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arXiv:quant-ph/0510131

Is the Adiabatic Approximation Inconsistent?

Solomon Duki, Harsh Mathur, Onuttom Narayan

Published Oct 17, 2005• 2 citations
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Abstract

Marzlin and Sanders \cite{marzlin} have shown rigorously that the adiabatic approximation can be very inaccurate when applied to a Hamiltonian $H(t)$ that generates the evolution $U^{\dagger} (t)$ even if it gives an excellent approximation to the evolution $U(t)$ generated by a dual Hamiltonian $h(t)$. We show that this is not inconsistent with the adiabatic theorem and find that in general even if $h(t)$ satisfies the conditions of the adiabatic theorem, $H(t)$ will likely violate those conditions.

DOI
10.48550/arxiv.quant-ph/0510131

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Marzlin and Sanders \cite{marzlin} have shown rigorously that the adiabatic approximation can be very inaccurate when applied to a Hamiltonian H(t)H(t)H(t) that generates the evolution U†(t)U^{\dagger} (t)U†(t) even if it gives an excellent approximation to the evolution U(t)U(t)U(t) generated by a dual Hamiltonian h(t)h(t)h(t). We show that this is not inconsistent with the adiabatic theorem and find that in general even if h(t)h(t)h(t) satisfies the conditions of the adiabatic theorem, H(t)H(t)H(t) will likely violate those conditions.
A consensus that questions the perfunctory use of the quantum adiabatic theorem has emerged since Marzlin and Sanders [Phys. Rev. Lett. 93, 160408 (2004)] showed the existence of an inconsistency in the applicability of the theorem. Further analysis proved that the inconsistency may arise from the existence of resonant terms in the Hamiltonian, but recent work indicates that the debate about the full extent of the problem remains open. Here, we first show that key premises required in the standard demonstration of the theorem do not hold for a dual Hamiltonian involved in the Marzlin-Sanders inconsistency. Also, we show that two simple conditions can identify systems for which the adiabatic approximation fails, in spite of satisfying traditional quantitative conditions that were believed to guarantee its validity. Finally, we prove that the inconsistency only arises for Hamiltonians that contain resonant terms whose amplitudes go asymptotically to zero.
We construct the adiabatic theorem independent of the quantum representations. The quantum-representation-independent geometrical phase, adiabatic condition, and fidelity of the adiabatic approximation are derived. As an example, we apply the quantum-representation-independent adiabatic theorem to provide a thorough resolution of the Marzlin-Sanders inconsistency. We show that the Marzlin-Sanders inconsistency results from the wrong identification of the time-dependent Hamiltonian as the instantaneous energy operator and the quantum representation dependence of the Marzlin-Sanders transformation.
The discovery of adiabatic quantum computing has revealed a need for rigorously proven bounds for the error in the adiabatic approximation. We present in this paper, a rigorous and elementary derivation of upper and lower bounds on the error incurred from using the adiabatic approximation for quantum systems that can be used to assess the runtime of adiabatic algorithms. Our bounds are often asymptotically tight in the limit of slow evolution for fixed Hamiltonians, and are used to provide sufficient conditions for the application of the adiabatic approximation. We show that our sufficiency criteria exclude the Marzlin–Sanders counterexample from the class of Hamiltonians that obey the adiabatic approximation. Finally, we demonstrate the existence of classes of Hamiltonians that resemble the Marzlin–Sanders counterexample Hamiltonian, but also obey the adiabatic approximation.