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Автор: Grafiati

Опубліковано: 14 березня 2023

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1

SILVER, R. N., and H. RÖDER. "DENSITIES OF STATES OF MEGA-DIMENSIONAL HAMILTONIAN MATRICES." International Journal of Modern Physics C 05, no. 04 (August 1994): 735–53. http://dx.doi.org/10.1142/s0129183194000842.

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Анотація:

We propose a statistical method to estimate densities of states (DOS) and thermodynamic functions of very large Hamiltonian matrices. Orthogonal polynomials are defined on the interval between lower and upper energy bounds. The DOS is represented by a kernel polynomial constructed out of polynomial moments of the DOS and modified to damp the Gibbs phenomenon. The moments are stochastically evaluated using matrixvector multiplications on Gaussian random vectors and the polynomial recurrence relations. The resulting kernel estimate is a controlled approximation to the true DOS, because it also provides estimates of statistical and systematic errors. For a given fractional energy resolution and statistical accuracy, the required cpu time and memory scale linearly in the number of states for sparse Hamiltonians. The method is demonstrated for the two-dimensional Heisenberg anti-ferromagnet with the number of states as large as 226. Results are compared to exact diagonalization where available.

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2

RÜHL, WERNER, and ALEXANDER TURBINER. "EXACT SOLVABILITY OF THE CALOGERO AND SUTHERLAND MODELS." Modern Physics Letters A 10, no. 29 (September 21, 1995): 2213–21. http://dx.doi.org/10.1142/s0217732395002374.

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Translationally invariant symmetric polynomials as coordinates for N-body problems with identical particles are proposed. It is shown that in those coordinates the Calogero and Sutherland N-body Hamiltonians, after appropriate gauge transformations, can be presented as a quadratic polynomial in the generators of the algebra sl N in finitedimensional degenerate representation. The exact solvability of these models follows from the existence of the infinite flag of such representation spaces, preserved by the above Hamiltonians. A connection with Jack polynomials is discussed.

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3

Sokolov, A. V. "Polynomial supersymmetry for matrix Hamiltonians." Physics Letters A 377, no. 9 (March 2013): 655–62. http://dx.doi.org/10.1016/j.physleta.2013.01.012.

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4

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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Анотація:

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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5

Lu, Kang. "Completeness of Bethe Ansatz for Gaudin Models with gl(1|1) Symmetry and Diagonal Twists." Symmetry 15, no. 1 (December 21, 2022): 9. http://dx.doi.org/10.3390/sym15010009.

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Анотація:

We studied the Gaudin models with gl(1|1) symmetry that are twisted by a diagonal matrix and defined on tensor products of polynomial evaluation gl(1|1)[t]-modules. Namely, we gave an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation gl(1|1)[t]-modules and showed that a bijection exists between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also gave dimensions of the generalized eigenspaces.

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6

UENO, YUICHI. "POLYNOMIAL HAMILTONIANS FOR QUANTUM PAINLEVÉ EQUATIONS." International Journal of Mathematics 20, no. 11 (November 2009): 1335–45. http://dx.doi.org/10.1142/s0129167x09005789.

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Recently, a quantum version of Painlevé equations from the point of view of their symmetries was proposed by Nagoya. These quantum Painlevé equations can be written as Hamiltonian systems with a (noncommutative) polynomial Hamiltonian H J . We give a characterization of the quantum Painlevé equations by certain holomorphic properties. Namely, we introduce canonical transformations such that the Painlevé Hamiltonian system is again transformed into a polynomial Hamiltonian system, and we show that the Hamiltonian can be uniquely characterized through this holomorphic property.

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7

Aharonov, Dorit, Michael Ben-Or, Fernando G. S. L. Brandão, and Or Sattath. "The Pursuit of Uniqueness: Extending Valiant-Vazirani Theorem to the Probabilistic and Quantum Settings." Quantum 6 (March 17, 2022): 668. http://dx.doi.org/10.22331/q-2022-03-17-668.

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Анотація:

Valiant-Vazirani showed in 1985 \cite{VV85} that solving NP with the promise that "yes" instances have only one witness is powerful enough to solve the entire NP class (under randomized reductions).We are interested in extending this result to the quantum setting. We prove extensions to the classes Merlin-Arthur MA and Quantum-Classical-Merlin-Arthur QCMA \cite{AN02}. Our results have implications for the complexity of approximating the ground state energy of a quantum local Hamiltonian with a unique ground state and an inverse polynomial spectral gap. We show that the estimation (to within polynomial accuracy) of the ground state energy of poly-gapped 1-D local Hamiltonians is QCMA-hard, under randomized reductions. This is in stark contrast to the case of constant gapped 1-D Hamiltonians, which is in NP \cite{Has07}. Moreover, it shows that unless QCMA can be reduced to NP by randomized reductions, there is no classical description of the ground state of every poly-gapped local Hamiltonian that allows efficient calculation of expectation values.Finally, we discuss a few of the obstacles to the establishment of an analogous result to the class Quantum-Merlin-Arthur (QMA). In particular, we show that random projections fail to provide a polynomial gap between two witnesses.

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8

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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Анотація:

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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9

Vigo-Aguiar, M. I., M. E. Sansaturio, and J. M. Ferrándiz. "Integrability of Hamiltonians with polynomial potentials." Journal of Computational and Applied Mathematics 158, no. 1 (September 2003): 213–24. http://dx.doi.org/10.1016/s0377-0427(03)00467-9.

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10

Mingalev, Oleg V., Yurii N. Orlov, and Victor V. Vedenyapin. "Conservation laws for polynomial quantum Hamiltonians." Physics Letters A 223, no. 4 (December 1996): 246–50. http://dx.doi.org/10.1016/s0375-9601(96)00680-9.

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11

Hussin, V., I. Marquette, and K. Zelaya. "Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials." Journal of Physics A: Mathematical and Theoretical 55, no. 4 (January 6, 2022): 045205. http://dx.doi.org/10.1088/1751-8121/ac43cc.

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Abstract We extend and generalize the construction of Sturm–Liouville problems for a family of Hamiltonians constrained to fulfill a third-order shape-invariance condition and focusing on the ‘−2x/3’ hierarchy of solutions to the fourth Painlevé transcendent. Such a construction has been previously addressed in the literature for some particular cases but we realize it here in the most general case. The corresponding potential in the Hamiltonian operator is a rationally extended oscillator defined in terms of the conventional Okamoto polynomials, from which we identify three different zero-modes constructed in terms of the generalized Okamoto polynomials. The third-order ladder operators of the system reveal that the complete set of eigenfunctions is decomposed as a union of three disjoint sequences of solutions, generated from a set of three-term recurrence relations. We also identify a link between the eigenfunctions of the Hamiltonian operator and a special family of exceptional Hermite polynomial.

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12

Palacián, Jesús, and Patricia Yanguas. "Equivariant N-Dof Hamiltonians Via Generalized Normal Forms." Communications in Contemporary Mathematics 05, no. 03 (June 2003): 449–80. http://dx.doi.org/10.1142/s0219199703001026.

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In the present paper we study polynomial Hamiltonian systems depending on one or various real parameters. We determine the values that these parameters should take in order to be able to construct formal (asymptotic) integrals of the system. In this respect, a method to calculate the formal integrals of a polynomial Hamiltonian vector field is presented. The original Hamilton function represents a family of dynamical systems composed by a principal part (quadratic terms) plus the perturbation (terms of degree three or bigger). We extend an integral of the principal part to the perturbed system by means of Lie transformations for autonomous Hamiltonian systems. Thus, the procedure is carried out order by order starting with polynomials of degree three. We obtain the conditions that the external parameters have to satisfy so that the integral of the quadratic terms persists for the whole system up to a certain order of approximation. Once the formal integral is computed the departure system has been transformed into a generalized normal form, i.e. a system which is equivalent to the initial one but easier to be analysed by making use of reduction theory. The truncated normal form defines a system with less degrees of freedom than the original Hamiltonian and is written exactly in terms of the polynomial first integrals associated to the quadratic part of the new integral and it contains the qualitative description of the initial system. The theory is illustrated with two examples borrowed from Physics.

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13

GÉRARD, C., and A. PANATI. "SPECTRAL AND SCATTERING THEORY FOR SOME ABSTRACT QFT HAMILTONIANS." Reviews in Mathematical Physics 21, no. 03 (April 2009): 373–437. http://dx.doi.org/10.1142/s0129055x09003645.

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We introduce an abstract class of bosonic QFT Hamiltonians and study their spectral and scattering theories. These Hamiltonians are of the form H = dΓ(ω) + V acting on the bosonic Fock space Γ(𝔥), where ω is a massive one-particle Hamiltonian acting on 𝔥 and V is a Wick polynomial Wick(w) for a kernel w satisfying some decay properties at infinity. We describe the essential spectrum of H, prove a Mourre estimate outside a set of thresholds and prove the existence of asymptotic fields. Our main result is the asymptotic completeness of the scattering theory, which means that the CCR representations given by the asymptotic fields are of Fock type, with the asymptotic vacua equal to the bound states of H. As a consequence, H is unitarily equivalent to a collection of second quantized Hamiltonians.

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14

Hall, Laurence S. "Invariants Polynomial in Momenta for Integrable Hamiltonians." Physical Review Letters 54, no. 7 (February 18, 1985): 614–15. http://dx.doi.org/10.1103/physrevlett.54.614.

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15

Matushko, M. G., and V. V. Sokolov. "Polynomial forms for quantum elliptic Calogero–Moser Hamiltonians." Theoretical and Mathematical Physics 191, no. 1 (April 2017): 480–90. http://dx.doi.org/10.1134/s004057791704002x.

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16

BORESKOV, KONSTANTIN G., JUAN CARLOS LOPEZ VIEYRA, and ALEXANDER V. TURBINER. "SOLVABILITY OF THE F4 INTEGRABLE SYSTEM." International Journal of Modern Physics A 16, no. 29 (November 20, 2001): 4769–801. http://dx.doi.org/10.1142/s0217751x0100550x.

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Анотація:

It is shown that the F4 rational and trigonometric integrable systems are exactly-solvable for arbitrary values of the coupling constants. Their spectra are found explicitly while eigenfunctions are by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)-triangular. These variables are invariant with respect to the Weyl group of F4 root system and can be obtained by averaging over an orbit of the Weyl group. An alternative way of finding these variables exploiting a property of duality of the F4 model is presented. It is demonstrated that in these variables the Hamiltonian of each model can be expressed as a quadratic polynomial in the generators of some infinite-dimensional Lie algebra of differential operators in a finite-dimensional representation. Both Hamiltonians preserve the same flag of spaces of polynomials and each subspace of the flag coincides with the finite-dimensional representation space of this algebra. Quasi-exactly-solvable generalization of the rational F4 model depending on two continuous and one discrete parameters is found.

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17

Bravyi, Sergey. "Monte Carlo simulation of stoquastic Hamiltonians." Quantum Information and Computation 15, no. 13&14 (October 2015): 1122–40. http://dx.doi.org/10.26421/qic15.13-14-3.

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Stoquastic Hamiltonians are characterized by the property that their off-diagonal matrix elements in the standard product basis are real and non-positive. Many interesting quantum models fall into this class including the Transverse field Ising Model (TIM), the Heisenberg model on bipartite graphs, and the bosonic Hubbard model. Here we consider the problem of estimating the ground state energy of a local stoquastic Hamiltonian $H$ with a promise that the ground state of $H$ has a non-negligible correlation with some `guiding' state that admits a concise classical description. A formalized version of this problem called Guided Stoquastic Hamiltonian is shown to be complete for the complexity class $\MA$ (a probabilistic analogue of $\NP$). To prove this result we employ the Projection Monte Carlo algorithm with a variable number of walkers. Secondly, we show that the ground state and thermal equilibrium properties of the ferromagnetic TIM can be simulated in polynomial time on a classical probabilistic computer. This result is based on the approximation algorithm for the classical ferromagnetic Ising model due to Jerrum and Sinclair (1993).

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18

Cao, Yudong, and Daniel Nagaj. "Perturbative gadgets without strong interactions." Quantum Information and Computation 15, no. 13&14 (October 2015): 1197–222. http://dx.doi.org/10.26421/qic15.13-14-7.

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Perturbative gadgets are used to construct a quantum Hamiltonian whose low-energy subspace approximates a given quantum $k$-local Hamiltonian up to an absolute error $\epsilon$. Typically, gadget constructions involve terms with large interaction strengths of order $\text{poly}(\epsilon^{-1})$. Here we present a 2-body gadget construction and prove that it approximates a Hamiltonian of interaction strength $\gamma = O(1)$ up to absolute error $\epsilon\ll\gamma$ using interactions of strength $O(\epsilon)$ instead of the usual inverse polynomial in $\epsilon$. A key component in our proof is a new condition for the convergence of the perturbation series, allowing our gadget construction to be applied in parallel on multiple many-body terms. We also discuss how to apply this gadget construction for approximating 3- and $k$-local Hamiltonians. The price we pay for using much weaker interactions is a large overhead in the number of ancillary qubits, and the number of interaction terms per particle, both of which scale as $O(\text{poly}(\epsilon^{-1}))$. Our strong-from-weak gadgets have their primary application in complexity theory (QMA hardness of restricted Hamiltonians, a generalized area law counterexample, gap amplification), but could also motivate practical implementations with several weak interactions simulating a much stronger quantum many-body interaction.

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19

Palacián, Jesús, and Patricia Yanguas. "Reduction of Polynomial Planar Hamiltonians with Quadratic Unperturbed Part." SIAM Review 42, no. 4 (January 2000): 671–91. http://dx.doi.org/10.1137/s0036144599362327.

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20

Shi, Jicong, and Yiton T. Yan. "Explicitly integrable polynomial Hamiltonians and evaluation of Lie transformations." Physical Review E 48, no. 5 (November 1, 1993): 3943–51. http://dx.doi.org/10.1103/physreve.48.3943.

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21

Kelbert, E., A. Hyder, F. Demir, Z. T. Hlousek, and Z. Papp. "Green's operator for Hamiltonians with Coulomb plus polynomial potentials." Journal of Physics A: Mathematical and Theoretical 40, no. 27 (June 19, 2007): 7721–28. http://dx.doi.org/10.1088/1751-8113/40/27/020.

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22

Ukolov, Yu A., N. A. Chekanov, A. A. Gusev, V. A. Rostovtsev, S. I. Vinitsky, and Y. Uwano. "A REDUCE program for the normalization of polynomial Hamiltonians." Computer Physics Communications 166, no. 1 (February 2005): 66–80. http://dx.doi.org/10.1016/j.cpc.2004.10.010.

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23

Letourneau, P., and L. Vinet. "Superintegrable Systems: Polynomial Algebras and Quasi-Exactly Solvable Hamiltonians." Annals of Physics 243, no. 1 (October 1995): 144–68. http://dx.doi.org/10.1006/aphy.1995.1094.

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24

Maniraguha, Jean de Dieu, Krzysztof Marciniak, and Célestin Kurujyibwami. "Transforming Stäckel Hamiltonians of Benenti type to polynomial form." Advances in Theoretical and Mathematical Physics 26, no. 3 (2022): 711–34. http://dx.doi.org/10.4310/atmp.2022.v26.n3.a5.

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25

Gu, Shouzhen, Rolando D. Somma, and Burak Şahinoğlu. "Fast-forwarding quantum evolution." Quantum 5 (November 15, 2021): 577. http://dx.doi.org/10.22331/q-2021-11-15-577.

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Анотація:

We investigate the problem of fast-forwarding quantum evolution, whereby the dynamics of certain quantum systems can be simulated with gate complexity that is sublinear in the evolution time. We provide a definition of fast-forwarding that considers the model of quantum computation, the Hamiltonians that induce the evolution, and the properties of the initial states. Our definition accounts for any asymptotic complexity improvement of the general case and we use it to demonstrate fast-forwarding in several quantum systems. In particular, we show that some local spin systems whose Hamiltonians can be taken into block diagonal form using an efficient quantum circuit, such as those that are permutation-invariant, can be exponentially fast-forwarded. We also show that certain classes of positive semidefinite local spin systems, also known as frustration-free, can be polynomially fast-forwarded, provided the initial state is supported on a subspace of sufficiently low energies. Last, we show that all quadratic fermionic systems and number-conserving quadratic bosonic systems can be exponentially fast-forwarded in a model where quantum gates are exponentials of specific fermionic or bosonic operators, respectively. Our results extend the classes of physical Hamiltonians that were previously known to be fast-forwarded, while not necessarily requiring methods that diagonalize the Hamiltonians efficiently. We further develop a connection between fast-forwarding and precise energy measurements that also accounts for polynomial improvements.

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26

Cervia, Michael J., Amol V. Patwardhan, and A. B. Balantekin. "Symmetries of Hamiltonians describing systems with arbitrary spins." International Journal of Modern Physics E 28, no. 05 (May 2019): 1950032. http://dx.doi.org/10.1142/s0218301319500320.

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Анотація:

We consider systems where dynamical variables are the generators of the SU(2) group. A subset of these Hamiltonians is exactly solvable using the Bethe ansatz techniques. We show that Bethe ansatz equations are equivalent to polynomial relationships between the operator invariants, or equivalently, between eigenvalues of those invariants.

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27

Mostafazadeh, Ali. "Parasupersymmetric Quantum Mechanics and Indices of Fredholm Operators." International Journal of Modern Physics A 12, no. 15 (June 20, 1997): 2725–39. http://dx.doi.org/10.1142/s0217751x9700150x.

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Анотація:

The general features of the degeneracy structure of (p = 2) parasupersymmetric quantum mechanics are employed to yield a classification scheme for the form of the parasupersymmetric Hamiltonians. The method is applied to parasupersymmetric systems whose Hamiltonian is the square root of a fourth order polynomial in the generators of the parasupersymmetry. These systems are interesting to study for they lead to the introduction of a set of topological invariants very similar to the Witten indices of ordinary supersymmetric quantum mechanics. The topological invariants associated with parasupersymmetry are shown to be related to a pair of Fredholm operators satisfying two compatibility conditions. An explicit algebraic expression for the topological invariants of a class of parasupersymmetric systems is provided.

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28

Palacián, Jesús, and Patricia Yanguas. "Reduction of polynomial Hamiltonians by the construction of formal integrals." Nonlinearity 13, no. 4 (May 8, 2000): 1021–54. http://dx.doi.org/10.1088/0951-7715/13/4/303.

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29

Rivera, A. L., N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf. "Evolution under polynomial Hamiltonians in quantum and optical phase spaces." Physical Review A 55, no. 2 (February 1, 1997): 876–89. http://dx.doi.org/10.1103/physreva.55.876.

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30

Ramani, A., B. Dorizzi, B. Grammaticos, and J. Hietarinta. "Linearization on a submanifold of integrable Hamiltonians with polynomial potentials." Physica D: Nonlinear Phenomena 18, no. 1-3 (January 1986): 171–79. http://dx.doi.org/10.1016/0167-2789(86)90174-0.

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31

Basios, V., N. A. Chekanov, B. L. Markovski, V. A. Rostovtsev, and S. I. Vinitsky. "GITA: A REDUCE program for the normalization of polynomial Hamiltonians." Computer Physics Communications 90, no. 2-3 (October 1995): 355–68. http://dx.doi.org/10.1016/0010-4655(95)00080-y.

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32

Mastroianni, Rita, and Christos Efthymiopoulos. "Kolmogorov algorithm for isochronous Hamiltonian systems." Mathematics in Engineering 5, no. 2 (2022): 1–35. http://dx.doi.org/10.3934/mine.2023035.

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Анотація:

<abstract><p>We present a Kolmogorov-like algorithm for the computation of a normal form in the neighborhood of an invariant torus in 'isochronous' Hamiltonian systems, i.e., systems with Hamiltonians of the form $ {\mathcal{H}} = {\mathcal{H}}_0+\varepsilon {\mathcal{H}}_1 $ where $ {\mathcal{H}}_0 $ is the Hamiltonian of $ N $ linear oscillators, and $ {\mathcal{H}}_1 $ is expandable as a polynomial series in the oscillators' canonical variables. This method can be regarded as a normal form analogue of a corresponding Lindstedt method for coupled oscillators. We comment on the possible use of the Lindstedt method itself under two distinct schemes, i.e., one producing series analogous to those of the Birkhoff normal form scheme, and another, analogous to the Kolomogorov normal form scheme in which we fix in advance the frequency of the torus.</p></abstract>

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33

Gharibian, Sevag, and Justin Yirka. "The complexity of simulating local measurements on quantum systems." Quantum 3 (September 30, 2019): 189. http://dx.doi.org/10.22331/q-2019-09-30-189.

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Анотація:

An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, [Ambainis, CCC 2014] defined the complexity class PQMA[log], and motivated its study by showing that the physical task of estimating the expectation value of a local observable against the ground state of a local Hamiltonian is PQMA[log]-complete. In this paper, we continue the study of PQMA[log], obtaining the following lower and upper bounds.Lower bounds (hardness results): - The PQMA[log]-completeness result of [Ambainis, CCC 2014] requires O(log⁡n)-local observables and Hamiltonians. We show that simulating even a single qubit measurement on ground states of 5-local Hamiltonians is PQMA[log]-complete, resolving an open question of Ambainis.- We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is similarly PQMA[log]-complete. - We identify a flaw in [Ambainis, CCC 2014] regarding a PUQMA[log]-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a ``query validation'' technique, we build on [Ambainis, CCC 2014] to obtain PUQMA[log]-hardness for estimating spectral gaps under polynomial-time Turing reductions. Upper bounds (containment in complexity classes): - PQMA[log] is thought of as ``slightly harder'' than QMA. We justify this formally by exploiting the hierarchical voting technique of [Beigel, Hemachandra, Wechsung, SCT 1989] to show PQMA[log]⊆PP. This improves the containment QMA⊆PP [Kitaev, Watrous, STOC 2000]. This work contributes a rigorous treatment of the subtlety involved in studying oracle classes in which the oracle solves a promise problem. This is particularly relevant for quantum complexity theory, where most natural classes such as BQP and QMA are defined as promise classes.

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34

Qi, Xiao-Liang, and Daniel Ranard. "Determining a local Hamiltonian from a single eigenstate." Quantum 3 (July 8, 2019): 159. http://dx.doi.org/10.22331/q-2019-07-08-159.

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We ask whether the knowledge of a single eigenstate of a local Hamiltonian is sufficient to uniquely determine the Hamiltonian. We present evidence that the answer is ``yes" for generic local Hamiltonians, given either the ground state or an excited eigenstate. In fact, knowing only the two-point equal-time correlation functions of local observables with respect to the eigenstate should generically be sufficient to exactly recover the Hamiltonian for finite-size systems, with numerical algorithms that run in a time that is polynomial in the system size. We also investigate the large-system limit, the sensitivity of the reconstruction to error, and the case when correlation functions are only known for observables on a fixed sub-region. Numerical demonstrations support the results for finite one-dimensional spin chains (though caution must be taken when extrapolating to infinite-size systems in higher dimensions). For the purpose of our analysis, we define the ``k-correlation spectrum" of a state, which reveals properties of local correlations in the state and may be of independent interest.

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35

DOLCINI, FABRIZIO, and ARIANNA MONTORSI. "INTEGRABLE EXTENDED HUBBARD HAMILTONIANS FROM SYMMETRIC GROUP EQUATIONS." International Journal of Modern Physics B 14, no. 17 (July 10, 2000): 1719–28. http://dx.doi.org/10.1142/s0217979200001540.

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We consider the most general form of extended Hubbard Hamiltonian conserving the total spin and number of electrons, and find all the 1-dimensional completely integrable models which can be derived from first degree polynomial solution of the Yang–Baxter equation. It is shown that such models are 96. They are identified with the 16-dimensional representations of the class of solutions of symmetric group relations acting as generalized permutators. As particular examples, the EKS and some other known models are obtained. A method for determining the physical features of the above models is outlined.

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36

Bibikov, Pavel Vitalievich. "On Classification of Polynomial Hamiltonians With Nondegenerate Linearly Stable Singular Point." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 1 (2019): 86–88. http://dx.doi.org/10.26907/0021-3446-2019-1-86-88.

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37

Bibikov, P. V. "On Classification of Polynomial Hamiltonians With Nondegenerate Linearly Stable Singular Point." Russian Mathematics 63, no. 1 (January 2019): 76–78. http://dx.doi.org/10.3103/s1066369x19010092.

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38

Leyvraz, F. "An approach for obtaining integrable Hamiltonians from Poisson-commuting polynomial families." Journal of Mathematical Physics 58, no. 7 (July 2017): 072902. http://dx.doi.org/10.1063/1.4996581.

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39

Rowe, D. J. "An algebraic approach to problems with polynomial Hamiltonians on Euclidean spaces." Journal of Physics A: Mathematical and General 38, no. 47 (November 9, 2005): 10181–201. http://dx.doi.org/10.1088/0305-4470/38/47/009.

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40

Ivanyos, G., A. B. Nagy, and L. Ronyai. "Constructions for quantum computing with symmetrized gates." Quantum Information and Computation 8, no. 5 (May 2008): 411–29. http://dx.doi.org/10.26421/qic8.5-4.

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We investigate constructions for simulating quantum computers with a polynomial slowdown on ensembles composed of qubits on which symmetrized versions of one- and two-qubit gates can be performed. The simulation is based on taking Lie commutators of symmetrized Hamiltonians to extract Hamiltonians at desired local positions. During the simulation, only a part of the qubits can be used for storing information, the others are left unchanged by the commutators. We propose constructions for various symmetry groups where a pretty large fraction of the qubits can be used. As a few of the other qubits need to be set to one, our construction requires individual initialization of some of the qubits.

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41

BRIHAYE, YVES. "QUASI-EXACTLY SOLVABLE MATRIX SCHRÖDINGER OPERATORS." Modern Physics Letters A 15, no. 26 (August 30, 2000): 1647–53. http://dx.doi.org/10.1142/s0217732300002073.

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Two families of quasi-exactly solvable 2×2 matrix Schrödinger operators are constructed. The first one is based on a polynomial matrix potential and depends on three parameters. The second is a one-parameter generalization of the scalar Lamé equation. The relationship between these operators and QES Hamiltonians already considered in the literature is pointed out.

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42

LIN, SHAO-SHIUNG, and SHI-SHYR ROAN. "ALGEBRAIC GEOMETRY AND HOFSTADTER TYPE MODEL." International Journal of Modern Physics B 16, no. 14n15 (June 20, 2002): 2097–106. http://dx.doi.org/10.1142/s0217979202011846.

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In this report, we study the algebraic geometry aspect of Hofstadter type models through the algebraic Bethe equation. In the diagonalization problem of certain Hofstadter type Hamiltonians, the Bethe equation is constructed by using the Baxter vectors on a high genus spectral curve. When the spectral variables lie on rational curves, we obtain the complete and explicit solutions of the polynomial Bethe equation; the relation with the Bethe ansatz of polynomial roots is discussed. Certain algebraic geometry properties of Bethe equation on the high genus algebraic curves are discussed in cooperation with the consideration of the physical model.

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43

Wahlberg, Patrik. "Propagation of polynomial phase space singularities for Schrödinger equations with quadratic Hamiltonians." MATHEMATICA SCANDINAVICA 122, no. 1 (February 20, 2018): 107. http://dx.doi.org/10.7146/math.scand.a-97187.

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We study propagation of phase space singularities for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. Phase space singularities are measured by the lack of polynomial decay of given order in open cones in the phase space, which gives a parametrized refinement of the Gabor wave front set. The main result confirms the fundamental role of the singular space associated to the quadratic form for the propagation of phase space singularities. The singularities are contained in the singular space, and propagate in the intersection of the singular space and the initial datum singularities along the flow of the Hamilton vector field associated to the imaginary part of the quadratic form.

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44

Gusev, A. A., N. A. Chekanov, V. A. Rostovtsev, S. I. Vinitsky, and Y. Uwano. "A Comparison of Algorithms for the Normalization and Quantization of Polynomial Hamiltonians." Programming and Computer Software 30, no. 2 (March 2004): 75–82. http://dx.doi.org/10.1023/b:pacs.0000021264.38623.52.

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45

Znojil, Miloslav. "Perturbation method for non-square Hamiltonians and its application to polynomial oscillators." Physics Letters A 341, no. 1-4 (June 2005): 67–80. http://dx.doi.org/10.1016/j.physleta.2005.04.061.

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46

Daubechies, Ingrid, and John R. Klauder. "Quantum‐mechanical path integrals with Wiener measure for all polynomial Hamiltonians. II." Journal of Mathematical Physics 26, no. 9 (September 1985): 2239–56. http://dx.doi.org/10.1063/1.526803.

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47

BAGCHI, BIJAN, A. BANERJEE, EMANUELA CALICETI, FRANCESCO CANNATA, HENDRIK B. GEYER, CHRISTIANE QUESNE, and MILOSLAV ZNOJIL. "${\mathcal{CPT}}$-CONSERVING HAMILTONIANS AND THEIR NONLINEAR SUPERSYMMETRIZATION USING DIFFERENTIAL CHARGE-OPERATORS ${\mathcal C}$." International Journal of Modern Physics A 20, no. 30 (December 10, 2005): 7107–28. http://dx.doi.org/10.1142/s0217751x05022901.

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A brief overview is given of recent developments and fresh ideas at the intersection of [Formula: see text]- and/or [Formula: see text]-symmetric quantum mechanics with supersymmetric quantum mechanics (SUSY QM). Within the framework of the resulting supersymmetric version of [Formula: see text]-symmetric quantum mechanics we study the consequences of the assumption that the "charge" operator [Formula: see text] is represented in a differential-operator form of the second or higher order. Besides the freedom allowed by the Hermiticity constraint for the operator [Formula: see text], encouraging results are obtained in the second-order case. In particular, the integrability of intertwining relations proves to match the closure of our nonlinear (viz., polynomial) SUSY algebra. In a particular illustration, our form of [Formula: see text]-symmetric SUSY QM leads to a new class of non-Hermitian polynomial oscillators with real spectrum which turn out to be [Formula: see text]-asymmetric.

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48

Cruise, Joseph R., and Alexander Seidel. "Sequencing the Entangled DNA of Fractional Quantum Hall Fluids." Symmetry 15, no. 2 (January 21, 2023): 303. http://dx.doi.org/10.3390/sym15020303.

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We introduce and prove the “root theorem”, which establishes a condition for families of operators to annihilate all root states associated with zero modes of a given positive semi-definite k-body Hamiltonian chosen from a large class. This class is motivated by fractional quantum Hall and related problems, and features generally long-ranged, one-dimensional, dipole-conserving terms. Our theorem streamlines analysis of zero-modes in contexts where “generalized” or “entangled” Pauli principles apply. One major application of the theorem is to parent Hamiltonians for mixed Landau-level wave functions, such as unprojected composite fermion or parton-like states that were recently discussed in the literature, where it is difficult to rigorously establish a complete set of zero modes with traditional polynomial techniques. As a simple application, we show that a modified V1 pseudo-potential, obtained via retention of only half the terms, stabilizes the ν=1/2 Tao–Thouless state as the unique densest ground state.

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49

Palacián, J., P. Yanguas, and S. Ferrer. "Simple Periodic Orbits in Elliptical Galaxies Modelled by Hamiltonians in 1-1-1 Resonance." International Astronomical Union Colloquium 172 (1999): 411–12. http://dx.doi.org/10.1017/s0252921100072948.

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AbstractWe consider elliptical galactic models, whose dynamical system consists of a three-dimensional isotropic harmonic oscillator plus a potential given by a homogeneous polynomial of degree four with an additional discrete symmetry. We identify families of simple periodic orbits by studying the reduced phase space.

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50

Vedenyapin, V. V., and Yu N. Orlov. "Conservation laws for polynomial Hamiltonians and for discrete models of the Boltzmann equation." Theoretical and Mathematical Physics 121, no. 2 (November 1999): 1516–23. http://dx.doi.org/10.1007/bf02557222.

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