This study presents a novel technique to estimate the computational complexity of sequential decoding using the Berry-Esseen theorem. Unlike the theoretical bounds determined by the conventional central limit theorem argument, which often holds only for sufficiently large codeword length, the new bound obtained from the Berry-Esseen theorem is valid for any blocklength. The accuracy of the new ...

To lower bound the effective dimension for local Hamiltonian models, we will use a theorem from [1, 2] as a stepping stone. This is a quantum version of the BerryEsseen theorem. The Berry-Esseen theorem is a more powerful statement than the central limit theorem, as it gives the rate of convergence of a distribution to a Gaussian. Let H = ∑ ν Eν |ν〉〈ν| be k-local, i.e., let H be of the form H =...

We consider the random walk among random conductances on Z. We assume that the conductances are independent, identically distributed and uniformly bounded away from 0 and infinity. We obtain a quantitative version of the central limit theorem for this random walk, which takes the form of a Berry-Esseen estimate with speed t−1/10 for d 6 2, and speed t−1/5 for d > 3, up to logarithmic corrections.

We consider the problem of whether the canonical and microcanonical ensembles are locally equivalent for short-ranged quantum Hamiltonians of N spins arranged on a ddimensional lattices. For any temperature for which the system has a finite correlation length, we prove that the canonical and microcanonical state are approximately equal on regions containing up toO(N) spins. The proof rests on a...