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Journal articles on the topic 'Hamiltonian problems'

Author: Grafiati

Published: 10 March 2023

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1

Bravyi, S., D. P. DiVincenzo, R. Oliveira, and B. M. Terhal. "The complexity of stoquastic local Hamiltonian problems." Quantum Information and Computation 8, no. 5 (May 2008): 361–85. http://dx.doi.org/10.26421/qic8.5-1.

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We study the complexity of the Local Hamiltonian Problem (denoted as LH-MIN) in the special case when a Hamiltonian obeys the condition that all off-diagonal matrix elements in the standard basis are real and non-positive. We will call such Hamiltonians, which are common in the natural world, stoquastic. An equivalent characterization of stoquastic Hamiltonians is that they have an entry-wise non-negative Gibbs density matrix for any temperature. We prove that LH-MIN for stoquastic Hamiltonians belongs to the complexity class \AM{}--- a probabilistic version of \NP{} with two rounds of communication between the prover and the verifier. We also show that $2$-local stoquastic LH-MIN is hard for the class \MA. With the additional promise of having a polynomial spectral gap, we show that stoquastic LH-MIN belongs to the class \POSTBPP=\BPPpath --- a generalization of \BPP{} in which a post-selective readout is allowed. This last result also shows that any problem solved by adiabatic quantum computation using stoquastic Hamiltonians is in PostBPP.

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2

Elyseeva, Julia. "The Oscillation Numbers and the Abramov Method of Spectral Counting for Linear Hamiltonian Systems." EPJ Web of Conferences 248 (2021): 01002. http://dx.doi.org/10.1051/epjconf/202124801002.

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In this paper we consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. For the Hamiltonian problems we do not assume any controllability and strict normality assumptions which guarantee that the classical eigenvalues of the problems are isolated. We also omit the Legendre condition for their Hamiltonians. We show that the Abramov method of spectral counting can be modified for the more general case of finite eigenvalues of the Hamiltonian problems and then the constructive ideas of the Abramov method can be used for stable calculations of the oscillation numbers and finite eigenvalues of the Hamiltonian problems.

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3

Pannell, William H. "The intersection between dual potential and sl(2) algebraic spectral problems." International Journal of Modern Physics A 35, no. 32 (November 20, 2020): 2050208. http://dx.doi.org/10.1142/s0217751x20502085.

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The relation between certain Hamiltonians, known as dual, or partner Hamiltonians, under the transformation [Formula: see text] has long been used as a method of simplifying spectral problems in quantum mechanics. This paper seeks to examine this further by expressing such Hamiltonians in terms of the generators of sl(2) algebra, which provides another method of solving spectral problems. It appears that doing so greatly restricts the set of allowable potentials, with the only nontrivial potentials allowed being the Coulomb [Formula: see text] potential and the harmonic oscillator [Formula: see text] potential, for both of which the sl(2) expression is already known. It also appears that, by utilizing both the partner potential transformation and the formalism of the Lie-algebraic construction of quantum mechanics, it may be possible to construct part of a Hamiltonian’s spectrum from the quasi-solvability of its partner Hamiltonian.

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4

Iserles, A., J. M. Sanz-Serna, and M. P. Calvo. "Numerical Hamiltonian Problems." Mathematics of Computation 64, no. 211 (July 1995): 1346. http://dx.doi.org/10.2307/2153506.

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5

Bassour, Mustapha. "Hamiltonian Polynomial Eigenvalue Problems." Journal of Applied Mathematics and Physics 08, no. 04 (2020): 609–19. http://dx.doi.org/10.4236/jamp.2020.84047.

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6

Sattath, Or, Siddhardh C. Morampudi, Chris R. Laumann, and Roderich Moessner. "When a local Hamiltonian must be frustration-free." Proceedings of the National Academy of Sciences 113, no. 23 (May 19, 2016): 6433–37. http://dx.doi.org/10.1073/pnas.1519833113.

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A broad range of quantum optimization problems can be phrased as the question of whether a specific system has a ground state at zero energy, i.e., whether its Hamiltonian is frustration-free. Frustration-free Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms to, at least, partially answer this question. Here we prove a general criterion—a sufficient condition—under which a local Hamiltonian is guaranteed to be frustration-free by lifting Shearer’s theorem from classical probability theory to the quantum world. Remarkably, evaluating this condition proceeds via a fully classical analysis of a hardcore lattice gas at negative fugacity on the Hamiltonian’s interaction graph, which, as a statistical mechanics problem, is of interest in its own right. We concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass physics that permit us to obtain new bounds on the satisfiable to unsatisfiable transition in random quantum satisfiability. We are then led to natural conjectures for when such bounds will be tight, as well as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this work underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory.

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7

Masanes, Ll, G. Vidal, and J. I. Latorre. "Time--optimal Hamiltonian simulation and gate synthesis using homogeneous local unitaries." Quantum Information and Computation 2, no. 4 (June 2002): 285–96. http://dx.doi.org/10.26421/qic2.4-2.

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Motivated by experimental limitations commonly met in the design of solid state quantum computers, we study the problems of non--local Hamiltonian simulation and non--local gate synthesis when only {\em homogeneous} local unitaries are performed in order to tailor the available interaction. Homogeneous (i.e. identical for all subsystems) local manipulation implies a more refined classification of interaction Hamiltonians than the inhomogeneous case, as well as the loss of universality in Hamiltonian simulation. For the case of symmetric two--qubit interactions, we provide time--optimal protocols for both Hamiltonian simulation and gate synthesis.

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8

Zhao, Qi, and Xiao Yuan. "Exploiting anticommutation in Hamiltonian simulation." Quantum 5 (August 31, 2021): 534. http://dx.doi.org/10.22331/q-2021-08-31-534.

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Quantum computing can efficiently simulate Hamiltonian dynamics of many-body quantum physics, a task that is generally intractable with classical computers. The hardness lies at the ubiquitous anti-commutative relations of quantum operators, in corresponding with the notorious negative sign problem in classical simulation. Intuitively, Hamiltonians with more commutative terms are also easier to simulate on a quantum computer, and anti-commutative relations generally cause more errors, such as in the product formula method. Here, we theoretically explore the role of anti-commutative relation in Hamiltonian simulation. We find that, contrary to our intuition, anti-commutative relations could also reduce the hardness of Hamiltonian simulation. Specifically, Hamiltonians with mutually anti-commutative terms are easy to simulate, as what happens with ones consisting of mutually commutative terms. Such a property is further utilized to reduce the algorithmic error or the gate complexity in the truncated Taylor series quantum algorithm for general problems. Moreover, we propose two modified linear combinations of unitaries methods tailored for Hamiltonians with different degrees of anti-commutation. We numerically verify that the proposed methods exploiting anti-commutative relations could significantly improve the simulation accuracy of electronic Hamiltonians. Our work sheds light on the roles of commutative and anti-commutative relations in simulating quantum systems.

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9

Amodio, P., F. Iavernaro, and D. Trigiante. "Symmetric schemes and Hamiltonian perturbations of linear Hamiltonian problems." Numerical Linear Algebra with Applications 12, no. 2-3 (2005): 171–79. http://dx.doi.org/10.1002/nla.408.

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10

Sanz-Serna, J. M. "Symplectic integrators for Hamiltonian problems: an overview." Acta Numerica 1 (January 1992): 243–86. http://dx.doi.org/10.1017/s0962492900002282.

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In the sciences, situations where dissipation is not significant may invariably be modelled by Hamiltonian systems of ordinary, or partial, differential equations. Symplectic integrators are numerical methods specifically aimed at advancing in time the solution of Hamiltonian systems. Roughly speaking, ‘symplecticness’ is a characteristic property possessed by the solutions of Hamiltonian problems. A numerical method is called symplectic if, when applied to Hamiltonian problems, it generates numerical solutions which inherit the property of symplecticness.

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11

Lynch, Mark A. M. "Creating recreational Hamiltonian cycle problems." Mathematical Gazette 88, no. 512 (July 2004): 215–18. http://dx.doi.org/10.1017/s0025557200174935.

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In this paper graphs that contain unique Hamiltonian cycles are introduced. The graphs are of arbitrary size and dense in the sense that their average vertex degree is greater than half the number of vertices that make up the graph. The graphs can be used to generate challenging puzzles. The problem is particularly challenging when the graph is large and the ‘method’ of solution is unknown to the solver.

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12

Bohner, M. "Discrete linear Hamiltonian eigenvalue problems." Computers & Mathematics with Applications 36, no. 10-12 (November 1998): 179–92. http://dx.doi.org/10.1016/s0898-1221(98)80019-9.

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13

BERA, P. K., M. M. PANJA, and B. TALUKDAR. "ISOSPECTRAL INTERACTIONS FOR THREE-BODY PROBLEMS ON THE LINE." Modern Physics Letters A 11, no. 26 (August 30, 1996): 2129–38. http://dx.doi.org/10.1142/s0217732396002113.

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The algebraic methods of supersymmetric quantum mechanics are used to construct isospectral Hamiltonians for the three-particle Calogero problem [F. Calogero, J. Math. Phys. 10, 2191 (1969)]. The similarity and points of contrast of the present study with the corresponding two-body problem are discussed. It is found that the family of isospectral interactions is determined essentially by the angular part of the potential in the basic Hamiltonian. A case study is presented to investigate the nature of the individual member in the family.

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14

Stogiannos, Evangelos, Christos Papalitsas, and Theodore Andronikos. "Experimental Analysis of Quantum Annealers and Hybrid Solvers Using Benchmark Optimization Problems." Mathematics 10, no. 8 (April 13, 2022): 1294. http://dx.doi.org/10.3390/math10081294.

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This paper studies the Hamiltonian cycle problem (HCP) and the traveling salesman problem (TSP) on D-Wave quantum systems. Motivated by the fact that most libraries present their benchmark instances in terms of adjacency matrices, we develop a novel matrix formulation for the HCP and TSP Hamiltonians, which enables the seamless and automatic integration of benchmark instances in quantum platforms. We also present a thorough mathematical analysis of the precise number of constraints required to express the HCP and TSP Hamiltonians. This analysis explains quantitatively why, almost always, running incomplete graph instances requires more qubits than complete instances. It turns out that QUBO models for incomplete graphs require more quadratic constraints than complete graphs, a fact that has been corroborated by a series of experiments. Moreover, we introduce a technique for the min-max normalization for the coefficients of the TSP Hamiltonian to address the problem of invalid solutions produced by the quantum annealer, a trend often observed. Our extensive experimental tests have demonstrated that the D-Wave Advantage_system4.1 is more efficient than the Advantage_system1.1, both in terms of qubit utilization and the quality of solutions. Finally, we experimentally establish that the D-Wave hybrid solvers always provide valid solutions, without violating the given constraints, even for arbitrarily big problems up to 120 nodes.

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15

ANDRIANOV, A. A., M. V. IOFFE, F. CANNATA, and J. P. DEDONDER. "SUSY QUANTUM MECHANICS WITH COMPLEX SUPERPOTENTIALS AND REAL ENERGY SPECTRA." International Journal of Modern Physics A 14, no. 17 (July 10, 1999): 2675–88. http://dx.doi.org/10.1142/s0217751x99001342.

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We extend the standard intertwining relations used in supersymmetrical (SUSY) quantum mechanics which involve real superpotentials to complex superpotentials. This allows us to deal with a large class of non-Hermitian Hamiltonians and to study in general the isospectrality between complex potentials. In very specific cases we can construct in a natural way "quasicomplex" potentials which we define as complex potentials having a global property so as to lead to a Hamiltonian with real spectrum. We also obtained a class of complex transparent potentials whose Hamiltonian can be intertwined to a free Hamiltonian. We provide a variety of examples both for the radial problem (half axis) and for the standard one-dimensional problem (the whole axis), including remarks concerning scattering problems.

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16

Udriste, Constantin, and Ionel Tevy. "Properties of Hamiltonian in free final multitime problems." Studia Universitatis Babes-Bolyai Matematica 66, no. 1 (March 20, 2021): 223–40. http://dx.doi.org/10.24193/subbmath.2021.1.18.

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"In single-time autonomous optimal control problems, the Hamiltonian is constant on optimal evolution. In addition, if the final time is free, the optimal Hamiltonian vanishes on the hole interval of evolution. The purpose of this paper is to extend some of these results to the case of multitime optimal control. The original results include: anti-trace problem, weak and strong multitime maximum principles, multitime-invariant systems and change rate of Hamiltonian, the variational derivative of volume integral, necessary conditions for a free final multitime expressed with the Hamiltonian tensor that replaces the energy-momentum tensor, change of variables in multitime optimal control, conversion of free final multitime problems to problems over fixed interval."

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17

Kurzhanskii, A. B. "Hamiltonian Formalism in Team Control Problems." Differential Equations 55, no. 4 (April 2019): 532–40. http://dx.doi.org/10.1134/s0012266119040116.

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18

SANZ-SERNA, J. M., and M. P. CALVO. "SYMPLECTIC NUMERICAL METHODS FOR HAMILTONIAN PROBLEMS." International Journal of Modern Physics C 04, no. 02 (April 1993): 385–92. http://dx.doi.org/10.1142/s0129183193000410.

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We consider symplectic methods for the numerical integration of Hamiltonian problems, i.e. methods that preserve the Poincaré integral invariants. Examples of symplectic methods are given and numerical experiments are reported.

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19

Cubitt, Toby, and Ashley Montanaro. "Complexity Classification of Local Hamiltonian Problems." SIAM Journal on Computing 45, no. 2 (January 2016): 268–316. http://dx.doi.org/10.1137/140998287.

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20

Subbotina, Nina N. "Hamiltonian Systems in Dynamic Reconstruction Problems." IFAC-PapersOnLine 51, no. 32 (2018): 136–40. http://dx.doi.org/10.1016/j.ifacol.2018.11.368.

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21

Watkinson, L. R., A. S. Lawless, N. K. Nichols, and I. Roulstone. "Variational data assimilation for Hamiltonian problems." International Journal for Numerical Methods in Fluids 47, no. 10-11 (2005): 1361–67. http://dx.doi.org/10.1002/fld.844.

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22

Wu, Xin, Ying Wang, Wei Sun, Fu-Yao Liu, and Wen-Biao Han. "Explicit Symplectic Methods in Black Hole Spacetimes." Astrophysical Journal 940, no. 2 (December 1, 2022): 166. http://dx.doi.org/10.3847/1538-4357/ac9c5d.

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Abstract Many Hamiltonian problems in the solar system are separable into two analytically solvable parts, and thus serve as a great chance to develop and apply explicit symplectic integrators based on operator splitting and composing. However, such constructions are not in general available for curved spacetimes in general relativity and modified theories of gravity because these curved spacetimes correspond to nonseparable Hamiltonians without the two-part splits. Recently, several black hole spacetimes such as the Schwarzschild black hole were found to allow for the construction of explicit symplectic integrators, since their corresponding Hamiltonians are separable into more than two explicitly integrable pieces. Although some other curved spacetimes including the Kerr black hole do not have such multipart splits, their corresponding appropriate time-transformation Hamiltonians do. In fact, the key problem in obtaining symplectic analytically integrable decomposition algorithms is how to split these Hamiltonians or time-transformation Hamiltonians. Considering this idea, we develop explicit symplectic schemes in curved spacetimes. We introduce a class of spacetimes whose Hamiltonians are directly split into several explicitly integrable terms. For example, the Hamiltonian of a rotating black ring has a 13-part split. We also present two sets of spacetimes whose appropriate time-transformation Hamiltonians have the desirable splits. For instance, an eight-part split exists in a time-transformed Hamiltonian of a Kerr–Newman solution with a disformal parameter. In this way, the proposed symplectic splitting methods can be used widely for long-term integrations of orbits in most curved spacetimes we know of.

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23

Fuqara, Anoud K., Amer D. Al-Oqali, and Khaled I. Nawafleh. "Hamilton-Jacobi Equation of Time Dependent Hamiltonians." Oriental Journal of Physical Sciences 5, no. 1-2 (December 30, 2020): 09–15. http://dx.doi.org/10.13005/ojps05.01-02.04.

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In this work, we apply the geometric Hamilton-Jacobi theory to obtain solution of Hamiltonian systems in classical mechanics that are either compatible with two structures: the first structure plays a central role in the theory of time- dependent Hamiltonians, whilst the second is used to treat classical Hamiltonians including dissipation terms. It is proved that the generalization of problems from the calculus of variation methods in the nonstationary case can be obtained naturally in Hamilton-Jacobi formalism.

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24

Zhang, Hairui, and Yongxin Yuan. "Generalized inverse eigenvalue problems for Hermitian and J-Hamiltonian/skew-Hamiltonian matrices." Applied Mathematics and Computation 361 (November 2019): 609–16. http://dx.doi.org/10.1016/j.amc.2019.06.004.

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25

BECKWITH, A. W. "AN OPEN QUESTION: ARE TOPOLOGICAL ARGUMENTS HELPFUL IN SETTING INITIAL CONDITIONS FOR TRANSPORT PROBLEMS IN CONDENSED MATTER PHYSICS?" Modern Physics Letters B 20, no. 05 (February 20, 2006): 233–43. http://dx.doi.org/10.1142/s0217984906010585.

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The tunneling Hamiltonian is a proven method to treat particle tunneling between different states represented as wavefunctions in many-body physics. Our problem is how to apply a wave functionals formulation of tunneling Hamiltonians to a driven sine-Gordon system. We apply a generalization of the tunneling Hamiltonian to charge density wave (CDW) transport problems in which we consider tunneling between states that are wave functionals of a scalar quantum field ϕ. We present derived I–E curves that match Zenier curves used to fit data experimentally with wave functionals congruent with the false vacuum hypothesis. The open question is whether the coefficients picked in both the wave functionals and the magnitude of the coefficients of the driven sine Gordon physical system should be picked by topological charge arguments that in principle appear to assign values that have a tie in with the false vacuum hypothesis first presented by Sidney Coleman.

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26

Li, Xiao Chuan, and Jin Shuang Zhang. "Hamiltonian Duality Equation on Three-Dimensional Problems of Magnetoelectroelastic Solids." Applied Mechanics and Materials 268-270 (December 2012): 1099–104. http://dx.doi.org/10.4028/www.scientific.net/amm.268-270.1099.

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Hamiltonian system used in dynamics is introduced to formulate the three-dimensional problems of the transversely isotropic magnetoelectroelastic solids. The Hamiltonian dual equations in magnetoelectroelastic solids are developed directly from the modified Hellinger-Reissner variational principle derived from generalized Hellinger-Ressner variational principle with two classes of variables. These variables not only include such origin variables as displaces, electric potential and magnetic potential, but also include such their dual variables as lengthways stress, electric displacement and magnetic induction in the symplectic space. Similar to the Hamiltonian formulation in classic dynamics, the z coordinate is treated analogous to the time coordinate so that the method of separation of variables can be used. The governing equations are a set of first order differential equations in z, and the coefficient matrix of the differential equations is Hamiltonian in (x, y).

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27

Amodio, Pierluigi, Luigi Brugnano, and Felice Iavernaro. "Continuous-Stage Runge–Kutta Approximation to Differential Problems." Axioms 11, no. 5 (April 21, 2022): 192. http://dx.doi.org/10.3390/axioms11050192.

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In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge–Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge–Kutta methods. In this review paper, we recall this aspect and extend it to higher-order differential problems.

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28

Mielke, Alexander. "Weak-convergence methods for Hamiltonian multiscale problems." Discrete & Continuous Dynamical Systems - A 20, no. 1 (2008): 53–79. http://dx.doi.org/10.3934/dcds.2008.20.53.

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29

Chan, R. P. K., and A. Murua. "Extrapolation of symplectic methods for Hamiltonian problems." Applied Numerical Mathematics 34, no. 2-3 (July 2000): 189–205. http://dx.doi.org/10.1016/s0168-9274(99)00127-0.

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30

Buttà, Paolo, and Silvia Noschese. "Structured maximal perturbations for Hamiltonian eigenvalue problems." Journal of Computational and Applied Mathematics 272 (December 2014): 304–12. http://dx.doi.org/10.1016/j.cam.2013.04.031.

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31

VIGO-AGUIAR, JESÚS, T. E. SIMOS, and A. TOCINO. "AN ADAPTED SYMPLECTIC INTEGRATOR FOR HAMILTONIAN PROBLEMS." International Journal of Modern Physics C 12, no. 02 (February 2001): 225–34. http://dx.doi.org/10.1142/s0129183101001626.

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In this paper, a new procedure for deriving efficient symplectic integrators for Hamiltonian problems is introduced. This procedure is based on the combination of the trigonometric fitting technique and symplecticness conditions. Based on this procedure, a simple modified Runge–Kutta–Nyström second algebraic order trigonometrically fitted method is developed. We present explicity the symplecticity conditions for the new modified Runge–Kutta–Nyström method. Numerical results indicate that the new method is much more efficient than the "classical" symplectic Runge–Kutta–Nyström second algebraic order method introduced in Ref. 1.

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32

Baryshnikov, Yu M. "Hamiltonian form of non-holonomic variational problems." Russian Mathematical Surveys 45, no. 1 (February 28, 1990): 198–99. http://dx.doi.org/10.1070/rm1990v045n01abeh002307.

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33

Castrillón, López, and Masqué Muñoz. "Hamiltonian structure of gauge-invariant variational problems." Advances in Theoretical and Mathematical Physics 16, no. 1 (2012): 39–63. http://dx.doi.org/10.4310/atmp.2012.v16.n1.a2.

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34

Faibusovich, L. E. "Collective Hamiltonian method in optimal control problems." Cybernetics 25, no. 2 (1989): 230–37. http://dx.doi.org/10.1007/bf01070131.

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35

Qian, Jiang, and Roger C. E. Tan. "On some inverse eigenvalue problems for Hermitian and generalized Hamiltonian/skew-Hamiltonian matrices." Journal of Computational and Applied Mathematics 250 (October 2013): 28–38. http://dx.doi.org/10.1016/j.cam.2013.02.023.

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36

SHIFMAN, M. A. "NEW FINDINGS IN QUANTUM MECHANICS (PARTIAL ALGEBRAIZATION OF THE SPECTRAL PROBLEM)." International Journal of Modern Physics A 04, no. 12 (July 20, 1989): 2897–952. http://dx.doi.org/10.1142/s0217751x89001151.

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We discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly-solvable problems (like the famous harmonic oscillator) and all others. The problems belonging to this class are distinguished by the fact that an (arbitrary) part of the eigenvalues and eigenfunctions can be found algebraically, but not the whole spectrum. The reason explaining the existence of the quasi-exactly-solvable problems is a hidden dynamical symmetry present in the Hamiltonian. For one-dimensional motion, this hidden symmetry is SU(2). The simplest one-dimensional system admitting algebraization for a part of the spectrum is the anharmonic oscillator with the x6 anharmonicity and a relation between the coefficients in front of x2 and x6. We review also more complicated cases with the emphasis on pedagogical aspects. The groups SU (2)× SU (2), SO(3) and SU(3) generate two-dimensional problems with the partial algebraization of the spectrum. Typically we get Schrödinger-type equations in curved space. An intriguing relation between the algebraic structure of the Hamiltonian and the geometry of the space emerges. Another interesting development is the use of the graded algebras which allow one to construct multi-component quasi-exactly-solvable Hamiltonians.

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37

Evangelisti, S., J. P. Daudey, and J. P. Malrieu. "Qualitative intruder-state problems in effective Hamiltonian theory and their solution through intermediate Hamiltonians." Physical Review A 35, no. 12 (June 1, 1987): 4930–41. http://dx.doi.org/10.1103/physreva.35.4930.

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38

Novo, Leonardo, and Dominic Berry. "Improved Hamiltonian simulation via a truncated Taylor series and corrections." Quantum Information and Computation 17, no. 7&8 (May 2017): 623–35. http://dx.doi.org/10.26421/qic17.7-8-5.

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We describe an improved version of the quantum algorithm for Hamiltonian simulation based on the implementation of a truncated Taylor series of the evolution operator. The idea is to add an extra step to the previously known algorithm which implements an operator that corrects the weightings of the Taylor series. This way, the desired accuracy is achieved with an improvement in the overall complexity of the algorithm. This quantum simulation method is applicable to a wide range of Hamiltonians of interest, including to quantum chemistry problems.

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39

Paulraja, P., and Kumar Sampath. "On hamiltonian decompositions of tensor products of graphs." Applicable Analysis and Discrete Mathematics 13, no. 1 (2019): 178–202. http://dx.doi.org/10.2298/aadm170803003p.

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Finding a hamiltonian decomposition of G is one of the challenging problems in graph theory. We do not know for what classes of graphs G and H, their tensor product G x H is hamiltonian decomposable. In this paper, we have proved that, if G is a hamiltonian decomposable circulant graph with certain properties and H is a hamiltonian decomposable multigraph, then G x H is hamiltonian decomposable. In particular, tensor products of certain sparse hamiltonian decomposable circulant graphs are hamiltonian decomposable.

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40

Zhang, Lina, Xin Wu, and Enwei Liang. "Adjustment of Force–Gradient Operator in Symplectic Methods." Mathematics 9, no. 21 (October 27, 2021): 2718. http://dx.doi.org/10.3390/math9212718.

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Many force–gradient explicit symplectic integration algorithms have been designed for the Hamiltonian H=T(p)+V(q) with kinetic energy T(p)=p2/2 in the existing references. When a force–gradient operator is appropriately adjusted as a new operator, it is still suitable for a class of Hamiltonian problems H=K(p,q)+V(q) with integrable part K(p,q)=∑i=1n∑j=1naijpipj+∑i=1nbipi, where aij=aij(q) and bi=bi(q) are functions of coordinates q. The newly adjusted operator is not a force–gradient operator but is similar to the momentum-version operator associated to the potential V. The newly extended (or adjusted) algorithms are no longer solvers of the original Hamiltonian, but are solvers of slightly modified Hamiltonians. They are explicit symplectic integrators with symmetry or time reversibility. Numerical tests show that the standard symplectic integrators without the new operator are generally poorer than the corresponding extended methods with the new operator in computational accuracies and efficiencies. The optimized methods have better accuracies than the corresponding non-optimized counterparts. Among the tested symplectic methods, the two extended optimized seven-stage fourth-order methods of Omelyan, Mryglod and Folk exhibit the best numerical performance. As a result, one of the two optimized algorithms is used to study the orbital dynamical features of a modified Hénon–Heiles system and a spring pendulum. These extended integrators allow for integrations in Hamiltonian problems, such as the spiral structure in self-consistent models of rotating galaxies and the spiral arms in galaxies.

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41

Meissner, Leszek, and JarosłaW Gryniaków. "Effective Hamiltonian and Intermediate Hamiltonian Formulations of the Fock-Space Coupled-Cluster Method." Collection of Czechoslovak Chemical Communications 68, no. 1 (2003): 105–38. http://dx.doi.org/10.1135/cccc20030105.

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Various aspects of the effective Hamiltonian and intermediate Hamiltonian formulations are discussed in the context of the Fock-space coupled-cluster method. Problems that occur when single-reference methods of solving the Schrödinger equation need to be generalized to the multireference (MR) cases are pointed out. These problems make the generalization nontrivial, especially in the case of the most powerful coupled-cluster (CC) method. It is shown how some specific features of one of the basic MR-CC schemes, the Fock-space CC method, can be used to obtain a simple, yet very effective version of the method. This requires, however, switching from the effective Hamiltonian to the intermediate Hamiltonian formulation. The intermediate Hamiltonian version of the Fock-space CC method is discussed in detail and all its advantages over the standard one are emphasized.

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SZPAK, B., J. DUDEK, M. G. PORQUET, K. RYBAK, H. MOLIQUE, and B. FORNAL. "NUCLEAR MEAN-FIELD HAMILTONIANS AND FACTORS LIMITING THEIR SPECTROSCOPIC PREDICTIVE POWER: ILLUSTRATIONS." International Journal of Modern Physics E 19, no. 04 (April 2010): 665–71. http://dx.doi.org/10.1142/s0218301310015072.

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Determination of the mean-field Hamiltonian parameters can be seen as gathering information about all the single-particle states out of a very partial information on only a few experimentally known levels. This is exactly what the inverse problem in applied mathematics is about. We illustrate some of the related concepts in view of a preparation of the fully statistically significant parameter adjustment procedures. For this purpose we construct the exactly soluble inverse problems associated with the realistic and phenomenologically powerful nuclear Woods-Saxon Hamiltonian and we analyse a few both physical and mathematical aspects of such procedures. Presented illustrations suggest that to be able to discuss the predictive power of the mean-field Hamiltonians the parameter adjustment procedures must be based on a relatively complex statistical analysis partially addressed in Ref.1

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Fu, Jingli, Lijun Zhang, Shan Cao, Chun Xiang, and Weijia Zao. "A Symplectic Algorithm for Constrained Hamiltonian Systems." Axioms 11, no. 5 (May 7, 2022): 217. http://dx.doi.org/10.3390/axioms11050217.

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In this paper, a symplectic algorithm is utilized to investigate constrained Hamiltonian systems. However, the symplectic method cannot be applied directly to the constrained Hamiltonian equations due to the non-canonicity. We firstly discuss the canonicalization method of the constrained Hamiltonian systems. The symplectic method is used to constrain Hamiltonian systems on the basis of the canonicalization, and then the numerical simulation of the system is carried out. An example is presented to illustrate the application of the results. By using the symplectic method of constrained Hamiltonian systems, one can solve the singular dynamic problems of nonconservative constrained mechanical systems, nonholonomic constrained mechanical systems as well as physical problems in quantum dynamics, and also available in many electromechanical coupled systems.

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WOCJAN, PAWEL, and THOMAS BETH. "THE 2-LOCAL HAMILTONIAN PROBLEM ENCOMPASSES NP." International Journal of Quantum Information 01, no. 03 (September 2003): 349–57. http://dx.doi.org/10.1142/s021974990300022x.

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We show that the NP-complete problems max cut and independent set can be formulated as the 2-local Hamiltonian problem as defined by Kitaev. The 5-local Hamiltonian problem was the first problem to be shown to be complete for the quantum complexity class QMA — the quantum analog of NP. Subsequently, it was shown that 3-locality is already sufficient for QMA-completeness. It is still not known whether the 2-local Hamiltonian problem is QMA-complete. Therefore it is interesting to determine what problems can be reduced to the 2-local Hamiltonian problem. Kitaev showed that 3-SAT can be formulated as a 3-local Hamiltonian problem. We extend his result by showing that 2-locality is sufficient in order to encompass NP.

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Bang-Jensen, J., M. El Haddad, Y. Manoussakis, and T. M. Przytycka. "Parallel algorithms for the hamiltonian cycle and hamiltonian path problems in semicomplete bipartite digraphs." Algorithmica 17, no. 1 (January 1997): 67–87. http://dx.doi.org/10.1007/bf02523239.

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SHINJO, Kazumasa. "Hamiltonian Algorithm : A Method to Solve Optimization Problems." Journal of Japan Society for Fuzzy Theory and Systems 11, no. 3 (1999): 382–95. http://dx.doi.org/10.3156/jfuzzy.11.3_24.

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Krasovskii, Andrey A., and Alexander M. Tarasyev. "Conjugation of Hamiltonian Systems in Optimal Control Problems." IFAC Proceedings Volumes 41, no. 2 (2008): 7784–89. http://dx.doi.org/10.3182/20080706-5-kr-1001.01316.

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Fichtner, Andreas, Andrea Zunino, and Lars Gebraad. "Hamiltonian Monte Carlo solution of tomographic inverse problems." Geophysical Journal International 216, no. 2 (November 22, 2018): 1344–63. http://dx.doi.org/10.1093/gji/ggy496.

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Pester, C. "Hamiltonian Eigenvalue Symmetry for Quadratic Operator Eigenvalue Problems." Journal of Integral Equations and Applications 17, no. 1 (March 2005): 71–89. http://dx.doi.org/10.1216/jiea/1181075311.

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McLachlan, R. I., and C. Offen. "Bifurcation of solutions to Hamiltonian boundary value problems." Nonlinearity 31, no. 6 (May 9, 2018): 2895–927. http://dx.doi.org/10.1088/1361-6544/aab630.

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