Quantum Hamiltonian complexity studies questions at the intersection of condensed matter physics and complexity theory. In this talk I will focus on three basic questions:
1. Do 'typical' quantum states that occur in Nature have succinct (polynomial) description?
2. Can quantum systems at room temperature exhibit exponential complexity?
3. Is the scientific method sufficiently powerful to comprehend general quantum systems?
Each of these issues is best studied through the computational lens as a question about computation. The resulting questions lie at the core of computational complexity theory. The first asks about the structure of solutions to the quantum analog of SAT. The second asks whether there is a quantum analog of the PCP theorem. And the third can be formulated as a question about interactive proof systems with BQP provers.
The computational lens is a major theme for the new Simons Institute at Berkeley, and I will briefly describe a special semester-long program on Quantum Hamiltonian Complexity that we are organizing at the Simons Institute in Spring 2014.
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