Bounds for the adiabatic approximation with applications to quantum computation

Jansen, Sabine; Ruskai, Mary-Beth; Seiler, R.

doi:10.1063/1.2798382

Cited by 356 publications

(458 citation statements)

References 26 publications

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“…One possibility of such a stationary state is to assume the thermal equilibrium state (β −1 = k B T at temperature T ) [4] is performed by replacing (in the interaction picture) ρ S (t ′ ) → ρ S (t) under the integral, substituting τ = t − t ′ and extending the integration to infinity. This is usually motivated by fastly decaying reservoir correlation functions (9). By doing so, we obtain the BM Master Equatioṅ…”

Section: Discussionmentioning

confidence: 99%

“…Consequently, the maximum transformation rate (where the final excitations are acceptably small) corresponds to the computational complexity of the adiabatic algorithm. For closed systems, it is related to the spectral properties of the time-dependent system Hamiltonian [9,10]. For a reservoir at sufficiently low temperatures, this scheme is thought to be robust against decoherence [6] and might even be aided by it [11].…”

Section: Introductionmentioning

confidence: 99%

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Preservation of positivity by dynamical coarse graining

2008

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We compare different quantum Master equations for the time evolution of the reduced density matrix. The widely applied secular approximation (rotating wave approximation) applied in combination with the Born-Markov approximation generates a Lindblad type master equation ensuring for completely positive and stable evolution and is typically well applicable for optical baths. For phonon baths however, the secular approximation is expected to be invalid. The usual Markovian master equation does not generally preserve positivity of the density matrix. As a solution we propose a coarse-graining approach with a dynamically adapted coarse graining time scale. For some simple examples we demonstrate that this preserves the accuracy of the integro-differential Born equation. For large times we analytically show that the secular approximation master equation is recovered. The method can in principle be extended to systems with a dynamically changing system Hamiltonian, which is of special interest for adiabatic quantum computation. We give some numerical examples for the spin-boson model of cases where a spin system thermalizes rapidly, and other examples where thermalization is not reached.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Preservation of positivity by dynamical coarse graining

2008

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“…Recent work [19] has shown that the adiabatic condition may be a more complicated function of these parameters, but all of this recent work has the running time scaling like an inverse polynomial in g min . Since the norm of the Hamiltonian's derivative is usually independent of parameters such as our α and β, typically the gap is taken as the important part of this expression.…”

Section: Quantum Adiabatic Optimization Of Symmetric Functionsmentioning

confidence: 99%

Spectral-gap analysis for efficient tunneling in quantum adiabatic optimization

Brady

Dam

2016

Phys. Rev. A

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We investigate the efficiency of Quantum Adiabatic Optimization when overcoming potential barriers to get from a local to a global minimum. Specifically we look at n qubit systems with symmetric cost functions f : {0, 1} n → R where the ground state must tunnel through a potential barrier of width n α and height n β . By the quantum adiabatic theorem the time delay sufficient to ensure tunneling grows quadratically with the inverse spectral gap during this tunneling process. We analyze barrier sizes with 1/2 ≤ α + β and α < 1/2 and show that the minimum gap scales polynomially as n 1/2−α−β when 2α + β ≤ 1 and exponentially as n −β/2 exp(−Cn (2α+β−1)/2 ) when 1 < 2α + β. Our proof uses elementary techniques and confirms and extends an unpublished folklore result by Goldstone, which used large spin and instanton methods. Parts of our result also refine recent results by Kong and Crosson and Jiang et al. about the exponential gap scaling.

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“…Recently, two of the main results of this article, i.e., the optimal run-time scaling T = O(∆E −1 min ) and the fasterthan-polynomial decrease of the final error a 1 (T ), have been demonstrated rigorously for a class of Hamiltonians using methods of spectral analysis [20].…”

Section: Note Addedmentioning

confidence: 99%

General error estimate for adiabatic quantum computing

2006

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Most investigations devoted to the conditions for adiabatic quantum computing are based on the first-order correction Ψ ground (t)|Ḣ(t)|Ψ excited (t) /∆E 2 (t) ≪ 1. However, it is demonstrated that this first-order correction does not yield a good estimate for the computational error. Therefore, a more general criterion is proposed, which includes higher-order corrections as well and shows that the computational error can be made exponentially small -which facilitates significantly shorter evolution times than the above first-order estimate in certain situations. Based on this criterion and rather general arguments and assumptions, it can be demonstrated that a run-time T of order of the inverse minimum energy gap ∆Emin is sufficient and necessary, i.e., T = O(∆E −1 min ). For some examples, these analytical investigations are confirmed by numerical simulations.

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Bounds for the adiabatic approximation with applications to quantum computation

Cited by 356 publications

References 26 publications

Preservation of positivity by dynamical coarse graining

Preservation of positivity by dynamical coarse graining

Spectral-gap analysis for efficient tunneling in quantum adiabatic optimization

General error estimate for adiabatic quantum computing

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