Tensor-Train Thermo-Field Memory Kernels for Generalized Quantum Master Equations

Abstract

The generalized quantum master equation (GQME) approach provides a rigorous framework for deriving the exact equation of motion for any subset of electronic reduced density matrix elements (e.g., the diagonal elements). In the context of electronic dynamics, the memory kernel and inhomogeneous term of the GQME introduce the implicit coupling to nuclear motion and dynamics of electronic density matrix elements that are projected out (e.g., the off-diagonal elements), allowing for efficient quantum dynamics simulations. Here, we focus on benchmark quantum simulations of electronic dynamics in a spin-boson model system described by various types of GQMEs. Exact memory kernels and inhomogeneous terms are obtained from short-time quantum-mechanically exact tensor-train thermo-field dynamics (TT-TFD) simulations and are compared with those obtained from an approximate linearized semiclassical method, allowing for assessment of the accuracy of these approximate memory kernels and inhomogeneous terms. Moreover, we have analyzed the computational cost of the full and reduced-dimensionality GQMEs. The scaling of the computational cost is dependent on several factors, sometimes with opposite scaling trends. The TT-TFD memory kernels can provide insights on the main sources of inaccuracies of GQME approaches when combined with approximate input methods and pave the road for the development of quantum circuits that implement GQMEs on digital quantum computers.

Simulating Open Quantum System Dynamics on NISQ Computers with Generalized Quantum Master Equations

“…Starting with the quantum-mechanically exact memory kernels (adopted from ref ), the time evolution superoperator for the electronic reduced density matrix scriptG ( τ ) was generated for the four models given in Table by solving the corresponding GQME (eq ).…”

“…To this end, we take inspiration from reduced-dimensionality GQMEs, which correspond to EoMs for subsets of the open quantum system’s reduced density matrix elements rather than the full reduced density matrix. , For example, for the spin-boson model described in section , the memory kernel in the GQME for the full reduced density matrix, σ̂ ( t ) , is a 4 × 4 matrix, while the memory kernel in the GQME for only the two populations (the diagonal elements of the reduced density matrix, σ 00 ( t ) and σ 11 ( t )) is a 2 × 2 matrix. , Below we demonstrate how one can take advantage of this reduced dimensionality to lower the circuit depth and thereby improve the accuracy of the simulation on quantum machines.…”

“…22,23,84,85 It should also be noted that quantum-mechanically exact memory kernels for this model are available. 78,86,87 The spin-boson Hamiltonian has the form of eq 1 with N e = 2…”

New Density Matrix Renormalization Group Approaches for Strongly Correlated Systems Coupled with Large Environments

“…However, due to its exponential form, the accurate construction of Û ( t ) on the large MPS space is highly challenging because in principle it requires the information on not only the ground state but also a number of excited states. Therefore, different TD-DMRG algorithms have been proposed to approximate the time evolution operator Û ( t ), based on split operators, − , series expansion − or time-dependent variational principle (TDVP). ,, Among the different available schemes, nowadays, TDVP is the most popular one, as it can be efficiently applied to general Hamiltonians with/without long-range interactions. In this paper, we only discuss the implementation of TDVP in TD-DMRG.…”

“…In recent years many algorithms have been proposed to accelerate TD-DMRG, such as increasing the basis number adaptively in the single-site algorithm − and utilizing the purification-projection (PP) to restore the U (1)-particle number conservation symmetry in the phononic system , as well as using the graphical processing units (GPUs) to accelerate the heavy tensor contractions, but nowadays it is still impossible to include all environmental phononic modes into TD-DMRG. Rather than including all environmental phononic modes into the embedded active subsystem explicitly, latest developments in TD-DMRG incline for methods based on open quantum systems or developing new algorithms to identify a small number of environmental modes which have strongest couplings with the interested subsystem. − For this purpose, we recently proposed a hierarchical mapping (HM) method which uses QIT to identify a small number of renormalized environmental phononic modes directly coupled to the quantum subsystem followed by successively transforming the vibronic Hamiltonian matrix to a nearly block-tridiagonal form by the block Lanczos algorithm. Numerical tests on model spin-boson systems and realistic singlet fission models in rubrene crystal environment with up to 7000 phononic modes and strong system–environment interactions indicate the HM can reduce the size of full quantum subsystem by 1–2 orders of magnitude and accelerate the calculation by ∼80% with negligible accuracy loss.…”