Ground state of Gaussian Hamiltons

Photonic Software Strawberry Fields
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Hello, I am trying to find a way to compute the covariance matrix and vector of first moments of the ground state of an arbitrary Gaussian Hamiltonian of an N-mode system such as

\hat{H}=\sum_{i=1}^N c_i\hat{q_i}^2 + d_i\hat{p_i}.

Does Pennylane or Starberryfields have a method that can accomplish this task? And if not are there any resources that explain how to compute the ground state of a Gaussian Hamiltonian? The problem should be able to be solved efficiently classical since the Hamiltonian is Gaussian but I haven’t found much information on it beyond the Williamson decomposition which I don’t think is what is needed to solve the problem. Thanks in advance for any help!

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Hi @Amanuel ,

There’s a Xanadu library called MrMustard, which may be able to do what you’re looking for. It has a function for calculating the covariance matrix of a Gaussian state.

You can also find the first moments by calculating the means as shown in this example in the docs.

For a general reference on how to use MrMustard I recommend following the Basic API Reference.

I hope this helps!

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Hi @CatalinaAlbornoz, thank you for the reply and resources! However, MrMustard only seems to compute the covariance matrix of a Gaussian state whose symplectic matrix you already know. What I am trying to do is actually find the ground state (which is some Gaussian state) of a Gaussian Hamiltonian and its corresponding covariance matrix and vector of first moments. Does that make sense?

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Hi @Amanuel ,

Unfortunately I don’t think we have any function or workflow to do what you’re looking for.

You could try creating a Hamiltonian in PennyLane and using the default.gaussian device.
You can also try using squeezed_cov if you have a squeezed state.

I don’t think this is exactly what you need but it’s all that I can think of.

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@CatalinaAlbornoz Thank you for the references.

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