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Journal articles on the topic 'Hamiltonian Problem'

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Author: Grafiati

Published: 10 March 2023

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1

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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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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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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4

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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Klassen, Joel, and Barbara M. Terhal. "Two-local qubit Hamiltonians: when are they stoquastic?" Quantum 3 (May 6, 2019): 139. http://dx.doi.org/10.22331/q-2019-05-06-139.

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We examine the problem of determining if a 2-local Hamiltonian is stoquastic by local basis changes. We analyze this problem for two-qubit Hamiltonians, presenting some basic tools and giving a concrete example where using unitaries beyond Clifford rotations is required in order to decide stoquasticity. We report on simple results for n-qubit Hamiltonians with identical 2-local terms on bipartite graphs. Our most significant result is that we give an efficient algorithm to determine whether an arbitrary n-qubit XYZ Heisenberg Hamiltonian is stoquastic by local basis changes.

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6

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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7

Gamboa, J., and F. Méndez. "Deformed quantum mechanics and the Landau problem." Modern Physics Letters A 36, no. 18 (June 14, 2021): 2150126. http://dx.doi.org/10.1142/s0217732321501261.

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A deformation of the Landau problem based on a modification of Fock algebra is considered. Systems with the Hamiltonians [Formula: see text] where [Formula: see text] is the Landau Hamiltonian in the lowest level are discussed. The case [Formula: see text] is studied and it is shown that in this particular example, parameters of the problem can be fixed by using the quadratic Zeeman effect data and the Breit–Rabi formula.

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8

NEIDHARDT, HAGEN, and VALENTIN ZAGREBNOV. "TOWARDS THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS VIA REGULARIZATION AND EXTENSION THEORY." Reviews in Mathematical Physics 08, no. 05 (July 1996): 715–40. http://dx.doi.org/10.1142/s0129055x96000251.

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For singular potentials in quantum mechanics it can happen that the Schrödinger operator is not esssentially self-adjoint on a natural domain, i.e., each self-adjoint extension is a candidate for the right physical Hamiltonian. Traditional way to single out this Hamiltonian is the removing cut-offs for regularizing potential. Connecting regularization and extension theory we develop an abstract operator method to treat the problem of the right Hamiltonian. We show that, using the notion of the maximal (with respect to the perturbation) Friedrichs extension of unperturbed operator, one can classify the above problem as wellposed or ill-posed depending on intersection of the quadratic form domain of perturbation and deficiency subspace corresponding to restriction of unperturbed operator to stability domain. If this intersection is trivial, then the right Hamiltonian is unique: it coincides with the form sum of perturbation and the Friedrich extension of the unperturbed operator restricted to the stability domain. Otherwise it is not unique: the family of “right Hamiltonians” can be described in terms of symmetric extensions reducing the ill-posed problem to the well-posed problem.

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LALONDE, FRANÇOIS, and ANDREI TELEMAN. "THE g-AREAS AND THE COMMUTATOR LENGTH." International Journal of Mathematics 24, no. 07 (June 2013): 1350057. http://dx.doi.org/10.1142/s0129167x13500572.

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The commutator length of a Hamiltonian diffeomorphism f ∈ Ham (M,ω) of a closed symplectic manifold (M,ω) is by definition the minimal k such that f can be written as a product of k commutators in Ham (M,ω). We introduce a new invariant for Hamiltonian diffeomorphisms, called the k+-area, which measures the "distance", in a certain sense, to the subspace [Formula: see text] of all products of k commutators. Therefore, this invariant can be seen as the obstruction to writing a given Hamiltonian diffeomorphism as a product of k commutators. We also consider an infinitesimal version of the commutator problem: what is the obstruction to writing a Hamiltonian vector field as a linear combination of k Lie brackets of Hamiltonian vector fields? A natural problem related to this question is to describe explicitly, for every fixed k, the set of linear combinations of k such Lie brackets. The problem can be obviously reformulated in terms of Hamiltonians and Poisson brackets. For a given Morse function f on a symplectic Riemann surface M (verifying a weak genericity condition) we describe the linear space of commutators of the form {f, g}, with [Formula: see text].

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10

Godinho, Leonor, and M. E. Sousa-Dias. "The Fundamental Group ofS1-manifolds." Canadian Journal of Mathematics 62, no. 5 (October 1, 2010): 1082–98. http://dx.doi.org/10.4153/cjm-2010-053-3.

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AbstractWe address the problem of computing the fundamental group of a symplecticS1-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a HamiltonianS1-action. Several examples are presented to illustrate our main results.

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11

Low, Guang Hao, and Isaac L. Chuang. "Hamiltonian Simulation by Qubitization." Quantum 3 (July 12, 2019): 163. http://dx.doi.org/10.22331/q-2019-07-12-163.

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We present the problem of approximating the time-evolution operatore−iH^tto errorϵ, where the HamiltonianH^=(⟨G|⊗I^)U^(|G⟩⊗I^)is the projection of a unitary oracleU^onto the state|G⟩created by another unitary oracle. Our algorithm solves this with a query complexityO(t+log⁡(1/ϵ))to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which ared-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as whereH^is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed anyH^in an invariantSU(2)subspace. A large class of operator functions ofH^can then be computed with optimal query complexity, of whiche−iH^tis a special case.

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12

H. Sierra, V., C. A. Aguirre, and José José Barba-Ortega. "Interpretación didáctica de la teoría de grupos aplicada en cristales." Respuestas 23, no. 1 (April 14, 2018): 68. http://dx.doi.org/10.22463/0122820x.1337.

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ResumenLa determinación del Hamiltoniano de una molécula o un cristal puede llegar a ser un problema muy complicado; sin embargo, las consideraciones de simetría sobre el problema pueden llegar a simplificarlo de manera sustancial. Razón por la cual, es pertinente buscar el mayor número de simetrías de un cristal. En este punto, se realza la importancia de la teoría de grupos como herramienta de cálculo, pues a través de ésta, se sintetizan todas las propiedades del cristal: las rotaciones, las inversiones y las reflexiones. Empero, el estudio realizado por muchos libros acerca de esta temática es demasiado confuso y complicado para los estudiantes de Licenciatura en Física, debido a la naturaleza abstracta del método de la teoría, y las relaciones que éste tiene con el Hamiltoniano. Lo anterior, motiva la realización de un estudio didáctico, así como detallado de los principios que rigen el uso del método. Además, se ilustra a través de un ejemplo detallado para el caso de un cristal ortorrómbico, procediendo a establecer los isomorfismos entre el álgebra utilizada en la teoría de grupos y la correspondiente representación de matrices, que permita efectuar la reducción del Hamiltoniano y los cálculos correspondientes.Palabras clave: Simetría, Hamiltoniano, Teoría de grupos.AbstractThe determination of the Hamiltonian of a molecule or a crystal can become a very complicated problem. However, considerations of symmetry of the problem may make it simpler. Therefore it is relevant seek the greatest number of symmetries of a crystal. At this point, it highlights the importance of group theory as a tool for calculation, then, through it synthesizes all of these properties of the crystal, like the rotations, inversions and reflections. However, the study present in many books on this subject is too confusing and complicated for the students of Bachelor in Physics, because of the abstract nature of theoretical method and the relationship it has with the Hamiltonian. This one motivates the realization of a didactic study, as well as detailed principles governing the use of the method. Also, a detailed example is present for the case of an orthorhombic crystal, proceeding to establish the isomorphism between the algebra used in group theory and the corresponding matrix representation, permitting a reduction in the Hamiltonian and the calculations.Keywords: Symmetry, Hamiltonian, Group TheoryResumoA determinação do Hamiltoniano de uma molécula ou cristal pode se tornar um problema muito complicado; No entanto, considerações de simetria sobre o problema podem simplificá-lo substancialmente. Pelo que, é pertinente procurar o maior número de simetrias de um cristal. Nesse ponto, enfatiza-se a importância da teoria dos grupos como ferramenta de cálculo, pois através dela todas as propriedades do cristal são sintetizadas: rotações, inversões e reflexões. No entanto, o estudo de muitos livros sobre este assunto é muito confuso e complicado para os estudantes de graduação em Física, por causa da natureza abstrata do método da teoria, e a relação que tem com o hamiltoniano. O exposto, motiva a realização de um estudo didático, bem como detalha os princípios que regem o uso do método. Além disso, é ilustrada através de um processo detalhado de um cristal ortorrômbico, prosseguir para estabelecer a isomorfismo entre álgebra utilizado na teoria de grupos e a representação correspondente de matrizes, permitindo a redução do exemplo Hamiltoniano e cálculos.Palabras chave: Simetria, hamiltoniana, teoria dos grupos

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13

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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14

Ferraz-Mello, S. "Do Average Hamiltonians Exist?" International Astronomical Union Colloquium 172 (1999): 243–48. http://dx.doi.org/10.1017/s0252921100072596.

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AbstractThe word “average” and its variations became popular in the sixties and implicitly carried the idea that “averaging” methods lead to “average” Hamiltonians. However, given the Hamiltonian H = H0(J) + ϵR(θ,J),(ϵ ≪ 1), the problem of transforming it into a new Hamiltonian H* (J*) (dependent only on the new actions J*), through a canonical transformation given by zero-average trigonometrical series has no general solution at orders higher than the first.

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15

Nishiyama, Hiroshi, Yusuke Kobayashi, Yukiko Yamauchi, Shuji Kijima, and Masafumi Yamashita. "The parity Hamiltonian cycle problem." Discrete Mathematics 341, no. 3 (March 2018): 606–26. http://dx.doi.org/10.1016/j.disc.2017.10.025.

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Liu, Muhuo, Hong-Jian Lai, and Kinkar Ch Das. "Spectral results on Hamiltonian problem." Discrete Mathematics 342, no. 6 (June 2019): 1718–30. http://dx.doi.org/10.1016/j.disc.2019.02.016.

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17

Branco, I. M., and J. D. Coelho. "The hamiltonian p-median problem." European Journal of Operational Research 47, no. 1 (July 1990): 86–95. http://dx.doi.org/10.1016/0377-2217(90)90092-p.

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18

SUN, JIE, and SONG-FENG LU. "ON THE ADIABATIC EVOLUTION OF ONE-DIMENSIONAL PROJECTOR HAMILTONIANS." International Journal of Quantum Information 10, no. 04 (June 2012): 1250046. http://dx.doi.org/10.1142/s0219749912500463.

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In this paper, we discuss the adiabatic evolution of one-dimensional projector Hamiltonians. Three kinds of adiabatic algorithms for this problem are shown, in which two of them provide a quadratic speedup over the other one. But when the ground state of the initial Hamiltonian and that of the final Hamiltonian have a zero overlap, the algorithms above all fail, in the sense of infinite time complexity. A corresponding revised method for this phenomenon through adding a driving Hamiltonian is also shown, from which a constant time complexity can be gained. But by a simple analysis, we find that the original time complexity for the adiabatic evolution is shifted to implement the driving Hamiltonian, which is supported by several early works in the literature.

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19

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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Pakrouski, Kiryl. "Automatic design of Hamiltonians." Quantum 4 (September 2, 2020): 315. http://dx.doi.org/10.22331/q-2020-09-02-315.

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We formulate an optimization problem of Hamiltonian design based on the variational principle. Given a variational ansatz for a Hamiltonian we construct a loss function to be minimised as a weighted sum of relevant Hamiltonian properties specifying thereby the search query. Using fractional quantum Hall effect as a test system we illustrate how the framework can be used to determine a generating Hamiltonian of a finite-size model wavefunction (Moore-Read Pfaffian and Read-Rezayi states), find optimal conditions for an experiment or "extrapolate" given wavefunctions in a certain universality class from smaller to larger system sizes. We also discuss how the search for approximate generating Hamiltonians may be used to find simpler and more realistic models implementing the given exotic phase of matter by experimentally accessible interaction terms.

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Choi, Jaeho, Seunghyeok Oh, and Joongheon Kim. "Quantum Approximation for Wireless Scheduling." Applied Sciences 10, no. 20 (October 13, 2020): 7116. http://dx.doi.org/10.3390/app10207116.

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This paper proposes an application algorithm based on a quantum approximate optimization algorithm (QAOA) for wireless scheduling problems. QAOA is one of the promising hybrid quantum-classical algorithms to solve combinatorial optimization problems and it provides great approximate solutions to non-deterministic polynomial-time (NP) hard problems. QAOA maps the given problem into Hilbert space, and then it generates the Hamiltonian for the given objective and constraint. Then, QAOA finds proper parameters from the classical optimization loop in order to optimize the expectation value of the generated Hamiltonian. Based on the parameters, the optimal solution to the given problem can be obtained from the optimum of the expectation value of the Hamiltonian. Inspired by QAOA, a quantum approximate optimization for scheduling (QAOS) algorithm is proposed. The proposed QAOS designs the Hamiltonian of the wireless scheduling problem which is formulated by the maximum weight independent set (MWIS). The designed Hamiltonian is converted into a unitary operator and implemented as a quantum gate operation. After that, the iterative QAOS sequence solves the wireless scheduling problem. The novelty of QAOS is verified with simulation results implemented via Cirq and TensorFlow-Quantum.

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Liang, Kai Fu, Ming Jun Li, and Ze Lin Zhu. "On Solutions of Inverse Problem for Hermitian Generalized Hamiltonian Matrices." Advanced Materials Research 860-863 (December 2013): 2727–31. http://dx.doi.org/10.4028/www.scientific.net/amr.860-863.2727.

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Hamiltonian matrices have many applications to design automation and autocontrol, in particular in the linear-quadratic autocontrol problem. This paper studies the inverse problems of generalized Hamiltonian matrices for matrix equations. By real representation of complex matrix, we give the necessary and sufficient conditions for the existence of a Hermitian generalized Hamiltonian solutions to the matrix equations, and then derive the representation of the general solutions.

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ARRIOLA, E. RUIZ, and L. L. SALCEDO. "SEMICLASSICAL EXPANSION FOR DIRAC HAMILTONIANS." Modern Physics Letters A 08, no. 22 (July 20, 1993): 2061–69. http://dx.doi.org/10.1142/s021773239300177x.

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A simple and efficient calculation of the level density of Dirac Hamiltonians in the semi-classical approximation is presented. The method is applied to compute the level density up to ħ4-order of a Dirac Hamiltonian with time independent scalar and electromagnetic external fields. The final expressions are explicitly gauge invariant and convergent at the turning points. As a byproduct, we obtain ħ-corrections to the semiclassical quantization rule of a Dirac Hamiltonian in D space dimensions. The result is illustrated in an exactly solvable problem.

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Bueno, Letícia, Luerbio Faria, Figueiredo De, and Fonseca Da. "Hamiltonian paths in odd graphs." Applicable Analysis and Discrete Mathematics 3, no. 2 (2009): 386–94. http://dx.doi.org/10.2298/aadm0902386b.

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Lov?sz conjectured that every connected vertex-transitive graph has a Hamiltonian path. The odd graphs Ok form a well-studied family of connected, k-regular, vertex-transitive graphs. It was previously known that Ok has Hamiltonian paths for k ? 14. A direct computation of Hamiltonian paths in Ok is not feasible for large values of k, because Ok has (2k - 1, k - 1) vertices and k/2 (2k - 1, k - 1) edges. We show that Ok has Hamiltonian paths for 15 ? k ? 18. Instead of directly running any heuristics, we use existing results on the middle levels problem, therefore further relating these two fundamental problems, namely finding a Hamiltonian path in the odd graph and finding a Hamiltonian cycle in the corresponding middle levels graph. We show that further improved results for the middle levels problem can be used to find Hamiltonian paths in Ok for larger values of k.

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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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Olszewski, S., T. Roliński, M. Baszczak, and R. Kozak. "Phase-space Symmetry and the Action Function of the Pendulum Problem." Zeitschrift für Naturforschung A 57, no. 11 (November 1, 2002): 888–96. http://dx.doi.org/10.1515/zna-2002-1108.

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An approach to the pendulum problem, which is an alternative to the well-known traditional treatment of that problem, has been formulated. An advantage of the new approach is provided by a full symmetry in the position and momentum variables of the Hamiltonian expression for the energy of the system. A similar symmetry holds for the Hamilton equations describing the motion of a pendulum-like point mass. Calculations of the action function forthe two kinds of pendulum Hamiltonians - the traditional one and the new one - are presented.

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Wocjan, P., D. Janzing, and T. Beth. "Simulating arbitrary pair-interactions by a given Hamiltonian: graph-theoretical bounds on the time-complexity." Quantum Information and Computation 2, no. 2 (February 2002): 117–32. http://dx.doi.org/10.26421/qic2.2-2.

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We consider a quantum computer consisting of n spins with an arbitrary but fixed pair-interaction Hamiltonian and describe how to simulate other pair-interactions by interspersing the natural time evolution with fast local transformations. Calculating the minimal time overhead of such a simulation leads to a convex optimization problem. Lower and upper bounds on the minimal time overhead are derived in terms of chromatic indices of interaction graphs and spectral majorization criteria. These results classify Hamiltonians with respect to their computational power. For a specific Hamiltonian, namely \sigma_z\otimes\sigma_z-interactions between all spins, the optimization is mathematically equivalent to a separability problem of n-qubit density matrices. We compare the complexity defined by such a quantum computer with the usual gate complexity.

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Childs, A. M., and R. Kothari. "Limitations on the simulation of non-sparse Hamiltonians." Quantum Information and Computation 10, no. 7&8 (July 2010): 669–84. http://dx.doi.org/10.26421/qic10.7-8-7.

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The problem of simulating sparse Hamiltonians on quantum computers is well studied. The evolution of a sparse $N \times N$ Hamiltonian $H$ for time $t$ can be simulated using $\O(\norm{Ht} \poly(\log N))$ operations, which is essentially optimal due to a no--fast-forwarding theorem. Here, we consider non-sparse Hamiltonians and show significant limitations on their simulation. We generalize the no--fast-forwarding theorem to dense Hamiltonians, ruling out generic simulations taking time $\o(\norm{Ht})$, even though $\norm{H}$ is not a unique measure of the size of a dense Hamiltonian $H$. We also present a stronger limitation ruling out the possibility of generic simulations taking time $\poly(\norm{Ht},\log N)$, showing that known simulations based on discrete-time quantum walk cannot be dramatically improved in general. On the positive side, we show that some non-sparse Hamiltonians can be simulated efficiently, such as those with graphs of small arboricity.

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SYLJUÅSEN, OLAV F. "RANDOM WALKS NEAR ROKHSAR–KIVELSON POINTS." International Journal of Modern Physics B 19, no. 12 (May 10, 2005): 1973–93. http://dx.doi.org/10.1142/s021797920502964x.

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There is a class of quantum Hamiltonians known as Rokhsar–Kivelson (RK)–Hamiltonians for which static ground state properties can be obtained by evaluating thermal expectation values for classical models. The ground state of an RK–Hamiltonian is known explicitly, and its dynamical properties can be obtained by performing a classical Monte Carlo simulation. We discuss the details of a Diffusion Monte Carlo method that is a good tool for studying statics and dynamics of perturbed RK–Hamiltonians without time discretization errors. As a general result we point out that the relation between the quantum dynamics and classical Monte Carlo simulations for RK–Hamiltonians follows from the known fact that the imaginary-time evolution operator describing optimal importance sampling, where the exact ground state is used as guiding function, is Markovian. Thus quantum dynamics can be studied by classical Monte Carlo for any Hamiltonian provided its ground state is known explicitly and that there is no sign problem.

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30

Rudnev, M., and V. Ten. "An inverse problem of Hamiltonian dynamics." Proceedings of the American Mathematical Society 134, no. 11 (May 8, 2006): 3295–99. http://dx.doi.org/10.1090/s0002-9939-06-08351-1.

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31

Knowles, Ian W. "An inverse problem for Hamiltonian systems." Journal of Computational and Applied Mathematics 148, no. 1 (November 2002): 99–113. http://dx.doi.org/10.1016/s0377-0427(02)00576-9.

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Chen, Zhi-Hong, Hong-Jian Lai, Wai-Chee Shiu, and Deying Li. "An s-Hamiltonian Line Graph Problem." Graphs and Combinatorics 23, no. 3 (June 2007): 241–48. http://dx.doi.org/10.1007/s00373-007-0727-y.

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33

Heinzl, T. "Hamiltonian approach to the Gribov problem." Nuclear Physics B - Proceedings Supplements 54, no. 1-2 (March 1997): 194–97. http://dx.doi.org/10.1016/s0920-5632(97)00039-x.

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34

Bagram Sibgatullovich, Kochkarev. "Problem of Recognition of Hamiltonian Graph." International Journal of Wireless Communications and Mobile Computing 4, no. 2 (2016): 52. http://dx.doi.org/10.11648/j.wcmc.20160402.17.

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35

Gould, Ronald J. "Updating the hamiltonian problem—A survey." Journal of Graph Theory 15, no. 2 (June 1991): 121–57. http://dx.doi.org/10.1002/jgt.3190150204.

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36

MARTÍNEZ, D., R. D. MOTA, and V. D. GRANADOS. "SYMMETRY AND SUPERSYMMETRY OF A NEUTRON IN THE MAGNETIC FIELD OF A LINEAR CURRENT." International Journal of Modern Physics A 21, no. 32 (December 30, 2006): 6621–28. http://dx.doi.org/10.1142/s0217751x06034446.

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We study a neutron in an external magnetic field in coordinate space and show that the 2 × 2 radial matrix operators that factorize the Hamiltonian are contained within the constants of motion of the problem. Also we show that the 2 × 2 partners Hamiltonians satisfy the shape invariance condition.

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37

Piddock, Stephen, and Ashley Montanaro. "Universal Qudit Hamiltonians." Communications in Mathematical Physics 382, no. 2 (February 23, 2021): 721–71. http://dx.doi.org/10.1007/s00220-021-03940-3.

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AbstractA family of quantum Hamiltonians is said to be universal if any other finite-dimensional Hamiltonian can be approximately encoded within the low-energy space of a Hamiltonian from that family. If the encoding is efficient, universal families of Hamiltonians can be used as universal analogue quantum simulators and universal quantum computers, and the problem of approximately determining the ground-state energy of a Hamiltonian from a universal family is QMA-complete. One natural way to categorise Hamiltonians into families is in terms of the interactions they are built from. Here we prove universality of some important classes of interactions on qudits (d-level systems): We completely characterise the k-qudit interactions which are universal, if augmented with arbitrary Hermitian 1-local terms. We find that, for all $$k \geqslant 2$$ k ⩾ 2 and all local dimensions $$d \geqslant 2$$ d ⩾ 2 , almost all such interactions are universal aside from a simple stoquastic class. We prove universality of generalisations of the Heisenberg model that are ubiquitous in condensed-matter physics, even if free 1-local terms are not provided. We show that the SU(d) and SU(2) Heisenberg interactions are universal for all local dimensions $$d \geqslant 2$$ d ⩾ 2 (spin $$\geqslant 1/2$$ ⩾ 1 / 2 ), implying that a quantum variant of the Max-d-Cut problem is QMA-complete. We also show that for $$d=3$$ d = 3 all bilinear-biquadratic Heisenberg interactions are universal. One example is the general AKLT model. We prove universality of any interaction proportional to the projector onto a pure entangled state.

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38

Wheeler, James T. "Not-so-classical mechanics: unexpected symmetries of classical motion." Canadian Journal of Physics 83, no. 2 (February 1, 2005): 91–138. http://dx.doi.org/10.1139/p05-003.

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A survey of topics of recent interest in Hamiltonian and Lagrangian dynamical systems, including accessible discussions of regularization of the central-force problem; inequivalent Lagrangians and Hamiltonians; constants of central-force motion; a general discussion of higher order Lagrangians and Hamiltonians, with examples from Bohmian quantum mechanics, the Kortewegde Vries equation, and the logistic equation; gauge theories of Newtonian mechanics; and classical spin, Grassmann numbers, and pseudomechanics. PACS No.: 45.25.Jj

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39

Renjith, P., and N. Sadagopan. "Hamiltonian Cycle in K1,r-Free Split Graphs — A Dichotomy." International Journal of Foundations of Computer Science 33, no. 01 (October 20, 2021): 1–32. http://dx.doi.org/10.1142/s0129054121500337.

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For an optimization problem known to be NP-Hard, the dichotomy study investigates the reduction instances to determine the line separating polynomial-time solvable vs NP-Hard instances (easy vs hard instances). In this paper, we investigate the well-studied Hamiltonian cycle problem (HCYCLE), and present an interesting dichotomy result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete on planar bipartite graphs with maximum degree [Formula: see text]. We use this result to show that HCYCLE is NP-complete for [Formula: see text]-free split graphs. Further, we present polynomial-time algorithms for Hamiltonian cycle in [Formula: see text]-free and [Formula: see text]-free split graphs. We believe that the structural results presented in this paper can be used to show similar dichotomy result for Hamiltonian path problem and other variants of Hamiltonian cycle (path) problems.

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40

Melnikov, B. F., and V. A. Dudnikov. "ON THE NP-COMPLETENESS OF THE PROBLEM OF PLACEMENT OF THE GRAPH." Informatization and communication, no. 1 (March 20, 2019): 51–54. http://dx.doi.org/10.34219/2078-8320-2019-10-1-51-54.

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This paper provides a proof of NP-completeness of the problem of the placement of the graph. Thus, it is concluded that the algorithmization of the graph placement problem requires an approach based on heuristic or stochastic methods. To prove the NP-completeness of the graph placement problem, we use the well-known NP-complete Hamiltonian cycle search problem. It is shown that the graph placement problem is a generalization over the Hamiltonian cycle problem. Also the problem of the placement of the graph as the optimization and presents some auxiliary results, which are set nalivayut the connection between the two problems: NP-complete optimization problem and embed the graph.

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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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Giordano, M., G. Marmo, and C. Rubano. "The inverse problem in the Hamiltonian formalism: integrability of linear Hamiltonian fields." Inverse Problems 9, no. 4 (August 1, 1993): 443–67. http://dx.doi.org/10.1088/0266-5611/9/4/001.

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43

Kohler, Tamara, Stephen Piddock, Johannes Bausch, and Toby Cubitt. "Translationally Invariant Universal Quantum Hamiltonians in 1D." Annales Henri Poincaré 23, no. 1 (October 23, 2021): 223–54. http://dx.doi.org/10.1007/s00023-021-01111-7.

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AbstractRecent work has characterized rigorously what it means for one quantum system to simulate another and demonstrated the existence of universal Hamiltonians—simple spin lattice Hamiltonians that can replicate the entire physics of any other quantum many-body system. Previous universality results have required proofs involving complicated ‘chains’ of perturbative ‘gadgets.’ In this paper, we derive a significantly simpler and more powerful method of proving universality of Hamiltonians, directly leveraging the ability to encode quantum computation into ground states. This provides new insight into the origins of universal models and suggests a deep connection between universality and complexity. We apply this new approach to show that there are universal models even in translationally invariant spin chains in 1D. This gives as a corollary a new Hamiltonian complexity result that the local Hamiltonian problem for translationally invariant spin chains in one dimension with an exponentially small promise gap is PSPACE-complete. Finally, we use these new universal models to construct the first known toy model of 2D–1D holographic duality between local Hamiltonians.

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44

Guzzo, Massimiliano. "Nekhoroshev Stability in Quasi-Integrable Degenerate Hamiltonian Systems." International Astronomical Union Colloquium 172 (1999): 443–44. http://dx.doi.org/10.1017/s0252921100073085.

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Many classical problems of Mechanics can be studied regarding them as perturbations of integrable systems; this is the case of the fast rotations of the rigid body in an arbitrary potential, the restricted three body problem with small values of the mass-ratio, and others. However, the application of the classical results of Hamiltonian Perturbation Theory to these systems encounters difficulties due to the presence of the so-called ‘degeneracy’. More precisely, the Hamiltonian of a quasi-integrable degenerate system looks likewhere (I, φ) є U × Tn, U ⊆ Rn, are action-angle type coordinates, while the degeneracy of the system manifests itself with the presence of the ‘degenerate’ variables (p, q) є B ⊆ R2m. The KAM theorem has been applied under quite general assumptions to degenerate Hamiltonians (Arnold, 1963), while the Nekhoroshev theorem (Nekhoroshev, 1977) provides, if h is convex, the following bounds: there exist positive ε0, a0, t0 such that if ε < ε0 then if where Te is the escape time of the solution from the domain of (1). An escape is possible because the motion of the degenerate variables can be bounded in principle only by , and so over the time they can experience large variations. Therefore, there is the problem of individuating which assumptions on the perturbation and on the initial data allow to control the motion of the degenerate variables over long times.

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45

Irsigler, Bernhard, and Tobias Grass. "The quantum annealing gap and quench dynamics in the exact cover problem." Quantum 6 (January 18, 2022): 624. http://dx.doi.org/10.22331/q-2022-01-18-624.

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Quenching and annealing are extreme opposites in the time evolution of a quantum system: Annealing explores equilibrium phases of a Hamiltonian with slowly changing parameters and can be exploited as a tool for solving complex optimization problems. In contrast, quenches are sudden changes of the Hamiltonian, producing a non-equilibrium situation. Here, we investigate the relation between the two cases. Specifically, we show that the minimum of the annealing gap, which is an important bottleneck of quantum annealing algorithms, can be revealed from a dynamical quench parameter which describes the dynamical quantum state after the quench. Combined with statistical tools including the training of a neural network, the relation between quench and annealing dynamics can be exploited to reproduce the full functional behavior of the annealing gap from the quench data. We show that the partial or full knowledge about the annealing gap which can be gained in this way can be used to design optimized quantum annealing protocols with a practical time-to-solution benefit. Our results are obtained from simulating random Ising Hamiltonians, representing hard-to-solve instances of the exact cover problem.

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46

Poznyak, Alex S. "Robust stochastic maximum principle: Complete proof and discussions." Mathematical Problems in Engineering 8, no. 4-5 (2002): 389–411. http://dx.doi.org/10.1080/10241230306722.

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This paper develops a version of Robust Stochastic Maximum Principle (RSMP) applied to the Minimax Mayer Problem formulated for stochastic differential equations with the control-dependent diffusion term. The parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding Hamiltonian formalism. The Hamiltonian function used for the construction of the robust optimal control is shown to be equal to the Lebesque integral over a parametric set of the standard stochastic Hamiltonians corresponding to a fixed value of the uncertain parameter. The paper deals with a cost function given at finite horizon and containing the mathematical expectation of a terminal term. A terminal condition, covered by a vector function, is also considered. The optimal control strategies, adapted for available information, for the wide class of uncertain systems given by an stochastic differential equation with unknown parameters from a given compact set, are constructed. This problem belongs to the class of minimax stochastic optimization problems. The proof is based on the recent results obtained for Minimax Mayer Problem with a finite uncertainty set [14,43-45] as well as on the variation results of [53] derived for Stochastic Maximum Principle for nonlinear stochastic systems under complete information. The corresponding discussion of the obtain results concludes this study.

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47

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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48

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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49

Lakhno, Victor. "Cooper pairs and bipolarons." Modern Physics Letters B 30, no. 31 (November 20, 2016): 1650365. http://dx.doi.org/10.1142/s0217984916503656.

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It is shown that Cooper pairs are a solution of the bipolaron problem for model Fröhlich Hamiltonian. The total energy of a pair for the initial Fröhlich Hamiltonian is found. Differences between the solutions for the model and initial two-particle problems are discussed.

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50

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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