## Title:

### Quantum State Transfer in Graphs

## Dr. Hiranmoy Pal

In light of the principles of quantum mechanics, we have witnessed a new way of considering

information processing. Classically information is encoded into sequence of zeros and ones. How-

ever, in this framework, information is encrypted with quantum states or qubits of some physical

systems, such as atoms, trapped ions, etc. Several interacting physical systems makes a quantum

web that are designed to perform several tasks, particularly, transferring quantum states between

two nodes of a network.

These physical systems are essentially static entities which are considered

as nodes of a graph having edges that indicate interactions between two nodes. Typically any

control to these systems induces signi

cant amount of noise in the passed out information, and

therefore systems are maintained in minimal dynamic control. Several physical models can be

engineered in such a way that the Heisenberg Hamiltonian [10] of the underlying network becomes

the adjacency matrix, laplacian matrix, etc. Mathematically a state is represented by an unit

vector in a complex Hilbert space of dimension equals the number of nodes in the network.

The

evolution of a quantum state j ( t) i is described by the time-dependent Schrodinger equation

## i } d dt

j

( t) i = H j ( t) i :

If the Hamiltonian His a time-independent operator then Schrodinger equation can easily be

solved to obtain j ( t) i = exp ( itH )j (0) iconsidering scaled Plank constant }= 1 :If jj i is the

state with one at j-th place and zero elsewhere, then fjjig forms a basis of the Hilbert space.

The probability that the initial state j (0) i= j1 i evolves to the state jN iafter time tis obtained

## by jp

N (

## t) j2

= jhN jexp ( itH )j1 ij 2

: The state transfer is called perfect if the

## delity of transfer

jh N jexp ( itH )j1 ij = 1 :The state transfer is considered pretty good if jhN jexp ( itH )j1 ij comes

arbitrarily close to unity. Perfect state transfer (PST) was originally introduced by Bose [4] in quantum spin chains

consisting two and three particles with nearest neighbour coupling. There it is observed that both

## path P

2 (the path on two vertices) and

P

3 (the path on three vertices) exhibit PST between the

### end vertices at time 2

## and p

2

, respectively. Later it was uncovered that such spin chains with

more than three particles never exhibits PST, however, Godsil et al. [9] surprisingly established

a number theoritic nature of state transfer classifying all such spin chains exhibiting pretty good

end to end transfer. More complex networks has been considered since the inception of quantum

state transfer. Christandl et al. [6, 7] have showed that PST occurs between particles at large

### distances in Cartesian powers of P

2 and

P

3 at time 2

## and p

2

, respectively. Natural questions

arise [19], whether state transfer occurs if we let more particles to interact in those Cartesian

## powers P

2 and

P

3. A NEPS (Non-complete Extended P-Sum) is a graph product generalizing few

well known graph products, such as Cartesian product, Kronecker product, etc. In this direction,

we see several partial characterizations of PST on cubelike graphs (or NEPS of P

2) in [3, 5]. In

addition, we have contributed few characterizations of PST and pretty good state transfer (PGST)

## on NEPS of P

3 in [14, 16]. Remarkably, when considering NEPS with factor graphs consisting

## both P

2 and

P

3, several graphs have emerged admitting PGST from a vertex to two di erent

vertices, which was never possible in case of PST. Beside this we have explored the possibility of

state transfer in several class of Cayley graphs. In contrust, an integral graph is which there is

only integer eigenvalues. A complete caharcterization of State transfer in integral circulant was

unearthed by Ba si c [1]. Among other circulant graphs, we have uncovered that a cycle exhibits

PGST if an only if the consisting number of vertices is a power of two (see [15]). We have further

built up results and obtained more general circulant graphs exhibiting state transfer in [12, 17].

In a more general scenario, we have observed in [11, 13] that there always exist a gcd-graph (a

### class of integral Caley graphs de

ned by gcd) exhibiting PST whenever the size of the underlying

### group is divisible by four.

1

1 Ob jectives

The area of research is interdisciplinary in nature. The main ob jective is to

## nd graphs exhibiting

quantum state transfer between distance vertices. There are few graphs known to exhibit state

transfer. Accordingly, there are immense scopes of research in this area. We brie

## y enlist some of

### the possibilities of future research.

1. We have established several characterizations of PST and PGST in NEPS of P

3 in [14, 16].

### It will be interesting to

nd if there is any other graphs in this class exhibiting PST or

PGST. We may also investigate NEPS of other type of graphs having state transfer.

2. In [11, 13], we have addressed few characterizations of PST in gcd-graphs, however, a com- plete classi

cation of all gcd-graphs admitting state transfer is highly desirable. In contrast,

we may investigate PST in those integral Cayley graphs which are in fact not gcd-graphs.

3. We have successively classi

ed a lot of circulant graphs exhibiting PGST in [12, 15]. We may continue our investigation and classify all circulant graphs exhibiting PGST. In general, it

is preferable to have a characterization of PGST in Cayley graphs.

4. There are few articles which discuss state transfer with respect to the Laplacian matrix instead of the adjacency matrix. Apparently, this consideration is equally relevant, and

therefore, we may examine the existence of state transfer on new families of graphs with

respect to both adjacency matrix and Laplacian matrix. If possible, we may try to uncover

new families of graphs having state transfer between vertices at distances.

2 Brief Outline of Methodology

The area of research is a relatively new, and consequently, there are very few techniques known

to

nd graphs exhibiting PST (or PGST). One way is to obtain explicitly the transition matrix of

a graph. After analysing the entries of the transition matrix it reveals whether the graph admits

PST (or PGST) between any pair of vertices. However, this process becomes quite impossible for

large graphs. The research mainly proceeds in the following two directions.

Spectral decomposition is often used to

nd the transition matrix of a graph. It is therefore

natural to assume that some relations on eigenvalues or eigenvectors are involved in char-

acterizing PST (or PGST) on a class graphs. A number of such results have already been

set-up for some well known class of graphs. We plan to

### nd such relations on eigenvalues or

eigenvectors classifying PST (or PGST) on a new class graphs.

Some algebraic, graph theoretic or combinatorial methods may help in characterizing PST

(or PGST). It is well known that if PST (or PGST) occurs between two vertices in a

graph then it also occurs between the images of those vertices under an automorphism.

An implication to this fact is that a vertex transitive graph on odd number of vertices does

not admit PST (or PGST). Likewise, we wish to obtain results on PST (or PGST) in various

### graphs using these methods.

Although the pro ject is theoretical in nature, it requires rigorous study into several examples

of graphs, which demands enormous computing power and usage of professional software like

Mathematica, Matlab, etc. Using computational data, we may gain clues to formulate new results

classifying PST (or PGST) in various graphs.

2

3 Signi

## cance

The study of state transfer is a rapidly growing area as it contributes to the research in quantum

information processing and cryptography (see [2, 18]). There are few graphs known to exhibit

PST or PGST, and therefore it is highly desirable to classify new families of graphs having state

transfer. Cayley graphs appear frequently in communication networks, therefore the research on

Cayley graphs in particular have tremendous importance. However, investigation into any well

known families of graphs shall produce remarkable results.

## References

[1] M. Ba si c. Characterization of circulant networks having perfect state transfer . Quantum Infor-

### mation Processing, 12:345-364 (2011).

[2] C. H. Bennett and G. Brassard. Quantum Cryptography: Public Key Distribution and Coin

Tossing . Proc. IEEE Int. Conf. Computers Systems and Signal Processing, Bangalore, India.

175-179 (1984).

[3] A. Bernasconi, C. Godsil and S. Severini. Quantum networks on cubelike graphs. Physical

### Review A, 78:052320 (2008).

[4] S. Bose. Quantum communication through an unmodulated spin chain . Physical Review Letters,

91 (20):207901 (2003).

[5] W. Cheung and C. Godsil. Perfect state transfer in cubelike graphs . Linear Algebra and Its

Applications, 435(10):2468-2474 (2011).

[6] M. Christandl, N. Datta, T. Dorlas, A Ekert, A. Kay and A. J. Landahl. Perfect transfer of

arbitrary states in quantum spin networks . Physical Review A,71:032312 (2005).

[7] M. Christandl, N. Datta, A. Ekert and A. J. Landahl. Perfect state transfer in quantum spin

networks . Physical Review Letters, 92:187902 (2004).

[8] C. Godsil. State transfer on graphs . Discrete Mathematics, 312(1): 129{147 (2012).

[9] C. Godsil, S. Kirkland, S. Severini and J. Smith. Number-theoretic nature of communication

in quantum spin systems. Physical review letters 109, 5:050502 (2012).

[10] G. M. Nikolopoulos and I. Jex (Editors). Quantum State Transfer and Network Engineering .

Springer Heidelberg New York Dordrecht London (2014).

[11] H. Pal and B. Bhattacharjya. A class of gcd-graphs having Perfect State Transfer . Electronic

Notes in Discrete Mathematics, 53:319-329 (2016).

[12] H. Pal. More Circulant Graphs Exhibiting Pretty Good State Transfer . Discrete Mathematics,

341 :889-895 (2018).

[13] H. Pal and B. Bhattacharjya. Perfect state transfer on gcd-graphs . Linear and Multilinear

### Algebra, 65(11):2245-2256 (2016).

[14] H. Pal and B. Bhattacharjya. Perfect state transfer on NEPS of the path on three vertices .

Discrete Mathematics, 339(2):831-838 (2016).

[15] H. Pal and B. Bhattacharjya. Pretty good state transfer on circulant graphs . The Electronic

Journal of Combinatorics, 24(2), #P2.23 (1-13) (2017).

[16] H. Pal and B. Bhattacharjya. Pretty Good State Transfer on Some NEPS . Discrete Mathe-

### matics, 340(4), 746-752 (2017).

[17] H. Pal. Quantum State Transfer on a Class of Circulant Graphs . arXiv:1901.01999v1 (2019).

[18] A. Kay. The Basics of Perfect Communication through Quantum Networks . arXiv:1102.2338

(2011).

[19] D. Stevanovi c. Application of graph spectra in quantum physics in: D. Cevtkovi c, I. Gutman

(Eds.), Selected Topics on Applications of Graph Spectra . Zbornik radova 14(22), Mathematical

Institute SANU, Belgrade, 85-111 (2011).

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