Several ground state calculations at once

In summary, the conversation discusses different methods for finding the ground states of Hamiltonian operators and proposes an approach that uses time evolution to compute the ground states of multiple Hamiltonians simultaneously. While there may not be a formal study on optimizing quantum physics or chemistry calculations in this way, the idea is not new and the accuracy of the results may vary depending on numerical errors.
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hilbert2
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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?
 
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  • #2
hilbert2 said:
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).
 
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Thanks. It's probably difficult to quantify "usefulness" of some data as that's a subjective thing, anyway.
 

Related to Several ground state calculations at once

1. What is the purpose of performing several ground state calculations at once?

The purpose of performing several ground state calculations at once is to compare and contrast the results of different methods and determine the most accurate and reliable approach for a given system. This can also help to identify any inconsistencies or errors in the calculations.

2. How does performing multiple ground state calculations improve the accuracy of the results?

By performing multiple calculations, it is possible to average out any errors or variations that may occur in individual calculations. This can help to provide a more accurate representation of the ground state energy and other properties of a system.

3. Can different methods be used for the simultaneous ground state calculations?

Yes, different methods can be used for simultaneous ground state calculations. This allows for a comprehensive analysis of the system using various techniques and can provide a more complete understanding of the system.

4. How do you choose which methods to use for simultaneous ground state calculations?

The choice of methods for simultaneous ground state calculations depends on the specific properties and characteristics of the system being studied. It is important to consider the strengths and limitations of each method and choose a combination that can provide the most accurate and reliable results.

5. Are there any potential challenges or limitations when performing several ground state calculations at once?

Yes, there can be potential challenges and limitations when performing several ground state calculations at once. These may include computational limitations, difficulty in comparing results from different methods, and the potential for inconsistencies in the calculations. It is important to carefully consider these factors and address any potential issues to ensure accurate and reliable results.

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