Abstract
We exploit insights into the geometry of bosonic and fermionic Gaussian states to develop an efficient local optimization algorithm to extremize arbitrary functions on these families of states. The method is based on notions of gradient descent attuned to the local geometry which also allows for the implementation of local constraints. The natural group action of the symplectic and orthogonal group enables us to compute the geometric gradient efficiently. While our parametrization of states is based on covariance matrices and linear complex structures, we provide compact formulas to easily convert from and to other parametrization of Gaussian states, such as wave functions for pure Gaussian states, quasiprobability distributions and Bogoliubov transformations. We review applications ranging from approximating ground states to computing circuit complexity and the entanglement of purification that have both been employed in the context of holography. Finally, we use the presented methods to collect numerical and analytical evidence for the conjecture that Gaussian purifications are sufficient to compute the entanglement of purification of arbitrary mixed Gaussian states.
Highlights
Gaussian states form one of the most prominently used and best understood families of quantum states
We have presented a geometric approach to optimize over arbitrary differentiable functions on the manifolds of pure bosonic or fermionic Gaussian states
Our method is based on the well-known gradient descent algorithm, but exploits the natural action of a Lie group onto these manifolds to move between different Gaussian states
Summary
Gaussian states form one of the most prominently used and best understood families of quantum states. The standard definition covers bosonic [1,2,3] and fermionic [4] Gaussian states both pure and mixed They naturally appear as ground and thermal states of quadratic Hamiltonians in physical systems and are ubiquitous in non-interacting quantum many-body systems in the condensed matter context and as vacua in free field theories. The inverse metric Gμν needs to be re-evaluated at every point of the manifold, but for Gaussian states we can explicitly construct a basis in which the matrix representation of Gμν is constant This provides a crucial speedup of the underlying algorithm. The package can be downloaded from our arXiv submission This manuscript is structured as follows: In section 2, we review a unified formalism to treat pure bosonic and fermionic Gaussian states and compute the resulting Kähler geometry (positive-definite metric, symplectic form) on the resulting state manifold.
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