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Description: Quantum computing is arguably one of the most revolutionary and disruptive technologies of this century. Due to the ever-increasing number of potential applications as well as the continuing rise in complexity, the development, simulation, optimization, and physical realization of quantum circuits is of utmost importance for designing novel algorithms. We show how matrix product states (MPSs) and matrix product operators (MPOs) can be used to express certain quantum states, quantum gates, and entire quantum circuits as low-rank tensors. This enables the analysis and simulation of complex quantum circuits on classical computers and to gain insight into the underlying structure of the system. We present different examples to demonstrate the advantages of MPO formulations and show that they are more efficient than conventional techniques if the bond dimensions of the wave function representation can be kept small throughout the simulation.

Description: We demonstrate how to apply the tensor-train format to solve the time-independent Schrödinger equation for quasi-one-dimensional excitonic chain systems with and without periodic boundary conditions. The coupled excitons and phonons are modeled by Fröhlich–Holstein type Hamiltonians with on-site and nearest-neighbor interactions only. We reduce the memory consumption as well as the computational costs significantly by employing efficient decompositions to construct low-rank tensor-train representations, thus mitigating the curse of dimensionality. In order to compute also higher quantum states, we introduce an approach that directly incorporates the Wielandt deflation technique into the alternating linear scheme for the solution of eigenproblems. Besides systems with coupled excitons and phonons, we also investigate uncoupled problems for which (semi-)analytical results exist. There, we find that in the case of homogeneous systems, the tensor-train ranks of state vectors only marginally depend on the chain length, which results in a linear growth of the storage consumption. However, the central processing unit time increases slightly faster with the chain length than the storage consumption because the alternating linear scheme adopted in our work requires more iterations to achieve convergence for longer chains and a given rank. Finally, we demonstrate that the tensor-train approach to the quantum treatment of coupled excitons and phonons makes it possible to directly tackle the phenomenon of mutual self-trapping. We are able to confirm the main results of the Davydov theory, i.e., the dependence of the wave packet width and the corresponding stabilization energy on the exciton–phonon coupling strength, although only for a certain range of that parameter. In future work, our approach will allow calculations also beyond the validity regime of that theory and/or beyond the restrictions of the Fröhlich–Holstein type Hamiltonians.