A quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors

We describe a new polynomial time quantum algorithm that uses the quantum fast fourier transform to find eigenvalues and eigenvectors of a Hamiltonian operator, and that can be applied in cases (commonly found in ab initio physics and chemistry problems) for which all known classical algorithms require exponential time. Applications of the algorithm to specific problems are considered, and we find that classically intractable and interesting problems from atomic physics may be solved with between 50 and 100 quantum bits.Comment: 10 page

Quantum interferometric optical lithography:towards arbitrary two-dimensional patterns

As demonstrated by Boto et al. [Phys. Rev. Lett. 85, 2733 (2000)], quantum lithography offers an increase in resolution below the diffraction limit. Here, we generalize this procedure in order to create patterns in one and two dimensions. This renders quantum lithography a potentially useful tool in nanotechnology.Comment: 9 pages, 5 figures Revte

Quantum algorithms

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Physics, 1999.Includes bibliographical references (leaves 89-94).by Daniel S. Abrams.Ph.D

Eigenvalue Estimation of Differential Operators

We demonstrate how linear differential operators could be emulated by a quantum processor, should one ever be built, using the Abrams-Lloyd algorithm. Given a linear differential operator of order 2S, acting on functions psi(x_1,x_2,...,x_D) with D arguments, the computational cost required to estimate a low order eigenvalue to accuracy Theta(1/N^2) is Theta((2(S+1)(1+1/nu)+D)log N) qubits and O(N^{2(S+1)(1+1/nu)} (D log N)^c) gate operations, where N is the number of points to which each argument is discretized, nu and c are implementation dependent constants of O(1). Optimal classical methods require Theta(N^D) bits and Omega(N^D) gate operations to perform the same eigenvalue estimation. The Abrams-Lloyd algorithm thereby leads to exponential reduction in memory and polynomial reduction in gate operations, provided the domain has sufficiently large dimension D > 2(S+1)(1+1/nu). In the case of Schrodinger's equation, ground state energy estimation of two or more particles can in principle be performed with fewer quantum mechanical gates than classical gates.Comment: significant content revisions: more algorithm details and brief analysis of convergenc

Nonlinear quantum mechanics implies polynomial-time solution for NP-complete and #P problems

If quantum states exhibit small nonlinearities during time evolution, then quantum computers can be used to solve NP-complete problems in polynomial time. We provide algorithms that solve NP-complete and #P oracle problems by exploiting nonlinear quantum logic gates. It is argued that virtually any deterministic nonlinear quantum theory will include such gates, and the method is explicitly demonstrated using the Weinberg model of nonlinear quantum mechanics.Comment: 10 pages, no figures, submitted to Phys. Rev. Let

Sources of Water for Communities in Northeastern Illinois

published or submitted for publicationis peer reviewedOpe

Quantum Clock Synchronization Based on Shared Prior Entanglement

We demonstrate that two spatially separated parties (Alice and Bob) can utilize shared prior quantum entanglement, and classical communications, to establish a synchronized pair of atomic clocks. In contrast to classical synchronization schemes, the accuracy of our protocol is independent of Alice or Bob's knowledge of their relative locations or of the properties of the intervening medium.Comment: 4 page

Solvable model for chimera states of coupled oscillators

Networks of identical, symmetrically coupled oscillators can spontaneously split into synchronized and desynchronized sub-populations. Such chimera states were discovered in 2002, but are not well understood theoretically. Here we obtain the first exact results about the stability, dynamics, and bifurcations of chimera states by analyzing a minimal model consisting of two interacting populations of oscillators. Along with a completely synchronous state, the system displays stable chimeras, breathing chimeras, and saddle-node, Hopf and homoclinic bifurcations of chimeras.Comment: 4 pages, 4 figures. This version corrects a previous error in Figure 3, where the sign of the phase angle psi was inconsistent with Equation 1

Groundwater Flow Models of Illinois: Data, Processes, Model Performance, and Key Results

The Illinois State Water Survey (ISWS) has a long history of developing groundwater flow models to simulate water supply and groundwater contamination issues in the state of Illinois. However, past local- and regional-scale models developed by the ISWS have traditionally been project based; thus models are archived when the project is completed and may not be updated for many years. This report presents the first version of the Evolving Network of Illinois Groundwater Monitoring and Modeling Analyses (ENIGMMA), which is the framework of data, procedures, protocols, and scripts that facilitate the development of a single, continuously updated groundwater flow model and other outputs (hydrographs, maps, animations of groundwater potentiometric surfaces). This report focuses on five aspects of ENIGMMA: 1. The archived models and high-resolution datasets that serve as inputs to ENIGMMA 2. The procedures for developing model-ready datasets from these inputs 3. The Illinois Groundwater Flow Model (IGWFM), which serves as the single model that will be continuously updated by ENIGMMA 4. The ISWS Calibration Toolbox, used to facilitate a transient calibration of the IGWFM 5. Animations of groundwater potentiometric surfaces using head-specified models This report is a living document that will be updated periodically. Future updates to this report will focus on additional aspects of ENIGMMA, including the automated development of model-ready inputs and display of model outputs. Updates to this report will also chronicle any additional geologic data added to ENIGMMA, and subsequently, to the Illinois Groundwater Flow Model. Updates will also highlight both local- and regional-scale advancements made with the model, including any key results from these models. The current version of the IGWFM combines and expands on two existing groundwater flow models: 1) the Northeastern Illinois Cambrian-Ordovician Sandstone Aquifer model and 2) the East-Central Illinois Mahomet Aquifer model. In addition, the model incorporates new geologic information developed by the Illinois State Geological Survey in the Middle Illinois Water Supply Planning region. The current model domain covers large portions of Illinois, Wisconsin, Indiana, and Michigan. This large spatial extent is necessary to capture the far-reaching regional head declines in the deep Cambrian-Ordovician sandstone aquifer system, which can extend beyond state boundaries. Depicting some shallow, unconsolidated aquifers also requires a simultaneous simulation of the deep sandstone to account for flow exchange between units. This is because the low-permeable stratigraphic units (aquitards) overlying the sandstone aquifers are absent over large areas of northern Illinois or are locally punctured by wells with long, open intervals. To capture these complex flow pathways, the three-dimensional IGWFM explicitly simulates all geologic materials from the land surface to the impermeable Pre-Cambrian crystalline bedrock. The IGWFM does not currently include a groundwater flow simulation of the southern portion of the state where the deep basin sandstones are highly saline and not used for water supply. Incorporating the shallow aquifers in the southern portion of the state into the IGWFM is a long-term goal. The primary datasets currently incorporated into IGWFM include surface water elevations, annual groundwater withdrawals, well information such as open intervals, geologic 2 surfaces, measured water levels, and aquifer properties inferred from previous modeling studies. These datasets are input at their best available spatial and temporal resolutions, allowing for the development of refined local-scale models. Such local-scale models are essential for simulating groundwater-surface water interactions, well interference, and contaminant transport. Major local-scale models already exist for the Mahomet Aquifer, Kane County, and McHenry County. The IGWFM can address a number of water supply planning questions, particularly the impacts of historic, modern, and future high-capacity groundwater withdrawals on heads and groundwater discharging to surface waters. In addition, where detailed geologic information of the shallow aquifers is available, the IGWFM can also simulate the subsurface migration of point (e.g., volatile organic compounds) and nonpoint (e.g., chloride and nitrate) contaminants.published or submitted for publicationis peer reviewedOpe