Chaos, Solitons and Fractals 39 (2009) 1936–1942

www.elsevier.com/locate/chaos

The improved quantum switching mechanismbased on contention

Jiang Min a,*, Zhang Zeng-ke a, Dong Dao-yi b, Liu Bin a, Tzyh-Jong Tarn a,c

a Department of Automation, Tsinghua University, Beijing 100084, Chinab Key Laboratory of Systems and Control, Chinese Academy of Sciences, Beijing 100080, China

c Department of Systems Science and Mathematics, Washington University, St. Louis, MO 63130, USA

Accepted 20 June 2007

Abstract

Quantum switching architecture is one of the promising schemes for transmitting quantum data to its destination. Inthis paper, we propose an improved quantum switching mechanism for solving the output contention problem. Firstlywe analyze the impropriety of Tsai and Kuo’s quantum switching scheme for multicast service. Secondly an improvedquantum switching scheme is presented for completing the switching task with output contention. Lastly an illustrativeexample is given to evaluate our method and the result shows that the present improved mechanism can ensure the datareliable in transmitting and is scalable for quantum switching with output contention.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

Digital switching technology plays a core role in network communications. Switching can be performed in the timedomain and wavelength domain or a combination of these mechanisms and some switching schemes have been devel-oped for realistic networks [1]. In parallel, a lot of attention has been focused on quantum information science since the1980s [2–6].

Thus the demand for establishing quantum communication network becomes more and more urgent [7–10]. In orderto avoid a fully connected quantum network, analogous to classical networks, quantum switching device is a veryattractive system to build a realistic network where microscopic particles work as information carriers in the quantumcommunications. Therefore, quantum information storage, transmission and processing must be developed within theframework of quantum principles.

Kuo and Tsai [11] first present a quantum switching architecture. As shown in Fig. 1, the proposed mechanismallows data to be switched using a series of quantum operations. This method has high efficiency and both classicaland quantum signals can be transmitted. Cheng and Wang [12] further describe the quantum merging sorting to dynam-ically permute each input quantum data to its destination port. However, their design only considers the one-to-onemapping between input and output data. In fact, contention for an input will occur when one input data is routed

0960-0779/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2007.06.080

* Corresponding author. Tel.: +86 10 62771392.E-mail address: jiangmin03@mails.tsinghua.edu.cn (M. Jiang).

QuantumDomainSwitchingModule

Quantum inputports

Quantum output ports

Control

P0P1P2

1

2

4

7

3

6

0

5

P3P4P5P6

P7

P0P1P2P3P4P5P6

P7

Fig. 1. Architecture of a digital quantum switch.

M. Jiang et al. / Chaos, Solitons and Fractals 39 (2009) 1936–1942 1937

to two destination ports. Similarly, contention for an output will also occur when two input data are routed to the samedestination port. In other words, both one-to-multitude mapping and multitude-to-one mapping exist in the quantumswitching. Hence it is an important task to improve quantum switching scheme for completing some switching tasksunder contention.

In this paper, we point out the impropriety of Tsai and Kuo’s scheme for multicast service and improve their quan-tum switching mechanism to adapt it to quantum switching with contention. In classical network, when we send twoquantum data to the same line, we generally drop one randomly chosen packet and route the other. However, consid-ering the quantum no-clone theorem, once the packet is dropped, the data cannot be recovered. Therefore, this willresult in an unreliable quantum network. We address the problem of contention for an output which can occur betweenthe input packets at an internal switch. Based on that, all quantum data can be kept integrality. To illustrate our point,we present an example of a quantum switch with 8 ports.

This paper is organized as follows. In Section 2, we give some related concepts of quantum network. In Section 3, weshow the impropriety of multicast service in Tsai and Kuo’s quantum switching. In Section 4, we deal with the outputcontention in detail. Finally, we give conclusions in Section 5.

2. Preliminaries and notations

In this section, we introduce the concepts of quantum computing and quantum switching related to this paper.Detailed discussion can be found in Refs. [8,11].

Analogous to classical bits, qubit is the underlying unit in quantum information science. A qubit is a two-level quan-tum system, which can be spin states of a particle, two energy levels in an atom or different polarizations of a photon.Base vectors can be introduced in the quantum computation:

j0i ¼1

0

� �; j1i ¼

0

1

� �ð1Þ

A qubit state is often represented as a linear combination of these two base vectors (so called superposition):jWi ¼ aj0i þ bj1i, where a and b are complex numbers and jaj2 þ jbj2 ¼ 1. However, if we measure a qubit in such asuperposition state, the qubit would immediately collapse into j0i with probability jaj2 or j1i with probability jbj2,respectively.

The state of qubits can be transformed via quantum gates. CNOT gate is the basis of quantum switching mechanism[11–13] and has been successfully obtained in different quantum systems. tcnot is the time to implement CNOT gate andcan work as time unit in our implementation scheme. The CNOT gate applied on a two-qubit system can be used toachieve the transposition. As shown in Fig. 2, a two-qubit system j/;ui can change to ju;/i through three CNOT

P1:

P2:

φ

ϕ φ

ϕ

Fig. 2. Switching two qubits with three CNOT gates.

1938 M. Jiang et al. / Chaos, Solitons and Fractals 39 (2009) 1936–1942

gates. Any permutation of n qubits can be performed in six layers of CNOT gates with no ancillae. Circuit of this typehas been explored in Ref. [13].

Quantum switching architecture is firstly presented by Tsai and Kuo [11] Quantum data from the input port can besent to the output port by the control. Ports in the switching mechanism can be divided into two categories: input portset Si and output port set So. As shown in Fig. 1, the input port set Si can be described as fp1; p2; . . . png and the outputport set So can be shown as fp10; p20; . . . pn0g.

Given an n · n switch, at time t the behavior of a switch can be represented as a mapping Mt:fp1; p2; . . . png ! fp10; p20; . . . pn0g.

3. The impropriety of multicast service in quantum switching

Unicast and multicast services are two common transmission ways in the classical communication network. Tsai andKuo’s quantum switching mechanism also involved the concepts of unicast and multicast services. As shown in Fig. 3[11], its multicast service was focused on that quantum information from one input port could be distributed to a num-ber of different output ports. In other words, multicasting refers to the input contention problem or one-to-multitudemapping.

Actually, due to the principle of quantum mechanics, multicast service can never be achieved in quantum switching.Here, we will point out the impropriety of Tsai and Kuo’s quantum switching mechanism for multicast service by anexample of simple 3 · 3 switch. As shown in Fig. 4, the aim of this quantum switch is to distribute the quantum datafrom ports 3 to the 1 and 2.

We assume the quantum data from port 3 is aj0i þ bj1i. After three layers of CNOT gates, aj0i þ bj1i can be trans-ferred to the 1. However, if the CNOT gate acts on the ports 1and 2, the state of the two ports will become an entangledone: aj00i þ bj11i. Consequently, the quantum data in output port 2 is not simply aj0i þ bj1i.

In essence, the linear nature of quantum mechanics prohibits cloning precisely any unknown quantum state. More-over, measuring the quantum state will fundamentally damage the original information. Since qubits cannot be repli-cated, the contention problem for an input cannot be solved and multicasting service is unavailable in the quantumswitching, which also tells us the connection graph presented by Tsai and Kuo [11] cannot include ‘‘tree’’ or ‘‘forest’’.

QuantumDomainSwitchingModule

Quantum input ports Quantum output ports

Control

P0P1P2

3&6

5&7

0&2

4

P3P4P5P6P7

P0P1P2P3P4P5P6P7

Fig. 3. Multicast service in quantum switching.

1

2

3

1

2

3a 0 +b1

a

0 00 11a +b

1+b0

Fig. 4. The quantum switching mechanism cannot support multicast service.

M. Jiang et al. / Chaos, Solitons and Fractals 39 (2009) 1936–1942 1939

4. Quantum circuit for output contention problem

Analogous to classical communication, as shown in Fig. 5, the quantum switching will also deal with the multitudeto one mapping, named as output contention problem. Considering the quantum data juai, jubi from the input portP a; P b, there exist four different types of output contention status (Fig. 6).

(1) At time t, the data from pa and pb are sent to the port pa or pb; the connection graph is fpa; pbg ! fpa; pag orfpa; pbg ! fpb; pbg.

QuantumDomainSwitchingModule

Quantum input ports Quantum output ports

P0P1P2

5

6

303

0

03

P3P4P5P6P7

P0P1P2P3P4P5P6P7

5P8 P8

Control

Fig. 5. Output contention in quantum switching.

Pa

(3)

Pa

Pb

Pk Pk

Pa

Pb

Pa

Pa

Pb

(4)

Pk Pk

Pf, )a b≠( f

Pa

Pb

(2)

Pk Pk, )a b≠(k

Pa

Pb

Pa

(1)

Pa

Pb

Pk

Pa

Pb

Pk

Pa

Pb

Pk Pf

Fig. 6. Four competition status existing in the quantum switching architecture.

1940 M. Jiang et al. / Chaos, Solitons and Fractals 39 (2009) 1936–1942

(2) At time t, both data from pa and pb are sent to the port pkðk–a; bÞ, and no useful data exists in the port pk ; theconnection graph is fpa; pbg ! fpk ; pkgðk–a; bÞ.

(3) At time t, the data from pa and pb are sent to the port pk, the data from pk port should be sent to the port pa or pb

simultaneously, the connection graph is fpa; pb; pkg ! fpk ; pk ; pagðk–a; bÞ.(4) At time t, the data from pa and pb are sent to the port pk, the data from pk port should be sent to the port

pf ðf –a; bÞ, the connection graph is fpa; pb; pkg ! fpk ; pk ; pf gðk–a; bÞ.

In classical network, one of the contending packets is randomly dropped. However, due to the quantum no-clonetheorem, the data cannot be recovered once it is dropped in quantum domain This will result in an unreliable quantumnetwork.

Taking into account the above limitation, we developed how multitude to one mapping can be executed efficiently onquantum switching.

Considering a quantum switch at time t, the behavior can be depicted as Mt: fp1; p2; . . . ; png ! fp10; p20; . . . ; pn0g.The output set So ¼ fp10; p20; . . . pn0g contains l different elements which can be indexed as pl1

0ð1Þ; pl10ð2Þ; . . . pl1

0ðl1Þ.We assume that the busiest output port has N input data simultaneously at time t. Therefore, our implementationscheme can be constructed by N steps.

Firstly, assume the output set fp10; p20; . . . ; pn0g has at most l1 different output ports indexed aspl10ð1Þ; pl1

0ð2Þ; . . . pl10ðl1Þ and the corresponding input ports are pð1Þl1

; pð2Þl1; . . . pðl1Þ

l1, respectively. The mapping

M1 : fpð1Þl1; pð2Þl1

; . . . pðl1Þl1g ! fpl1

0ð1Þ; p0ð2Þl1; . . . pl1

0ðl1Þg can be firstly executed by permutation circuit. The whole task canbe completed in six layers of CNOT gates [13]. Then, a quick quantum data retrieval can be performed on thepl10ð1Þ; pl1

0ð2Þ; . . . pl10ðl1Þ ports simultaneously.

Secondly, the different ones of the remaining output ports (assume l2) can be indexed as pl20ð1Þ; pl2

0ð2Þ; . . . pl20ðl2Þ and

the corresponding input ports are pð1Þl2; pð2Þl2

; . . . pðl2Þl2

, respectively. However, the information of the input port pðiÞl2has been

exchanged with the other port pl1ði0Þ previously. Thus, the input ports should be substituted with the port pði0Þl2

. The map-ping is M1 : fpð10Þl2

; pð20Þl2; . . . pðl2 0Þ

l2g ! pl2

0ð1Þ; pl20ð2Þ; . . . pl2

0ðl2Þg. Then the permutation circuit can be implemented fol-lowed by the data retrieval.

In a similar way, we can execute the above process recursively until Mn to complete all the switching work.Here, take an 8 ports switch as an example. At given time t (Fig. 7), the status of the switch is described as the fol-

lowed mapping . Firstly, the subset of the different ports is fp5; p6; p3; p0g whose corresponding output subset can bechosen from the input port set such as fp0; p1; p2; p5g. Hence, we should complete permutation circuit ofM1 : fp0; p1; p2; p5g ! fp5; p6; p3; p0g. The first step can be implemented by three layers of CNOT gates. Then a quickdata retrieval can be performed on the fp5; p6; p3; p0g. As shown in Fig. 8, data retrieval index as ‘‘·’’.

After that, the biggest subset is fp3; p0g in the remaining output ports whose corresponding input subset originallyhas four selections: fp3; p4g,fp3; p7g, fp4; p6g,fp6; p7g. Here we randomly choose fp4; p6g. However, since data from p6

has been sent to p1 by previous operation, the port p1 should be substituted by p6. fp4; p1g should correspond to outputports fp3; p0g. Finally, there is the last step for permutation: fp4; p7g ! fp0; p3g to substitute fp2; p7g ! fp0; p3g.

We can get several helpful conclusions as follows.Remark 1: In the recursive operation, we can index every step as S1; S2; . . . ; Sn. Every output set involved in the N

steps can be indexed as s1; s2; . . . sn. Then, there must be sn # sn�1 . . . # s1.Remark 2: Our improved implementation has a good scalability. For example, in the Fig. 4, if there is data from a

new input port P8 sent to the port P5, then such permutation can support parallel with the second step as depicted bythe dotted box in Figs. 7 and 8. The time duration for the whole switching architecture will not increase.

P0 P5

P1 P6

P2 P3

The first layer

P1 P0

P4 P3

P2 P0

The second layer The third layer

P7 P3

P8 P5

Fig. 7. Three steps in the example.

P1

P2

P7

P6

P5

P4

P3

P0

P8

Fig. 8. The permutation circuits for contention in the example.

M. Jiang et al. / Chaos, Solitons and Fractals 39 (2009) 1936–1942 1941

Remark 3: When one step is implemented by disjoint permutation, then the time is only 3tcnot. However, when that isimplemented by joint permutation, cycle or queue, this should be done in six layers of CNOT gates. If the busiest portshas N input data simultaneously, the time duration for the switching is 3ntcnot 6 t 6 6ntcnot. Quantum computationmust be done on a time-scale less than the time of decoherence [14]. So the total implementation time should satisfythat: 6ntcnot � td.

5. Summary and discussion

With the rapid development of quantum information technology, establishing quantum communication networkbecomes an important task. Quantum switching architecture is one of the promising schemes for applicable quantumcommunication network. In this paper, we firstly analyze the impropriety of Tsai and Kuo’s quantum switching schemefor multicast service. Then we propose an improved quantum switching mechanism which can accomplish some quan-tum switching tasks with output contention. The illustrative example with 8 output ports demonstrates the effectivenessof our method.

The features of our scheme are dual: (i) it can reliably complete quantum switching task with output contention; (ii)it is applicable and scalable by improving the layout of the quantum network topology. Currently, Tsai and Kuo havedemonstrated an experimental realization of a 2 · 2 quantum switching using nuclear spins and magnetic resonantpulses [15]. The physical implementation of our improved quantum switching scheme is also realizable and it is ourfuture work.

Acknowledgements

This research is supported in part by the National Natural Science Foundation of China under Grant Numbers60433050, 60274025 and 60635040. T.J. Tarn also acknowledges the partial support from the US Army Research Officeunder Grant 911NF-04-1–0386.

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