Filter functions are a simple computational heuristic employed to determine the sensitivity of an arbitrary control operation on a single or multi-qubit system to time-dependent noise.  

The filter function itself is represented as a compact graphical object, similar to a Bode plot in classical engineering.  

This formalism is efficient and effective for performance evaluation of complex controls as the average infidelity is approximated as the overlap integral of a noise power spectrum and the associated filter function.  

Tip:  Look for filters that are sloped downwards towards lower frequencies


Most filter functions behave as high-pass filters, in that they suppress low-frequency noise (correlated noise) in average Hamiltonian theory.  The slope of the filter function is a measure of the filter order; steeper slopes provide more effective noise suppression in the stop-band.  However, in Black Opal a control can also be optimized to possess a filter function which specifically targets certain frequency ranges where the noise is strong.

Above we show, in a simplified graphic, a visual representation of the filter function for different controls implementing a driven rotation.  In all cases the filter function is represented vs frequency. The standard or primitive control shows a flat response down to DC while noise-filtering controls decrease to the left of the graph. The shaded purple area represents the relative improvement in the noise susceptibility associated with using an effective noise filter. 

This entire framework is laboratory validated.  In the experiment below, performed using trapped ions, we see good agreement between experimental measurements of control performance and the predicted fidelity calculated using filter functions (down to measurement fidelity limits given by the shading).  In this plot we are studying both primitive and noise-suppressing driven controls found in the Black Opal package, and achieve this agreement using no free parameters.  Note that the shapes mimic what you see above - noise suppressing controls slope downwards while the primitive is flat across a wide range of frequencies.

Details on how the filter functions are calculated for single or multi-qubit operations in Black Opal are available from our technical documentation.

Primary references for the filter function formalism include:

Todd J. Green, Jarrah Sastrawan, Hermann Uys, and M.J. Biercuk, “Arbitrary quantum control of qubits in the presence of universal noise.”  New Journal of Physics 15, 095004 (2013).
General reference introducing the mathematical formalism

A. Soare, H. Ball, D. Hayes, M. C. Jarratt, J.J. McLoughlin, X. Zhen, T.J. Green and  M.J. Biercuk, “Experimental noise filtering by quantum control” Nature Physics 10, 825-829 (2014).
Experimental validation of the filter function formalism for non-trivial single-qubit unitaries

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