The filter function formalism was originally introduced to describe the decay of phase coherence under dynamical decoupling sequences. The formalism was later extended to fidelities of gate operations for single or multiple qubits using the Magnus expansion. In recent years, people became interested in the spectral characterization of environmental noise and cross-correlated properties during quantum processes. In this presentation, I will shortly summarize the history of filter function formalism and discuss an examplary work on two-qubit spectroscopy experiment conducted at 2020.
The first use of the term "filter function" in this context appears to be in a paper by Uhrig [17] published in 2007. However, similar concepts were already used by other authors before that [18–20].
(1) Filter-function formalism and software package to compute quantum .... https://link.aps.org/doi/10.1103/PhysRevResearch.3.043047 (2) Accessing the Full Capabilities of Filter Functions: A Tool for .... https://arxiv.org/abs/2303.01660
The filter function formalism is a useful tool in quantum control for designing control fields that can steer the system towards a desired target state. This formalism involves defining a filter function which characterizes the response of the system to the control field and is a complex-valued function. Specifically, the filter function is defined as the ratio of the Fourier transform of the desired output state to the Fourier transform of the current state. By applying a control field that is proportional to the inverse Fourier transform of the filter function, we can robustly manipulate the system towards the target state.
(1) Preparation - dynamics - measurements
E.g General experiment, quantum computing
(2) Drawbacks: Noisy, degrees of freedom matter
(3) This presentation, Filter function: Focuses on the noisy dynamics
$$ \rho_{SE}=\rho_{S}\otimes\rho_{E} $$
Noise act on dynamics
Property of Noise: X(t)