Abstract
Continuous-time quantum walks and adiabatic quantum evolution are two general techniques for quantum computing, both of which are described by Hamiltonians that govern their evolutions by Schrödinger's equation. In the former, the Hamiltonian is fixed, while in the latter, the Hamiltonian varies with time. As a result, their formulations of Grover's algorithm evolve differently through Hilbert space. We show that this difference is fundamental; they cannot be made to evolve along each other's path without introducing structure more powerful than the standard oracle for unstructured search. For an adiabatic quantum evolution to evolve like the quantum walk search algorithm, it must interpolate between three fixed Hamiltonians, one of which is complex and introduces structure that is stronger than the oracle for unstructured search. Conversely, for a quantum walk to evolve along the path of the adiabatic search algorithm, it must be a chiral quantum walk on a weighted, directed star graph with structure that is also stronger than the oracle for unstructured search. Thus, the two techniques, although similar in being described by Hamiltonians that govern their evolution, compute by fundamentally irreconcilable means.
| Original language | English (US) |
|---|---|
| Article number | 062313 |
| Journal | Physical Review A |
| Volume | 93 |
| Issue number | 6 |
| DOIs | |
| State | Published - Jun 13 2016 |
| Externally published | Yes |
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All Science Journal Classification (ASJC) codes
- Atomic and Molecular Physics, and Optics
Cite this
Irreconcilable difference between quantum walks and adiabatic quantum computing. / Wong, Thomas; Meyer, David A.
In: Physical Review A, Vol. 93, No. 6, 062313, 13.06.2016.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Irreconcilable difference between quantum walks and adiabatic quantum computing
AU - Wong, Thomas
AU - Meyer, David A.
PY - 2016/6/13
Y1 - 2016/6/13
N2 - Continuous-time quantum walks and adiabatic quantum evolution are two general techniques for quantum computing, both of which are described by Hamiltonians that govern their evolutions by Schrödinger's equation. In the former, the Hamiltonian is fixed, while in the latter, the Hamiltonian varies with time. As a result, their formulations of Grover's algorithm evolve differently through Hilbert space. We show that this difference is fundamental; they cannot be made to evolve along each other's path without introducing structure more powerful than the standard oracle for unstructured search. For an adiabatic quantum evolution to evolve like the quantum walk search algorithm, it must interpolate between three fixed Hamiltonians, one of which is complex and introduces structure that is stronger than the oracle for unstructured search. Conversely, for a quantum walk to evolve along the path of the adiabatic search algorithm, it must be a chiral quantum walk on a weighted, directed star graph with structure that is also stronger than the oracle for unstructured search. Thus, the two techniques, although similar in being described by Hamiltonians that govern their evolution, compute by fundamentally irreconcilable means.
AB - Continuous-time quantum walks and adiabatic quantum evolution are two general techniques for quantum computing, both of which are described by Hamiltonians that govern their evolutions by Schrödinger's equation. In the former, the Hamiltonian is fixed, while in the latter, the Hamiltonian varies with time. As a result, their formulations of Grover's algorithm evolve differently through Hilbert space. We show that this difference is fundamental; they cannot be made to evolve along each other's path without introducing structure more powerful than the standard oracle for unstructured search. For an adiabatic quantum evolution to evolve like the quantum walk search algorithm, it must interpolate between three fixed Hamiltonians, one of which is complex and introduces structure that is stronger than the oracle for unstructured search. Conversely, for a quantum walk to evolve along the path of the adiabatic search algorithm, it must be a chiral quantum walk on a weighted, directed star graph with structure that is also stronger than the oracle for unstructured search. Thus, the two techniques, although similar in being described by Hamiltonians that govern their evolution, compute by fundamentally irreconcilable means.
UR - http://www.scopus.com/inward/record.url?scp=84975168246&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84975168246&partnerID=8YFLogxK
U2 - 10.1103/PhysRevA.93.062313
DO - 10.1103/PhysRevA.93.062313
M3 - Article
AN - SCOPUS:84975168246
VL - 93
JO - Physical Review A - Atomic, Molecular, and Optical Physics
JF - Physical Review A - Atomic, Molecular, and Optical Physics
SN - 1050-2947
IS - 6
M1 - 062313
ER -