We discover and characterise strong quantum scars, or quantum eigenstates resembling classical periodic orbits, in two-dimensional quantum wells perturbed by local impurities. These scars are not explained by ordinary scar theory, which would require the existence of short, moderately unstable periodic orbits in the perturbed system. Instead, they are supported by classical resonances in the unperturbed system and the resulting quantum near-degeneracy. Even in the case of a large number of randomly scattered impurities, the scars prefer distinct orientations that extremise the overlap with the impurities. We demonstrate that these preferred orientations can be used for highly efficient transport of quantum wave packets across the perturbed potential landscape. Assisted by the scars, wave-packet recurrences are significantly stronger than in the unperturbed system. Together with the controllability of the preferred orientations, this property may be very useful for quantum transport applications.
Abstract. The work presented here extends upon the best known universal quantum circuit, the Quantum Shannon Decomposition proposed in [Vivek V. Shende, Stephen S. Bullock and Igor Markov, Synthesis of Quantum Logic Circuits, IEEE : 1000-1010 (2006)]. We obtain the basis of the circuit's design in a pair of Cartan decompositions. This insight gives a simple constructive algorithm for obtaining the Quantum Shannon Decomposition of a given unitary matrix in terms of the corresponding Cartan involutions.
In principle, learning can be increased by assessing the detailed state of student knowledge and mistaken knowledge with a pre-test and then optimizing instruction as measured by the post-test score. As a first step in this direction, we applied a Multidimensional Item Response Theory (MIRT) to 17,000 pre-instruction administrations of the Force Concept Inventory (FCI) to study students' initial knowledge in detail. Examination of Item Response Curves (IRCs) showed that even students scoring below chance are not randomly guessing, but instead preferentially select only one or two distractors. Two dimensional IRT applied to the entire set of 150 possible responses, rather than applied dichotomously to the thirty questions, revealed two skill dimensions of comparable variance. Perpendicular directions were identified within this space corresponding to Newtonian ability and propensity to select responses whose IRC's have a maximum at intermediate Newtonian ability rather than at the top of bottom of this dimension. These intermediate responses corresponded to known pre-Newtonian ideas, particularly the Medieval concept of impetus. The ability to measure the detailed misconceptions of individual students or classes will allow development and application of instructional interventions for such specific misunderstandings, which are typically unchanged by traditional instruction.
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