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arXiv:1103.4174 · Journal of Physics A Mathematical and Theoretical

Improved error bounds for the adiabatic approximation

Donny Cheung, Peter Høyer, Nathan Wiebe

Published Sep 20, 2011• 36 citations
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Abstract

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.

DOI
10.1088/1751-8113/44/41/415302

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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.
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 provide rigorous bounds for the error of the adiabatic approximation of quantum mechanics under four sources of experimental error: perturbations in the initial condition, systematic time-dependent perturbations in the Hamiltonian, coupling to low-energy quantum systems, and decoherent time-dependent perturbations in the Hamiltonian. For decoherent perturbations, we find both upper and lower bounds on the evolution time to guarantee that the adiabatic approximation performs within a prescribed tolerance. Our results include explicit definitions of constants, and we apply them to the spin-1/2 particle in a rotating magnetic field and to the superconducting flux qubit. We compare the theoretical bounds on the superconducting flux qubit to simulation results.
The quantum adiabatic theorem is a widely applied result in quantum mechanics. However, recently, counterexamples were produced, indicating that the transition probabilities in some systems do not converge as predicted by the adiabatic theorem [K. P. Marzlin and B. C. Sanders, Phys. Rev. Lett. 93, 160408 (2004)]. In this paper, we examine a proof of the adiabatic theorem, paying specific attention to the validity of the assumptions made in the proof. We show that the convergence of the transition probabilities is path dependent when the change in the Hamiltonian is large. Finally, we consider the application of this reformulated form of the adiabatic theorem to the adiabatic evolution paradigm in quantum computation.
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.
This thesis explores two mathematical aspects of adiabatic quantum computation. Adiabatic quantum computation depends on the adiabatic theorem of quantum mechanics, and (a) we provide a rigorous formulation of the adiabatic theorem with explicit definitions of constants, and (b) we bound error in the adiabatic approximation under conditions of noise and experimental error. We apply the new results to a standard example of violation of the adiabatic approximation, and to a superconducting flux qubit. Further, adiabatic quantum computation requires large ground-state energy gaps throughout a Hamiltonian evolution if it is to solve problems in polynomial time. We identify a class of random Hamiltonians with non-nearest-neighbor interactions and a ground-state energy gap of O(1/n)\mathcal{O}(1/\sqrt{n})O(1/n​), where nnn is the number of qubits. We also identify two classes of Hamiltonians with non-nearest-neighbor interactions whose ground state can be found in polynomial time with adiabatic quantum computing. We then use the Jordan-Wigner transformation to derive equivalent results for Hamiltonians defined using Pauli operators.