Leaf-to-leaf distances and their moments in finite and infinite ordered 
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math>
-ary tree graphs

Goldsborough, Andrew M.; Rautu, S. Alex; Römer, Rudolf A.

doi:10.1103/physreve.91.042133

Cited by 3 publications

(8 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…1. The absence of paths that would be present in a complete binary tree seems similar to the more random structure seen in random binary trees [16]. We might hence expect that Catalan trees should in many ways be similar to the tree structure found previously [1].…”

Section: Root Nodesupporting

confidence: 68%

“…Hence it is interesting to ask more generally about the path lengths of different tree graphs. In a previous work [16], three of us studied the case of full and complete m-ary tree graphs using a recursive approach. We found explicit expressions in terms of Hurwitz-Lerch transcendants.…”

Section: Root Nodementioning

confidence: 99%

“…Average length of a leaf to leaf path ℓ n (r) versus separation r for small n = 50 (solid black line), 100 (solid red), 200 (solid blue) and large n = 5, 000 (dashed black line), 10, 000 (dashed red), 20, 000 (dashed blue). The solid green line denotes ℓ ∞ (r), the dotted line corresponds to r while the dashed-dotted line shows the corresponding result for complete binary trees [16]. …”

mentioning

confidence: 99%

See 2 more Smart Citations

Diagrammatic Aproach to Leaf-to-Leaf Distances in Catalan Trees

Goldsborough¹,

Fellows²,

Bates³

et al. 2019

Jour. Pure Appl. Math. Adv. Appl.

View full text Add to dashboard Cite

We study the average leaf-to-leaf path lengths on ordered Catalan tree graphs with n nodes and show that these are equivalent to the average length of paths starting from the root node. We give an explicit analytic formula for the average leafto-leaf path length as a function of separation of the leaves and study its asymptotic properties. At the heart of our method is a strategy based on an abstract graph representation of generating functions. PACS numbers:

show abstract

Section: Root Nodesupporting

confidence: 68%

Section: Root Nodementioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Diagrammatic Aproach to Leaf-to-Leaf Distances in Catalan Trees

Goldsborough¹,

Fellows²,

Bates³

et al. 2019

Jour. Pure Appl. Math. Adv. Appl.

View full text Add to dashboard Cite

show abstract

“…In fact, it is much longer than the phase memory time of BDPA determined to be of the order of 100 ns between 77 K and room temperature. [62,63] Spin echoes have been observed (at millikelvin temperatures) under conditions where the single-spin-cavity coupling is large, but in most cases the cooperativity parameter C was small, meaning that the response of individual spins was measured. [52,64,65] Therefore, that scenario is not applicable to the present case.…”

Section: Investigation Of the Quantum Dynamics By Pulsed Measurementsmentioning

confidence: 99%

Room‐Temperature Quantum Memories Based on Molecular Electron Spin Ensembles

et al. 2021

View full text Add to dashboard Cite

Superconducting quantum bits are addressed by means of microwave radiation and quantum memories in this context thus need to be able to store microwave photon states. To this end, resonant structures for electromagnetic radiation can be used to strongly couple quantum bits and quantum memories to quantized cavity modes of the electromagnetic field. The strong coupling generates a hybrid quantum system that allows mapping the qubit state onto the cavity field state. [6] This strategy has given rise to the field of cavity quantum electrodynamics and has already been used to couple two superconducting qubits together via a cavity bus. [7] A second application of quantum memories is in quantum repeaters for quantum communication. Because telecom wavelengths are in the near-infrared, for this application, optical quantum memories are required that store optical photon states. [8] Considering the choice of material platform for designing microwave quantum memories, electron spins in solids can be easily addressed by microwave pulses and have been shown to possess excellent coherence times up to seconds, for some systems even up to room temperature. [9][10] Unfortunately, the coupling of a single electron spin to a photon is very weak. Although resonant structures can be tailored to improve single-spin coupling, [11][12][13] the weakness of the coupling renders addressing individual spins by means of coupling to cavity modes very challenging. This can be overcome by using an electron spin ensemble rather than single spins, because the coupling strength of a spin ensemble to an electromagnetic resonator mode is proportional to the square root of the number of electron spins in the ensemble. [14] This then leads to the general idea of the implementation of a microwave quantum memory using spin ensembles and microwave resonators (Figure 1). The state to be stored is encoded in a weak microwave pulse and sent to the hybrid quantum system constituted of an electron spin ensemble that is strongly coupled to a microwave resonator, where it is stored. When the quantum state is needed again, a strong microwave pulse is sent to the resonator, which leads to deterministic retrieval of the quantum state that is emitted as a microwave photon to be used as desired.Strong coupling between cavity and electron spin ensembles has been observed in a variety of spin systems, including organic radicals, [15][16][17][18][19][20] phosphorous dopants in silicon, [21,22] P1 and nitrogen vacancy (NV) defects in diamond, [23][24][25][26][27][28][29][30][31][32] N@ C 60 , [33] rare earth dopants in oxides, [34,35] transition metal Whilst quantum computing has recently taken great leaps ahead, the development of quantum memories has decidedly lagged behind. Quantum memories are essential devices in the quantum technology palette and are needed for intermediate storage of quantum bit states and as quantum repeaters in long-distance quantum communication. Current quantum memories operate at cryogenic, mostly sub-Kelvin temperatures and require extens...

show abstract

“…In the present work we apply a tree tensor network (TTN) method [37][38][39][40][41] (specifically our recently introduced gauge-adaptive TTN technique [42]), a natural ansatz for the simulation of one-dimensional quantum many-body systems with PBC, to the disordered Bose-Hubbard model. We compute the superfluid stiffness and correlation functions, avoiding the detrimental boundary effects arising in simulations with open boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Superfluid density and quasi-long-range order in the one-dimensional disordered Bose–Hubbard model

et al. 2016

View full text Add to dashboard Cite

We study the equilibrium properties of the one-dimensional disordered Bose-Hubbard model by means of a gauge-adaptive tree tensor network variational method suitable for systems with periodic boundary conditions. We compute the superfluid stiffness and superfluid correlations close to the superfluid to glass transition line, obtaining accurate locations of the critical points. By studying the statistics of the exponent of the power-law decay of the correlation, we determine the boundary between the superfluid region and the Bose glass phase in the regime of strong disorder and in the weakly interacting region, not explored numerically before. In the former case our simulations are in agreement with previous Monte Carlo calculations.

show abstract

Leaf-to-leaf distances and their moments in finite and infinite ordered m -ary tree graphs

Cited by 3 publications

References 18 publications

Diagrammatic Aproach to Leaf-to-Leaf Distances in Catalan Trees

Diagrammatic Aproach to Leaf-to-Leaf Distances in Catalan Trees

Room‐Temperature Quantum Memories Based on Molecular Electron Spin Ensembles

Superfluid density and quasi-long-range order in the one-dimensional disordered Bose–Hubbard model

Contact Info

Product

Resources

About