Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians

A key feature of ground states of gapped local 1D Hamiltonians is their relatively low entanglement --- they are well approximated by matrix product states (MPS) with bond dimension scaling polynomially in the length $N$ of the chain, while general states require a bond dimension scaling exponentially. We show that the bond dimension of these MPS approximations can be improved to a constant, independent of the chain length, if we relax our notion of approximation to be more local: for all length-$k$ segments of the chain, the reduced density matrices of our approximations are $\epsilon$-close to those of the exact state. If the state is a ground state of a gapped local Hamiltonian, the bond dimension of the approximation scales like $(k/\epsilon)^{1+o(1)}$, and at the expense of worse but still $\text{poly}(k,1/\epsilon)$ scaling of the bond dimension, we give an alternate construction with the additional features that it can be generated by a constant-depth quantum circuit with nearest-neighbor gates, and that it applies generally for any state with exponentially decaying correlations. For a completely general state, we give an approximation with bond dimension $\exp(O(k/\epsilon))$, which is exponentially worse, but still independent of $N$. Then, we consider the prospect of designing an algorithm to find a local approximation for ground states of gapped local 1D Hamiltonians. When the Hamiltonian is translationally invariant, we show that the ability to find $O(1)$-accurate local approximations to the ground state in $T(N)$ time implies the ability to estimate the ground state energy to $O(1)$ precision in $O(T(N)\log(N))$ time.Comment: 24 pages, 3 figures. v2: Theorem 1 extended to include construction for general states; Lemma 7 & Theorem 2 slightly improved; figures added; lemmas rearranged for clarity; typos fixed. v3: Reformatted & additional references inserte

A Revised Method of Condensed Tannin Analysis in Leucaena spp.

The proanthocyanidin (butanol-HCl) assay was used to measure condensed tannin (CT) in extracts from Leucaena spp. at the University of Queensland. Inconsistent results were found to be caused by the iron catalyst in the butanol/HCl reagent and the presence of ascorbic acid in the sample matrix which enhanced anthocyanidin development. In addition, preparation of sample extracts by back extraction with organic solvents reduced CT recoveries. An accurate and rapid technique was developed that measures CT directly in 70% aqueous acetone 0.1% (w/v) sodium metabisulphite plant extracts

Breeding a Psyllid-Resistant Interspecific Hybrid \u3cem\u3eLeucaena\u3c/em\u3e for Beef Cattle Production in Northern Australia

Production of the valuable fodder tree legume Leucaena leucocephala (leucaena) is limited to the subhumid (600-800 mm annual rainfall) areas of northern Australia by the psyllid insect pest Heteropsylla cubana. Defoliation caused by severe psyllid infestations can suppress forage yields of commercial leucaena varieties by 50-80%. Susceptibility to psyllid damage is a major impediment to grazier adoption of leucaena pastures in the more humid tropical areas of Australia. A comprehensive international agronomic evaluation of the entire Leucaena genus (Mullen et al., 2003) revealed that the artificial interspecific F1 hybrid of L. pallida x L. leucocephala ssp. glabrata (called KX2) had a high degree of psyllid resistance, excellent vigour and broad environmental adaptation. The KX2 F1 hybrid also had superior forage quality compared to other psylli-resistant taxa, such as L. pallida, L. trichandra and L. diversifolia. Commercial utilization of the KX2 F1 hybrid by Australian graziers has been prevented by a lack of planting material. To date, seed production of the F1 hybrid has only been possible by laborious hand pollination. The KX2 F1 hybrid has been successfully vegetatively propagated for smallholders in SE Asia, however cloned cuttings are expensive to produce and are not suited to broad acre leucaena planting in Australia. A recurrent selection breeding program was initiated to produce a genetically stable, advanced generation KX2 hybrid that breeds true-to-type and is suitable for commercial release. We anticipate that 4 cycles of selection will be required to achieve this objective. This paper reports the agronomic evaluation of the KX2 F2 generation

Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end

We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), Ising spin glasses ($p$-spin model), and $k$-local constraint satisfaction problems ($k$-CSP). We show that either (a) the algorithm finds the optimal solution in time $O^*(2^{(0.5-c)n})$ for an $n$-independent constant $c$, a $2^{cn}$ advantage over Grover's algorithm; or (b) there are sufficiently many low-cost solutions such that classical random guessing produces a $(1-\eta)$ approximation to the optimal cost value in sub-exponential time for arbitrarily small choice of $\eta$. Additionally, we show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case. The algorithm and its analysis is largely inspired by Hastings' short-path algorithm [$\textit{Quantum}$ $\textbf{2}$ (2018) 78].Comment: 49 pages, 3 figure

The Effect of Tree Densities on the Biomass of \u3cem\u3eLeucaena leucocephala\u3c/em\u3e and \u3cem\u3eChloris gayana\u3c/em\u3e Using a Nelder Fan Design

Leucaena leucocephala-grass pastures are widely used for ruminant feeding in tropical and subtropical regions. In Australia, over 200,000 ha of leucaena grass pasture have been planted with more plantings expected as it is recognized as the most productive, profitable and sustainable feeding system (Shelton and Dalzell, 2007). Planting densities and planting configurations for the leucaena component vary, ranging from single or double leucaena hedgerows 3 to 12 m apart (Radrizzani et al., 2010). There is little information about how tree/grass planting configurations and resulting inter- and intraspecific competition affect above and below-ground interactions. We hypothesise that individual leucaena tree biomass will be inversely related to leucaena tree density, with greatest competition at low density, while medium to high leucaena densities will reduce grass biomass production

Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians

A key feature of ground states of gapped local 1D Hamiltonians is their relatively low entanglement --- they are well approximated by matrix product states (MPS) with bond dimension scaling polynomially in the length N of the chain, while general states require a bond dimension scaling exponentially. We show that the bond dimension of these MPS approximations can be improved to a constant, independent of the chain length, if we relax our notion of approximation to be more local: for all length-k segments of the chain, the reduced density matrices of our approximations are ϵ-close to those of the exact state. If the state is a ground state of a gapped local Hamiltonian, the bond dimension of the approximation scales like (k/ϵ)^(1+o(1)), and at the expense of worse but still poly(k,1/ϵ) scaling of the bond dimension, we give an alternate construction with the additional features that it can be generated by a constant-depth quantum circuit with nearest-neighbor gates, and that it applies generally for any state with exponentially decaying correlations. For a completely general state, we give an approximation with bond dimension exp(O(k/ϵ)), which is exponentially worse, but still independent of N. Then, we consider the prospect of designing an algorithm to find a local approximation for ground states of gapped local 1D Hamiltonians. When the Hamiltonian is translationally invariant, we show that the ability to find O(1)-accurate local approximations to the ground state in T(N) time implies the ability to estimate the ground state energy to O(1) precision in O(T(N)log(N)) time

Efficient classical simulation of random shallow 2D quantum circuits

Random quantum circuits are commonly viewed as hard to simulate classically. In some regimes this has been formally conjectured, and there had been no evidence against the more general possibility that for circuits with uniformly random gates, approximate simulation of typical instances is almost as hard as exact simulation. We prove that this is not the case by exhibiting a shallow circuit family with uniformly random gates that cannot be efficiently classically simulated near-exactly under standard hardness assumptions, but can be simulated approximately for all but a superpolynomially small fraction of circuit instances in time linear in the number of qubits and gates. We furthermore conjecture that sufficiently shallow random circuits are efficiently simulable more generally. To this end, we propose and analyze two simulation algorithms. Implementing one of our algorithms numerically, we give strong evidence that it is efficient both asymptotically and, in some cases, in practice. To argue analytically for efficiency, we reduce the simulation of 2D shallow random circuits to the simulation of a form of 1D dynamics consisting of alternating rounds of random local unitaries and weak measurements -- a type of process that has generally been observed to undergo a phase transition from an efficient-to-simulate regime to an inefficient-to-simulate regime as measurement strength is varied. Using a mapping from quantum circuits to statistical mechanical models, we give evidence that a similar computational phase transition occurs for our algorithms as parameters of the circuit architecture like the local Hilbert space dimension and circuit depth are varied

A genetic linkage map for the saltwater crocodile (Crocodylus porosus)

<p>Abstract</p> <p>Background</p> <p>Genome elucidation is now in high gear for many organisms, and whilst genetic maps have been developed for a broad array of species, surprisingly, no such maps exist for a crocodilian, or indeed any other non-avian member of the Class Reptilia. Genetic linkage maps are essential tools for the mapping and dissection of complex quantitative trait loci (QTL), and in order to permit systematic genome scans for the identification of genes affecting economically important traits in farmed crocodilians, a comprehensive genetic linage map will be necessary.</p> <p>Results</p> <p>A first-generation genetic linkage map for the saltwater crocodile (<it>Crocodylus porosus</it>) was constructed using 203 microsatellite markers amplified across a two-generation pedigree comprising ten full-sib families from a commercial population at Darwin Crocodile Farm, Northern Territory, Australia. Linkage analyses identified fourteen linkage groups comprising a total of 180 loci, with 23 loci remaining unlinked. Markers were ordered within linkage groups employing a heuristic approach using CRIMAP v3.0 software. The estimated female and male recombination map lengths were 1824.1 and 319.0 centimorgans (cM) respectively, revealing an uncommonly large disparity in recombination map lengths between sexes (ratio of 5.7:1).</p> <p>Conclusion</p> <p>We have generated the first genetic linkage map for a crocodilian, or indeed any other non-avian reptile. The uncommonly large disparity in recombination map lengths confirms previous preliminary evidence of major differences in sex-specific recombination rates in a species that exhibits temperature-dependent sex determination (TSD). However, at this point the reason for this disparity in saltwater crocodiles remains unclear.</p> <p>This map will be a valuable resource for crocodilian researchers, facilitating the systematic genome scans necessary for identifying genes affecting complex traits of economic importance in the crocodile industry. In addition, since many of the markers placed on this genetic map have been evaluated in up to 18 other extant species of crocodilian, this map will be of intrinsic value to comparative mapping efforts aimed at understanding genome content and organization among crocodilians, as well as the molecular evolution of reptilian and other amniote genomes. As researchers continue to work towards elucidation of the crocodilian genome, this first generation map lays the groundwork for more detailed mapping investigations, as well as providing a valuable scaffold for future genome sequence assembly.</p