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 $Ntimes{N}$ matrix $A$ and a vector $vec b$, find the vector $vec x$ that satisfies $Avec x = vec b$. 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.
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