In this talk I will describe Decoded Quantum Interferometry (DQI), a quantum algorithm for reducing classical optimization problems to classical decoding problems by exploiting structure in the Fourier spectrum of the objective function. (See: https://arxiv.org/abs/2408.08292.) For a regression problem called optimal polynomial intersection, which has been previously studied in the contexts of coding theory and cryptanalysis, we believe DQI achieves an exponential quantum speedup. We also investigate the application of DQI to average-case instances of max-k-XORSAT. DQI reduces max-k-XORSAT to decoding LDPC codes, which can be achieved using powerful classical algorithms such as belief propagation. As an initial investigation, we benchmark DQI using belief propagation decoding against classical optimization via simulated annealing. In this setting we identify a family of max-XORSAT instances where DQI achieves a better approximation ratio than simulated annealing, although not better than specialized classical algorithms tailored to those instances. The recent quantum query complexity speedup of Yamakawa and Zhandry can also be obtained as a special case of DQI. This is joint work with Noah Shutty, Mary Wootters, Adam Zalcman, Alexander Schmidhuber, Robbie King, Sergei V. Isakov, and Ryan Babbush.

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