Solving gapped Hamiltonians locally
By M. B. Hastings. 2005
213 citations recorded by the source corpus.
We show that any short-range Hamiltonian with a gap between the ground and excited states can be written as a sum of local operators, such that the ground state is an approximate eigenvector of each operator separately. We then show that the ground state of any such Hamiltonian is close to a generalized matrix product state. The range of the given operators needed to obtain a good approximation to the ground state is proportional to the square of the logarithm of the system size times a characteristic ``factorization length.'' Applications to many-body quantum simulation are discussed. We also consider density matrices of systems at non zero temperature.
Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/63041ff4ee2a86777b55e2b6684d9757e3de2c03
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In Atomic and Molecular Physics, and Optics
- J. Dubail
- Frank Verstraete
- Sahel Ashhab
- Franco Nori
- I. Georgescu
- Jens Eisert
- Sergey Bravyi
- Pasquale Calabrese
In Condensed Matter Physics
- J. Dubail
- Frank Verstraete
- Sergey Bravyi
- J. I. Cirac
- Franco Nori
- Sahel Ashhab
- I. Georgescu
- Arnau Riera
In Geometry and Topology
- Sergey Bravyi
- Sahel Ashhab
- I. Georgescu
- Franco Nori
- J. Dubail
- Frank Verstraete
- Jens Eisert
- Christian Gogolin
Citations in the corpus
- Entanglement scaling of operators: a conformal field theory approach, with a glimpse of simulability of long-time dynamics in 1 + 1d (2016, background): “Let us also emphasize that the question of approximability of thermal density matrices by MPOs or their higher dimensional versions has been tackled directly, with rigorous statements and proofs, in [39, 40].”
- Area laws in quantum systems: mutual information and correlations. (2007, background): “[24] M.”
- Locality of temperature (2013, methodology): “[40] to establish a mathematically (not algorithmically) constructive version of Theorem 3, on MPO approximations, where the MPO in Eq.”
- Efficient quantum state tomography. (2010, background): “This insight is not new: researchers in many-body physics and quantum information theory have found many classes of states which are described by a number of parameters scaling polynomially in N [9–13] and which closely approximate the kind of states found in physical systems [14, 15].”
- Quantum Simulation (2013, methodology): “…studied, and several methods have been developed (see, e.g., (Berry et al., 2007; Bravyi et al., 2008; Bremner et al., 2005; Brown et al., 2011; Dodd et al., 2002; Dür et al., 2008; Hastings, 2006; Wang and Zanardi, 2002; Wocjan et al., 2002a,b,c)), but it still remains a difficult problem.”
- A Short Proof of Stability of Topological Order under Local Perturbations (2010, methodology): “The reduction from globally block-diagonal to locally block-diagonal perturbations is in two steps; in Lemma 7 we exploit the idea of [16, 17] to write a Hamiltonian of a gapped system as a sum of terms such that the ground states are (approximate) eigenvectors of each term separately, and then we s”