Quasi-Quantum States and the Quasi-Quantum PCP Theorem

Explore a groundbreaking mathematical framework that bridges quantum computing and classical complexity theory through this 53-minute conference talk. Discover k-local quasi-quantum states, a novel extension of regular quantum states created by relaxing positivity constraints, and learn how these states can be mapped one-to-one to distributions of assignments over classical variables with specific alphabet constraints. Examine the equivalence between solving k-local Hamiltonians over quasi-quantum states and optimizing distributions over classical k-local constraint satisfaction problems, while understanding why this optimization remains NP-complete despite its classical nature. Investigate the unique structural properties these distributions share with ordinary quantum states, including their lack of simple tensor-product structure and the resulting complexity in determining them from local marginals. Delve into the main theoretical contribution: a PCP theorem for k-local Hamiltonians over quasi-quantum states presented as a hardness-of-approximation result, and understand how the proof methodology suggests new promise-gap amplification procedures that provide valuable insights into the quantum PCP conjecture.