Unifying time evolution and optimization with matrix product states

We show that the time-dependent variational principle provides a unifying framework for time-evolution methods and optimization methods in the context of matrix product states. In particular, we introduce a new integration scheme for studying time evolution, which can cope with arbitrary Hamiltonians, including those with long-range interactions. Rather than a Suzuki-Trotter splitting of the Hamiltonian, which is the idea behind the adaptive time-dependent density matrix renormalization group method or time-evolving block decimation, our method is based on splitting the projector onto the matrix product state tangent space as it appears in the Dirac-Frenkel time-dependent variational principle. We discuss how the resulting algorithm resembles the density matrix renormalization group (DMRG) algorithm for finding ground states so closely that it can be implemented by changing just a few lines of code and it inherits the same stability and efficiency. In particular, our method is compatible with any Hamiltonian for which ground-state DMRG can be implemented efficiently. In fact, DMRG is obtained as a special case of our scheme for imaginary time evolution with infinite time step.

Figure 1

An arbitrary MPS tensor is represented as a shape with fat legs corresponding to the virtual indices of dimension $D$ and a normal leg corresponding to the physical index of dimension $d$ (a). The left-orthonormal [right-orthonormal] MPS tensors are represented using triangles (b) [(c)] and satisfies the condition (d) [(e)]. An MPS $|\mathrm{\Psi }\left[A\right]\rangle$ using general tensors (f) can be brought into a mixed-canonical form with a one-site center $A_C\left(n\right)$ at site $n$ (g) or even a zero-site center $C\left(n\right)$ (h), whose singular values correspond to the Schmidt coefficients of the state.

Figure 2

(a) The one-site effective Hamiltonian $H\left(n\right)$ plays a central role in DMRG and the current TDVP integrator, and can be efficiently computer when $\widehat{H}$ is, for example, represented as a matrix product operator. (b) Similarly, one can define a zero-site effective Hamiltonian $K\left(n\right)$, which can easily be computed from $H\left(n\right)$.

Figure 3

Right-hand side (up to the factor $-i)$ of the TDVP equation [Eq. (3)].

Figure 4

Right-hand side (up to the factor $-i)$ of the TDVP equation [Eq. (3)].

Figure 5

Absolute value of $\langle {\mathrm{\Psi }}_0\left|U^{†}{\sigma }_r^x\left(t\right)U\right|{\mathrm{\Psi }}_0\rangle -\langle {\mathrm{\Psi }}_0\left|{\sigma }_r^x\left(t\right)\right|{\mathrm{\Psi }}_0\rangle$ with $H$ the XY Hamiltonian in Eq. (9), $|{\mathrm{\Psi }}_0\rangle$ the ground state of $H$ and $U=exp(\mathrm{i}{\sigma }_{51}^y\pi /4)$ on a chain of $N=101$ spins, as function of $r$ and $t$ for various values of $\alpha$.