Time-dependent variational principle for quantum lattices.

By J. Haegeman, J. I. Cirac, Tobias J. Osborne, Iztok Pižorn, H. Verschelde, Frank Verstraete. 2011

761 citations recorded by the source corpus.

We develop a new algorithm based on the time-dependent variational principle applied to matrix product states to efficiently simulate the real- and imaginary-time dynamics for infinite one-dimensional quantum lattices. This procedure (i) is argued to be optimal, (ii) does not rely on the Trotter decomposition and thus has no Trotter error, (iii) preserves all symmetries and conservation laws, and (iv) has low computational complexity. The algorithm is illustrated by using both an imaginary-time and a real-time example.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Geometry and Topology

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Initially proposed for low-rank matrix manifolds in [166], dynamical low-rank methods have been extended to low-rank tensors in Tucker [167, 199], TT/MPS [118, 123, 163, 186], and HT [9, 186, 255] decomposition.”
Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, background): “It conserves the energy 22 and all symmetries that can be represented within the tangent plane, and it is possible to associate symplectic structure with a Poisson bracket to it (Haegeman et al., 2011).”
Tangent-space methods for uniform matrix product states (2018, methodology): “The MPS version of the TDVP [7, 15, 27] has been applied to spin chains [28], gauge theories [29, 30], and spin systems with long-range interactions [27, 31–34].”
Complexity and entanglement for thermofield double states (2018, background): “rges as the spatial volume occupied by a boundary quantum eld theory measured in the units of the UV cut-o with a non-universal 48The similarity becomes even more apparent with the cMERA construction [112] for free scalars or fermions which also acts directly on the normal modes. 77 prefactor [29]. ”
Time-evolution methods for matrix-product states (2019, methodology): “[28, 29] was presented in Refs.”
Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy) (2018, background): “ks down". For example, TEBD does not even preserve the energy when the truncation is large. An improved algorithm based on the time dependent variational principle (TDVP) was introduced in Refs. [28,29] which performs a unitary evolution in the space of MPS with given bond dimension ˜ max. If w”
Low-Rank Tensor Learning with Discriminant Analysis for Action Classification and Image Recovery (2014, background): “Many tensor representation learning approaches have been proposed (Chen and Saad 2009; Koch and Lubich 2010; Haegeman et al. 2011; Holtz, Rohwedder, and Schneider 2012; Khoromskij, Oseledets, and Schneider 2012; Arnold and Jahnke 2012; Lubich et al. 2013; Uschmajew and Vandereycken 2013; Mu et al.…”
At the limits of criticality-based quantum metrology: apparent super-Heisenberg scaling revisited (2017, methodology): “The time evolution is simulated using the time-dependent variational principle [80], which projects the Schrödinger equation onto the tangent space of the manifold of the MPS.”