  Quantum computers have the potential of solving certain problems
exponentially faster than classical computers. Recently, Harrow, Hassidim and
Lloyd proposed a quantum algorithm for solving linear systems of equations:
given an $N\times{N}$ matrix $A$ and a vector $\vec b$, find the vector $\vec
x$ that satisfies $A\vec x = \vec b$. It has been shown that using the
algorithm one could obtain the solution encoded in a quantum state $|x$ using
$O(\log{N})$ quantum operations, while classical algorithms require at least
O(N) steps. If one is not interested in the solution $\vec{x}$ itself but
certain statistical feature of the solution ${x}|M|x$ ($M$ is some quantum
mechanical operator), the quantum algorithm will be able to achieve exponential
speedup over the best classical algorithm as $N$ grows. Here we report a
proof-of-concept experimental demonstration of the quantum algorithm using a
4-qubit nuclear magnetic resonance (NMR) quantum information processor. For all
the three sets of experiments with different choices of $\vec b$, we obtain the
solutions with over 96% fidelity. This experiment is a first implementation of
the algorithm. Because solving linear systems is a common problem in nearly all
fields of science and engineering, we will also discuss the implication of our
results on the potential of using quantum computers for solving practical
linear systems.
