMM98]. Accordingly, we give two definitions of quantum mixing times via controlled quantum walks: instant quantum mixing time and average quantum mixing time. We later prove the bounds of these two quantum mixing times on complete graphs. We also prove the upper bound of the average quantum mixing times on cycle graphs. To have clearer views on the bounds of instant and average quantum mixing times, we then discuss and present the simulations of quantum mixing processes, with four types of graphs (complete graphs, cycle graphs, 2-dimensional tori and dumbbell graphs) as examples. The simulations include the quantum mixing processes with the same ~θ (an important parameter in the controlled quantum walk operator) as in [DH17a], and the suitable range of ~θ that achieves the quadratic speed-up for different types of graphs. Inspired by the quantum search algorithm via controlled quantum walks in [DH17a], we adapt this algorithm to a new algorithm of quantum mixing, which generates samples from the stationary distribution of the Markov chain, with its runtime upper bounded by the quantum hitting time, which is quadratically faster than the classical hitting process.
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Quantum, Quantum Computing, Quantum Algorithm, Quantum Walks
Citation
Li, S. (2020). Bounds of Quantum Mixing Processes via Controlled Walks (Master's thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.