Traditional single-site density matrix renormalization group (DMRG) does not allow bond expansion; multi-site DMRG does but at much higher computational costs. For matrix product states (MPS), we present a controlled bond expansion (CBE) algorithm that yields two-site accuracy at single-site costs.

Given an MPS, $\Psi$, with a confined variational space, CBE explores its orthogonal subspaces and finds components carrying significant weight in $H \Psi$ to enrich it. This CBE approach in contrast with the previous perturbative expansion approaches is fully variational. Moreover, to simulate quantum dynamics using the time-dependent variational principle (TDVP), the same strategy applies. CBE alleviates the numerical difficulties of the standard, fixed-rank one-site TDVP integrator, and thereby greatly extends the time window over which time evolution can be simulated accurately and reliably.

The implementation of CBE–DMRG/TDVP does not require a high degree of technical sophistication. Hence, existing, optimized MPS codes can be easily adapted. We will illustrate the performance of CBE with several numerical examples on finite quantum lattices which carry interesting physics.

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