STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Quantum computation and quantum information
Quantum computation and quantum information
An Example of the Difference Between Quantum and Classical Random Walks
Quantum Information Processing
Quantum Walks on the Hypercube
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
Exponential algorithmic speedup by a quantum walk
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
How Powerful is Adiabatic Quantum Computation?
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Quantum Speed-Up of Markov Chain Based Algorithms
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Quantum Walk Algorithm for Element Distinctness
SIAM Journal on Computing
Any AND-OR Formula of Size N can be Evaluated in time N^{1/2 + o(1)} on a Quantum Computer
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Decoherence in quantum walks – a review
Mathematical Structures in Computer Science
Mixing and decoherence in continuous-time quantum walks on cycles
Quantum Information & Computation
Quantum walks: a comprehensive review
Quantum Information Processing
Mixing-time and large-decoherence in continuous-time quantum walks on one-dimension regular networks
Quantum Information Processing
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The n-dimensional hypercube quantum random walk (QRW) is a particularilyappealing example of a quantum walk because it has a natural implementationon a register on n qubits. However, any real implementation will encounterdecoherence effects due to interactions with uncontrollable degrees of freedom.We present a complete characterization of the mixing properties of the hypercubeQRW under a physically relevant Markovian decoherence model. In the localdecoherence model considered the non-unitary dynamics are modeled as a sum ofprojections on individual qubits to an arbitrary direction on the Bloch sphere. Weprove that there is always classical (asymptotic) mixing in this model and specifythe conditions under which instantaneous mixing always exists. And we show thatthe latter mixing property, as well as the classical mixing time, depend heavilyon the exact environmental interaction and its strength. Therefore, algorithmicapplications of the QRW on the hypercube, if they intend to employ mixingproperties, need to consider both the walk dynamics and the precise decoherencemodel.