# Optimal Quantum Spatial Search on Random Temporal Networks.

@article{Chakraborty2017OptimalQS, title={Optimal Quantum Spatial Search on Random Temporal Networks.}, author={Shantanav Chakraborty and Leonardo Novo and Serena Di Giorgio and Yasser Omar}, journal={Physical review letters}, year={2017}, volume={119 22}, pages={ 220503 } }

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…

## 31 Citations

### Quantum spatial search on graphs subject to dynamical noise

- PhysicsPhysical Review A
- 2018

We address quantum spatial search on graphs and its implementation by continuous-time quantum walks in the presence of dynamical noise. In particular, we focus on search on the complete graph and on…

### Optimality of spatial search via continuous-time quantum walks

- Computer Science, Mathematics
- 2020

This work derives general expressions, depending on the spectral properties of the Hamiltonian driving the walk, that predict the performance of this quantum search algorithm provided certain spectral conditions are fulfilled and shows the optimality of quantum search for certain graphs with very small spectral gaps, such as graphs that can be efficiently partitions into clusters.

### Finding a marked node on any graph by continuous time quantum walk

- Computer SciencePhysical Review A
- 2020

This article presents a modified version of the Childs and Goldstone algorithm which can search for a marked element for any ergodic, reversible Markov chain $P$ by performing a quantum walk on its edges and establishes a connection between discrete-time and continuous-time quantum walks.

### Quadratic speedup for spatial search by continuous-time quantum walk

- MathematicsPhysical review letters
- 2022

This article provides a new continuous-time quantum walk search algorithm that can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks.

### How fast do quantum walks mix?

- PhysicsPhysical review letters
- 2020

An upper bound on the quantum mixing time for almost all networks is proved, i.e., the fraction of networks for which this bound holds, goes to one in the asymptotic limit.

### Feedback‐Assisted Quantum Search by Continuous‐Time Quantum Walks

- Computer ScienceAdvanced Quantum Technologies
- 2022

A dynamical oracle implemented through a feedback Hamiltonian is considered, based on continuously monitoring the position of the quantum walker on the graph and then to apply a unitary feedback operation based on the information obtained from measurement.

### Vertices cannot be hidden from quantum spatial search for almost all random graphs

- Computer ScienceQuantum Inf. Process.
- 2018

It is shown that all nodes can be found optimally for almost all random Erdős–Rényi graphs using continuous-time quantum spatial search procedure and the derivation concerning Laplacian matrix is tight.

### Continuous-time quantum walks on dynamic graphs

- MathematicsPhysical Review A
- 2019

Continuous-time quantum walks (CTQWs) on static graphs provide efficient methods for search and sampling as well as a model for universal quantum computation. We consider an extension of CTQWs to the…

### Simplifying continuous-time quantum walks on dynamic graphs

- Mathematics, Computer ScienceQuantum Inf. Process.
- 2022

Six scenarios under which a dynamic graph can be simplified are given, and they exploit commuting graphs, identical graphs, perfect state transfer, complementary graphs, isolated vertices, and uniform mixing on the hypercube.

### On analog quantum algorithms for the mixing of Markov chains

- MathematicsPhysical Review A
- 2020

An analog quantum algorithm is provided that given a Markov chain, allows us to sample from its stationary distribution in a time that scales as the sum of thesquare root of the classical mixing time and the square root ofThe classical hitting time.

## References

SHOWING 1-10 OF 29 REFERENCES

### Spatial Search by Quantum Walk is Optimal for Almost all Graphs.

- Computer Science, MathematicsPhysical review letters
- 2016

It is proved 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, and that quantum spatial search is in fact optimal for almost all graphs.

### Spatial search by quantum walk

- Computer Science
- 2004

This work considers an alternative search algorithm based on a continuous-time quantum walk on a graph and shows that full {radical}(N) speedup can be achieved on a d-dimensional periodic lattice for d>4.

### Continuous Time Quantum Walks in finite Dimensions

- Physics
- 2016

We consider the quantum search problem with a continuous time quantum walk for networks of finite spectral dimension ds of the network Laplacian. For general networks of fractal (integer or…

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

- MathematicsScientific reports
- 2015

This work uses 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.

### Degree distribution in quantum walks on complex networks

- Computer Science
- 2013

This theoretical study analytically relates the average long time probability distribution for the location of a unitary quantum walker to that of a corresponding classical walker, which measures the connectivity of the network nodes and underlies many methods for analyzing classical networks including website ranking.

### Time evolution of continuous-time quantum walks on dynamical percolation graphs

- Mathematics
- 2013

We study the time evolution of continuous-time quantum walks on randomly changing graphs. At certain moments, edges of the graph appear or disappear with a given probability, as in percolation. We…

### Connectivity is a poor indicator of fast quantum search.

- PhysicsPhysical review letters
- 2015

This Letter shows intuition that graph connectivity is an indicator of fast quantum search on complete graphs, strongly regular graphs, and hypercubes to be false by giving two examples of graphs for which the opposite holds true: one with low connectivity but fast search, and one with high connectivity but slow search.

### Random walks on temporal networks.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2012

It is shown that the random walk exploration is slower on temporal networks than it is on the aggregate projected network, even when the time is properly rescaled, and a fundamental role is played by the temporal correlations between consecutive contacts present in the data.

### Coined quantum walks on percolation graphs

- Physics
- 2010

Quantum walks, both discrete (coined) and continuous time, form the basis of several quantum algorithms and have been used to model processes such as transport in spin chains and quantum chemistry.…

### Asymptotic dynamics of coined quantum walks on percolation graphs.

- Mathematics, PhysicsPhysical review letters
- 2012

It is found that a rich variety of asymptotic evolutions occur: not only the fully mixed state, but other stationary states; stable periodic and quasiperiodic oscillations can emerge, depending on the coin operator, the initial state, and the topology of the underlying graph.