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.


Can controllers for quantum systems beat classical limits of robustness?, Advances in Engineering

chains and rings Advances in Engineering

Can controllers for quantum systems beat classical limits of robustness?, Advances in Engineering

Feedback_structure Advances in Engineering

Can controllers for quantum systems beat classical limits of robustness?, Advances in Engineering
Logarithmic sensitivity versus error plots (plots are in logarithmic scale for better clarity). Observe that the error and the logarithmic sensitivity are concordant, in defiance of the classical control limitation calling for trade-off between such classically conflicting objectives. (The abscissa is the controller index.)


About the author

Edmond Jonckheere received his Ph. D. in Electrical Engineering from the University of Southern California in 1978. He is currently a (full) Professor of Electrical Engineering and Mathematics at the University of Southern California, where he is a member of the Center for Applied Mathematical Sciences and the Center for Quantum Information Science and Technology. He has had short-term academic appointments with the Australian National University, Canberra, Australia, the University of Namur, Belgium, the Max-Planck Institute, Gottingen, Germany, and Cardiff University, UK. His industrial experience includes Lockheed, Philips, and Honeywell.

His current research interests include quantum control, wireless networking and energy systems, and biomedical signal processing. Edmond Jonckheere is a Life Fellow of the Institute of Electrical and Electronic Engineers (IEEE) and an Associate Member of the IOP, Institute of Physics, London, UK.

About the author

Sophie Schirmer is an Associate Professor in Physics at Swansea University, Wales, UK. Sophie Schirmer previously held positions as a Cambridge-MIT (CMI) Research Fellow and Advanced Research Fellow of the UK Engineering and Physical Sciences Research Council (EPSRC) at the University of Cambridge, as a Marie Curie Visiting Professor at Kuopio University, Finland, as well as research positions at the Open University and the University of Oregon. Sophie Schirmerʼs research interests range from nano-science at the quantum edge and quantum engineering, especially the design, modeling and characterization of quantum devices and new paradigms for optimal and robust control, to medical physics and imaging.

About the author

Frank C. Langbein received his Mathematics degree from Stuttgart University, Germany, in 1998 and a PhD on “Beautification of Reverse Engineered Geometric Models” from Cardiff University, Wales, UK, in 2003. He is currently a senior lecturer at the School of Computer Science and Informatics, Cardiff University where he is a member of the Visual Computing research group and leads the Quantum Technologies and Engineering research priority area.

His research interests include modeling, simulation, control and machine learning applied to quantum technologies, geometric modeling, computer graphics, computer vision and healthcare problems. He is a member of the Institute of Electrical and Electronics Engineers and the American Mathematical Society.


E. Jonckheere, S. Schirmer, F. Langbein. Jonckheere-Terpstra test for non-classical error versus log-sensitivity relationship of quantum spin network controllers. International Journal of Robust Nonlinear Control. 2018; volume 28: page 2383–2403.

Go To International Journal of Robust Nonlinear Control

Check Also

On the trade-off between timeliness and accuracy for low voltage distribution system grid monitoring utilizing smart meter data