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Annals of Physics

Hamiltonian formulation of time-dependent variational principles for the many-body system

Arthur K. Kerman, S. E. Koonin

Published Sep 1, 1976• 315 citations
Publisher
DOI
10.1016/0003-4916(76)90065-8

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Clifford Circuits Augmented Time-Dependent Variational Principle

Xiangjian Qian, Jiale Huang, Mingpu Qin

The recently proposed Clifford circuits augmented matrix product states (CA-MPSs) [X. Qian, J. Huang, and M. Qin, Phys. Rev. Lett. 133, 190402 (2024)PRLTAO0031-900710.1103/PhysRevLett.133.190402] seamlessly augment density matrix renormalization groups with Clifford circuits. In CA-MPSs, the entanglement from stabilizers is transferred to the Clifford circuits, which can be easily handled according to the Gottesman-Knill theorem. As a result, an MPS needs only to deal with the nonstabilizer entanglement, which largely reduces the bond dimension and the resource required for the accurate simulation of many-body systems. In this Letter, we generalize CA-MPSs to the framework of the time-dependent variational principle (TDVP) for time evolution simulations. In this method, we apply Clifford circuits to the resulting MPS in each TDVP step with a two-site sweeping process similar as in density matrix renormalization groups, aiming at reducing the entanglement entropy in the MPS, and the Hamiltonian is transformed accordingly using the chosen Clifford circuits. Similar as in CA-MPSs, the Clifford circuits does not increase the number of terms in the Hamiltonian, which makes the overhead very small in the new method. We test this method in XXZ chain, 2D Heisenberg model, and Kitaev honeycomb model. The results show that the Clifford circuits augmented TDVP method can reduce the entanglement entropy in the time evolution process and hence makes the simulation reliable for a longer time. The Clifford circuits augmented time-dependent variational principle provides a useful tool for the simulation of the time evolution process of many-body systems in the future.

arXiv:2407.03202•Physical Review Letters•Apr 15, 2025•22 citations

Time-independent perturbation theory and the variational principle

Institute of Physics eBooks•Nov 26, 2004•0 citations
The anharmonic lattice is a representative example of an interacting, bosonic, many-body system.The selfconsistent harmonic approximation has proven versatile for the study of the equilibrium properties of anharmonic lattices.However, the study of dynamical properties therein resorts to an ansatz, whose validity has not yet been theoretically proven.Here we apply the time-dependent variational principle, a recently emerging useful tool for studying the dynamic properties of interacting many-body systems, to the anharmonic lattice Hamiltonian at finite temperature using the Gaussian states as the variational manifold.We derive an analytic formula for the position-position correlation function and the phonon self-energy, proving the dynamical ansatz of the self-consistent harmonic approximation.We establish a fruitful connection between time-dependent variational principle and the anharmonic lattice Hamiltonian, providing insights in both fields.Our work expands the range of applicability of the time-dependent variational principle to first-principles lattice Hamiltonians and lays the groundwork for the study of dynamical properties of the anharmonic lattice using a fully variational framework.
We study low-rank tensor methods for the numerical solution of Schrödinger's equation with time-independent and explicitly time-dependent Hamiltonians, motivated by large-scale simulations of many-body quantum systems and quantum computing devices subject to time-dependent control pulses. We outline the recent application of the "basis update and Galerkin" (BUG) method for tensor trains, and describe the established TDVP and TDVP-2 algorithms based on the time-dependent variational principle. For comparison, we also consider the BUG method in the Tucker format. All these approaches enable memory efficient representations of partially entangled quantum states and thereby mitigate the exponential cost of conventional state-vector formulations. The rank-adaptivity relies on the truncated singular value decomposition, in which the rank of a matrix is reduced by setting its smallest singular values to zero, based on a threshold parameter that controls the truncation error. Numerical experiments on representative time-independent and time-dependent Hamiltonian models quantify the tradeoff between accuracy and compression across methods, with particular attention to the interplay between the time-step and the truncation threshold, and how the computational effort scales with the number of sub-systems in the quantum system.
The recently proposed Clifford circuits augmented matrix product states (CA-MPSs) [X. Qian, J. Huang, and M. Qin, Phys. Rev. Lett. 133, 190402 (2024)PRLTAO0031-900710.1103/PhysRevLett.133.190402] seamlessly augment density matrix renormalization groups with Clifford circuits. In CA-MPSs, the entanglement from stabilizers is transferred to the Clifford circuits, which can be easily handled according to the Gottesman-Knill theorem. As a result, an MPS needs only to deal with the nonstabilizer entanglement, which largely reduces the bond dimension and the resource required for the accurate simulation of many-body systems. In this Letter, we generalize CA-MPSs to the framework of the time-dependent variational principle (TDVP) for time evolution simulations. In this method, we apply Clifford circuits to the resulting MPS in each TDVP step with a two-site sweeping process similar as in density matrix renormalization groups, aiming at reducing the entanglement entropy in the MPS, and the Hamiltonian is transformed accordingly using the chosen Clifford circuits. Similar as in CA-MPSs, the Clifford circuits does not increase the number of terms in the Hamiltonian, which makes the overhead very small in the new method. We test this method in XXZ chain, 2D Heisenberg model, and Kitaev honeycomb model. The results show that the Clifford circuits augmented TDVP method can reduce the entanglement entropy in the time evolution process and hence makes the simulation reliable for a longer time. The Clifford circuits augmented time-dependent variational principle provides a useful tool for the simulation of the time evolution process of many-body systems in the future.