Search for: All records

Quantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit Λ on each local quantum number, and if the initial state has low local quantum numbers, then an error at most ϵ can be achieved by choosing Λ to scale polylogarithmically with ϵ − 1 , an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on Λ that achieves accuracy ϵ , obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error in time evolution, we also prove that spectrally isolated energy eigenstates can be approximated with accuracy ϵ by truncating local quantum numbers at Λ = polylog ( ϵ − 1 ) . 

While genome size limits the minimum sizes and maximum numbers of cells that can be packed into a given leaf volume, mature cell sizes can be substantially larger than their meristematic precursors and vary in response to abiotic conditions. Mangroves are iconic examples of how abiotic conditions can influence the evolution of plant phenotypes.

Here, we examined the coordination between genome size, leaf cell sizes, cell packing densities and leaf size in 13 mangrove species across four sites in China. Four of these species occurred at more than one site, allowing us to test the effect of climate on leaf anatomy.

We found that genome sizes of mangroves were very small compared to other angiosperms, but, like other angiosperms, mangrove cells were always larger than the minimum size defined by genome size. Increasing mean annual temperature of a growth site led to higher packing densities of veins (Dv) and stomata (Ds) and smaller epidermal cells but had no effect on stomatal size. In contrast to other angiosperms, mangroves exhibited (1) a negative relationship between guard cell size and genome size; (2) epidermal cells that were smaller than stomata; and (3) coordination between Dv and Ds that was not mediated by epidermal cell size. Furthermore, mangrove epidermal cell sizes and packing densities covaried with leaf size.

While mangroves exhibited coordination between veins and stomata and attained a maximum theoretical stomatal conductance similar to that of other angiosperms, the tissue-level tradeoffs underlying these similar relationships across species and environments were markedly different, perhaps indicative of the unique structural and physiological adaptations of mangroves to their stressful environments.

 

In this article, we present two new concepts related to subgraph counting where the focus is not on the number of subgraphs that are isomorphic to some fixed graph $H$, but on the frequency with which a vertex or an edge belongs to such subgraphs. In particular, we are interested in the case where $H$ is a complete graph. These new concepts are termed vertex participation and edge participation, respectively. We combine these concepts with that of the rich-club to identify what we call a Super rich-club and rich edge-club. We show that the concept of vertex participation is a generalization of the rich-club. We present experimental results on randomized Erdös–Rényi and Watts–Strogatz small-world networks. We further demonstrate both concepts on a complex brain network and compare our results to the rich-club of the brain.

 

We consider simulating quantum systems on digital quantum computers. We show that the performance of quantum simulation can be improved by simultaneously exploiting commutativity of the target Hamiltonian, sparsity of interactions, and prior knowledge of the initial state. We achieve this using Trotterization for a class of interacting electrons that encompasses various physical systems, including the plane-wave-basis electronic structure and the Fermi-Hubbard model. We estimate the simulation error by taking the transition amplitude of nested commutators of the Hamiltonian terms within the η -electron manifold. We develop multiple techniques for bounding the transition amplitude and expectation of general fermionic operators, which may be of independent interest. We show that it suffices to use ( n 5 / 3 η 2 / 3 + n 4 / 3 η 2 / 3 ) n o ( 1 ) gates to simulate electronic structure in the plane-wave basis with n spin orbitals and η electrons, improving the best previous result in second quantization up to a negligible factor while outperforming the first-quantized simulation when n = η 2 − o ( 1 ) . We also obtain an improvement for simulating the Fermi-Hubbard model. We construct concrete examples for which our bounds are almost saturated, giving a nearly tight Trotterization of interacting electrons. 

The difficulty of simulating quantum dynamics depends on the norm of the Hamiltonian. When the Hamiltonian varies with time, the simulation complexity should only depend on this quantity instantaneously. We develop quantum simulation algorithms that exploit this intuition. For sparse Hamiltonian simulation, the gate complexity scales with the L 1 norm ∫ 0 t d τ ‖ H ( τ ) ‖ max , whereas the best previous results scale with t max τ ∈ [ 0 , t ] ‖ H ( τ ) ‖ max . We also show analogous results for Hamiltonians that are linear combinations of unitaries. Our approaches thus provide an improvement over previous simulation algorithms that can be substantial when the Hamiltonian varies significantly. We introduce two new techniques: a classical sampler of time-dependent Hamiltonians and a rescaling principle for the Schrödinger equation. The rescaled Dyson-series algorithm is nearly optimal with respect to all parameters of interest, whereas the sampling-based approach is easier to realize for near-term simulation. These algorithms could potentially be applied to semi-classical simulations of scattering processes in quantum chemistry.