Characterization of the critical sets of quantum unitary control landscapes

Jason M. Dominy, Tak San Ho, Herschel A. Rabitz

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


This work considers various families of quantum control landscapes (i.e. objective functions for optimal control) for obtaining target unitary transformations as the general solution of the controlled Schrödinger equation. We examine the critical point structure of the kinematic landscapes JF (U2) = ||(U -W)A||2 and JP (U) = ||A||4 - |Tr(AA†W†U)|2 defined on the unitary group U(H) of a finite-dimensional Hilbert space H. The parameter operator A ∈ B(H) is allowed to be completely arbitrary, yielding an objective function that measures the difference in the actions of U and the target W on a subspace of state space, namely the column space of A. The analysis of this function includes a description of the structure of the critical sets of these kinematic landscapes and characterization of the critical points as maxima, minima, and saddles. In addition, we consider the question of whether these landscapes are Morse-Bott functions on U (H). Landscapes based on the intrinsic (geodesic) distance on U (H) and the projective unitary group PU (H) are also considered. These results are then used to deduce properties of the critical set of the corresponding dynamical landscapes.

Original languageEnglish (US)
Article number6807674
Pages (from-to)2083-2098
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number8
StatePublished - Aug 2014

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


  • Optimization
  • quantum control
  • quantum information


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