An **Ansatz **is a starting point or basic approach to a problem. In mathematics, it is a statement not based on any formal principle or theory. It’s a guess or approach (especially with reference to an unknown function) that you test; it can also be an additional assumption or piece of information that you add, such as adding a prior in Bayesian statistics.

This educated guess leads to a verifiable result. If you can’t verify the result, you start over.

“If you don’t like the answer, change the question” ~ (Martin, 2004)

The ansatz is a starting point for a more precise solution. It is not necessarily the correct solution, but it can be used to guide the search for a better solution.

## Ansatz vs Hypothesis

In some ways, the ansatz is similar to hypothesis testing in statistics: you state a problem, and test your theory. The main difference though is that in hypothesis testing, you’re taking a *known hypothesis * (something that’s generally accepted as true) and you’re trying to falsify it (i.e. reject the null hypothesis). With an ansatz, you’re taking something you know is probably *not *true (at least, not when taken as a whole). Then along the way, you tweak it, hopefully leading to a true result. The concept then, is probably closer to priors in Bayesian theory; a prior is a “best guess” about a situation, before you actually have any evidence. Once you gather evidence, the prior gets updated.

## When is it NOT an Ansatz?

Let’s say you were trying to find a solution to a differential equation. These equations can’t be solved by a single method, so you could use a **guess and check **method. If your guess and check results in a solution for the differential equation, then you’ve just used an ansatz. Note though, that your initial guess *can’t be based on a known theory or principle*. So if you’re following, for example, the separation of variables method, that isn’t an ansatz.

## Ansatz Examples

- The
**Bethe Ansatz**is an exact method (developed by Cornell physicist Hans Bethe) for calculating eigenvalues and eigenvectors for a select class of quantum many-body systems. It is a way to solve the quantum inverse scattering problem — a set of equations that describe the scattering of particles in a one-dimensional system. The Bethe ansatz is an ansatz for the wavefunction of the particles, and it has been used to solve a wide variety of problems in quantum physics, including the Heisenberg spin chain and the Hubbard model. - The coupled cluster ansatz is a method for approximating the ground state of a many-body quantum system. The ansatz is based on a product of exponentials of one-body operators, and it has been used to study a wide variety of systems, including atoms, molecules, and solids.
- The variational autoencoder is a type of machine learning model that can be used to learn the latent structure of a dataset. The model is an ansatz for the probability distribution of the data, and it can be used to generate new data that is similar to the training data.

## References

Martin, R. Electronic Structure Basic Theory and Practical Methods. Cambridge University Press, 2004.

Müller, G. “Introduction to the Bethe….” https://aip.scitation.org/doi/pdf/10.1063/1.4822511

Sandia National Laboratories. Statistical modeling for quantum information processing.