  In this brief paper, we go through the theoretical steps of the spectral
clustering on quantum computers by employing the phase estimation and the
amplitude amplification algorithms. We discuss circuit designs for each step
and show how to obtain the clustering solution from the output state. In
addition, we introduce a biased version of the phase estimation algorithm which
significantly speeds up the amplitude amplification process. The complexity of
the whole process is analyzed: it is shown that when the circuit representation
of a data matrix of order $N$ is produced through an ancilla based circuit in
which the matrix is written as a sum of $L$ number of Householder matrices; the
computational complexity is bounded by $O(2^mLN)$ number of quantum gates.
Here, $m$ represents the number of qubits (e.g., 6) involved in the phase
register of the phase estimation algorithm.
