Drawing together control landscape and tomography principles

Abstract

The ability to control quantum systems using shaped fields as well as to infer the states of such controlled systems from measurement data are key tasks in the design and operation of quantum devices. Here we associate the success of performing both tasks to the structure of the underlying control landscape. We relate the ability to control and reconstruct the full state of the system to the absence of singular controls, and show that for sufficiently long evolution times singular controls rarely occur. Based on these findings, we describe a learning algorithm for finding optimal controls that makes use of measurement data obtained from partially accessing the system. Open challenges stemming from the concentration of measure phenomenon in high dimensional systems are discussed.

From Pulses to Circuits and Back Again: A Quantum Optimal Control Perspective on Variational Quantum Algorithms

“…Extensive QOC simulation and laboratory studies indicate the relative ease of satisfying all three assumptions [44]. Furthermore, it has been shown that assumptions (1) and ( 3) are almost always satisfied in a measure theoretic sense [68,110,111], thereby implying that QOC landscapes are almost always free of local minima under the premise of sufficient control field resources [112]. While the precise meaning of "sufficient" is application-dependent and remains an open challenge to systematically assess, it has recently been shown that local surjectivity is almost always met when the control fields allow for approximating Haar random evolutions [111,113] within the time interval of interest: [0, T ].…”

“…Furthermore, it has been shown that assumptions (1) and ( 3) are almost always satisfied in a measure theoretic sense [68,110,111], thereby implying that QOC landscapes are almost always free of local minima under the premise of sufficient control field resources [112]. While the precise meaning of "sufficient" is application-dependent and remains an open challenge to systematically assess, it has recently been shown that local surjectivity is almost always met when the control fields allow for approximating Haar random evolutions [111,113] within the time interval of interest: [0, T ]. As such, even though performing the optimization over a set of parametrized control fields does not avoid the non-convexity of the problem, provided the assumptions (1)-(3) hold, the interior of the optimization landscape provably contains saddles only.…”

“…We also remark that as the dimension d of the quantum system increases the landscape flattens out, such that for certain objective functionals, gradients become exponentially small as a function of the number of qubits, making these regions difficult to leave with local optimization algorithms. These so-called barren plateaus [19] are a consequence of a concentration of measure phenomenon, and occur in both VQA implementations and QOC [111]. Understanding how to avoid them in each context independently could yield useful and transferable techniques benefiting both areas of study, and will likely involve careful design of ansatz, initialization, and training methods, e.g., ref.…”