quspin.tools.lanczos.expm_lanczos

quspin.tools.lanczos.expm_lanczos(E, V, Q_T, a=1.0, out=None)[source]

Calculates action of matrix exponential on vector using Lanczos algorithm.

The Lanczos decomposition (E,V,Q) with initial state v0 of a hermitian matrix A can be used to compute the matrix exponential \(\mathrm{exp}(aA)|v_0\rangle\) applied to the quantum state \(|v_0\rangle\), without actually computing the exact matrix exponential:

Let \(A \approx Q T Q^\dagger\) with \(T=V \mathrm{diag}(E) V^T\). Then, we can compute an approximation to the matrix exponential, applied to a state \(|\psi\rangle\) as follows:

\[\exp(a A)|v_0\rangle \approx Q \exp(a T) Q^\dagger |v_0\rangle = Q V \mathrm{diag}(e^{a E}) V^T Q^\dagger |v_0\rangle.\]

If we use \(|v_0\rangle\) as the (nondegenerate) initial state for the Lanczos algorithm, then \(\sum_{j,k}V^T_{ij}Q^\dagger_{jk}v_{0,k} = \sum_{j}V_{ji}\delta_{0,j} = V_{i,0}\) [by construction, \(|v_{0}\rangle\) is the zero-th row of \(Q\) and all the rows are orthonormal], and the expression simplifies further.

Parameters:

E(m,) np.ndarray

eigenvalues of Krylov subspace tridiagonal matrix \(T\).

V(m,m) np.ndarray

eigenvectors of Krylov subspace tridiagonal matrix \(T\).

Q_T(m,n) np.ndarray, generator

Matrix containing the m Lanczos vectors in the rows.

ascalar, optional

Scale factor a for the generator of the matrix exponential \(\mathrm{exp}(aA)\).

out(n,) np.ndarray()

Array to store the result in.

Returns:

(n,) np.ndarray

Matrix exponential applied to a state, evaluated using the Lanczos method.

Notes

• uses precomputed Lanczos data (E,V,Q_T), see e.g., lanczos_full and lanczos_iter functions.

• the initial state v0 used in lanczos_full and lanczos_iter is the state the matrix exponential is evaluated on.

Examples

>>> E, V, Q_T = lanczos_iter(H,v0,20)
>>> expH_v0 = expm_lanczos(E,V,Q_T,a=-1j)