Can controllers for quantum systems beat classical limits of robustness?

Amongst the established fundamental facts of classical linear single-degree-of-freedom (SDoF) multivariable control, one stands out. This unique law dictates that the two fundamental figures of merit, i.e., the logarithmic sensitivity to model uncertainty and the tracking error, are in conflict. Some general studies have already established that the SDoF drawback can be resolved by a 2–degree-of-freedom configuration. Specifically in quantum control, selective information transmission in spin ring networks by means of energy landscape shaping control has the unique attribute that the error, 1-prob, where prob is the transfer success probability, and the sensitivity of the error to spin coupling uncertainties are statistically increasing across a family of controllers of increasing error. However, the noisy behavior of the sensitivity versus the error as a consequence of the optimization of the controllers in a challenging error landscape has made it imperative to develop a statistical hypothesis test of a concordant trend.

Recently, Professor Edmond Jonckheere from the Department of Electrical Engineering, University of Southern California in collaboration with Dr. Sophie Schirmer at Swansea University and Dr. Frank Langbein at Cardiff University cross-examined the concordant trend between the error and the measure of performance, i.e., the logarithmic sensitivity, used in robust control to formulate a well-known fundamental limitation. Fundamentally, they investigated whether the outlined fundamental limitations could survive in the quantum world. Their work is currently published in the International Journal of Robust Nonlinear Control.

Briefly, the research method employed entailed: first, undertaking a detailed review of the spin network concept and the single excitation subspace, where the three researchers defined the quantum excitation transport as the problem of having the solution to Schrödinger’s equation move from an initial state of excitation to a target state of excitation. Next, the researchers contrasted the quantum excitation transport with classical tracking control. As a substitute to the analytical method, they proceeded to introduce two statistical rank correlation tests, i.e., the Kendall τ and the Jonckheere-Terpstra tests, which they utilized to assess whether the error and the logarithmic sensitivity were positively correlated. Eventually, they employed the formal statistical method and also introduced the Type II error in the test.

The authors observed that contrary to error versus sensitivity, the error-versus-logarithmic-sensitivity trend was less obvious due to the amplification of the noise because of the logarithmic normalization. Additionally, this observation instigated that the Kendall τ test for rank correlation between the error and the log sensitivity was in a way pessimistic with marginal significance levels. Other observations as recorded from the Jonckheere-Terpstra test were seen to alleviate the inherent statistical problem encountered in the Kendall τ test. Furthermore, further analysis showed that for transfers between nearby spins, the classical limitation did not hold, whereas it tends to be recovered for transfers between distant spins.

In summary, Edmond Jonckheere and colleagues presented the successful development of a statistical approach based on a sample set of numerically optimized controllers for a quantum transport problem. To actualize their study, they constructed a fairly large dataset of locally optimal controllers, arranged by increasing order of their transfer errors after which they investigated whether the logarithmic sensitivity anticlassically increased with the error using the two tests stated earlier. It was seen that both the anticlassical behavior and to a lesser extent the classical behavior were present in the dataset investigated and were associated to some factors. Altogether, their work identifies cases of the concordant trend between the error and the logarithmic sensitivity, i.e., a highly anticlassical feature that goes against the well-known sensitivity versus the complementary sensitivity limitation.