Exactly Solvable Schrodinger Problems, Perturbation Theory, Quantum Gate Design

2021 ◽  
pp. 7-16
Author(s):  
Harish Parthasarathy
Keyword(s):  
Perturbation Theory ◽  
Quantum Gate ◽  
Exactly Solvable ◽  
Gate Design

scholarly journals SYSTEMATIC SEARCH OF EXACTLY SOLVABLE NON-CENTRAL POTENTIALS

2005 ◽  
Vol 20 (05) ◽  
pp. 355-361 ◽  
Author(s):  
B. GÖNÜL ◽  
M. KOÇAK
Keyword(s):  
Perturbation Theory ◽  
Mathematical Analysis ◽  
Systematic Search ◽  
Exact Solvability ◽  
Exactly Solvable ◽  
Present Formalism

Recently developed supersymmetric perturbation theory has been successfully employed to make a complete mathematical analysis of the reason behind exact solvability of some non-central potentials. This investigation clarifies once more the effectiveness of the present formalism.

Quantum gate design: A perspective

2009 ◽  
Vol 246 (5) ◽  
pp. 972-974
Author(s):  
Michael R. Geller ◽  
Emily J. Pritchett ◽  
Andrei Galiautdinov ◽  
John M. Martinis
Keyword(s):  
Quantum Gate ◽  
Gate Design

scholarly journals A NEW ALGEBRAIC APPROACH TO PERTURBATION THEORY

2005 ◽  
Vol 20 (22) ◽  
pp. 1683-1694 ◽  
Author(s):  
B. GÖNÜL ◽  
N. ÇELİK ◽  
E. OLĞAR
Keyword(s):  
Perturbation Theory ◽  
Excited States ◽  
Large Range ◽  
Algebraic Approach ◽  
Coupling Constant ◽  
Anharmonic Oscillators ◽  
Analytical Treatment ◽  
Exactly Solvable ◽  
Recursion Formulas ◽  
Range Of Values

An algebraic nonperturbative approach is proposed for the analytical treatment of Schrödinger equations with a potential that can be expressed in terms of an exactly solvable piece with an additional potential. Avoiding disadvantages of standard approaches, new handy recursion formulas with the same simple form both for ground and excited states have been obtained. As an illustration the procedure, well adapted to the use of computer algebra, is successfully applied to quartic anharmonic oscillators by means of very simple algebraic manipulations. The trend of the exact values of the energies is rather well reproduced for a large range of values of the coupling constant (g = 0.001–10000).

scholarly journals Treating divergent perturbation theory: Lessons from exactly solvable 2D models at large N

2021 ◽  
Vol 104 (8) ◽  
Author(s):  
Daniel Schubring ◽  
Chao-Hsiang Sheu ◽  
Mikhail Shifman
Keyword(s):  
Perturbation Theory ◽  
Large N ◽  
Exactly Solvable

scholarly journals Perturbation theory for arbitrary coupling strength?

2018 ◽  
Vol 33 (07n08) ◽  
pp. 1850037
Author(s):  
Bimal P. Mahapatra ◽  
Noubihary Pradhan
Keyword(s):  
Perturbation Theory ◽  
Coupling Strength ◽  
Mean Field ◽  
Power Series Expansion ◽  
Feedback Mechanism ◽  
Analytic Structure ◽  
Exactly Solvable ◽  
Arbitrary Strength ◽  
System Properties ◽  
New Formulation

We present a new formulation of perturbation theory for quantum systems, designated here as: “mean field perturbation theory” (MFPT), which is free from power-series-expansion in any physical parameter, including the coupling strength. Its application is thereby extended to deal with interactions of arbitrary strength and to compute system-properties having non-analytic dependence on the coupling, thus overcoming the primary limitations of the “standard formulation of perturbation theory” (SFPT). MFPT is defined by developing perturbation about a chosen input Hamiltonian, which is exactly solvable but which acquires the nonlinearity and the analytic structure (in the coupling strength) of the original interaction through a self-consistent, feedback mechanism. We demonstrate Borel-summability of MFPT for the case of the quartic- and sextic-anharmonic oscillators and the quartic double-well oscillator (QDWO) by obtaining uniformly accurate results for the ground state of the above systems for arbitrary physical values of the coupling strength. The results obtained for the QDWO may be of particular significance since “renormalon”-free, unambiguous results are achieved for its spectrum in contrast to the well-known failure of SFPT in this case.

Two-qubit quantum gates construction via unitary factorization

2017 ◽  
Vol 17 (9&10) ◽  
pp. 721-746
Author(s):  
Francisco Delgado
Keyword(s):  
Quantum Computer ◽  
Fault Tolerant ◽  
Pulse Sequence ◽  
Quantum Gate ◽  
Magnetic Systems ◽  
Research Areas ◽  
Non Local ◽  
Physical Interactions ◽  
Physical Elements ◽  
Gate Design

Quantum information and quantum computation are emerging research areas based on the properties of quantum resources, such as superposition and entanglement. In the quantum gate array version, the use of convenient and proper gates is essential. While these gates adopt theoretically convenient forms to reproduce computational algorithms, their design and feasibility depend on specific quantum systems and physical resources used in their setup. These gates should be based on systems driven by physical interactions ruled by a quantum Hamiltonian. Then, the gate design is restricted to the properties and the limitations imposed by the interactions and the physical elements involved. This work shows how anisotropic Heisenberg-Ising interactions, written in a non-local basis, allow the reproduction of quantum computer operations based on unitary processes. We show that gates can be generated by a pulse sequence of driven magnetic fields. This fact states alternative techniques in quantum gate design for magnetic systems. A brief final discussion around associated fault tolerant extensions to the current work is included.

Perturbation Theory for Quantum Mechanics in its Hessenberg-Matrix Representation

1997 ◽  
Vol 12 (01) ◽  
pp. 299-304 ◽  
Author(s):  
Miloslav Znojil
Keyword(s):  
Perturbation Theory ◽  
Quantum Mechanics ◽  
Matrix Representation ◽  
Triangular Matrix ◽  
Model Space ◽  
Zero Order ◽  
Matrix Structure ◽  
Hessenberg Matrix ◽  
Exactly Solvable ◽  
Partial Integrability

For years, the partial integrability in quantum mechanics [e.g., the exceptional terminating Lanczos solutions or the so called quasi-exactly solvable "next-to-elementary" systems] represented a challenge in perturbation theory. The main difficulty lied in an incompleteness of the available zero-order wavefunctions and in the related impossibility of an easy construction of the necessary zero-order propagators. We describe a solution of this problem, based on an unusual choice of model space. A few examples illustrate the underlying technicalities: Paying detailed attention to the Hessenberg-matrix Hamiltonians, our formalism compensates the incompleteness of integrability by a slightly less elementary (viz., non-diagonal but still triangular) matrix structure of its propagators.

scholarly journals A NEW GENERAL APPROXIMATION SCHEME IN QUANTUM THEORY: APPLICATION TO THE ANHARMONIC AND THE DOUBLE WELL OSCILLATORS

2005 ◽  
Vol 20 (12) ◽  
pp. 2687-2714 ◽  
Author(s):  
B. P. MAHAPATRA ◽  
N. SANTI ◽  
N. B. PRADHAN
Keyword(s):  
Perturbation Theory ◽  
Approximation Scheme ◽  
Energy Levels ◽  
Exact Results ◽  
Exact Solvability ◽  
Exactly Solvable ◽  
Anharmonic Potential ◽  
Theory Application ◽  
Suitable Potential ◽  
Partner Potential

A self-consistent, nonperturbative approximation scheme is proposed which is potentially applicable to arbitrary interacting quantum systems. For the case of self-interaction, the scheme consists in approximating the original interaction HI(ϕ) by a suitable "potential" V(ϕ) which satisfies the following two basic requirements, (i) exact solvability (ES): the "effective" Hamiltonian H0 generated by V(ϕ) is exactly solvable i.e., the spectrum of states |n〉 and the eigenvalues En are known and (ii) equality of quantum averages (EQA): 〈n|HI(ϕ)|n〉 = 〈n|V(ϕ)|n〉 for arbitrary n. The leading order (LO) results for |n〉 and En are thus readily obtained and are found to be accurate to within a few percent of the "exact" results. These LO-results are systematically improvable by the construction of an improved perturbation theory (IPT) with the choice of H0 as the unperturbed Hamiltonian and the modified interaction, λH′(ϕ)≡λ(HI(ϕ) - V(ϕ)), as the perturbation where λ is the coupling strength. The condition of convergence of the IPT for arbitrary λ is satisfied due to the EQA requirement which ensures that 〈n|λH′(ϕ)|n〉 = 0for arbitrary λ and n. This is in contrast to the divergence (which occurs even for infinitesimal λ!) in the naive perturbation theory where the original interaction λHI(ϕ) is chosen as the perturbation. We apply the method to the different cases of the anharmonic and the double well potentials, e.g. quartic-, sextic- and octic-anharmonic oscillators and quartic-, sextic-double well oscillators. Uniformly accurate results for the energy levels over the full allowed range of λ and n are obtained. The results compare well with the exact results predicted by supersymmetry for the case of the sextic anharmonic potential and the double well partner potential. Further improvement in the accuracy of the results by the use of IPT, is demonstrated. We also discuss the vacuum structure and stability of the resulting theory in the above approximation scheme.

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