An alternating phase-walk on a star graph with percolation
R.S.H. Steller (TU Delft - Applied Sciences)
M. Blaauboer – Mentor (TU Delft - QN/Blaauboer Group)
R.H.M. Smit – Graduation committee member (TU Delft - QN/Smit Lab)
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Abstract
Since their theoretical introduction in 1980, quantum computers have progressed significantly in
both theory and experimentation. Quantum computers have the potential to solve certain prob-
lems significantly faster than their classical counterparts. One particularly promising area is search
algorithms. In 1996, Lov Grover introduced a quantum search algorithm that laid the founda-
tion for numerous variants. Among them, the alternating phase-walk stands out as a method for
searching an element in an ordered list, which is modeled as a graph. However, a major challenge
in practical quantum computing remains quantum decoherence.
The objective of this project was to model decoherence as bond percolation on a star graph
during an alternating phase-walk, and to investigate its resulting effects. The introduction of
decoherence transformed the search algorithm from a deterministic process into a probabilistic
one, as the probability of measuring the marked state is no longer guaranteed to be 100% on every
run of the algorithm. The most significant results were observed when varying the walk time in
the continuous-time quantum walk (CTQW), both with and without a varying initial state. Here,
the optimal time topt and its corresponding maximum average value μmax are presented for each
value of p.
Static Initial State
• N = 3 star graph:
[p = 0.1, topt = 2.46 τ3, μmax = 0.67]
[p = 0.5, topt = 0.62 τ3, μmax = 0.91]
[p = 0.9, topt = 0.37 τ3, μmax = 1.00]
• N = 7 star graph:
[p = 0.1, topt = 2.21 τ7, μmax = 0.56]
[p = 0.5, topt = 0.59 τ7, μmax = 0.96]
[p = 0.9, topt = 0.34 τ7, μmax = 0.99]
• N = 14 star graph:
[p = 0.1, topt = 5.50 τ14, μmax = 0.55]
[p = 0.5, topt = 0.62 τ14, μmax = 0.96]
[p = 0.9, topt = 0.35 τ14, μmax = 0.99]
Varying Initial State
• N = 3 star graph:
[p = 0.1, topt = 2.30 τ3, μmax = 0.68]
[p = 0.5, topt = 0.37 τ3, μmax = 0.70]
[p = 0.9, topt = 0.36 τ3, μmax = 0.99]
• N = 7 star graph:
[p = 0.1, topt = 2.30 τ7, μmax = 0.57]
[p = 0.5, topt = 0.37 τ7, μmax = 0.66]
[p = 0.9, topt = 0.33 τ7, μmax = 0.99]
• N = 14 star graph:
[p = 0.1, topt = 6.42 τ14, μmax = 0.58]
[p = 0.5, topt = 0.39 τ14, μmax = 0.60]
[p = 0.9, topt = 0.34 τ14, μmax = 0.98]