Abstract
The lazy random walk, where the walker has some probability of staying put, is a useful tool in classical algorithms. We propose a quantum analogue, the lackadaisical quantum walk, where each vertex is given l self-loops, and we investigate its effects on Grover's algorithm when formulated as search for a marked vertex on the complete graph of N vertices. For the discrete-time quantum walk using the phase flip coin, adding a self-loop to each vertex boosts the success probability from 1/2 to 1. Additional self-loops, however, decrease the success probability. Using instead the Shenvi, Kempe, and Whaley (2003) coin, adding self-loops simply slows down the search. These coins also differ in that the first is faster than classical when l scales less than N, while the second requires that l scale less than N 2. Finally, continuous-time quantum walks differ from both of these discrete-time examples - the self-loops make no difference at all. These behaviors generalize to multiple marked vertices.
| Original language | English (US) |
|---|---|
| Article number | 435304 |
| Journal | Journal of Physics A: Mathematical and Theoretical |
| Volume | 48 |
| Issue number | 43 |
| DOIs | |
| State | Published - Oct 7 2015 |
| Externally published | Yes |
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All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Statistics and Probability
- Modeling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)
Cite this
Grover search with lackadaisical quantum walks. / Wong, Thomas.
In: Journal of Physics A: Mathematical and Theoretical, Vol. 48, No. 43, 435304, 07.10.2015.Research output: Contribution to journal › Article
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TY - JOUR
T1 - Grover search with lackadaisical quantum walks
AU - Wong, Thomas
PY - 2015/10/7
Y1 - 2015/10/7
N2 - The lazy random walk, where the walker has some probability of staying put, is a useful tool in classical algorithms. We propose a quantum analogue, the lackadaisical quantum walk, where each vertex is given l self-loops, and we investigate its effects on Grover's algorithm when formulated as search for a marked vertex on the complete graph of N vertices. For the discrete-time quantum walk using the phase flip coin, adding a self-loop to each vertex boosts the success probability from 1/2 to 1. Additional self-loops, however, decrease the success probability. Using instead the Shenvi, Kempe, and Whaley (2003) coin, adding self-loops simply slows down the search. These coins also differ in that the first is faster than classical when l scales less than N, while the second requires that l scale less than N 2. Finally, continuous-time quantum walks differ from both of these discrete-time examples - the self-loops make no difference at all. These behaviors generalize to multiple marked vertices.
AB - The lazy random walk, where the walker has some probability of staying put, is a useful tool in classical algorithms. We propose a quantum analogue, the lackadaisical quantum walk, where each vertex is given l self-loops, and we investigate its effects on Grover's algorithm when formulated as search for a marked vertex on the complete graph of N vertices. For the discrete-time quantum walk using the phase flip coin, adding a self-loop to each vertex boosts the success probability from 1/2 to 1. Additional self-loops, however, decrease the success probability. Using instead the Shenvi, Kempe, and Whaley (2003) coin, adding self-loops simply slows down the search. These coins also differ in that the first is faster than classical when l scales less than N, while the second requires that l scale less than N 2. Finally, continuous-time quantum walks differ from both of these discrete-time examples - the self-loops make no difference at all. These behaviors generalize to multiple marked vertices.
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UR - http://www.scopus.com/inward/citedby.url?scp=84946051064&partnerID=8YFLogxK
U2 - 10.1088/1751-8113/48/43/435304
DO - 10.1088/1751-8113/48/43/435304
M3 - Article
AN - SCOPUS:84946051064
VL - 48
JO - Journal of Physics A: Mathematical and Theoretical
JF - Journal of Physics A: Mathematical and Theoretical
SN - 1751-8113
IS - 43
M1 - 435304
ER -