Gradient descent#
This is a very simple iterative algorithm to solve the problem of minimizing the energy associated to a Hamiltonian \(H\), over the space of matrix-product states \(\psi\). In other words, given the definition
\[E[\psi] := \frac{\langle{\psi|H|\psi}\rangle}{\langle{\psi|\psi}\rangle},\]
we want to find \(\mathrm{argmin}_\psi E[\psi]\). The gradient of this functional is simply
\[\nabla E[\psi] = (H - E[\psi])|\psi\rangle := \tilde{H}|\psi\rangle.\]
The algorithm proceeds in discrete steps, where given a state \(\psi_k\), it finds the next state that minimizes the energy along the gradient direction:
\[|\psi_{k+1}\rangle \propto |\psi_k\rangle - a \tilde{H}|\psi_k\rangle,\]
where
\[a = \mathrm{argmin} E[(1- a \tilde{H})\psi_k]\]
The optimum of this descent is given by
\[a = \frac{E_3 - \sqrt{E_3^2 + 4 \Delta H^6}}{2 \Delta H^2}\]
with the definitions
\[\Delta H^2 = \langle\psi|(H-E[\psi])^2|\psi\rangle,\;
E_3 = \langle\psi|(H-E[\psi])^3|\psi\rangle.\]
This formulation, used in Ref. García-Molina et al. [GMTGR24], is implemented by the function gradient_descent().
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Ground state search of Hamiltonian H by gradient descent. |