Abstract
Quantum walks are standard tools for searching graphs for marked vertices, and they often yield quadratic speedups over a classical random walk’s hitting time. In some exceptional cases, however, the system only evolves by sign flips, staying in a uniform probability distribution for all time. We prove that the one-dimensional periodic lattice or cycle with any arrangement of marked vertices is such an exceptional configuration. Using this discovery, we construct a search problem where the quantum walk’s random sampling yields an arbitrary speedup in query complexity over the classical random walk’s hitting time. In this context, however, the mixing time to prepare the initial uniform state is a more suitable comparison than the hitting time, and then, the speedup is roughly quadratic.
| Original language | English (US) |
|---|---|
| Article number | 154 |
| Journal | Quantum Information Processing |
| Volume | 16 |
| Issue number | 6 |
| DOIs | |
| State | Published - Jun 1 2017 |
| Externally published | Yes |
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All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Theoretical Computer Science
- Electronic, Optical and Magnetic Materials
- Signal Processing
- Modeling and Simulation
- Electrical and Electronic Engineering
Cite this
Exceptional quantum walk search on the cycle. / Wong, Thomas; Santos, Raqueline A.M.
In: Quantum Information Processing, Vol. 16, No. 6, 154, 01.06.2017.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Exceptional quantum walk search on the cycle
AU - Wong, Thomas
AU - Santos, Raqueline A.M.
PY - 2017/6/1
Y1 - 2017/6/1
N2 - Quantum walks are standard tools for searching graphs for marked vertices, and they often yield quadratic speedups over a classical random walk’s hitting time. In some exceptional cases, however, the system only evolves by sign flips, staying in a uniform probability distribution for all time. We prove that the one-dimensional periodic lattice or cycle with any arrangement of marked vertices is such an exceptional configuration. Using this discovery, we construct a search problem where the quantum walk’s random sampling yields an arbitrary speedup in query complexity over the classical random walk’s hitting time. In this context, however, the mixing time to prepare the initial uniform state is a more suitable comparison than the hitting time, and then, the speedup is roughly quadratic.
AB - Quantum walks are standard tools for searching graphs for marked vertices, and they often yield quadratic speedups over a classical random walk’s hitting time. In some exceptional cases, however, the system only evolves by sign flips, staying in a uniform probability distribution for all time. We prove that the one-dimensional periodic lattice or cycle with any arrangement of marked vertices is such an exceptional configuration. Using this discovery, we construct a search problem where the quantum walk’s random sampling yields an arbitrary speedup in query complexity over the classical random walk’s hitting time. In this context, however, the mixing time to prepare the initial uniform state is a more suitable comparison than the hitting time, and then, the speedup is roughly quadratic.
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U2 - 10.1007/s11128-017-1606-y
DO - 10.1007/s11128-017-1606-y
M3 - Article
VL - 16
JO - Quantum Information Processing
JF - Quantum Information Processing
SN - 1570-0755
IS - 6
M1 - 154
ER -