This review gives an account of the major theoretical developments in the field, while focusing on the closed-system setting. This theorem is then used to derive the adiabatic theorem, do the scattering theory for such Hamiltonians, and prove some classical propagation estimates and asymptotic completeness.Īdiabatic quantum computing (AQC) started as an approach to solving optimization problems and has evolved into an important universal alternative to the standard circuit model of quantum computing, with deep connections to both classical and quantum complexity theory and condensed matter physics. We prove a new uniform ergodic theorem for slowly changing unitary operators. We treat some known cases in a new way, such as the Zener problem, and we give another proof of the adiabatic theorem in the gapless case. Slowly changing potential problems in Quantum Mechanics: Adiabatic theorems, ergodic theorems, and scatteringįishman, S., E-mail: Soffer, A., E-mail: employ the recently developed multi-time scale averaging method to study the large time behavior of slowly changing (in time) Hamiltonians. In general, this is a long-standing mathematical problem, which can be solved in the present particular case of a gapped system, relevant e.g. One important application is the proof of the validity of linear response theory for such extended, genuinely interacting systems. Our setup is that of quantum spin systems where the manifold of ground states is separated from the rest of the spectrum by a spectral gap. In the present paper, we prove an adiabatic theorem with an error bound that is independent of the number of degrees of freedom. Under suitable assumptions the solution of the time-inhomogenous equation stays close to an instantaneous fixpoint. The adiabatic theorem refers to a setup where an evolution equation contains a time-dependent parameter whose change is very slow, measured by a vanishing parameter É›. The Adiabatic Theorem and Linear Response Theory for Extended Quantum Systemsīachmann, Sven De Roeck, Wojciech Fraas, Martin We have chosen to use only the minimum geometric structure needed for the understanding of the adiabatic theorem for this case.… We present a simple and pedagogical derivation of the quantum adiabatic theorem for two-level systems (a single qubit) based on geometrical structures of quantum mechanics developed by Anandan and Aharonov, among others. Lobo, Augusto Cesar Ribeiro, Rafael Antunes Ribeiro, Clyffe de Assis Dieguez, Pedro Ruas Implications for adiabatic quantum computation are discussed.ĮRIC Educational Resources Information Center In the absence of fast driven oscillations the traditional adiabatic theorem holds. Here we show that the reported violations of the adiabatic theorem all arise from resonant transitions between energy levels. Recently the conditions required for the adiabatic theorem to hold have become a subject of some controversy. The adiabatic theorem provides the basis for the adiabatic model of quantum computation. We predict that the density of nonadiabatic excitations is exponentially small in the driving rate and the scaling of the exponent depends on the dimension. As an application, we give a mathematical proof of Kubo's linear response formula for a broad class of gapped interacting systems. In this Letter, we prove a version of the adiabatic theorem for gapped ground states of interacting quantum spin systems, under assumptions that remain valid in the thermodynamic limit. Therefore, applications to many-body systems are not covered by the proofs and arguments in the literature. In this setup, the rate of variation É› of local terms is indeed small compared to the gap, but the rate of variation of the total, extensive Hamiltonian, is not. Today, this theorem is increasingly applied in a many-body context, e.g., in quantum annealing and in studies of topological properties of matter. The first proof of the quantum adiabatic theorem was given as early as 1928. ![]() Adiabatic Theorem for Quantum Spin Systems
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