Adiabatic approximation in open quantum systems

Sarandy, M. S.; Lidar, Daniel A.

doi:10.1103/physreva.71.012331

Cited by 193 publications

(316 citation statements)

References 32 publications

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“…Moreover, a realistic adiabatic quantum computer will always couple to an environment. Therefore, other methods [20,26] are necessary to study the evolution of such open quantum systems. To summarize, we have shown that the inconsistencies in the traditional adiabatic theorem reported in the literature are all closely related to the fact that for systems subject to fast driven oscillations, resonant transitions between energy levels cannot be suppressed by just reducing the amplitude of oscillations, although the adiabatic condition (2) can be satisfied.…”

Section: H(t) = −U † (T)h(t)u (T)mentioning

confidence: 99%

“…Recently, the validity of the adiabatic theorem was experimentally examined [8], and (2) was reported to be neither sufficient nor necessary condition for adiabaticity. These inconsistencies have created debates among researchers [9,10,11,12] and motivated a search for alternative conditions [13,14,15,16,17,18], reexamination of some adiabatic algorithms [19], or generalizations of the adiabatic theorem to open quantum systems [20].…”

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Consistency of the Adiabatic Theorem

2009

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The adiabatic theorem provides the basis for the adiabatic model of quantum computation. Recently the conditions required for the adiabatic theorem to hold have become a subject of some controversy. Here we show that the reported violations of the adiabatic theorem all arise from resonant transitions between energy levels. In the absence of fast driven oscillations the traditional adiabatic theorem holds. Implications for adiabatic quantum computation is discussed.A statement of the traditional adiabatic theorem [1,2,3], as described in most recent publications, is as follows: Consider a system with a time dependent Hamiltonian H(t) and a wave function |ψ(t) , which is the solution of the Schrödinger equation (h = 1)Let |E n (t) be the instantaneous eigenstates of H(t) with eigenvalues E n (t). If at an initial time t = 0 the system starts in an eigenstate |E n (0) of the Hamiltonian H(0), it will remain in the same instantaneous eigenstate, |E n (t) , at a later time t = T , as long as the evolution of the Hamiltonian is slow enough to satisfywhereThe adiabatic theorem has recently gained renewed attention as it provides the basis for one of the important schemes for quantum computation, i.e., adiabatic quantum computation [4,5]. Recently, the adiabatic condition (2) has become a subject of controversy. It was first shown by Marzlin and Sanders [6] and then by Tong et al. [7] that if a first system with Hamiltonian H(t) follows an adiabatic evolution, a second system defined by Hamiltoniancannot have an adiabatic evolution unlesseven if both systems satisfy the same condition (2). Here, T denotes time ordering operator. Recently, the validity of the adiabatic theorem was experimentally examined [8], and (2) was reported to be neither sufficient nor necessary condition for adiabaticity. These inconsistencies have created debates among researchers [9,10,11,12] and motivated a search for alternative conditions [13,14,15,16,17,18], reexamination of some adiabatic algorithms [19], or generalizations of the adiabatic theorem to open quantum systems [20].While it is valuable to find new conditions that guarantee adiabaticity in general, it is important to understand why the traditional adiabatic condition (2) is sufficient for some Hamiltonians, but not for others. Moreover, from the practical point of view it is much easier to work with a simple condition like (2) than some other more sophisticated ones. In this letter, we relate the reported violations of the traditional adiabatic theorem to resonant transitions between energy levels. We further show that in the absence of such resonant oscillations, the traditional adiabatic condition is sufficient to assure adiabaticity. Our line of thought is close to that of Duki et al. [9], but largely extended with rigorous mathematical proofs.It is well known that fast driven oscillations invalidate the adiabatic theorem [21]. Consider a simple example of a two-state system driven by an oscillatory force:We take V to be a small positive number. The exact instantaneous eigen...

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Section: H(t) = −U † (T)h(t)u (T)mentioning

confidence: 99%

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confidence: 99%

Consistency of the Adiabatic Theorem

2009

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“…II A). The case of an adiabatic approximation where the superoperator can be transformed only in a Jordan canonical form can be found in [51].…”

Section: Strong-noise Limitmentioning

confidence: 99%

Effect of Poisson noise on adiabatic quantum control

2017

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Type of publicationArticle (peer-reviewed) We present a detailed derivation of the master equation describing a general time-dependent quantum system with classical Poisson white noise and outline its various properties. We discuss the limiting cases of Poisson white noise and provide approximations for the different noise strength regimes. We show that using the eigenstates of the noise superoperator as a basis can be a useful way of expressing the master equation. Using this, we simulate various settings to illustrate different effects of Poisson noise. In particular, we show a dip in the fidelity as a function of noise strength where high fidelity can occur in the strong-noise regime for some cases. We also investigate recent claims [J. Jing et al., Phys. Rev. A 89, 032110 (2014)] that this type of noise may improve rather than destroy adiabaticity.

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“…For example, in Refs. [12,13,14] the concept of adiabaticity is based upon Jordan block decompositions of the superoperator that describe the evolution of the open system. A different approach, designed for systems that are weakly open, has been put forward in Ref.…”

Section: Introductionmentioning

confidence: 99%

Quantum adiabatic search with decoherence in the instantaneous energy eigenbasis

2005

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In Phys. Rev. A 71, 060312(R) (2005) the robustness of the local adiabatic quantum search to decoherence in the instantaneous eigenbasis of the search Hamiltonian was examined. We expand this analysis to include the case of the global adiabatic quantum search. As in the case of the local search the asymptotic time complexity for the global search is the same as for the ideal closed case, as long as the Hamiltonian dynamics is present. In the case of pure decoherence, where the environment monitors the search Hamiltonian, we find that the time complexity of the global quantum adiabatic search scales like N 3/2 , where N is the list length. We moreover extend the analysis to include success probabilities p < 1 and prove bounds on the run time with the same scaling as in the conditions for the p → 1 limit. We supplement the analytical results by numerical simulations of the global and local search.

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Adiabatic approximation in open quantum systems

Cited by 193 publications

References 32 publications

Consistency of the Adiabatic Theorem

Consistency of the Adiabatic Theorem

Effect of Poisson noise on adiabatic quantum control

Quantum adiabatic search with decoherence in the instantaneous energy eigenbasis

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