Connecting global and local energy distributions in quantum spin models on a lattice

Generally, the local interactions in a many-body quantum spin system on a lattice do not commute with each other. Consequently, the Hamiltonian of a local region will generally not commute with that of the entire system, and so the two cannot be measured simultaneously. The connection between the probability distributions of measurement outcomes of the local and global Hamiltonians will depend on the angles between the diagonalizing bases of these two Hamiltonians. In this paper we characterize the relation between these two distributions. On one hand, we upperbound the probability of measuring an energy $\tau$ in a local region, if the global system is in a superposition of eigenstates with energies $\epsilon<\tau$. On the other hand, we bound the probability of measuring a global energy $\epsilon$ in a bipartite system that is in a tensor product of eigenstates of its two subsystems. Very roughly, we show that due to the local nature of the governing interactions, these distributions are identical to what one encounters in the commuting case, up to some exponentially small corrections. Finally, we use these bounds to study the spectrum of a locally truncated Hamiltonian, in which the energies of a contiguous region have been truncated above some threshold energy $\tau$. We show that the lower part of the spectrum of this Hamiltonian is exponentially close to that of the original Hamiltonian. A restricted version of this result in 1D was a central building block in a recent improvement of the 1D area-law.Comment: 23 pages, 2 figures. A new version with tigheter bounds and a re-written introductio

The Feynman-Kitaev computer's clock: bias, gaps, idling and pulse tuning

We present a collection of results about the clock in Feynman's computer construction and Kitaev's Local Hamiltonian problem. First, by analyzing the spectra of quantum walks on a line with varying endpoint terms, we find a better lower bound on the gap of the Feynman Hamiltonian, which translates into a less strict promise gap requirement for the QMA-complete Local Hamiltonian problem. We also translate this result into the language of adiabatic quantum computation. Second, introducing an idling clock construction with a large state space but fast Cesaro mixing, we provide a way for achieving an arbitrarily high success probability of computation with Feynman's computer with only a logarithmic increase in the number of clock qubits. Finally, we tune and thus improve the costs (locality, gap scaling) of implementing a (pulse) clock with a single excitation.Comment: 32 pages; v2: typos, references, and minor fixes, close to the published versio

An improved 1D area law for frustration-free systems

We present a new proof for the 1D area law for frustration-free systems with a constant gap, which exponentially improves the entropy bound in Hastings' 1D area law, and which is tight to within a polynomial factor. For particles of dimension $d$, spectral gap $\epsilon>0$ and interaction strength of at most $J$, our entropy bound is S_{1D}\le \orderof{1}X^3\log^8 X where X\EqDef(J\log d)/\epsilon. Our proof is completely combinatorial, combining the detectability lemma with basic tools from approximation theory. Incorporating locality into the proof when applied to the 2D case gives an entanglement bound that is at the cusp of being non-trivial in the sense that any further improvement would yield a sub-volume law.Comment: 15 pages, 6 figures. Some small style corrections and updated ref