On Complexity, Energy- and Implementation-Efficiency of Channel Decoders

Future wireless communication systems require efficient and flexible baseband receivers. Meaningful efficiency metrics are key for design space exploration to quantify the algorithmic and the implementation complexity of a receiver. Most of the current established efficiency metrics are based on counting operations, thus neglecting important issues like data and storage complexity. In this paper we introduce suitable energy and area efficiency metrics which resolve the afore-mentioned disadvantages. These are decoded information bit per energy and throughput per area unit. Efficiency metrics are assessed by various implementations of turbo decoders, LDPC decoders and convolutional decoders. New exploration methodologies are presented, which permit an appropriate benchmarking of implementation efficiency, communications performance, and flexibility trade-offs. These exploration methodologies are based on efficiency trajectories rather than a single snapshot metric as done in state-of-the-art approaches.Comment: Submitted to IEEE Transactions on Communication

The computational complexity of density functional theory

Density functional theory is a successful branch of numerical simulations of quantum systems. While the foundations are rigorously defined, the universal functional must be approximated resulting in a `semi'-ab initio approach. The search for improved functionals has resulted in hundreds of functionals and remains an active research area. This chapter is concerned with understanding fundamental limitations of any algorithmic approach to approximating the universal functional. The results based on Hamiltonian complexity presented here are largely based on \cite{Schuch09}. In this chapter, we explain the computational complexity of DFT and any other approach to solving electronic structure Hamiltonians. The proof relies on perturbative gadgets widely used in Hamiltonian complexity and we provide an introduction to these techniques using the Schrieffer-Wolff method. Since the difficulty of this problem has been well appreciated before this formalization, practitioners have turned to a host approximate Hamiltonians. By extending the results of \cite{Schuch09}, we show in DFT, although the introduction of an approximate potential leads to a non-interacting Hamiltonian, it remains, in the worst case, an NP-complete problem.Comment: Contributed chapter to "Many-Electron Approaches in Physics, Chemistry and Mathematics: A Multidisciplinary View

Approximating Gibbs states of local Hamiltonians efficiently with PEPS

We analyze the error of approximating Gibbs states of local quantum spin Hamiltonians on lattices with Projected Entangled Pair States (PEPS) as a function of the bond dimension ($D$), temperature ($\beta^{-1}$), and system size ($N$). First, we introduce a compression method in which the bond dimension scales as $D=e^{O(\log^2(N/\epsilon))}$ if $\beta<O(\log (N))$. Second, building on the work of Hastings [Phys. Rev. B 73, 085115 (2006)], we derive a polynomial scaling relation, $D=\left(N/\epsilon\right)^{O(\beta)}$. This implies that the manifold of PEPS forms an efficient representation of Gibbs states of local quantum Hamiltonians. From those bounds it also follows that ground states can be approximated with $D=N^{O(\log(N))}$ whenever the density of states only grows polynomially in the system size. All results hold for any spatial dimension of the lattice.Comment: 12 pages, 1 figur

Entanglement phases as holographic duals of anyon condensates

Anyon condensation forms a mechanism which allows to relate different topological phases. We study anyon condensation in the framework of Projected Entangled Pair States (PEPS) where topological order is characterized through local symmetries of the entanglement. We show that anyon condensation is in one-to-one correspondence to the behavior of the virtual entanglement state at the boundary (i.e., the entanglement spectrum) under those symmetries, which encompasses both symmetry breaking and symmetry protected (SPT) order, and we use this to characterize all anyon condensations for abelian double models through the structure of their entanglement spectrum. We illustrate our findings with the Z4 double model, which can give rise to both Toric Code and Doubled Semion order through condensation, distinguished by the SPT structure of their entanglement. Using the ability of our framework to directly measure order parameters for condensation and deconfinement, we numerically study the phase diagram of the model, including direct phase transitions between the Doubled Semion and the Toric Code phase which are not described by anyon condensation.Comment: 20+7 page