Displaced Fermionic Gaussian States and their Classical Simulation

This work is inspired by our previous work on fermionic convolution. We found that fermionic convolution retains desirable properties only for a subclass of special even quantum states. The even constraint also makes the math much easier work with, and previous works on fermionic Gaussian theory has predominantly focused on the zero-mean (even) case.

Many important questions are not well understood about the general displaced Gaussian states. We were originally motivated to answer them to obtain a general theory of fermionic convolution, but the study itself was bulky enough to warrant this separate work. The following motivating questions are answered in this work:

1. How to describe the action of displaced fermionic unitaries?
2. Zero-mean Gaussian states have a nice unified characterization from physical, computational, and Fourier perspectives. Do displaced Gaussian states have the same coherent definition?
3. The physical transformations of zero-mean fermionic Gaussian unitaries correspond to computational circuits of nearest-neighbor matchgates. What circuits to displaced Gaussian unitaries correspond to?
4. Can they be efficiently classically simulated, and how?
5. How to build a bridge between results which work for even operators (e.g. fermionic convolution) and the general case without parity constraint?

Building on a Lie algebra reduction identified by Knill in 2001, we obtain the following results in response to the questions above:

1. The extra mean of the fermionic Gaussian unitary effectively acts as covariance on one more mode.
2. Displaced Gaussian states have a unified characterization from physical, computational, and Fourier perspectives, generalizing the even Gaussian case.
3. Displaced Gaussian unitaries are equivalent to nearest-neighbor matchgates plus arbitrary single-qubit gate on the first line.
4. Efficient simulation can be conducted using Grassmann Gaussian integral overlap.
5. We proposed a unitary embedding of displaced Gaussian states and unitaries into even Gaussian counterparts.

Apart from its immediate contribution to fermionic Gaussian theory, this work sheds light on how super-selection constraints presents themselves mathematically through commutativity requirements, and provides one valuable example to generalize the existing theory beyond the parity constraint. The fermionic example is an important case-study for the more general Gaussian theories of parafermions, or even abelian and non-abelian anyons we wish to later pursue.