DMRG (Overview)

The density-matrix renormalization group (DMRG), introduced by Steven White in 1992, is the most successful numerical method for finding ground states of one-dimensional quantum systems. In its modern formulation it is a variational algorithm over matrix product states: it searches the space of MPS with bounded bond dimension for the one that minimises the energy.

The variational principle it exploits

For any Hamiltonian $H$ and any normalised state $|\psi\rangle$ , the energy expectation value bounds the ground-state energy from above:

E[\psi] = \langle \psi | H | \psi \rangle \ge E_0 ,

with equality only when $|\psi\rangle$ is the ground state. DMRG restricts $|\psi\rangle$ to be an MPS of bond dimension $\chi$ and minimises $E[\psi]$ over all such MPS. Because gapped 1D ground states obey an area law (previous lesson), a modest $\chi$ already contains an excellent approximation to the true ground state — the variational class is small but well-chosen.

Sweeping: optimise one tensor at a time

Minimising over all the MPS tensors at once is non-linear and hard. DMRG's trick is to optimise one (or two) site tensors at a time, holding the rest fixed. With the neighbours frozen, the energy is a quadratic function of the active tensor, and minimising it reduces to a standard eigenvalue problem:

H_{\text{eff}}\, A^{[k]} = E\, A^{[k]} ,

where $H_{\text{eff}}$ is the Hamiltonian projected onto the variational neighbourhood of site $k$ . You solve this small eigenproblem (e.g. with Lanczos) for its lowest eigenvalue, replace $A^{[k]}$ with the eigenvector, then move to the next site. Sweeping left-to-right and back, repeatedly, drives the energy down monotonically until it converges.

Two-site DMRG and adaptive bond dimension

The standard two-site variant optimises a merged pair of neighbouring tensors, then splits them with an SVD (just like applying a two-qubit gate to an MPS). The SVD step lets the algorithm grow or shrink the bond dimension adaptively, keeping enough singular values to hit a target truncation error. This is what makes DMRG robust: it discovers how much entanglement the ground state needs rather than requiring you to guess $\chi$ in advance.

Cost and reach

A DMRG sweep costs $O(n\,\chi^3\,d)$ per pass for local Hamiltonians (with $d$ the physical dimension per site, $d = 2$ for qubits), and only a handful of sweeps are typically needed. This polynomial scaling is why DMRG routinely handles chains of hundreds or thousands of sites with chemical-level accuracy — wildly beyond exact diagonalisation, which is capped near $2^{20}$ basis states.

Where it shines and where it strains

DMRG is the gold standard for gapped 1D ground states, and it extends usefully to gapless 1D systems (with larger $\chi$ ), narrow 2D strips (mapped to a 1D chain with long-range terms), and ground states in quantum chemistry. It strains exactly where the area law breaks: genuinely 2D systems, critical points, and dynamics that build up volume-law entanglement. Recognising those limits — the subject of the next lessons — is as important as knowing the method works.