The summary cleverly omits that solving a linear equation is neither NP complete nor NP hard, the speed up is from O(n) to O(log n). I think you’d need a ginormous matrix for this to be useful depending on the constants and such (Of course it’d be crime for me to read paper instead of the abstract to actually find out the details). They already have the applications for quantum computing, it will be much more interesting when they actually figure out a way to build the damn things. In this case, when A is sparse and well conditioned, with largest dimension n, the best classical algorithms can x and estimate xMx in O(n) time. Here, we exhibit a quantum algorithm for solving linear sets of equations that runs in O(log n) time, an exponential improvement over the best classical algorithm.
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