Generating a state t-design by diagonal quantum circuits

We investigate protocols for generating a state t-design by using a fixed separable initial state and a diagonal-unitary t-design in the computational basis, which is a t-design of an ensemble of diagonal unitary matrices with random phases as their eigenvalues. We first show that a diagonal-unitary t-design generates a O (1/2(N))-approximate state t-design, where N is the number of qubits. We then discuss a way of improving the degree of approximation by exploiting non-diagonal gates after applying a diagonal-unitary t-design. We also show that it is necessary and sufficient to use O (log(2)(t)) -qubit gates with random phases to generate a diagonal-unitary t-design by diagonal quantum circuits, and that each multi-qubit diagonal gate can be replaced by a sequence of multi-qubit controlled-phase-type gates with discrete-valued random phases. Finally, we analyze the number of gates for implementing a diagonal-unitary t-design by non-diagonal two- and one-qubit gates. Our results provide a concrete application of diagonal quantum circuits in quantum informational tasks.

Pre-announcement of symbiotic guests: transcriptional reprogramming by mycorrhizal lipochitooligosaccharides shows a strict co-dependency on the GRAS transcription factors NSP1 and RAM1

BACKGROUND: More than 80 % of all terrestrial plant species establish an arbuscular mycorrhiza (AM) symbiosis with Glomeromycota fungi. This plant-microbe interaction primarily improves phosphate uptake, but also supports nitrogen, mineral, and water aquisition. During the pre-contact stage, the AM symbiosis is controled by an exchange of diffusible factors from either partner. Amongst others, fungal signals were identified as a mix of sulfated and non-sulfated lipochitooligosaccharides (LCOs), being structurally related to rhizobial nodulation (Nod)-factor LCOs that in legumes induce the formation of nitrogen-fixing root nodules. LCO signals are transduced via a common symbiotic signaling pathway (CSSP) that activates a group of GRAS transcription factors (TFs). Using complex gene expression fingerprints as molecular phenotypes, this study primarily intended to shed light on the importance of the GRAS TFs NSP1 and RAM1 for LCO-activated gene expression during pre-symbiotic signaling. RESULTS: We investigated the genome-wide transcriptional responses in 5 days old primary roots of the Medicago truncatula wild type and four symbiotic mutants to a 6 h challenge with LCO signals supplied at 10(-7/-8) M. We were able to show that during the pre-symbiotic stage, sulfated Myc-, non-sulfated Myc-, and Nod-LCO-activated gene expression almost exclusively depends on the LysM receptor kinase NFP and is largely controled by the CSSP, although responses independent of this pathway exist. Our results show that downstream of the CSSP, gene expression activation by Myc-LCOs supplied at 10(-7/-8) M strictly required both the GRAS transcription factors RAM1 and NSP1, whereas those genes either co- or specifically activated by Nod-LCOs displayed a preferential NSP1-dependency. RAM1, a central regulator of root colonization by AM fungi, controled genes activated by non-sulfated Myc-LCOs during the pre-symbiotic stage that are also up-regulated in areas with early physical contact, e.g. hyphopodia and infecting hyphae; linking responses to externally applied LCOs with early root colonization. CONCLUSIONS: Since both RAM1 and NSP1 were essential for the pre-symbiotic transcriptional reprogramming by Myc-LCOs, we propose that downstream of the CSSP, these GRAS transcription factors act synergistically in the transduction of those diffusible signals that pre-announce the presence of symbiotic fungi.