Circuit Hamiltonians, Hamiltonian complexity, and approximate error correction.

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In this talk, I will discuss some interesting connections between Hamiltonian complexity, error correction, and quantum circuits. First, motivated by the Quantum PCP Conjecture, I will describe a construction of a family of local Hamiltonians where the complexity of ground states — even when subject to large amounts of noise — is superpolynomial (under plausible complexity assumptions). The construction is simple, making use of the well-known Feynman-Kitaev circuit Hamiltonian construction.

Next, I will describe how this complexity result can be repurposed to construct local approximate error correcting codes with (nearly) linear distance and (nearly) constant rate. By contrast, it remains an open problem to construct quantum low-density parity check (QLDPC) stabilizer codes that achieve similarly good parameters.

An interesting component of our code construction is the Brueckmann-Terhal spacetime circuit Hamiltonian construction, and a novel analysis of its spectral gap. Our codes are ground spaces of non-commuting  — but frustration-free -- local Hamiltonians. We believe that our results present additional evidence that the study of approximate error correction and non-stabilizer codes may be much richer than previously thought.

This talk will have more questions than answers, and is based on two papers:

· (joint with Chinmay Nirkhe and Umesh Vazirani)

· (joint work with Thom Bohdanowicz, Elizabeth Crosson, and Chinmay Nirkhe)