Shantanav Chakraborty

Spatial search by quantum walk is optimal for almost all graphs

The problem of finding a marked node in a graph can be solved by the spatial search algorithm based on continuous-time quantum walks (CTQW). However, this algorithm is known to run in optimal time only for a handful of graphs. In this work, we prove that for Erdös-Renyi random graphs, i.e., graphs of n vertices where each edge exists with probability p, search by CTQW is almost surely optimal as long as p≥log^{3/2}(n)/n. Consequently, we show that quantum spatial search is in fact optimal for almost all graphs, meaning that the fraction of graphs of n vertices for which this optimality holds tends to one in the asymptotic limit. We obtain this result by proving that search is optimal on graphs where the ratio between the second largest and the largest eigenvalue is bounded by a constant smaller than 1. Finally, we show that we can extend our results on search to establish high fidelity quantum communication between two arbitrary nodes of a random network of interacting qubits, namely, to perform quantum state transfer, as well as entanglement generation. Our work shows that quantum information tasks typically designed for structured systems retain performance in very disordered structures.

Systematic Dimensionality Reduction for Quantum Walks: Optimal Spatial Search and Transport on Non-Regular Graphs.

Continuous time quantum walks provide an important framework for designing new algorithms and modelling quantum transport and state transfer problems. Often, the graph representing the structure of a problem contains certain symmetries that confine the dynamics to a smaller subspace of the full Hilbert space. In this work, we use invariant subspace methods, that can be computed systematically using the Lanczos algorithm, to obtain the reduced set of states that encompass the dynamics of the problem at hand without the specific knowledge of underlying symmetries. First, we apply this method to obtain new instances of graphs where the spatial quantum search algorithm is optimal: complete graphs with broken links and complete bipartite graphs, in particular, the star graph. These examples show that regularity and high-connectivity are not needed to achieve optimal spatial search. We also show that this method considerably simplifies the calculation of quantum transport efficiencies. Furthermore, we observe improved efficiencies by removing a few links from highly symmetric graphs. Finally, we show that this reduction method also allows us to obtain an upper bound for the fidelity of a single qubit transfer on an XY spin network.

DNA sequence evolution through Integral Value Transformations.

In deciphering the DNA structures, evolutions and functions, Cellular Automata (CA) plays a significant role. DNA can be thought as a one-dimensional multi-state CA, more precisely four states of CA namely A, T, C, and G which can be taken as numerals 0, 1, 2 and 3. Earlier, Sirakoulis et al. (2003) reported the DNA structure, evolution and function through quaternary logic one dimensional CA and the authors have found the simulation results of the DNA evolutions with the help of only four linear CA rules. The DNA sequences which are produced through the CA evolutions, however, are seen by us not to exist in the established databases of various genomes although the initial seed (initial global state of CA) was taken from the database. This problem motivated us to study the DNA evolutions from more fundamental point of view. Parallel to CA paradigm we have devised an enriched set of discrete transformations which have been named as Integral Value Transformations (IVT). Interestingly, on applying the IVT systematically, we have been able to show that each of the DNA sequence at various discrete time instances in IVT evolutions can be directly mapped to a specific DNA sequence existing in the database. This has been possible through our efforts of getting quantitative mathematical parameters of the DNA sequences involving fractals. Thus we have at our disposal some transformational mechanism between one DNA to another.

Quantitative Description of Genomic Evolution of Olfactory Receptors

We investigate how the evolutionary network is associated among Human, Chimpanzee and Mouse with regards to their genomic information. We provide a quantitative description of genomic evolution through indexes based on fractals and mathematical morphology. These indexes are carefully chosen and reveal quantitatively the similarity or differences among the structure of the concerned sequences. They also reveal how the sequences have evolved over the course of time. We have considered olfactory receptors (ORs) as our case study. These ORs do function in different species with the subtle differences in between the structures of DNA sequences. Such differences are quantified in this paper.

Optimal Quantum Spatial Search on Random Temporal Networks

To investigate the performance of quantum information tasks on networks whose topology changes in time, we study the spatial search algorithm by continuous time quantum walk to find a marked node on a random temporal network. We consider a network of n nodes constituted by a time-ordered sequence of Erdös-Rényi random graphs G(n,p), where p is the probability that any two given nodes are connected: After every time interval τ, a new graph G(n,p) replaces the previous one. We prove analytically that, for any given p, there is always a range of values of τ for which the running time of the algorithm is optimal, i.e., O(sqrt[n]), even when search on the individual static graphs constituting the temporal network is suboptimal. On the other hand, there are regimes of τ where the algorithm is suboptimal even when each of the underlying static graphs are sufficiently connected to perform optimal search on them. From this first study of quantum spatial search on a time-dependent network, it emerges that the nontrivial interplay between temporality and connectivity is key to the algorithmic performance. Moreover, our work can be extended to establish high-fidelity qubit transfer between any two nodes of the network. Overall, our findings show that one can exploit temporality to achieve optimal quantum information tasks on dynamical random networks.

Controlled secret sharing protocol using a quantum cloning circuit

We demonstrate the possibility of controlling the success probability of a secret sharing protocol using a quantum cloning circuit. The cloning circuit is used to clone the qubits containing the encoded information and en route to the intended recipients. The success probability of the protocol depends on the cloning parameters used to clone the qubits. We also establish a relation between the concurrence of initially prepared state, entanglement of the mixed state received by the receivers after cloning scheme and the cloning parameters of cloning machine.