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Eigenstates from time-averaged evolution: a sinc filter, phase projection, and a squared-Hamiltonian comparison

Teemu Isojärvi

Oct 23, 20260 citations

Construction of quantum states with special properties by projection methods

Xuefeng Duan, Chi-Kwong Li, Diane Christine Pelejo

arXiv:1604.08289Quantum Information ProcessingNov 1, 20201 citation

Possible quantum eigenstates in a given system

pLalitha Nathp

Journal of Lasers Optics & PhotonicsFeb 2, 20190 citations

Eigenstate preparation by phase decoherence

Rolando D. Somma, Sergio Boixo, Emanuel H. Knill

A computation in adiabatic quantum computing is implemented by traversing a path of nondegenerate eigenstates of a continuous family of Hamiltonians. We introduce a method that traverses a discretized form of the path: at each step we apply the instantaneous Hamiltonian for a random time. The resulting decoherence approximates a projective measurement onto the desired eigenstate, achieving a version of the quantum Zeno effect. The average cost of our method is O(L2/Delta) for constant error probability, where L is the length of the path of eigenstates and Delta is the minimum spectral gap of the Hamiltonian. For many cases of interest, L does not depend on Delta so the scaling of the cost with the gap is better than the one obtained in rigorous proofs of the adiabatic theorem. We give an example where this situation occurs.A computation in adiabatic quantum computing is implemented by traversing a path of nondegenerate eigenstates of a continuous family of Hamiltonians. We introduce a method that traverses a discretized form of the path: at each step we apply the instantaneous Hamiltonian for a random time. The resulting decoherence approximates a projective measurement onto the desired eigenstate, achieving a version of the quantum Zeno effect. The average cost of our method is O(L2/Delta) for constant error probability, where L is the length of the path of eigenstates and Delta is the minimum spectral gap of the Hamiltonian. For many cases of interest, L does not depend on Delta so the scaling of the cost with the gap is better than the one obtained in rigorous proofs of the adiabatic theorem. We give an example where this situation occurs.

Canadian Workshop on Information Theory (CWIT)May 1, 20090 citations

Quantum-mechanical energy eigenstates and their superpositions

Mario Belloni, Wolfgang Christian

May 26, 20110 citations

Effects of the interplay between initial state and Hamiltonian on the thermalization of isolated quantum many-body systems

Eduardo Jonathan Torres-Herrera, Léa Ferreira dos Santos

We explore the role of the initial state on the onset of thermalization in isolated quantum many-body systems after a quench. The initial state is an eigenstate of an initial Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}$ and it evolves according to a different final Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$. If the initial state has a chaotic structure with respect to ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$, i.e., if it fills the energy shell ergodically, thermalization is certain to occur. This happens when ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}$ is a full random matrix, because its states projected onto ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$ are fully delocalized. The results for the observables then agree with those obtained with thermal states at infinite temperature. However, finite real systems with few-body interactions, as the ones considered here, are deprived of fully extended eigenstates, even when described by a nonintegrable Hamiltonian. We examine how the initial state delocalizes as it gets closer to the middle of the spectrum of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$, causing the observables to approach thermal averages, be the models integrable or chaotic. Our numerical studies are based on initial states with energies that cover the entire lower half of the spectrum of one-dimensional Heisenberg spin-1/2 systems.We explore the role of the initial state on the onset of thermalization in isolated quantum many-body systems after a quench. The initial state is an eigenstate of an initial Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}$ and it evolves according to a different final Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$. If the initial state has a chaotic structure with respect to ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$, i.e., if it fills the energy shell ergodically, thermalization is certain to occur. This happens when ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}$ is a full random matrix, because its states projected onto ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$ are fully delocalized. The results for the observables then agree with those obtained with thermal states at infinite temperature. However, finite real systems with few-body interactions, as the ones considered here, are deprived of fully extended eigenstates, even when described by a nonintegrable Hamiltonian. We examine how the initial state delocalizes as it gets closer to the middle of the spectrum of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$, causing the observables to approach thermal averages, be the models integrable or chaotic. Our numerical studies are based on initial states with energies that cover the entire lower half of the spectrum of one-dimensional Heisenberg spin-1/2 systems.

arXiv:1305.6937Physical Review EOct 11, 201349 citations

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