CMT Forum: Andrew Lucas

Low-density parity check (LDPC) codes are a powerful way to robustly protect classical, or quantum, information.  Although the Ising model is already the simplest kind of LDPC code, mathematicians have discovered more useful LDPC codes, in which n physical bits store O(n) logical bits, and protect them against O(n) errors.   I will summarize what LDPC codes are and then explain how we have used the structure of LDPC codes to prove two intriguing results in quantum statistical mechanics.  First, I will show that quantum LDPC codes are counterexamples to the intuition that passive quantum error correction only exists in systems with a finite temperature phase transition to topological order.  LDPC codes can have trivial free energy, like the 1d Ising model, yet also be superior quantum memories than high-dimensional toric codes.  This suggests that qLDPC codes could enable new experimental techniques for quantum error correction based only on few-body measurements and feedback.  Secondly, I will present extensive many-body Hamiltonians based on LDPC codes that have a many-body mobility edge:  all eigenstates below a critical energy density are localized in an exponentially small fraction of the "energetically-accessible" state space.  Localization does not require strong disorder, but is guaranteed by the “infinite spatial dimensions” of the code.