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Quantum computers have the potential of solving certain problemsexponentially faster than classical computers. Recently, Harrow, Hassidim andLloyd 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 $\vecx$ that satisfies $A\vec x = \vec b$. It has been shown that using thealgorithm one could obtain the solution encoded in a quantum state $|x$ using$O(\log{N})$ quantum operations, while classical algorithms require at leastO(N) steps. If one is not interested in the solution $\vec{x}$ itself butcertain statistical feature of the solution ${x}|M|x$ ($M$ is some quantummechanical operator), the quantum algorithm will be able to achieve exponentialspeedup over the best classical algorithm as $N$ grows. Here we report aproof-of-concept experimental demonstration of the quantum algorithm using a4-qubit nuclear magnetic resonance (NMR) quantum information processor. For allthe three sets of experiments with different choices of $\vec b$, we obtain thesolutions with over 96% fidelity. This experiment is a first implementation ofthe algorithm. Because solving linear systems is a common problem in nearly allfields of science and engineering, we will also discuss the implication of ourresults on the potential of using quantum computers for solving practicallinear systems.

Experimental realization of quantum algorithm for solving linear systems of equations