Fig. 10: A time-dependent MPS \(\left\vert \Psi (t)\right\rangle\) with fixed bond dimensions χ can be viewed as a point on a manifold \({{{{\mathcal{M}}}}}_{\chi }\in {{{\mathcal{H}}}}\).
From: Large-scale stochastic simulation of open quantum systems
The time-evolution caused by \(-iH\left\vert \Psi (t)\right\rangle\) will generally leave the manfiold, requiring a larger bond dimension to represent. TDVP projects this time-evolution to the tangent space of the original manifold (shown by the projector \({{{{\mathcal{P}}}}}_{{{{{\mathcal{M}}}}}_{\chi },\left\vert \Psi (t)\right\rangle }\)). Each point in the tangent space can be described as a linear combination of partial derivatives of the original point such that by solving a set of site-wise coupled differential equations, we can evolve purely on the original manifold and limit the growth in bond dimension to represent \(\left\vert \Psi (t+\delta t)\right\rangle\).
