  Recently, it is shown that quantum computers can be used for obtaining
certain information about the solution of a linear system Ax=b exponentially
faster than what is possible with classical computation. Here we first review
some key aspects of the algorithm from the standpoint of finding its efficient
quantum circuit implementation using only elementary quantum operations, which
is important for determining the potential usefulness of the algorithm in
practical settings. Then we present a small-scale quantum circuit that solves a
2x2 linear system. The quantum circuit uses only 4 qubits, implying a tempting
possibility for experimental realization. Furthermore, the circuit is
numerically simulated and its performance under different circuit parameter
settings is demonstrated.
