Power method

The power method, or power iteration, is an algorithm for approximating the largest eigenvalue of an operator and its corresponding eigenvector. Starting from an initial MPS \(\psi_0\), each iteration applies the operator and normalizes the result:

\[\psi_{k+1} = \frac{H \psi_k}{\| H \psi_k \|}\]

Convergence to the dominant eigenvector requires that the largest eigenvalue is non-degenerate and that the initial state has nonzero overlap with it. The convergence rate depends on the spectral gap between the first two eigenvalues, so a larger gap generally yields faster convergence.

Inverse power method

To target the smallest eigenvalue of an operator, the method can be combined with operator inversion. Instead of applying \(H\) directly, we solve \(H \psi_{k+1} = \psi_k\) at each step. This approach effectively applies \(H^{-1}\) to the state, causing the eigenvector with the smallest eigenvalue to become dominant.

For a shifted operator \((H - \epsilon)\), the inverse power method converges to the eigenvalue closest to \(\epsilon\). In SeeMPS, each iteration of the inverse power method involves solving a linear system using the conjugate gradient solver, which adds computational cost compared to the standard power iteration but enables targeting of the ground state.

power_method(H[, inverse, shift, guess, ...])

Ground state search of Hamiltonian H by power method.

See also

• Gradient descent - Optimization using the energy gradient

• Restarted Arnoldi iteration - Krylov-based optimization with faster convergence

• Density-Matrix Renormalization Group - Local tensor optimization algorithm