t b a b a

2. Partial Quantum Search: Partial Quantum Search considers the search problem
where given a quantum database f : {0, 1} n → {0, 1} such that f(x) = 1 for a unique
x ∈ {0, 1} n , we are required to determine only the first k bits of the address x. The
quantum algorithms and low bound can be found in [GR05, KG06].
3. Quantum Counting: To determine the number of solutions, M, to an N item search
problem, it takes O(N) consultations on a classical computer. On a quantum computer
it is possible to estimate the number of solutions much more quickly by combining the
Grover’s algorithm with the phase estimation based on quantum Fourier transform.
[BHT98]
4. Quantum Walk: Quantum random walks on graphs have been shown to display
many interesting properties, including exponentially fast hitting times when compared
with their classical counterparts [AAK01,Kem03]. It also provides an intuitive frame-
work on which to build novel quantum algorithms which run faster than their classical
counterparts. It can be divided into two models: discrete time quantum random walk
and continuous-time quantum random walk. Many applications such as search algo-
rithm [SKW03], element distinctness [Amb07] are developed upon quantum walk.
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