We improve Knabe's spectral gap bound for frustration-free translation-invariant local Hamiltonians in 1D. The bound is based on a relationship between global and local gaps. The global gap is the spectral gap of a size-m chain with periodic boundary conditions, while the local gap is that of a subchain of size n < m with open boundary conditions. Knabe proved that if the local gap is larger than the threshold value 1/(n − 1) for some n > 2, then the global gap is lower bounded by a positive constant in the thermodynamic limit m → ∞. Here we improve the threshold to 6 n(n+1) , which is better (smaller) for all n > 3 and which is asymptotically optimal. As a corollary we establish a surprising fact about 1D translation-invariant frustration-free systems that are gapless in the thermodynamic limit: for any such system the spectral gap of a size-n chain with open boundary conditions is upper bounded as O(n −2 ). This contrasts with gapless frustrated systems where the gap can be Θ(n −1 ). It also limits the extent to which the area law is violated in these frustration-free systems, since it implies that the half-chain entanglement entropy is O(1/ √ ϵ) as a function of spectral gap ϵ. We extend our results to frustration-free systems on a 2D square lattice. Published by AIP Publishing.[http://dx

Markovian master equations are a ubiquitous tool in the study of open quantum systems, but deriving them from first principles involves a series of compromises. On the one hand, the Redfield equation is valid for fast environments (whose correlation function decays much faster than the system relaxation time) regardless of the relative strength of the coupling to the system Hamiltonian, but is notoriously non-completely-positive. On the other hand, the Davies equation preserves complete positivity but is valid only in the ultra-weak coupling limit and for systems with a finite level spacing, which makes it incompatible with arbitrarily fast time-dependent driving. Here we show that a recently derived Markovian coarse-grained master equation (CGME), already known to be completely positive, has a much expanded range of applicability compared to the Davies equation, and moreover, is locally generated and can be generalized to accommodate arbitrarily fast driving. This generalization, which we refer to as the time-dependent CGME, is thus suitable for the analysis of fast operations in gate-model quantum computing, such as quantum error correction and dynamical decoupling. Our derivation proceeds directly from the Redfield equation and allows us to place rigorous error bounds on all three equations: Redfield, Davies, and coarse-grained. Our main result is thus a completely positive Markovian master equation that is a controlled approximation to the true evolution for any time-dependence of the system Hamiltonian, and works for systems with arbitrarily small level spacing. We illustrate this with an analysis showing that dynamical decoupling can extend coherence times even in a strictly Markovian setting.

In quantum many-body systems, the existence of a spectral gap above the ground state has far-reaching consequences. In this paper, we discuss "finite-size" criteria for having a spectral gap in frustration-free spin systems and their applications.We extend a criterion that was originally developed for periodic systems by Knabe and Gosset-Mozgunov to systems with a boundary. Our finite-size criterion says that if the spectral gaps at linear system size n exceed an explicit threshold of order n −3/2 , then the whole system is gapped. The criterion takes into account both "bulk gaps" and "edge gaps" of the finite system in a precise way. The n −3/2 scaling is robust: it holds in 1D and 2D systems, on arbitrary lattices and with arbitrary finite-range interactions. One application of our results is to give a rigorous foundation to the folklore that 2D frustration-free models cannot host chiral edge modes (whose finite-size spectral gap would scale like n −1 ). arXiv:1801.08915v2 [quant-ph] 3 May 2019It is well-known that the existence of a spectral gap may depend on the imposed boundary conditions, and this fact is at the core of our work.In general, the question whether a quantum spin system is gapped or gapless is difficult: In 1D, the Haldane conjecture [15,16] ("antiferromagnetic, integer-spin Heisenberg chains are gapped") remains open after 30 years of investigation. In 2D, the "gapped versus gapless" dichotomy is in fact undecidable in general [10], even among the class of translation-invariant, nearest-neighbor Hamiltonians.This paper studies the spectral gaps of a comparatively simple class of models: 1D and 2D frustration-free (FF) quantum spin systems with a non-trivial boundary (i.e., open boundary conditions). (A famous FF spin system is the AKLT chain [1]. One general reason why FF systems arise is that any quantum state which is only locally correlated can be realized as the ground state of an appropriate FF "parent Hamiltonian" [12,32,35].)Specifically, we are interested in finite-size criteria for having a spectral gap in such systems. Let us explain what we mean by this.Let γ m denote the spectral gap of the Hamiltonian of interest, when it acts on systems of linear size m. Letγ n be the "local gap", i.e., the spectral gap of a subsystem of linear size up to n. (We will be more precise later.) The finite-size criterion is a bound of the form γ m ≥ c n (γ n − t n ), (1.1) for all m sufficiently large compared to n (say m ≥ 2n).Here c n > 0 is an unimportant constant, but the value of t n (in particular its ndependence) is critical. Indeed, if for some fixed n 0 , we know from somewhere that γ n 0 > t n 0 , then (1.1) gives a uniform lower bound on the spectral gap γ m for all sufficiently large m. Accordingly, we call t n the "local gap threshold".The general idea to prove a finite-size criterion like (1.1) is that the Hamiltonian on systems of linear size m can be constructed out of smaller Hamiltonians acting on subsystems of linear size up to n, and these can be controlled in terms ofγ n . We emph...