A scheme for distributed quantum search through simultaneous state transfer mechanism

Ann. Phys. (Leipzig) 16, No. 12, 791 – 797 (2007) / DOI 10.1002/andp.200710265

A scheme for distributed quantum searchthrough simultaneous state transfer mechanism

Manu Gupta∗ and Anirban Pathak∗∗

JIIT University, A-10, Sector-62, Noida, UP-201307, India

Received 26 January 2007, accepted 15 August 2007 by U. EckernPublished online 29 October 2007

Key words Distributed quantum computing, quantum algorithm, Grover’s search algorithm.PACS 03.67.-a, 03.65.Ud, 03.67.Lx

Using a quantum network model, we present a scheme for distributed implementation of Grover’s algo-rithm. The proposed scheme can implement a quantum search over data bases stored in different computers.Entanglement is used to carry out different non-local operations over the spatially distributed quantum com-puters. A method to transfer the combined state of many qubits over the entanglement and subsequentlyrefreshing the entangled pair is presented. This method of simultaneous state transfer from one computer tothe other, is shown to result in a constant communication complexity.

c© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

One of the practical problems with the quantum computers is their scalability. This arises due to the dif-ficulties encountered in the present day implementation techniques such as decoherence, dissipation andexperimental imperfections [1–3]. A possible way to overcome this scalability problem is to combine alarge number of quantum computers, each having a small number of qubits, to simulate a quantum com-puter having a large number of qubits. This technique of distributed quantum computing (DQC) will beuseful in future, (even if we succeed to devise a practically large quantum computer) to perform searcheson distributed databases and to perform other complex tasks. At this point, we would like to note that thedistributed classical computing (DCC) is a matured subject but the existing protocols of DCC can not begeneralized to DQC. Actually, decoherence and the other difficulties related to scalability of a quantumcomputer also appear in the implementation of DQC. This is because in order to implement DQC, thequantum computers present in the network need to talk among themselves and we need to implement somenon-local quantum gates. Recent observations (see [4] and references therein) of the possibility of achiev-ing robust nonlocal gates and Eisert’s scheme [5] for distributed implementation of Shor’s algorithm [6] bytransferring a single qubit state over the entangled channel [5] have motivated us to study the possibilityof distributed implementation of Grover’s algorithm [7]. In Eisert’s scheme only the state of one qubit ispassed and then the entanglement between the computers is refreshed. This leads to a significant increase inthe communication complexity. In the present work, we have modified Eisert’s scheme and have provided ascheme to pass the combined states of many qubits over the entanglement and consequently demonstrateda new way to refresh the entanglement. Moreover, we have used this concept to implement Grover’s searchalgorithm on a distributed database.

In Sect. 2 we discuss the concept of DQC. We also describe a way to establish and refresh the entangledchannel between the two computers and a method of simultaneous state transfer of large number of qubits,from one computer to the other. Sect. 3 presents the generalized implementation of the distributed search

∗ E-mail: er [email protected]∗∗ Corresponding author E-mail: [email protected], Phone: +91 986 833 0126, Fax: +91 120 240 0986

c© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

792 M. Gupta and A. Pathak: Distributed quantum search

Fig. 1 (A) Transferring information over an entangled channel, (B) Taking measurement, transferring thestate classically and refreshing the bit, (C) Restoration to get the resultant state.

algorithm, while in Sect. 4 we analyze the complexity of the presented algorithm. Lastly in Sect. 5 wesummarize and conclude our findings.

2 Distributed quantum computing

DQC refers to a network of small quantum computers, connected by classical and quantum channels.Thus it provides a means for small capacity quantum computers to work together, to simulate a largecapacity quantum computer. Here each quantum computer optimally divide it’s set of qubits, betweenthe computation qubits and channel qubits. Computation qubits (generally large in number) are used forcarrying out different local operations, while the channel qubits (small in number) are used as a quantumchannel between the distributed computers. These channel qubits can be sent back and forth over thechannel. Qubits of a given computer can freely interact with all the other (computation or channel) qubitsof that computer; however to interact with the qubits of a connected remote computer it has to use thechannel qubits of that computer [8, 9].

DQC can be efficiently implemented using the model of a non-local CNOT. A non-local CNOT isactually an operation in which a CNOT gate is applied between two different computers i.e., one computercontains the control qubit and the other contains the target qubit. Practically, one shared entangled pair(ebit) and one bit of classical communication in each direction is necessary and sufficient to implementa non-local CNOT gate [5]. Since the CNOT gate together with the one-qubit gates constitute a universalset [6], the distributed implementation of any unitary transformation must be reduced to the implementationof non-local CNOT gates.

2.1 Establishing and refreshing an entangled pair

To perform a non-local operation a quantum channel has to be established between the interacting comput-ers; here we will entangle one of the channel qubits of both these computers to act as the quantum channelbetween them. To establish an entanglement between a remote and a local computer, the remote computersends a channel qubit to the local machine. Subsequently, the local machine entangles the remote channelqubit with one of its channel qubits and then sends one qubit of the pair back to the remote machine. Thisprocess establishes one ebit between the two computers. If one needs two ebits simultaneously betweentwo computers, then each of the computers entangle two of their own channel qubits and exchange onlyone qubit of the pairs with the other computer. As a result, two ebits are established with the cost of sendingone qubit, asymptotically.

During the application of non-local gates one needs to perform measurement (represented as M-gate)on the ebits and has to send some classical information (large dotted lines), resulting in the perturbation

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Ann. Phys. (Leipzig) 16, No. 12 (2007) 793

of entanglement. Hence the entangled pair must be refreshed and the channel qubits must be reset to state|0〉 so that they can be re-entangled maximally for further usage. In Fig. 1(A) qubits ‘2’ and ‘3’ pass someinformation to the qubit ‘1’, which in turn changes the entanglement present between ‘1’ and ‘1∗’. This ishow the scheme communicates the information from one computer to the other computer. To refresh theentangled bit a measurement on the ebit is made and based on the measured value a X-gate (selective π

2pulse) is applied on this bit (small dotted line) as shown in Fig. 1(B). This measured value is also passedon to the connected computer through the classical channel as shown by the large dotted line. However, toget the final result, a controlled-Z gate is applied to all the contributing qubits in the same way as they havecontributed (Fig. 1(C)), with the controlling bit being the measured ebit of the other computer. ComparingFigs. 1(A) and 1(C), a CNOT gate is applied on the ebit from both the computation bits to pass the requiredinformation to the other computer, while to reach the final state controlled-Z gate is applied to both thesebits in a similar order.

3 Distributed quantum search

As it is well known, Grover search algorithm [7,10] consists of two main parts: oracle and inversion aboutthe mean. In this presented algorithm we have implemented these two operations on a distributed platform,thus one has to use both classical and quantum channel for transferring the data between the spatiallyseparated computers. In general, with the increase in the number of qubits to be searched, the amount ofdata transferred between the two computers also increases, resulting in a large number of classical andquantum communications. A better way to deal with this problem is to jointly transmit the state of all thecomputational qubits to the entangled pair in a single operation. This will make the number of non-localoperations constant i.e., it will not depend on the number of qubits to be searched.

The method presented here is implemented on two spatially separated quantum computers having threequbits each. One qubit of each computer is used as the channel qubit, thus resulting in a four qubit dis-tributed quantum search (Fig. 2(A)). The initial input is a ‘N+4’ (N = 2n) state register initialized to|0〉⊗(n+2), where ‘n’ is 4 for our presentation, the total number of qubits to be searched on both the com-puters. To start with, the channel qubits should be entangled to establish a quantum channel between thetwo computers. Then Hadamard transformation is applied to all the qubits, to be searched. This producesan equal superposition state,

|ψ〉 =1√N

N−1∑

x=0

|x〉. (1)

Now the following steps are followed (Fig. 2);

1. The oracle

(a) Operator ‘UOX ’ is applied first, it flips the qubit |x〉4 according to the following equation,

|x〉4 → (σx)f(x)|x〉4 (2)

where f(x) is either 1 or 0. For the particular group of searched state, f(x) = 1 will only satisfy forthe states having desired last two bits, for all other states it will be 0, as shown by Table 1. Here thesearched state |x〉 is the given state, to be searched by the algorithm and |x〉5|x〉6 refers to the last twovalues of the states present in the quantum register.

(b) Subsequently the qubit |x〉4 is measured, and the flip gate is applied through classical communi-cation.

(c) The next step is to apply the operator ‘UOF ’, which changes the phase of the resultant searchedstate

|x〉123 → (−1)f(x)|x〉123 (3)

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Table 1

Searched state |x〉 f(x) = 1 if

|0000〉, |0100〉, |1000〉, |1100〉 |x5x6〉 �= |00〉|0001〉, |0101〉, |1001〉, |1101〉 |x5x6〉 = |01〉|0010〉, |0110〉, |1010〉, |1110〉 |x5x6〉 = |10〉|0011〉, |0111〉, |1011〉, |1111〉 |x5x6〉 = |11〉

where f(x) = 1, if the first two bits ‘|x〉1|x〉2’ of the states in the quantum register coincides with thefirst two bits of the given searched state ‘|x〉1s|x〉2s’, and the third bit matches with the defined bit ofthat group of searched state, as shown in Table 2.

Table 2

Searched state |x〉 f(x) = 1 if

|xx00〉 |x1x2x3〉 = |x1sx2s0〉|xx01〉 |x1x2x3〉 = |x1sx2s1〉|xx10〉 |x1x2x3〉 = |x1sx2s1〉|xx11〉 |x1x2x3〉 = |x1sx2s1〉

(d) Then the Hadamard gate is applied on qubit |x〉3, after which it is measured and based on themeasured value, operator ‘ROZ ’ is applied through classical communication i.e., if the |x〉3 = |1〉then only the operator is applied

|x〉56 → (−1)f(x)|x〉56 (4)

where f(x) = 1 according to the Table 1.

* At this point channel qubits are re-entangled and a fresh quantum channel is established betweenthe two computers.

2. Then Hadamard transformationH⊗n is applied on all the qubits to be searched.

3. The conditional phase shift

(a) Now perform the operation ‘UPO’, in which |x〉4 is not flipped only when all the other qubitsunder operation are |0〉

|x〉4 → (σx)δx0 |x〉4 (5)

(b) Subsequently the qubit |x〉4 is measured again, and the flip gate is applied through classical com-munication.

(c) Then the operator ‘UPF ’ is applied, in which every computational basis state except |0〉 receive aphase shift of -1

|x〉 → −(−1)δx0 |x〉 (6)

(d) Then the hadamard gate is again applied on qubit |x〉3, after which it is measured and based onthe measured value, operator ‘UPZ’ is applied through classical communication i.e., if the |x〉3 = |1〉then only the operator is applied



Fig. 2 (A) The circuit for four qubit distributed quantum search, (B) Initial density matrix, (C) Finaldensity matrix, for four qubit distributed quantum search.

|x〉56 → (−1)f(x)|x〉56 (7)

f(x) = 1 for all the state except |x〉56 = |00〉.4. And finally the Hadamard transformationH⊗n is applied.

Steps 1 to 4 are iterated√N times to get the desired result.



The simulation results of initial density matrix as shown in Fig. 2(B) marks only the initial state of|000000〉, while the final density matrix in Fig. 2(C) has the major trace of the state |110011〉 with somenegligible traces of other states, when the desired state of search is |1111〉. For the higher bit searchesthe number of states increases and proportionally the number of operations of the operator ‘UOX ’ also in-creases, thus behaving as a distributed oracle which operates, considering only the later half of the searchedstate. While the operator ‘UOF ’ acts, considering the former half of the searched state, and completes theoracle function by flipping only the searched state from the database.

4 Complexity analysis

The complexity of our algorithm for distributed quantum search remains the same as in Grover’s search

algorithm i.e.,√

NM , where ‘N’ is the number of states to be searched (N = 2n) and ‘M’ is the number

of solution states. This is because the number of iterations remains the same for both these algorithms.However the number of operations in the presented algorithm is more than the Grover’s case.

To find out the additional cost of the algorithm presented here, we must consider the complexity of theindividual operations. Our search algorithm consists of four major steps, similar to Grover’s algorithm.The Hadamard transformation in step 2 and 4 requires n = log(N) operations each, which remains samein both the algorithms. The conditional phase shift in step 3 can be originally implemented using O(n)gates, while we are using 2( n

2 + 1) gates. The cost of the Oracle depends on the specific applications; aswe require only two Oracle calls with half the number of qubits for each iteration, the complexity will beapproximately of the same order.

However, our implementation does have an extra burden of communication complexity; by communi-cation we refer here to the classical one. As the quantum communication is carried out by an entangledpair and by applying a local gate on one ebit the information reaches the other ebit, so it does not add up tocommunication complexity. Although establishing an entangled pair needs two classical communications,one in each direction. Both the oracle and the conditional phase shift operations need two classical commu-nications, independent of the number of qubits searched. Hence the total communication complexity of theimplementation is 4

√N which is clearly O(

√N). This shows that the number of qubits are just increasing

the iterations and not the classical communications per iteration.

5 Conclusion

We have presented a method to implement a quantum search based on a distributed computing paradigm. Toimplement this distributed search algorithm we need to connect the computers with one shared entangledpair and one bit of classical communication in each direction. The algorithm presented here have the uniquefeature that it makes the communication complexity independent of the number of qubits to be searched.The communication complexity depends only on the number of computers where the search is taking place.This work can be extended to a generalized network with search taking place on ‘n’ number of computerssimultaneously. Moreover the process can be extended to implement other distributed operations.

Acknowledgements MG thanks Prof. A. Kumar and Avik Mitra for helpful discussions and he is also grateful to DrPrasanta K. Panigrahi for his valuable advice and support. AP thanks DST, India for partial financial support throughSR/FTP/PS-13/2004.

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A scheme for distributed quantum search through simultaneous state transfer mechanism