Graduate Random Quantum Walk: Learn & Use w/ Quantum Gates

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A random quantum walk is a quantum analog of the classical random walk, where a quantum particle explores a graph or lattice through superposition and interference. It can be utilized in quantum computing to enhance algorithms and improve the efficiency of quantum logical gates. Understanding random quantum walks is essential for grasping more complex concepts like continuous-time quantum walks. While some may consider the topic impractical for undergraduates, it is relevant to current research in quantum mechanics. Mastery of this subject could provide valuable insights into quantum information processing.
Frank Schroer
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explanation of a random quantum walk with regards to quantum logical gates
I am an undergraduate doing research on QC/QI. My current topic to learn is continuous-time quantum walks, but first I must learn the random quantum walk. That being said, I was wondering if someone could simply explain what a random quantum walk is and then explain how they could be useful with quantum logical gates. I am familiar with the classical random walk already.
 
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Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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