Coherent states for arbitrary Lie group

Perelomov, A. M.

doi:10.1007/bf01645091

Cited by 1,264 publications

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“…Here, we focus on a Lie algebraic analysis to obtain other situations where quantum algorithms can be efficiently simulated by CCs. The so-called generalized coherent states (GCSs) [7] play a decisive role in our analysis.The algorithms considered here make use of the Liealgebraic model of quantum computing (LQC). An LQC algorithm begins with the specification of a semisimple, compact M -dimensional real Lie algebraĥ of skew-Hermitian operators acting on a finite-dimensional Hilbert space H, with Lie bracket [X,Ŷ ] :=XŶ −ŶX.…”

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Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models

et al. 2006

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We consider quantum computational models defined via a Lie-algebraic theory. In these models, specified initial states are acted on by Lie-algebraic quantum gates and the expectation values of Lie algebra elements are measured at the end. We show that these models can be efficiently simulated on a classical computer in time polynomial in the dimension of the algebra, regardless of the dimension of the Hilbert space where the algebra acts. Similar results hold for the computation of the expectation value of operators implemented by a gatesequence. We introduce a Lie-algebraic notion of generalized mean-field Hamiltonians and show that they are efficiently (exactly) solvable by means of a Jacobi-like diagonalization method. Our results generalize earlier ones on fermionic linear optics computation and provide insight into the source of the power of the conventional model of quantum computation. Quantum models of computation are widely believed to be more powerful than classical ones. Although this has been shown to be true in a few cases, it is still important to determine when a quantum algorithm for a given problem is more resource efficient than any classical one, or, conversely, when a classical algorithm is just as efficient as any quantum counterpart. In general, one needs to know whether it is worth investing in building a quantum computer (QC) and what is required for success. In this paper, we show close connections between these issues and the efficient (or exact) solvability of Hamiltonians. In particular, we show that a class of quantum models we call generalized mean-field Hamiltonians (GMFHs) [1] is efficiently solvable and furthermore does not provide a stronger-than-classical model of computation: A quantum device engineered to have dynamical gates generated by Hamiltonians from such a set cannot directly simulate universal efficient quantum computation and can be efficiently simulated by a classical computer (CC).An algorithm is a sequence of elementary instructions that solves instances of a problem. It is said to be efficient if the resources required to solve problem instances of size N are polynomial in N (poly(N )) resources. Typically, the size of a problem instance is the number of bits required to represent it, and the relevant resources are time and space. In the last few years it has been shown that many pure-state quantum algorithms can be efficiently simulated on a CC when the extent of entanglement is limited (e.g., [2,3]) or when the quantum gates available are far from allowing us to build a set of universal gates [4,5,6]. Here, we focus on a Lie algebraic analysis to obtain other situations where quantum algorithms can be efficiently simulated by CCs. The so-called generalized coherent states (GCSs) [7] play a decisive role in our analysis.The algorithms considered here make use of the Liealgebraic model of quantum computing (LQC). An LQC algorithm begins with the specification of a semisimple, compact M -dimensional real Lie algebraĥ of skew-Hermitian operators acting on a finite-...

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Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models

et al. 2006

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“…. Let us to conclude this section presenting the following comments: one can generate the called canonical coherent states, which are defined as the eigenstates of the lowering operator B − ( c 2 − 1) of the bosonic sector, according to the Barut-Girardello approach [12,13] and generalized coherent states according to Perelomov [14,15]. Results of our investigations on these coherent states will be reported separately.…”

Section: The Abstract Wh Algebra and Its Super-realisationmentioning

confidence: 99%

The Wigner-Heisenberg Algebra in Quantum Mechanics

Rodrigues¹

2013

Advances in Quantum Mechanics

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“…These three construction methods become inequivalent when considering generalized CS associated with other algebras than the oscillator one, defined in (1.1). One may therefore distinguish between annihilation-operator CS (often called Barut-Girardello CS [5]), displacement-operator CS (often called Perelomov CS [6]), and minimum-uncertainty or "intelligent" CS [7], but this in no way exhausts all the possibilities of defining generalized CS.…”

Section: Introductionmentioning

confidence: 99%

“…, λ − α − 1, satisfying Klauder's conditions [2] in some subspace and for some appropriate values of the C λ -extended oscillator parameters. This family will include not only the multiphoton nonlinear CS |z, µ of [33], corresponding to α = 0, but also the Perelomov su(1,1) CS [6], corresponding to λ = 2 and α = 1.…”

Section: Introductionmentioning

confidence: 99%

Generalized Coherent States Associated with the Cλ-Extended Oscillator

Quesne¹

2001

Annals of Physics

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Two new types of coherent states associated with the C λ -extended oscillator, where C λ is the cyclic group of order λ, are introduced. The first ones include as special cases both the Barut-Girardello and the Perelomov su(1,1) coherent states for λ = 2, as well as the annihilation-operator coherent states of the C λ -extended oscillator spectrum generating algebra for higher λ values. The second ones, which are eigenstates of the C λ -extended oscillator annihilation operator, extend to higher λ values the paraboson coherent states, to which they reduce for λ = 2. All these states satisfy a unity resolution relation in the C λ -extended oscillator Fock space (or in some subspace thereof). They give rise to Bargmann representations of the latter, wherein the generators of the C λ -extended oscillator algebra are realized as differential-operatorvalued matrices (or differential operators). The statistical and squeezing properties of the new coherent states are investigated over a wide range of parameters and some interesting nonclassical features are exhibited.

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Coherent states for arbitrary Lie group

Abstract: The concept of coherent states originally closely related to the nilpotent group of Weyl is generalized to arbitrary Lie group. For the simplest Lie groups the system of coherent states is constructed and its features are investigated.

Cited by 1,264 publications

References 9 publications

Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models

Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models

The Wigner-Heisenberg Algebra in Quantum Mechanics

Generalized Coherent States Associated with the Cλ-Extended Oscillator

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