Strachey Lecture 2016, generously supported by Oxford Asset Management. In the near future, it will likely become possible to perform special-purpose quantum computations that, while not immediately useful for anything, are plausibly hard to simulate using a classical computer. These "quantum supremacy experiments" would be a scientific milestone---decisively answering quantum computing skeptics, while casting doubt on one of the foundational tenets of computer science, the Extended Church-Turing Thesis. At the same time, these experiments also raise fascinating questions for computational complexity theorists: for example, on what grounds should we believe that a given quantum system really is hard to simulate classically?.
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