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Journal articles on the topic 'Hamiltonians'

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Author: Grafiati

Published: 4 June 2021

Last updated: 7 September 2023

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1

Hiroshima, Fumio. "Weak Coupling Limit with a Removal of an Ultraviolet Cutoff for a Hamiltonian of Particles Interacting with a Massive Scalar Field." Infinite Dimensional Analysis, Quantum Probability and Related Topics 01, no. 03 (July 1998): 407–23. http://dx.doi.org/10.1142/s0219025798000211.

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A Hamiltonian of an interaction system between N-particles and a massive scalar field is considered. The Hamiltonian with an ultraviolet cutoff is defined as a self-adjoint operator acting in a Hilbert space. Renormalized Hamiltonians are defined by subtracting renormalization terms from the Hamiltonian. It is shown that N-body Schrödinger Hamiltonians can be derived from taking a weak coupling limit and removing the ultraviolet cutoff simultaneously for the renormalized Hamiltonians. In particular, in the case where the space dimension equals three, the Yukawa potential appears in the N-body Schrödinger Hamiltonian. It is also shown that, in the case where the space dimensions are one or two, infimum of the spectra of the renormalized Hamiltonians converge to those of the N-body Schrödinger Hamiltonians.

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2

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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3

Hastings, Matthew. "Trivial low energy states for commuting Hamiltonians, and the quantum PCP conjecture." Quantum Information and Computation 13, no. 5&6 (May 2013): 393–429. http://dx.doi.org/10.26421/qic13.5-6-3.

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We consider the entanglement properties of ground states of Hamiltonians which are sums of commuting projectors (we call these commuting projector Hamiltonians), in particular whether or not they have ``trivial" ground states, where a state is trivial if it is constructed by a local quantum circuit of bounded depth and range acting on a product state. It is known that Hamiltonians such as the toric code only have nontrivial ground states in two dimensions. Conversely, commuting projector Hamiltonians which are sums of two-body interactions have trivial ground states\cite{bv}. Using a coarse-graining procedure, this implies that any such Hamiltonian with bounded range interactions in one dimension has a trivial ground state. In this paper, we further explore the question of which Hamiltonians have trivial ground states. We define an ``interaction complex" for a Hamiltonian, which generalizes the notion of interaction graph and we show that if the interaction complex can be continuously mapped to a $1$-complex using a map with bounded diameter of pre-images then the Hamiltonian has a trivial ground state assuming one technical condition on the Hamiltonians holds (this condition holds for all stabilizer Hamiltonians, and we additionally prove the result for all Hamiltonians under one assumption on the $1$-complex). While this includes the cases considered by Ref.~\onlinecite{bv}, we show that it also includes a larger class of Hamiltonians whose interaction complexes cannot be coarse-grained into the case of Ref.~\onlinecite{bv} but still can be mapped continuously to a $1$-complex. One motivation for this study is an approach to the quantum PCP conjecture. We note that many commonly studied interaction complexes can be mapped to a $1$-complex after removing a small fraction of sites. For commuting projector Hamiltonians on such complexes, in order to find low energy trivial states for the original Hamiltonian, it would suffice to find trivial ground states for the Hamiltonian with those sites removed. Such trivial states can act as a classical witness to the existence of a low energy state. While this result applies for commuting Hamiltonians and does not necessarily apply to other Hamiltonians, it suggests that to prove a quantum PCP conjecture for commuting Hamiltonians, it is worth investigating interaction complexes which cannot be mapped to $1$-complexes after removing a small fraction of points. We define this more precisely below; in some sense this generalizes the notion of an expander graph. Surprisingly, such complexes do exist as will be shown elsewhere\cite{fh}, and have useful properties in quantum coding theory.

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4

Liu, Yu, Jin Liu, and Da-jun Zhang. "On New Hamiltonian Structures of Two Integrable Couplings." Symmetry 14, no. 11 (October 27, 2022): 2259. http://dx.doi.org/10.3390/sym14112259.

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In this paper, we present new Hamiltonian operators for the integrable couplings of the Ablowitz–Kaup–Newell–Segur hierarchy and the Kaup–Newell hierarchy. The corresponding Hamiltonians allow nontrivial degeneration. Multi-Hamiltonian structures are investigated. The involutive property is proven for the new and known Hamiltonians with respect to the two Poisson brackets defined by the new and known Hamiltonian operators.

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5

Orlov, Yu N., V. Zh Sakbaev, and O. G. Smolyanov. "Randomizes hamiltonian mechanics." Доклады Академии наук 486, no. 6 (June 28, 2019): 653–58. http://dx.doi.org/10.31857/s0869-56524866653-658.

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Randomized Hamiltonian mechanics is the Hamiltonian mechanics which is determined by a time-dependent random Hamiltonian function. Corresponding Hamiltonian system is called random Hamiltonian system. The Feynman formulas for the random Hamiltonian systems are obtained. This Feynman formulas describe the solutions of Hamilton equation whose Hamiltonian is the mean value of random Hamiltonian function. The analogs of the above results is obtained for a random quantum system (which is a random infinite dimensional Hamiltonian system). This random quantum Hamiltonians are the part of Hamiltonians of open quantum system.

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6

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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7

Liu, Yingkai, and Emil Prodan. "A computer code for topological quantum spin systems over triangulated surfaces." International Journal of Modern Physics C 31, no. 07 (June 26, 2020): 2050091. http://dx.doi.org/10.1142/s0129183120500916.

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We derive explicit closed-form matrix representations of Hamiltonians drawn from tensored algebras, such as quantum spin Hamiltonians. These formulas enable us to soft-code generic Hamiltonian systems and to systematize the input data for uniformly structured as well as for un-structured Hamiltonians. The result is an optimal computer code that can be used as a black box that takes in certain input files and returns spectral information about the Hamiltonian. The code is tested on Kitaev’s toric model deployed on triangulated surfaces of genus 0 and 1. The efficiency of our code enables these simulations to be performed on an ordinary laptop. The input file corresponding to the minimal triangulation of genus 2 is also supplied.

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8

Konig, R. "Simplifying quantum double Hamiltonians using perturbative gadgets." Quantum Information and Computation 10, no. 3&4 (March 2010): 292–334. http://dx.doi.org/10.26421/qic10.3-4-9.

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Perturbative gadgets were originally introduced to generate effective $k$-local interactions in the low-energy sector of a $2$-local Hamiltonian. Extending this idea, we present gadgets which are specifically suited for realizing Hamiltonians exhibiting non-abelian anyonic excitations. At the core of our construction is a perturbative analysis of a widely used hopping-term Hamiltonian. We show that in the low-energy limit, this Hamiltonian can be approximated by a certain ordered product of operators. In particular, this provides a simplified realization of Kitaev's quantum double Hamiltonians.

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9

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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10

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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11

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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12

DEY, B., and C. N. KUMAR. "NEW SETS OF KINK-BEARING HAMILTONIANS." International Journal of Modern Physics A 09, no. 15 (June 20, 1994): 2699–705. http://dx.doi.org/10.1142/s0217751x94001096.

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Given a kink-bearing Hamiltonian, the isospectral Hamiltonian approach is used in generating new sets of Hamiltonians which also admit kink solutions. We use the sine-Gordon model as an example and explicitly work out new sets of potentials and solutions.

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13

GÖNÜL, BEŞİ, BÜLENT GÖNÜL, DİLEK TUTCU, and OKAN ÖZER. "SUPERSYMMETRIC APPROACH TO EXACTLY SOLVABLE SYSTEMS WITH POSITION-DEPENDENT EFFECTIVE MASSES." Modern Physics Letters A 17, no. 31 (October 10, 2002): 2057–66. http://dx.doi.org/10.1142/s0217732302008563.

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We discuss the relationship between exact solvability of the Schrödinger equation with a position-dependent mass and the ordering ambiguity in the Hamiltonian operator within the framework of supersymmetric quantum mechanics. The one-dimensional Schrödinger equation, derived from the general form of the effective mass Hamiltonian, is solved exactly for a system with exponentially changing mass in the presence of a potential with similar behaviour, and the corresponding supersymmetric partner Hamiltonians are related to the effective-mass Hamiltonians proposed in the literature.

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14

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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15

Crosson, Elizabeth, Tameem Albash, Itay Hen, and A. P. Young. "De-Signing Hamiltonians for Quantum Adiabatic Optimization." Quantum 4 (September 24, 2020): 334. http://dx.doi.org/10.22331/q-2020-09-24-334.

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Quantum fluctuations driven by non-stoquastic Hamiltonians have been conjectured to be an important and perhaps essential missing ingredient for achieving a quantum advantage with adiabatic optimization. We introduce a transformation that maps every non-stoquastic adiabatic path ending in a classical Hamiltonian to a corresponding stoquastic adiabatic path by appropriately adjusting the phase of each matrix entry in the computational basis. We compare the spectral gaps of these adiabatic paths and find both theoretically and numerically that the paths based on non-stoquastic Hamiltonians have generically smaller spectral gaps between the ground and first excited states, suggesting they are less useful than stoquastic Hamiltonians for quantum adiabatic optimization. These results apply to any adiabatic algorithm which interpolates to a final Hamiltonian that is diagonal in the computational basis.

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16

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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17

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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18

Cabedo-Olaya, Marina, Juan Gonzalo Muga, and Sofía Martínez-Garaot. "Shortcut-to-Adiabaticity-Like Techniques for Parameter Estimation in Quantum Metrology." Entropy 22, no. 11 (November 3, 2020): 1251. http://dx.doi.org/10.3390/e22111251.

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Quantum metrology makes use of quantum mechanics to improve precision measurements and measurement sensitivities. It is usually formulated for time-independent Hamiltonians, but time-dependent Hamiltonians may offer advantages, such as a T4 time dependence of the Fisher information which cannot be reached with a time-independent Hamiltonian. In Optimal adaptive control for quantum metrology with time-dependent Hamiltonians (Nature Communications 8, 2017), Shengshi Pang and Andrew N. Jordan put forward a Shortcut-to-adiabaticity (STA)-like method, specifically an approach formally similar to the “counterdiabatic approach”, adding a control term to the original Hamiltonian to reach the upper bound of the Fisher information. We revisit this work from the point of view of STA to set the relations and differences between STA-like methods in metrology and ordinary STA. This analysis paves the way for the application of other STA-like techniques in parameter estimation. In particular we explore the use of physical unitary transformations to propose alternative time-dependent Hamiltonians which may be easier to implement in the laboratory.

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19

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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20

Yeşiltaş, Özlem. "Non-Hermitian Dirac Hamiltonian in Three-Dimensional Gravity and Pseudosupersymmetry." Advances in High Energy Physics 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/484151.

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The Dirac Hamiltonian in the(2+1)-dimensional curved space-time has been studied with a metric for an expanding de Sitter space-time which is two spheres. The spectrum and the exact solutions of the time dependent non-Hermitian and angle dependent Hamiltonians are obtained in terms of the Jacobi and Romanovski polynomials. Hermitian equivalent of the Hamiltonian obtained from the Dirac equation is discussed in the frame of pseudo-Hermiticity. Furthermore, pseudosupersymmetric quantum mechanical techniques are expanded to a curved Dirac Hamiltonian and a partner curved Dirac Hamiltonian is generated. Usingη-pseudo-Hermiticity, the intertwining operator connecting the non-Hermitian Hamiltonians to the Hermitian counterparts is found. We have obtained a new metric tensor related to the new Hamiltonian.

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21

Zhou, Naying, Hongxing Zhang, Wenfang Liu, and Xin Wu. "A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes." Astrophysical Journal 927, no. 2 (March 1, 2022): 160. http://dx.doi.org/10.3847/1538-4357/ac497f.

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Abstract In recent publications, the construction of explicit symplectic integrators for Schwarzschild- and Kerr-type spacetimes is based on splitting and composition methods for numerical integrations of Hamiltonians or time-transformed Hamiltonians associated with these spacetimes. Such splittings are not unique but have various options. A Hamiltonian describing the motion of charged particles around the Schwarzschild black hole with an external magnetic field can be separated into three, four, and five explicitly integrable parts. It is shown through numerical tests of regular and chaotic orbits that the three-part splitting method is the best of the three Hamiltonian splitting methods in accuracy. In the three-part splitting, optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators exhibit the best accuracies. In fact, they are several orders of magnitude better than the fourth-order Yoshida algorithms for appropriate time steps. The first two algorithms have a small additional computational cost compared with the latter ones. Optimized sixth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators have no dramatic advantages over the optimized fourth-order ones in accuracy during long-term integrations due to roundoff errors. The idea of finding the integrators with the best performance is also suitable for Hamiltonians or time-transformed Hamiltonians of other curved spacetimes including Kerr-type spacetimes. When the numbers of explicitly integrable splitting sub-Hamiltonians are as small as possible, such splitting Hamiltonian methods would bring better accuracies. In this case, the optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström methods are worth recommending.

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22

Gosset, David, Jenish C. Mehta, and Thomas Vidick. "QCMA hardness of ground space connectivity for commuting Hamiltonians." Quantum 1 (July 14, 2017): 16. http://dx.doi.org/10.22331/q-2017-07-14-16.

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In this work we consider the ground space connectivity problem for commuting local Hamiltonians. The ground space connectivity problem asks whether it is possible to go from one (efficiently preparable) state to another by applying a polynomial length sequence of 2-qubit unitaries while remaining at all times in a state with low energy for a given HamiltonianH. It was shown in [Gharibian and Sikora, ICALP15] that this problem is QCMA-complete for general local Hamiltonians, where QCMA is defined as QMA with a classical witness and BQP verifier. Here we show that the commuting version of the problem is also QCMA-complete. This provides one of the first examples where commuting local Hamiltonians exhibit complexity theoretic hardness equivalent to general local Hamiltonians.

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23

Pasquotto, Federica, and Jagna Wiśniewska. "Bounds for tentacular Hamiltonians." Journal of Topology and Analysis 12, no. 01 (September 10, 2018): 209–65. http://dx.doi.org/10.1142/s179352531950047x.

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This paper represents a first step toward the extension of the definition of Rabinowitz Floer homology to non-compact energy hypersurfaces in exact symplectic manifolds. More concretely, we study under which conditions it is possible to establish [Formula: see text]-bounds for the Floer trajectories of a Hamiltonian with non-compact energy levels. Moreover, we introduce a class of Hamiltonians, called tentacular Hamiltonians which satisfy the conditions: how to define Rabinowitz Floer homology for these examples will be the subject of a follow-up paper.

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24

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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25

Childs, Andrew M., Debbie Leung, Laura Mancinska, and Maris Ozols. "Characterization of universal two-qubit Hamiltonians." Quantum Information and Computation 11, no. 1&2 (January 2011): 19–39. http://dx.doi.org/10.26421/qic11.1-2-3.

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Suppose we can apply a given 2-qubit Hamiltonian H to any (ordered) pair of qubits. We say H is n-universal if it can be used to approximate any unitary operation on n qubits. While it is well known that almost any 2-qubit Hamiltonian is 2-universal, an explicit characterization of the set of non-universal 2-qubit Hamiltonians has been elusive. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. In particular, there are three ways that a 2-qubit Hamiltonian $H$ can fail to be universal: (1) H shares an eigenvector with the gate that swaps two qubits, (2) H acts on the two qubits independently (in any of a certain family of bases), or (3) H has zero trace (with the third condition relevant only when the global phase of the unitary matters). A 2-non-universal 2-qubit Hamiltonian can still be n-universal for some n \geq 3. We give some partial results on 3-universality.

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26

Härkönen, Ville J., and Ivan A. Gonoskov. "On the diagonalization of quadratic Hamiltonians." Journal of Physics A: Mathematical and Theoretical 55, no. 1 (December 13, 2021): 015306. http://dx.doi.org/10.1088/1751-8121/ac3da5.

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Abstract A new procedure to diagonalize quadratic Hamiltonians is introduced. We show that one can establish the diagonalization of a quadratic Hamiltonian by changing the frame of reference by a unitary transformation. We give a general method to diagonalize an arbitrary quadratic Hamiltonian and derive a few of the simplest special cases in detail.

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27

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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28

Kieu, Tien D. "A class of time-energy uncertainty relations for time-dependent Hamiltonians." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2226 (June 2019): 20190148. http://dx.doi.org/10.1098/rspa.2019.0148.

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A new class of time-energy uncertainty relations is directly derived from the Schrödinger equations for time-dependent Hamiltonians. Only the initial states and the Hamiltonians, but neither the instantaneous eigenstates nor the full time-dependent wave funct- ions, which would demand a full solution for a time-dependent Hamiltonian, are required for our time-energy relations. Explicit results are then presented for particular subcases of interest for time-independent Hamiltonians and also for time-varying Hamiltonians employed in adiabatic quantum computation. Some estimates of the lower bounds on computational time are given for general adiabatic quantum algorithms, with Grover's search as an illustration. We particularly emphasize the role of required energy resources, besides the space and time complexity, for the physical process of (quantum) computation, in general.

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Urbanowski, Krzysztof. "Effective Hamiltonians for Complexes of Unstable Particles." Open Systems & Information Dynamics 20, no. 03 (September 2013): 1340008. http://dx.doi.org/10.1142/s1230161213400088.

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Effective Hamiltonians governing the time evolution in a subspace of unstable states can be found using more or less accurate approximations. A convenient tool for deriving them is the evolution equation for a subspace of state space sometime called the Królikowski–Rzewuski (KR) equation. KR equation results from the Schrödinger equation for the total system under considerations. We will discuss properties of approximate effective Hamiltonians derived using KR equation for n-particle, two-particle and for one-particle subspaces. In a general case these effective Hamiltonians depend on time t. We show that at times much longer than times at which the exponential decay take place the real part of the exact effective Hamiltonian for the one-particle subsystem (that is the instantaneous energy) tends to the minimal energy of the total system when t → ∞ whereas the imaginary part of this effective Hamiltonian tends to zero as t → ∞.

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Solivérez, Carlos E., Eduardo R. Gagliano, and Gustavo A. Arteca. "Derivation of model Hamiltonians: Exchange Hamiltonian forH2." Physical Review A 32, no. 1 (July 1, 1985): 81–92. http://dx.doi.org/10.1103/physreva.32.81.

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Zimborás, Zoltán, Terry Farrelly, Szilárd Farkas, and Lluis Masanes. "Does causal dynamics imply local interactions?" Quantum 6 (June 29, 2022): 748. http://dx.doi.org/10.22331/q-2022-06-29-748.

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We consider quantum systems with causal dynamics in discrete spacetimes, also known as quantum cellular automata (QCA). Due to time-discreteness this type of dynamics is not characterized by a Hamiltonian but by a one-time-step unitary. This can be written as the exponential of a Hamiltonian but in a highly non-unique way. We ask if any of the Hamiltonians generating a QCA unitary is local in some sense, and we obtain two very different answers. On one hand, we present an example of QCA for which all generating Hamiltonians are fully non-local, in the sense that interactions do not decay with the distance. We expect this result to have relevant consequences for the classification of topological phases in Floquet systems, given that this relies on the effective Hamiltonian. On the other hand, we show that all one-dimensional quasi-free fermionic QCAs have quasi-local generating Hamiltonians, with interactions decaying exponentially in the massive case and algebraically in the critical case. We also prove that some integrable systems do not have local, quasi-local nor low-weight constants of motion; a result that challenges the standard definition of integrability.

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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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33

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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34

Balajany, Hamideh, and Mohammad Mehrafarin. "Geometric phase of cosmological scalar and tensor perturbations in f(R) gravity." Modern Physics Letters A 33, no. 14 (May 10, 2018): 1850077. http://dx.doi.org/10.1142/s0217732318500773.

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By using the conformal equivalence of f(R) gravity in vacuum and the usual Einstein theory with scalar-field matter, we derive the Hamiltonian of the linear cosmological scalar and tensor perturbations in f(R) gravity in the form of time-dependent harmonic oscillator Hamiltonians. We find the invariant operators of the resulting Hamiltonians and use their eigenstates to calculate the adiabatic Berry phase for sub-horizon modes as a Lewis–Riesenfeld phase.

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GERMINET, FRANÇOIS, ABEL KLEIN, and JEFFREY H. SCHENKER. "QUANTIZATION OF THE HALL CONDUCTANCE AND DELOCALIZATION IN ERGODIC LANDAU HAMILTONIANS." Reviews in Mathematical Physics 21, no. 08 (September 2009): 1045–80. http://dx.doi.org/10.1142/s0129055x09003815.

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We prove quantization of the Hall conductance for continuous ergodic Landau Hamiltonians under a condition on the decay of the Fermi projections. This condition and continuity of the integrated density of states are shown to imply continuity of the Hall conductance. In addition, we prove the existence of delocalization near each Landau level for these two-dimensional Hamiltonians. More precisely, we prove that for some ergodic Landau Hamiltonians, there exists an energy E near each Landau level where a "localization length" diverges. For the Anderson–Landau Hamiltonian, we also obtain a transition between dynamical localization and dynamical delocalization in the Landau bands, with a minimal rate of transport, even in cases when the spectral gaps are closed.

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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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Bogolubov, N. N., E. N. Bogolubova, and S. P. Kruchinin. "Modern Approach to the Calculation of the Correlation Function in Superconductivity Models." Modern Physics Letters B 17, no. 10n12 (May 20, 2003): 709–24. http://dx.doi.org/10.1142/s0217984903005743.

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38

Pepe, Francesco V. "Multipartite entanglement and few-body Hamiltonians." International Journal of Quantum Information 12, no. 02 (March 2014): 1461003. http://dx.doi.org/10.1142/s0219749914610036.

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We investigate the possibility to obtain higly multipartite-entangled states as non-degenerate eigenstates of Hamiltonians that involve only short-range and few-body interactions. We study small-size systems (with a number of qubits ranging from three to five) and search for Hamiltonians with a maximally multipartite entangled state (MMES) as a non-degenerate eigenstate. We then find conditions, including bounds on the number of coupled qubits, to build a Hamiltonian with a Greenberger–Horne–Zeilinger (GHZ) state as a non-degenerate eigenstate. We finally comment on possible applications.

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SAKBAEV, VSEVOLOD Zh. "STOCHASTIC PROPERTIES OF DEGENERATED QUANTUM SYSTEMS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 13, no. 01 (March 2010): 65–85. http://dx.doi.org/10.1142/s0219025710003948.

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We study Schrödinger equation with degenerated symmetric but not self-adjoint Hamiltonian. The above properties of the quantum Hamiltonian arise in the description of the asymptotic behavior of the regularizing self-adjoint Hamiltonians sequence. A quantum dynamical semigroup corresponding to degenerated Hamiltonian is defined by means of the passage to the limit for the sequence of the regularizing dynamical semigroups. These semigroups are generated by the regularizing self-adjoint Hamiltonians. The necessary and sufficient conditions are obtained for the convergence of the regularizing semigroups sequence. The description of the divergent sequence of semigroups requires the extension of the stochastic process concept. We extend the stochastic process concept onto the family of measurable functions defined on the space endowed with finite additive measure. The above extension makes it possible to describe the structure of the partial limits set of the regularizing semigroups sequence.

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Berry, Dominic W., Andrew M. Childs, Yuan Su, Xin Wang, and Nathan Wiebe. "Time-dependent Hamiltonian simulation with L1-norm scaling." Quantum 4 (April 20, 2020): 254. http://dx.doi.org/10.22331/q-2020-04-20-254.

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The difficulty of simulating quantum dynamics depends on the norm of the Hamiltonian. When the Hamiltonian varies with time, the simulation complexity should only depend on this quantity instantaneously. We develop quantum simulation algorithms that exploit this intuition. For sparse Hamiltonian simulation, the gate complexity scales with the L1 norm ∫0tdτ‖H(τ)‖max, whereas the best previous results scale with tmaxτ∈[0,t]‖H(τ)‖max. We also show analogous results for Hamiltonians that are linear combinations of unitaries. Our approaches thus provide an improvement over previous simulation algorithms that can be substantial when the Hamiltonian varies significantly. We introduce two new techniques: a classical sampler of time-dependent Hamiltonians and a rescaling principle for the Schrödinger equation. The rescaled Dyson-series algorithm is nearly optimal with respect to all parameters of interest, whereas the sampling-based approach is easier to realize for near-term simulation. These algorithms could potentially be applied to semi-classical simulations of scattering processes in quantum chemistry.

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41

Scheie, A. "PyCrystalField: software for calculation, analysis and fitting of crystal electric field Hamiltonians." Journal of Applied Crystallography 54, no. 1 (February 1, 2021): 356–62. http://dx.doi.org/10.1107/s160057672001554x.

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PyCrystalField is a Python software package for calculating single-ion crystal electric field (CEF) Hamiltonians. This software can calculate a CEF Hamiltonian ab initio from a point charge model for any transition or rare earth ion in either the J basis or the LS basis, perform symmetry analysis to identify nonzero CEF parameters, calculate the energy spectrum and observables such as neutron spectrum and magnetization, and fit CEF Hamiltonians to any experimental data. The theory, implementation and examples of its use are discussed.

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Arenz, Christian, Denys I. Bondar, Daniel Burgarth, Cecilia Cormick, and Herschel Rabitz. "Amplification of quadratic Hamiltonians." Quantum 4 (May 25, 2020): 271. http://dx.doi.org/10.22331/q-2020-05-25-271.

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Speeding up the dynamics of a quantum system is of paramount importance for quantum technologies. However, in finite dimensions and without full knowledge of the details of the system, it is easily shown to be impossible. In contrast we show that continuous variable systems described by a certain class of quadratic Hamiltonians can be sped up without such detailed knowledge. We call the resultant procedure Hamiltonian amplification (HA). The HA method relies on the application of local squeezing operations allowing for amplifying even unknown or noisy couplings and frequencies by acting on individual modes. Furthermore, we show how to combine HA with dynamical decoupling to achieve amplified Hamiltonians that are free from environmental noise. Finally, we illustrate a significant reduction in gate times of cavity resonator qubits as one potential use of HA.

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CASASAYAS, J., J. MARTINEZ ALFARO, and A. NUNES. "KNOTS AND LINKS IN INTEGRABLE HAMILTONIAN SYSTEMS." Journal of Knot Theory and Its Ramifications 07, no. 02 (March 1998): 123–53. http://dx.doi.org/10.1142/s0218216598000097.

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The main purpose of this paper is to prove that Bott integrable Hamiltonian flows and non-singular Morse-Smale flows are closely related. As a consequence, we obtain a classification of the knots and links formed by periodic orbits of Bott integrable Hamiltonians on the 3-sphere and on the solid torus. We also show that most of Fomenko's theory on the topology of the energy levels of Bott integrable Hamiltonians can be derived from Morgan's results on 3-manifolds that admit non-singular Morse-Smale flows.

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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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Ouyang, Yingkai, David R. White, and Earl T. Campbell. "Compilation by stochastic Hamiltonian sparsification." Quantum 4 (February 27, 2020): 235. http://dx.doi.org/10.22331/q-2020-02-27-235.

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Simulation of quantum chemistry is expected to be a principal application of quantum computing. In quantum simulation, a complicated Hamiltonian describing the dynamics of a quantum system is decomposed into its constituent terms, where the effect of each term during time-evolution is individually computed. For many physical systems, the Hamiltonian has a large number of terms, constraining the scalability of established simulation methods. To address this limitation we introduce a new scheme that approximates the actual Hamiltonian with a sparser Hamiltonian containing fewer terms. By stochastically sparsifying weaker Hamiltonian terms, we benefit from a quadratic suppression of errors relative to deterministic approaches. Relying on optimality conditions from convex optimisation theory, we derive an appropriate probability distribution for the weaker Hamiltonian terms, and compare its error bounds with other probability ansatzes for some electronic structure Hamiltonians. Tuning the sparsity of our approximate Hamiltonians allows our scheme to interpolate between two recent random compilers: qDRIFT and randomized first order Trotter. Our scheme is thus an algorithm that combines the strengths of randomised Trotterisation with the efficiency of qDRIFT, and for intermediate gate budgets, outperforms both of these prior methods.

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Buhovsky, Lev, and Sobhan Seyfaddini. "Uniqueness of generating Hamiltonians for topological Hamiltonian flows." Journal of Symplectic Geometry 11, no. 1 (2013): 37–52. http://dx.doi.org/10.4310/jsg.2013.v11.n1.a3.

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Evans, L. C., and D. Gomes. "Effective Hamiltonians and Averaging for Hamiltonian Dynamics I." Archive for Rational Mechanics and Analysis 157, no. 1 (March 2001): 1–33. http://dx.doi.org/10.1007/pl00004236.

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Evans, L. C., and D. Gomes. "Effective Hamiltonians and Averaging¶for Hamiltonian Dynamics II." Archive for Rational Mechanics and Analysis 161, no. 4 (March 1, 2002): 271–305. http://dx.doi.org/10.1007/s002050100181.

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49

ŠAMAJ, L. "EVOLUTION OF QUANTUM SYSTEMS WITH A SCALING TYPE TIME-DEPENDENT HAMILTONIANS." International Journal of Modern Physics B 16, no. 26 (October 20, 2002): 3909–14. http://dx.doi.org/10.1142/s0217979202013158.

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We introduce a new class of quantum models with time-dependent Hamiltonians of a special scaling form. By using a couple of time-dependent unitary transformations, the time evolution of these models is expressed in terms of related systems with time-independent Hamiltonians. The mapping of dynamics can be performed in any dimension, for an arbitrary number of interacting particles and for any type of the scaling interaction potential. The exact solvability of a "dual" time-independent Hamiltonian automatically means the exact solvability of the original problem with model time-dependence.

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Valverde, Clodoaldo, and Basílio Baseia. "On the paradoxical evolution of the number of photons in a new model of interpolating Hamiltonians." Modern Physics Letters B 32, no. 03 (January 29, 2018): 1850026. http://dx.doi.org/10.1142/s0217984918500264.

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We introduce a new Hamiltonian model which interpolates between the Jaynes–Cummings model (JCM) and other types of such Hamiltonians. It works with two interpolating parameters, rather than one as traditional. Taking advantage of this greater degree of freedom, we can perform continuous interpolation between the various types of these Hamiltonians. As applications, we discuss a paradox raised in literature and compare the time evolution of the photon statistics obtained in the various interpolating models. The role played by the average excitation in these comparisons is also highlighted.

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