Decomposition of Unitary Matrices for Finding Quantum Circuits: Application to Molecular Hamiltonians

Constructing appropriate unitary matrix operators for new quantum algorithms and finding the minimum cost gate sequences for the implementation of these unitary operators is of fundamental importance in the field of quantum information and quantum computation. Evolution of quantum circuits faces two major challenges: complex and huge search space and the high costs of simulating quantum circuits on classical computers. Here, we use the group leaders optimization algorithm to decompose a given unitary matrix into a proper-minimum cost quantum gate sequence. We test the method on the known decompositions of Toffoli gate, the amplification step of the Grover search algorithm, the quantum Fourier transform, and the sender part of the quantum teleportation. Using this procedure, we present the circuit designs for the simulation of the unitary propagators of the Hamiltonians for the hydrogen and the water molecules. The approach is general and can be applied to generate the sequence of quantum gates for larger molecular systems

Supersymmetric factorization yields exact solutions to the molecular Stark effect problem for "stretched" states

By invoking supersymmetry, we found a condition under which the Stark effect problem for a polar and polarizable molecule subject to nonresonant electric fields becomes exactly solvable. The exact solvability condition for the interaction parameters involved yields exact wavefunction for the "stretched" states, $|J=m,m>$, and for the $|1,1>$ state in the case of a purely induced-dipole interaction. The analytic expressions for the eigenenergy, the space-fixed dipole moment, the alignment cosine, and the expectation value of the angular momentum allow to readily reverse-engineer the problem of finding the values of the interaction parameters required for creating quantum states with preordained characteristics.Comment: 4 pages, 2 figure

Finite size scaling for quantum criticality using the finite-element method

Finite size scaling for the Schr\"{o}dinger equation is a systematic approach to calculate the quantum critical parameters for a given Hamiltonian. This approach has been shown to give very accurate results for critical parameters by using a systematic expansion with global basis-type functions. Recently, the finite element method was shown to be a powerful numerical method for ab initio electronic structure calculations with a variable real-space resolution. In this work, we demonstrate how to obtain quantum critical parameters by combining the finite element method (FEM) with finite size scaling (FSS) using different ab initio approximations and exact formulations. The critical parameters could be atomic nuclear charges, internuclear distances, electron density, disorder, lattice structure, and external fields for stability of atomic, molecular systems and quantum phase transitions of extended systems. To illustrate the effectiveness of this approach we provide detailed calculations of applying FEM to approximate solutions for the two-electron atom with varying nuclear charge; these include Hartree-Fock, density functional theory under the local density approximation, and an "exact"' formulation using FEM. We then use the FSS approach to determine its critical nuclear charge for stability; here, the size of the system is related to the number of elements used in the calculations. Results prove to be in good agreement with previous Slater-basis set calculations and demonstrate that it is possible to combine finite size scaling with the finite-element method by using ab initio calculations to obtain quantum critical parameters. The combined approach provides a promising first-principles approach to describe quantum phase transitions for materials and extended systems.Comment: 15 pages, 19 figures, revision based on suggestions by referee, accepted in Phys. Rev.

Multipartite quantum entanglement evolution in photosynthetic complexes

We investigate the evolution of entanglement in the Fenna-Matthew-Olson (FMO) complex based on simulations using the scaled hierarchical equations of motion approach. We examine the role of entanglement in the FMO complex by direct computation of the convex roof. We use monogamy to give a lower bound for entanglement and obtain an upper bound from the evaluation of the convex roof. Examination of bipartite measures for all possible bipartitions provides a complete picture of the multipartite entanglement. Our results support the hypothesis that entanglement is maximum primary along the two distinct electronic energy transfer pathways. In addition, we note that the structure of multipartite entanglement is quite simple, suggesting that there are constraints on the mixed state entanglement beyond those due to monogamy. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4742333

Movements and spatial use of odontocetes in the western main Hawaiian Islands: results from satellite-tagging and photo-identification off Kaua‘i and Ni‘ihau in July/August 2011

Although considerable information is available on residency patterns and spatial use of odontocetes in the eastern half of the Hawai‘i Range Complex (HRC), much less is known about odontocetes in the western half of the HRC. In the second year of a three-year effort in the western main Hawaiian Islands we undertook surveys off Kaua‘i and Ni‘ihau in July/August 2011, to examine spatial use and residency patterns using satellite tags, to provide visual verification of acoustically-detected odontocetes on the Pacific Missile Range Facility (PMRF), and to obtain individual identification photographs and biopsy samples for assessment of population identity and structure. During 18 days of field effort we covered 1,972 km of trackline and had 65 encounters with five species of odontocetes. Twenty-four of the encounters, of three species, were cued by acoustic detections from the Marine Mammal Monitoring on Navy Ranges (M3R) system, thus providing species verifications for future use of the M3R system on the PMRF range. During the 65 encounters we obtained 22,645 photos for individual and species identification, and collected 48 biopsy samples for genetic analyses. One encounter with a group of four killer whales was only the second encounter with this species in 12 years of directed field surveys in Hawaiian waters. Photos from that encounter were compared to our photo-identification catalog but no matches were found, further suggesting that there is no population of this species resident to the Hawaiian Islands. There were three encounters with a lone pantropical spotted dolphin, each time in association with a group of spinner dolphins. Photos of this individual matched to a spotted dolphin identified off Kaua‘i in 2004 and in 2005, both times with spinner dolphins, suggesting this individual may be part of a long-term association with spinner dolphins. Four satellite tags were deployed; three on rough-toothed dolphins and one on a bottlenose dolphin. These are the first tag deployments on either species in Hawaiian waters and the first deployments of satellite tags on free-ranging rough-toothed dolphins anywhere in the world. Rough-toothed dolphin tag data were obtained over periods from 7.6 to 18.5 days. Over these periods the three rough-toothed dolphins moved cumulative horizontal distances ranging from 573 to 1,295 km, yet remained an average distance from the tagging locations of from 10.4 to 13.9 km. Median depths used by the three rough-toothed dolphins ranged from 816 to 1,107 m, with median distance from shore ranging from 11.6 to 12.2 km. Two of the three individuals had been previously photo-identified off Kaua‘i (in 2007 or 2008), and all link by association with the resident population from Kaua‘i and Ni‘ihau. Movement and habitat use data were obtained over a 34-day period for the satellite-tagged bottlenose dolphin. During this time the individual remained associated with the island of Kaua‘i using waters with a median depth of 82 m. Although this individual had not been previously photo-identified, others from the group it was in had been previously documented off Kaua‘i and/or Ni‘ihau in 2003-2005, suggesting it is part of the island-resident population. Overall these efforts provide the first unbiased movement and habitat use data for both species in Hawaiian waters.Grant No. N00244-10-1-004

Population and Coherence Dynamics in Light Harvesting Complex II (LH2)

The electronic excitation population and coherence dynamics in the chromophores of the photosynthetic light harvesting complex 2 (LH2) B850 ring from purple bacteria (Rhodopseudomonas acidophila) have been studied theoretically at both physiological and cryogenic temperatures. Similar to the well-studied Fenna-Matthews-Olson (FMO) protein, oscillations of the excitation population and coherence in the site basis are observed in LH2 by using a scaled hierarchical equation of motion (HEOM) approach. However, this oscillation time (300 fs) is much shorter compared to the FMO protein (650 fs) at cryogenic temperature. Both environment and high temperature are found to enhance the propagation speed of the exciton wave packet yet they shorten the coherence time and suppress the oscillation amplitude of coherence and the population. Our calculations show that a long-lived coherence between chromophore electronic excited states can exist in such a noisy biological environment.Comment: 21 pages, 9 figure

A SURVEY FOR ODONTOCETE CETACEANS OFF KAUA‘I AND NI‘IHAU, HAWAI‘I, DURING OCTOBER AND NOVEMBER 2005: EVIDENCE FOR POPULATION STRUCTURE AND SITE FIDELITY

Considerable uncertainty exists regarding population structure and population sizes of most species of odontocetes in the Hawaiian Islands. A small-boat based survey for odontocetes was undertaken off the islands of Kaua‘i and Ni‘ihau in October and November 2005 to photoidentify individuals and collect genetic samples for examining stock structure. Field effort on 24 days covered 2,194 km of trackline. Survey coverage was from shallow coastal waters out to over 3,000 m depth, though almost half (47%) was in waters less than 500 m in depth. There were 56 sightings of five species of odontocetes: spinner dolphins (30 sightings); bottlenose dolphins (14 sightings); short-finned pilot whales (6 sightings); rough-toothed dolphins (5 sightings); and pantropical spotted dolphins (1 sighting). One hundred and five biopsy samples were collected and 14,960 photographs were taken to document morphology and for individual photo-identification. Photographs of distinctive individuals of three species (bottlenose dolphins, 76 identifications; rough-toothed dolphins, 157 identifications; short-finned pilot whales, 68 identifications) were compared to catalogs of these species from a survey off Kaua‘i and Ni‘ihau in 2003, as well as from efforts off O‘ahu, Maui/Lana‘i and the island of Hawai‘i. Within- and between-year matches were found for all three species with individuals previously identified off Kaua‘i and Ni‘ihau, though no matches were found with individuals off any of the other islands. This suggests site fidelity to specific island areas, and population structure among island areas for all three species. Movements of photographically identified bottlenose dolphins were documented between deep water areas off the islands of Kaua‘i and Ni‘ihau, as well as between shallow (\u3c350 m) and deep (\u3e350 m) waters. A lack of sightings or reports of false killer whales off Kaua‘i or Ni‘ihau during our study, combined with documented movements among the other main Hawaiian Islands, suggest that there is no “resident” population of false killer whales that inhabits waters only off Kaua‘i or Ni‘iha

Entanglement, Berry Phases, and Level Crossings for the Atomic Breit-Rabi Hamiltonian

The relation between level crossings, entanglement, and Berry phases is investigated for the Breit-Rabi Hamiltonian of hydrogen and sodium atoms, describing a hyperfine interaction of electron and nuclear spins in a magnetic field. It is shown that the entanglement between nuclear and electron spins is maximum at avoided crossings. An entangled state encircling avoided crossings acquires a marginal Berry phase of a subsystem like an instantaneous eigenstate moving around real crossings accumulates a Berry phase. Especially, the nodal points of a marginal Berry phase correspond to the avoided crossing points.Comment: 5 figures, submitted to a journa

Supersymmetry identifies molecular Stark states whose eigenproperties can be obtained analytically

We made use of supersymmetric (SUSY) quantum mechanics to find a condition under which the Stark effect problem for a polar and polarizable closed-shell diatomic molecule subject to collinear electrostatic and nonresonant radiative fields becomes exactly solvable. The condition, $\Delta \omega = \frac{\omega^2}{4 (m+1)^2}$, connects values of the dimensionless parameters $\omega$ and $\Delta \omega$ that characterize the strengths of the permanent and induced dipole interactions of the molecule with the respective fields. The exact solutions are obtained for the $|\tilde{J}=m,m;\omega,\Delta \omega>$ family of "stretched" states. The field-free and strong-field limits of the combined-fields problem were found to exhibit supersymmetry and shape-invariance, which is indeed the reason why they are analytically solvable. By making use of the analytic form of the $|\tilde{J}=m,m;\omega,\Delta \omega>$ wavefunctions, we obtained simple formulae for the expectation values of the space-fixed electric dipole moment, the alignment cosine, the angular momentum squared, and derived a "sum rule" which combines the above expectation values into a formula for the eigenenergy. The analytic expressions for the characteristics of the strongly oriented and aligned states provide a direct access to the values of the interaction parameters required for creating such states in the laboratory.Comment: 12 pages, 4 figures, 1 tabl

Quantum algorithm and circuit design solving the Poisson equation

The Poisson equation occurs in many areas of science and engineering. Here we focus on its numerical solution for an equation in d dimensions. In particular we present a quantum algorithm and a scalable quantum circuit design which approximates the solution of the Poisson equation on a grid with error \varepsilon. We assume we are given a supersposition of function evaluations of the right hand side of the Poisson equation. The algorithm produces a quantum state encoding the solution. The number of quantum operations and the number of qubits used by the circuit is almost linear in d and polylog in \varepsilon^{-1}. We present quantum circuit modules together with performance guarantees which can be also used for other problems.Comment: 30 pages, 9 figures. This is the revised version for publication in New Journal of Physic