Graduate Several ground state calculations at once

[hilbert2]

Suppose I want to find the ground states corresponding to several Hamiltonian operators $\left\{ \hat{H}_i \right\}$, which are similar to each other. As an example, let's take the $\hat{H}_i$:s to be anharmonic oscillator Hamiltonians, written in nondimensional form ($\hbar = m = 1$) as

$\hat{H}_1 = -\frac{1}{2}\frac{\partial^2}{\partial x^2} + x^2 + 0.10x^3$
$\hat{H}_2 = -\frac{1}{2}\frac{\partial^2}{\partial x^2} + x^2 + 0.20x^3$
$\hat{H}_3 = -\frac{1}{2}\frac{\partial^2}{\partial x^2} + x^2 + 0.30x^3$

Now, one way to solve the ground states would be to form the evolution operators $\hat{U}_i (\Delta t) = e^{-i\hat{H}_ i \Delta t}$ and propagate a given initial trial state over a large imaginary time interval $\Delta t = is$, $s\in\mathbb{R}$.

A dumb way to do this would be to do three computations, using the same initial trial function in all of them and just use different evolution operators. A smarter way would obviously be to first find the approximate ground state of $\hat{H}_1$, and then use that as the trial function for computing the ground state of $\hat{H}_2$ and so on, as the ground states are probably more similar to each other than to any randomly chosen trial function.

But what about first calculating the ground state of $\hat{H}_1$, and then propagating that in real time with a time dependent Hamiltonian $\hat{H}_{td}(t) = -\frac{1}{2}\frac{\partial^2}{\partial x^2} + x^2 + \left(0.10 + \frac{0.20t}{T}\right) x^3$, where $T$ is a large quantity, through the time interval $t\in [0,T]$ ? By the adiabatic theorem, if the time $T$ is large enough, the state $\Psi (x,t)$ at moment $t$ will approximately be the ground state of $\hat{H}_{td} (t)$. So, the calculation would not only produce the approximate ground states of $\hat{H}_1 ,\hat{H}_2$ and $\hat{H}_3$, but also the ground states of many Hamiltonians "lying between them", because the time evolution would continuously take the state from an eigenstate of $\hat{H}_1$ to an eigenstate of $\hat{H}_3$.

This would be a bit similar to doing many Monte Carlo calculations at the same time and using the same random numbers in all of them, to save processing time by having to use your rand. num. generator less times.

Has there been any formal study on optimizing a quantum physics or chemistry calculation in the sense of producing as much useful information as possible at the same time per unit of computation hours?

 

[DrClaude]

Has there been any formal study on optimizing a quantum physics or chemistry calculation in the sense of producing as much useful information as possible at the same time per unit of computation hours?

I don't remember seeing any formal study of this, but the idea is not new. The quality of the result will of course depend on numerical errors, which might be greater for the time evolution approach (where there is an accumulation from time step to time step).

 

[hilbert2]

Thanks. It's probably difficult to quantify "usefulness" of some data as that's a subjective thing, anyway.