Digital Switching in Quantum Domain I. –Ming Tsai and Sy-Yen Kuo Presented by Chin-Yi Tsai

2 Outline Introduction Introduction Notation and Preliminaries Notation and Preliminaries Digital Switching Networks Digital Switching Networks Digital Quantum Switching Digital Quantum Switching Conclusions Conclusions

3 Introduction A switching architecture such that digital data can be switched in the quantum domain. A switching architecture such that digital data can be switched in the quantum domain. The proposed mechanism supports unicasting and multicasting. The proposed mechanism supports unicasting and multicasting. (interface conversion) The quantum switch can be used to build classical and quantum information networks. (interface conversion) The quantum switch can be used to build classical and quantum information networks. To define the connection digraph which can be used to describe the behavior of a switch at a given time. To define the connection digraph which can be used to describe the behavior of a switch at a given time. The connection digraph can be implemented using elementary quantum gates. The connection digraph can be implemented using elementary quantum gates.

4 Introduction (cont ’ d) Compared with a traditional space or time domain switch, the proposed switching mechanism is much more scalable. Compared with a traditional space or time domain switch, the proposed switching mechanism is much more scalable.

5 Notations and Preliminaries control target

6 Qubit Permutation and Replication A typical permutation P is represented using the symbol A typical permutation P is represented using the symbol A cycle is basically an ordered list, which is represented as C=(e 1, e 2, …, e n-1, e n ). A cycle is basically an ordered list, which is represented as C=(e 1, e 2, …, e n-1, e n ). The number of elements in a cycle is called length. The number of elements in a cycle is called length. Length 1 ： trivial cycle Length 1 ： trivial cycle Length 2 ： transposition Length 2 ： transposition P = (a, d )(c )(b, e, f )=(a, d )(b, e, f ) P = (a, d )(c )(b, e, f )=(a, d )(b, e, f )

7 Qubit Permutation and Replication (cont ’ d) (transposition circuit)

8 Qubit Permutation and Replication (cont ’ d) For a general n-qubit cycle C=(q 0, q 1, q 2, …, q n-1 ), it can be done by six layers of CN gates with ancillary qubits. For a general n-qubit cycle C=(q 0, q 1, q 2, …, q n-1 ), it can be done by six layers of CN gates with ancillary qubits. For an even n (n=2m, m=2, 3, … ), we define the following nonoverlapping qubit transpositions as: For an even n (n=2m, m=2, 3, … ), we define the following nonoverlapping qubit transpositions as: The cycle can be implemented using The cycle can be implemented using

9 For the odd n(n=2m+1, m=1, 2, 3, … ) For the odd n(n=2m+1, m=1, 2, 3, … ) n=6 X=(2, 4)(1, 5) Y=(3, 4)(2, 5)(1, 2) n=5 X=(2, 3)(1, 4) Y=(2, 4)(1, 0)

10 Qubit Replication (FANOUT) Qubit replication takes one bit as input and gives two copies of the same bit value as output. Qubit replication takes one bit as input and gives two copies of the same bit value as output.

11 Digital Switching Networks In classical digital communication, switching is needed in order to avoid a fully meshed transmission network. In classical digital communication, switching is needed in order to avoid a fully meshed transmission network. Digital switching technologies fall under two broad categories: Digital switching technologies fall under two broad categories: Circuit switching Circuit switching Packet switching Packet switching In both circuit switching and packet switching, the control subsystem needs to specify the switching configuration In both circuit switching and packet switching, the control subsystem needs to specify the switching configuration

12 Digital Switching Networks (cont ’ d) The switching configuration can be described using a connection digraph. The switching configuration can be described using a connection digraph. Definition 1: Given an n x n switch, the connection digraph at time t, G t ={V, E t }, is a digraph such that: Definition 1: Given an n x n switch, the connection digraph at time t, G t ={V, E t }, is a digraph such that: Each represents an I/O port Each represents an I/O port if and only if a connection exists from the input port v m to the output port v n at time t. if and only if a connection exists from the input port v m to the output port v n at time t. A digraph G t describes the connection status of a switch at a specific time, and is called the connection digraph at time t A digraph G t describes the connection status of a switch at a specific time, and is called the connection digraph at time t

13 Elementary Topologies The connection digraph can be built from a set of elementary topologies The connection digraph can be built from a set of elementary topologies null point, loopback, queue, cycle, tree, forest null point, loopback, queue, cycle, tree, forest

14 Connection with null points and loopbacks Connection digraph Connection

15 Queue connection and its connection digraph

16 Cycle connection and its connection digraph

17 Tree connection and its connection digraph

18 Forest connection and its connection digraph

19 Digital Quantum Switching The proposed architecture for building a digital quantum switching The proposed architecture for building a digital quantum switching 0 -> |0> 1 -> |1> |0> -> 0 |1> -> 1

20 Connection Digraph Implementation A connection digraph can be implemented using CN gates. A connection digraph can be implemented using CN gates. Transformation guideline can be used to implement a connection digraph. Transformation guideline can be used to implement a connection digraph.

21 Transformation Guideline Unicasting and multicasting have different types of connection digraphs Unicasting and multicasting have different types of connection digraphs The digraph of a unicast connection has a connection of disjointed null points, loopbacks, queues, and/or cycles as subdigraphs. The digraph of a unicast connection has a connection of disjointed null points, loopbacks, queues, and/or cycles as subdigraphs. However, in the digraph of a multicast connection, subdigraph such as trees and forests are possible. However, in the digraph of a multicast connection, subdigraph such as trees and forests are possible.

22 Interrelated Connection Topologies forest tree Cycle U=YX

23 Cycle Extraction The process of cycle extraction detaches all the null points, queues The process of cycle extraction detaches all the null points, queues This procedure transforms a forest into one cycle and a collection of null point, queues, and/or trees. This procedure transforms a forest into one cycle and a collection of null point, queues, and/or trees. null point and queues  loopback and cycles null point and queues  loopback and cycles Tree  forest Tree  forest forest

24 Link Recovery After each cycle has been implemented, the links that had been cut must be recovered. After each cycle has been implemented, the links that had been cut must be recovered.

25 Unicast Quantum Switching G C =(q 3, q 4, q 6, q 7, q 5 ) G Q =[q 0, q 1, q 2 ] G C’ =(q0, q1, q2)

26 G C =(q 3, q 4, q 6, q 7, q 5 ) X=(q 6, q 7 )(q 4, q 5 ) Y=(q 6, q 5 )(q 4, q 3 ) (q 4, q 5 )=CN(q 4, q 5 ). CN(q 5, q 4 ). CN(q 4, q 5 ) G C’ =(q 0, q 1, q 2 ) X=(q 1, q 2 ) Y=(q 1, q 0 )

27 Multicast Quantum Switching tree G T =[q 0, q 1 ][q 1, q 4 ][q 1, q 3 ][q 3, q 5, q 2 ][q 3, q 6, q 7 ]

28 Multicast Quantum Switching tree forest

29

30 Conclusions An architecture of digital quantum switching. An architecture of digital quantum switching. The proposed mechanism allows digital data to be switched using a series of quantum operations. The proposed mechanism allows digital data to be switched using a series of quantum operations. Connection digraph Connection digraph Null point, queue, cycle, tree, forest Null point, queue, cycle, tree, forest