Faster search by lackadaisical quantum walk

  • Thomas G. Wong


In the typical model, a discrete-time coined quantum walk searching the 2D grid for a marked vertex achieves a success probability of \(O(1/\log N)\) in \(O(\sqrt{N \log N})\) steps, which with amplitude amplification yields an overall runtime of \(O(\sqrt{N} \log N)\). We show that making the quantum walk lackadaisical or lazy by adding a self-loop of weight 4 / N to each vertex speeds up the search, causing the success probability to reach a constant near 1 in \(O(\sqrt{N \log N})\) steps, thus yielding an \(O(\sqrt{\log N})\) improvement over the typical, loopless algorithm. This improved runtime matches the best known quantum algorithms for this search problem. Our results are based on numerical simulations since the algorithm is not an instance of the abstract search algorithm.


Quantum walk Lackadaisical quantum walk Spatial search 



Thanks to Scott Aaronson for useful discussions. This work was supported by the U.S. Department of Defense Vannevar Bush Faculty Fellowship of Scott Aaronson.


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Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Texas at AustinAustinUSA
  2. 2.Department of PhysicsCreighton UniversityOmahaUSA

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