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Register a new userMatrix product state approach for a quantum system at finite temperatures using random phases and Trotter gates
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We develop a numerical method based on matrix product states for simulating quantum many-body systems at finite temperatures without importance sampling and evaluate its performance in spin 1/2 systems. Our method is an extension of the random phase product state (RPPS) approach introduced recently [T. Iitaka, arXiv:2006.14459]. We show that the original RPPS approach often gives unphysical values for thermodynamic quantities even in the Heisenberg chain. We find that by adding the operation of Trotter gates to the RPPS, the sampling efficiency of the approach significantly increases and its results are consistent with those of the purification approach. We also apply our method to a frustrated spin 1/2 system to exemplify that it can simulate a system in which the purification approach fails.
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We present a unified framework for renormalization group methods, including Wilsons numerical renormalization group (NRG) and Whites density-matrix renormalization group (DMRG), within the language of matrix product states. This allows improvements over Wilsons NRG for quantum impurity models, as we illustrate for the one-channel Kondo model. Moreover, we use a variational method for evaluating Greens functions. The proposed method is more flexible in its description of spectral properties at finite frequencies, opening the way to time-dependent, out-of-equilibrium impurity problems. It also substantially improves computational efficiency for one-channel impurity problems, suggesting potentially emph{linear} scaling of complexity for $n$-channel problems.
We prove that ground states of gapped local Hamiltonians on an infinite spin chain can be efficiently approximated by matrix product states with a bond dimension which scales as D~(L-1)/epsilon, where any local quantity on L consecutive spins is approximated to accuracy epsilon.
While general quantum many-body systems require exponential resources to be simulated on a classical computer, systems of non-interacting fermions can be simulated exactly using polynomially scaling resources. Such systems may be of interest in their own right, but also occur as effective models in numerical methods for interacting systems, such as Hartree-Fock, density functional theory, and many others. Often it is desirable to solve systems of many thousand constituent particles, rendering these simulations computationally costly despite their polynomial scaling. We demonstrate how this scaling can be improved by adapting methods based on matrix product states, which have been enormously successful for low-dimensional interacting quantum systems, to the case of free fermions. Compared to the case of interacting systems, our methods achieve an exponential speedup in the entanglement entropy of the state. We demonstrate their use to solve systems of up to one million sites with an effective MPS bond dimension of 10^15.
Quantum trajectories and superoperator algorithms implemented within the matrix product state (MPS) framework are powerful tools to simulate the real-time dynamics of open dissipative quantum systems. As for the unitary case, the reachable time-scales as well as system sizes are limited by the (possible) build-up of entanglement entropy. The aforementioned methods constitute complementary approaches how Lindblad master equations can be integrated relying either on a quasi-exact representation of the full density matrix or a stochastic unraveling of the density matrix in terms of pure states. In this work, we systematically benchmark both methods by studying the dynamics of a Bose-Hubbard chain in the presence of local as well as global dephasing. The build-up as well as system-size scaling of entanglement entropy strongly depends on the method and the parameter regime and we discuss the applicability of the methods for these cases as well as study the distribution of observables and time discretization errors that can become a limiting factor for global dissipation.
We study measures of decoherence and thermalization of a quantum system $S$ in the presence of a quantum environment (bath) $E$. The whole system is prepared in a canonical thermal state at a finite temperature. Applying perturbation theory with respect to the system-environment coupling strength, we find that under common Hamiltonian symmetries, up to first order in the coupling strength it is sufficient to consider the uncoupled system to predict decoherence and thermalization measures of $S$. This decoupling allows closed form expressions for perturbative expansions for the measures of decoherence and thermalization in terms of the free energies of $S$ and of $E$. Numerical results for both coupled and decoupled systems with up to 40 quantum spins validate these findings.
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