In this talk we'll examine a problem in topology, and we will uncover quantum mechanical structure in this seemingly classical setting. The connection can be formalized by complexity theory: the hardness of deciding whether or not a space has a hole is characterized by QMA. We will look at implications in quantum algorithms, supersymmetric quantum mechanics, and quantum PCPs. We will also see some of the key ideas behind the quantum hardness proof, which involves a tool from algebraic topology known as spectral sequences.

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