For detailed descriptions of each method, see:

# Overview of Time-Evolution Methods for Matrix-Product States

On the individual pages of time-evolution methods for MPS, we will discuss in detail five different time-evolution methods for MPS which are currently in use to solve the time-dependent Schrödinger equation (TDSE). Each of them has different strengths and weaknesses, requiring a careful consideration of each individual problem to determine the most promising approach.

The first two methods (TEBD and MPO $W^\mathrm{II}$) approximate the time-evolution operator $\hat U(\delta) = e^{-\mathrm{i} \delta \hat H}$, which is then repeatedly applied to the MPS $\ket{\psi(t)}$ to yield time-evolved states $\ket{\psi(t+\delta)}$, $\ket{\psi(t+2\delta)}$ etc.

The time-evolving block decimation, also known as the Trotter method and abbreviated here as TEBD, was developed around 2004 both for MPS[1][2][3] and the original classical DMRG[4][5]. It uses a Trotter-Suzuki decomposition of $\hat U(\delta)$, which is in particular applicable to short-ranged Hamiltonians. The time evolution is unitary up to the inherent truncation error, but the energy is typically not conserved. The main variants are using a second-order decomposition (TEBD2 in the following) and a fourth-order decomposition (TEBD4) to minimise the error due to the finite time step. While TEBD2 is essentially always useful, as it produces only two- to three times as many exponential terms as a first-order decomposition, TEBD4 produces four to five times as many exponentials as TEBD2. Depending on the desired error, this may or may not be worthwhile.

In contrast, the MPO $W^\mathrm{II}$ method was introduced only recently[6] and relies on a different decomposition scheme for the time-evolution operator $\hat U(\delta)$ particularly suited to construct an efficient representation as a matrix-product operator. It can directly deal with long-range interactions and generally generates smaller MPOs than TEBD. Its primary downside is that the evolution is not exactly unitary.

The time step error of both TEBD and the MPO $W^\mathrm{II}$ method is larger than in the other methods described below.

Compared to these methods, algorithms based on Krylov subspaces[7][8] directly approximate the action of $\hat U(\delta)$ onto $\ket{\psi}$, i.e. produce a state $\ket{\psi(t+\delta)}$ without explicitly constructing $\hat U(\delta)$ in the full Hilbert space while preserving the unitarity of the time evolution. The main advantage lies in a very precise evaluation of $\ket{\psi(t+\delta)}$ with a very small inherent finite time-step error[9].

The global Krylov algorithm is related to the time-dependent Lanczos method in exact diagonalization approaches to time evolution[10][11][12] and has only partially[13][14][15] been adapted to the MPS setting. Interestingly, evaluation of observables on a very fine time grid (e.g.~$\delta = 10^{-5}$) is possible, which would be prohibitively expensive using any of the time-stepping methods. The downside of this global Krylov method is its need to represent potentially highly entangled Krylov vectors as matrix-product states.

These highly-entangled global Krylov vectors do not need to be represented if instead of working globally, one works in the local reduced basis. From a DMRG perspective, this can be seen as a variant of the time-step targeting method[16][17][18][19]. Its primary objective is to solve the TDSE locally on each pair of lattice sites to construct an optimal MPS for the time-evolved state. Conversely, this local method can no longer evaluate observables precisely at arbitrarily-small time steps $\delta' \ll \delta$ (as no complete MPS is available at such intermediate times) but works much like TEBD2 as a time-stepper, producing a single final state $|\psi(t+\delta)\rangle$. In contrast to TEBD and the MPO \wii it allows for the treatment of arbitrary Hamiltonians. An uncontrolled projection error may, however, lead to incorrect results as the MPS-projected Hamiltonian cannot represent the actual physical Hamiltonian well if the MPS used for the projection is very small. A further development of this approach is the time-dependent variational principle[20][21] (TDVP). The TDVP can be considered an approach to remedy the dominant errors in the local Krylov approach by a thorough theoretical analysis leading to an optimized evolution method. Its implementation is nearly identical to the local Krylov method, but the different derivation of the local equations leads to smaller errors because it arrives at a closed solution of a series of coupled equations. In particular, using the two-site variant of TDVP (2TDVP), we know that nearest-neighbor Hamiltonians do not incur a projection error which is often the primary error source in the local methods. The single-site variant (1TDVP) has a larger projection error and also always works at constant MPS bond dimension but violates neither unitarity nor energy conservation during the time evolution.

Finally, a subjective selection of useful tricks which are applicable to most of the time-evolution methods and which can help in the treatment of complicated systems may be helpful to the reader. We will discuss in some detail: (i) how to combine the time evolution in the Heisenberg and the Schrödinger picture, respectively, to reach longer times; (ii) how to select time steps to increase the accuracy of the integrator; (iii) how removing low-lying eigenstates and the application of linear prediction helps in calculating Green’s functions; (iv) how to specifically select the optimal choice of purification schemes for finite-temperature calculations; and (v) briefly summarize the local basis optimization technique to treat systems with many bosonic degrees of freedom.

## References

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