Simulating sparse Hamiltonians with star decompositions

We present an efficient algorithm for simulating the time evolution due to a sparse Hamiltonian. In terms of the maximum degree d and dimension N of the space on which the Hamiltonian H acts for time t, this algorithm uses (d^2(d+log* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders, which scales like (d^4(log* N)||Ht||)^{1+o(1)}. To achieve this, we decompose a general sparse Hamiltonian into a small sum of Hamiltonians whose graphs of non-zero entries have the property that every connected component is a star, and efficiently simulate each of these pieces.Comment: 11 pages. v2: minor correction

Quantum query complexity of minor-closed graph properties

We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an $n$-vertex graph is planar, is a forest, or does not contain a path of a given length. We show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity \Theta(n^{3/2}). To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.Comment: v1: 25 pages, 2 figures. v2: 26 page

Hamiltonian simulation with nearly optimal dependence on all parameters

We present an algorithm for sparse Hamiltonian simulation whose complexity is optimal (up to log factors) as a function of all parameters of interest. Previous algorithms had optimal or near-optimal scaling in some parameters at the cost of poor scaling in others. Hamiltonian simulation via a quantum walk has optimal dependence on the sparsity at the expense of poor scaling in the allowed error. In contrast, an approach based on fractional-query simulation provides optimal scaling in the error at the expense of poor scaling in the sparsity. Here we combine the two approaches, achieving the best features of both. By implementing a linear combination of quantum walk steps with coefficients given by Bessel functions, our algorithm's complexity (as measured by the number of queries and 2-qubit gates) is logarithmic in the inverse error, and nearly linear in the product $\tau$ of the evolution time, the sparsity, and the magnitude of the largest entry of the Hamiltonian. Our dependence on the error is optimal, and we prove a new lower bound showing that no algorithm can have sublinear dependence on $\tau$.Comment: 21 pages, corrects minor error in Lemma 7 in FOCS versio

Exponential improvement in precision for simulating sparse Hamiltonians

We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a $d$-sparse Hamiltonian $H$ acting on $n$ qubits can be simulated for time $t$ with precision $\epsilon$ using $O\big(\tau \frac{\log(\tau/\epsilon)}{\log\log(\tau/\epsilon)}\big)$ queries and $O\big(\tau \frac{\log^2(\tau/\epsilon)}{\log\log(\tau/\epsilon)}n\big)$ additional 2-qubit gates, where $\tau = d^2 \|{H}\|_{\max} t$. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuous- and fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. We also simplify the analysis of this conversion, avoiding the need for a complex fault correction procedure. Our simplification relies on a new form of "oblivious amplitude amplification" that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error.Comment: v1: 27 pages; Subsumes and improves upon results in arXiv:1308.5424. v2: 28 pages, minor change

Simulating Hamiltonian dynamics with a truncated Taylor series

We describe a simple, efficient method for simulating Hamiltonian dynamics on a quantum computer by approximating the truncated Taylor series of the evolution operator. Our method can simulate the time evolution of a wide variety of physical systems. As in another recent algorithm, the cost of our method depends only logarithmically on the inverse of the desired precision, which is optimal. However, we simplify the algorithm and its analysis by using a method for implementing linear combinations of unitary operations to directly apply the truncated Taylor series.Comment: 5 page

Native Plasmid-Encoded Mercury Resistance Genes Are Functional and Demonstrate Natural Transformation in Environmental Bacterial Isolates.

Plasmid-mediated horizontal gene transfer (HGT) is a major driver of genetic diversity in bacteria. We experimentally validated the function of a putative mercury resistance operon present on an abundant 8-kbp native plasmid found in groundwater samples without detectable levels of mercury. Phylogenetic analyses of the plasmid-encoded mercury reductases from the studied groundwater site show them to be distinct from those reported in proximal metal-contaminated sites. We synthesized the entire native plasmid and demonstrated that the plasmid was sufficient to confer functional mercury resistance in Escherichia coli Given the possibility that natural transformation is a prevalent HGT mechanism in the low-cell-density environments of groundwaters, we also assayed bacterial strains from this environment for competence. We used the native plasmid-encoded metal resistance to design a screen and identified 17 strains positive for natural transformation. We selected 2 of the positive strains along with a model bacterium to fully confirm HGT via natural transformation. From an ecological perspective, the role of the native plasmid population in providing advantageous traits combined with the microbiome's capacity to take up environmental DNA enables rapid adaptation to environmental stresses.IMPORTANCE Horizontal transfer of mobile genetic elements via natural transformation has been poorly understood in environmental microbes. Here, we confirm the functionality of a native plasmid-encoded mercury resistance operon in a model microbe and then query for the dissemination of this resistance trait via natural transformation into environmental bacterial isolates. We identified 17 strains including Gram-positive and Gram-negative bacteria to be naturally competent. These strains were able to successfully take up the plasmid DNA and obtain a clear growth advantage in the presence of mercury. Our study provides important insights into gene dissemination via natural transformation enabling rapid adaptation to dynamic stresses in groundwater environments