Production of multipartite entanglement for electron spins in quantum dots

Production of multipartite entanglement for electron spins in quantum dots

F. Bodoky and M. Blaauboer

Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

共Received 9 July 2007; published 13 November 2007兲

We propose how to generate genuine multipartite entanglement of electron spin qubits in a chain of quantum dots using the naturally available single-qubit rotations and two-qubit Heisenberg exchange interaction in the system. We show that the minimum number of required operations to generate entangled states of the GHZ, cluster, and W type scales linearly with the number of qubits and estimate the fidelities of the generated entangled cluster states. As the required single and two-qubit operations have recently been realized, our proposed scheme opens the way for experimental investigation of multipartite entanglement with electron spin qubits.

DOI:10.1103/PhysRevA.76.052309 PACS number共s兲: 03.67.Mn, 03.67.Lx, 73.21.La, 03.65.Ud I. INTRODUCTION

A. Multipartite entanglement

Bipartite entanglement refers to nonclassical correlations 关1,2兴 between two quantum particles, and multipartite en-tanglement to nonclassical correlations between three or more quantum particles. The characterization and quantifica-tion of the latter is far less understood than for bipartite en-tanglement关3兴. In particular, in the case of multipartite en-tanglement it is no longer sufficient to ask if the qubits are entangled, but one needs to know how they are entangled as there are different ways—known as entanglement classes—in which three or more qubits can be entangled. For three qubits, there are two different equivalence classes of genuine tripartite entanglement 关4兴, for four qubits already nine关5兴 or eight 关6兴, and the number of classes is growing with the number of qubits. Two entangled states belong to the same equivalence class and are called locally equivalent if it is possible to transform between them using local opera-tions and classical communication共LOCC兲 only, i.e., without interactions between two or more qubits. The two classes of entanglement for three qubits are the GHZ and the W class 关4,7兴, with representative members 关8兴 兩GHZ3典=冑1₂共兩000典

+兩111典兲 and 兩W3典=冑1₃共兩001典+兩010典+兩100典兲 共the subscript

in-dicating the number of involved qubits兲, which are both go-ing to be addressed in this paper. Both of these classes can be generalized to arbitrary numbers of qubits. Another interest-ing class of multipartite entanglement for four or more qubits is the cluster class关9兴, which forms the basis of proposals to implement a measurement-only quantum computer, the one-way quantum computing scheme 关10兴. These states maxi-mize mutual bipartite entanglement and its four-partite rep-resentative is兩␾4典=1₂共兩0000典+兩0011典+兩1100典−兩1111典兲.

Multiqubit entanglement is thus not a straightforward ex-tension of bipartite entanglement and gives rise to new phe-nomena which can be exploited in quantum information and quantum computing processes. For example, there are quan-tum communication protocols that require multiparty en-tanglement such as universal error correction关11兴, quantum secret sharing 关12兴, and telecloning 关13兴. Also, highly en-tangled multipartite states are needed for efficient quantum computing—all known quantum algorithms共such as Shor’s factorization关14兴 and Grover’s search 关15兴 algorithm兲 work

with multipartite entanglement—and GHZ states can be used to construct a universal quantum computer关16兴. In addition, multiqubit entangled states provide a stronger test of local realism关17兴 which is based on individual 共rather than statis-tical, as in the bipartite case兲 measurement results. As a gen-eral rule, one can say that the more particles are entangled, the more clearly nonclassical effects are exhibited and the more useful the states are for quantum applications.

So far, multipartite entanglement has been realized in a number of experiments, using liquid-state NMR 关18兴, pho-tons关19–22兴, cold atoms 关23,24兴, and ions 关25,26兴. The latter two experiments for trapped ions have demonstrated the de-terministic creation of a GHZ and a W sate. Tripartite, and more generally multipartite, entanglement has not yet been realized for qubits in a solid-state environment. The latter type of qubit systems, consisting of, e.g., electrons confined in quantum dots关27兴 or superconducting Josephson junctions 关28兴, are attractive since they are in principle scalable to an arbitrary number of qubits. A number of ideas have been suggested for the creation of tripartite entangled states, using exciton states in coupled quantum dots 关29兴, electron-hole entanglement in the Fermi sea 关30兴, and superconducting charge and flux qubits关31–33兴.

In this paper, we present schemes for deterministic cre-ation of GHZ, W, and cluster states for electron spin qubits in quantum dots using the naturally available two-qubit 共Heisenberg exchange兲 interaction and single-spin rotations. This choice of system is motivated by the fact that both single-qubit rotations and tunable two-qubit Heisenberg ex-change interactions have already been demonstrated experi-mentally for these qubits关34,35兴. However, our scheme can easily be used for other types of qubits as well, e.g., super-conducting qubits for which tunable coupling has also very recently been realized关36兴. We show that the required num-ber of two-qubit interactions for the generation of N-partite entangled states and for the transformation from a disen-tangled to a maximally endisen-tangled basis scales linearly with N for all types of entangled states considered here. We also present arguments that the total number of single- and two-qubit operations that our schemes predict is in fact the mini-mum number required to create these multipartite entangled states using single-qubit rotations and Heisenberg exchange interactions.

B. Electron spin qubits

An electron spin qubit关37兴 consists of a single electron confined in a quantum dot共QD兲, an island in a semiconduct-ing nanostructure关27兴. The electron occupies discrete energy levels in the quantum dot which split into separate levels for spin up and spin down due to Zeeman splitting when the quantum dot is placed in an external magnetic field. The qubit is encoded in the spin degree of freedom, with the ground state spin up共denoted as 兩↑典 and defined along the direction of the magnetic field, which we assume to be the z axis兲 corresponding to the logical bit 兩0典 and spin down 共兩↓典兲 corresponding to the logical bit兩1典. Electron spin qubits are attractive candidates for quantum computing since they are in principle scalable, relatively robust against decoherence 共as compared to, e.g., charge qubits兲, and allow for a high level of control over individual qubits关38兴.

Two basic operations are available to manipulate the state of the qubit: First, coherent rotation of a spin around an axis in the 共x,y兲 plane using electron spin resonance 共ESR兲, which consists of applying an oscillating time-dependent magnetic field B共t兲 in this plane whose frequency is on reso-nance with the transition frequency between兩↑典 and 兩↓典. A rotation around a certain angle is controlled by the time of application and the strength of the magnetic field, and is described by the evolution operator

UR共t兲 = exp

冋

共i␥/2兲

冕

0

t

B共␶兲kជ·␴ជd␶

册

, 共1兲 which corresponds to the Hamiltonian HR共t兲

= −共1/2兲ប␥B共t兲kជ·␴ជ, where␥denotes the gyromagnetic ratio,

␴ជ=共␴x,␴y,␴z兲, and kជ⬅共sin␪cos␾, sin␪sin␾, cos␪兲

rep-resents a unit vector on the Bloch sphere

[␪僆关0,␲兲, ␾僆关0,2␲兲] in the direction of the magnetic

field. The evolution operator 共1兲 corresponds to a rotation R_k共n兲共␤兲=exp关−共1/2兲i␤kជ·␴ជ共n兲兴 of the nth qubit with angle ␤ ⬅−␥兰0

t

B共␶兲d␶around axis kជ, where␴ជ共n兲⬅1丢¯丢␴ជ丢¯1. These ESR-induced rotations have recently been experimen-tally observed in quantum dots关34兴.

The second available operation is Heisenberg interaction between two spins described by the evolution operator UEX共t兲=exp共−iប/4兲兰0

t

J共␶兲␴ជ共n兲·␴ជ共n+1兲d␶. Here, J共␶兲 is the time-dependent exchange energy. By tuning the interaction time t with a gate voltage, the共SWAP兲␣ gate can be directly generated as共SWAP兲␣⬅UEX共t兲, and we will denote it from now on as 共USWAP兲␣_{= e}−␣/4i␲

冤

e␣/2i␲ 0 0 0 0 cos

共

␣₂␲

兲

i sin

共

␣₂␲

兲

0 0 i sin

共

␣₂␲

兲

cos

共

␣₂␲

兲

0 0 0 0 e␣/2i␲

冥

, 共2兲 where␣共t兲⬅−_␲ប兰₀tJ共␶兲d␶. For ␣=1₂ the

冑

USWAP gate maxi-mally entangles two spins of opposite directions. A

冑

USWAP operation has also recently been demonstrated for spin qubits 关35兴. Together, single-qubit rotations and the

冑

U_SWAP-gate

form a universal set of quantum gates, into which any quan-tum operation can be decomposed关37兴.

C. Outline

This paper is organized as follows: in Sec. II we show how to generate N-partite entangled cluster states共Sec. II A兲, GHZ states 共Sec. II B兲, and W states 共Sec. II C兲 using the smallest number of two-qubit 共USWAP兲␣ operations and single-qubit rotations. In Sec. III we analyze the effects of errors in the timing of the共USWAP兲␣- and single-qubit opera-tions on the generation of N-partite cluster states, quantified by the fidelity. Finally, in Sec. IV, we discuss the feasibility of the multipartite entanglement generation scheme that we propose in the context of present-day experimental tech-niques, followed by conclusions.

II. GENERATION OF MULTIPARTITE ENTANGLED STATES

In this section we describe the generation of N-qubit en-tangled states in a chain of quantum dots where each dot is occupied by one electron, using single-qubit rotations and pairwise exchange interactions between nearest-neighbor spins. Starting from the ground state which consists of N disentangled up-spins 兩00¯0典 and using a recursive ap-proach in N we derive sequences of single-qubit rotations and two-qubit共USWAP兲␣ operations which, when applied to 兩00¯0典, yield a N-partite cluster, GHZ, or W state. We begin by briefly recounting the generation of entanglement and the implementation of a basis transformation for two qubits, and then present our main results in Secs. II A–II C below.

For two qubits, the shortest sequence required to trans-form the ground state兩00典 into a maximally entangled state is

冑

U_SWAP共1,2兲 R˜_k共i兲共␲兲, 共3兲

with k˜ an arbitrary axis in the 共x-y兲 plane and i=1,2. It has also been shown that the shortest sequence required to implement the transformation from the standard共or com-putational兲 basis to a maximally entangled basis consisting of Bell states is given by关39兴

E˜共1,2,i兲_k ⬅

冑

U_SWAP共1,2兲 R˜_k共i兲共␲兲

冑

U_SWAP共1,2兲 共4兲

with i = 1 , 2. Two

冑

USWAP operations are needed in Eq.共4兲, since one

冑

U_SWAP-interaction only entangles two of the four standard basis states. Given a linear array of quantum dots in which each dot is occupied by a single spin qubit in the兩0典 or 兩1典 state, entangled states of three or more qubits can be generated by pairwise application of the sequence共4兲, as we show in the next subsection. By applying local operations in between, one can control to which class the generated en-tangled states belong. This forms the basis of our calcula-tions in the next two subseccalcula-tions. Without loss of generality, we choose the axis k˜ as x and i=1 in Eq. 共4兲, and omit the indices k˜ and i in E˜_k

共1,2,i兲

in the following. A. Cluster states

either in the兩0典 or 兩1典 state, results in an N-partite entangled cluster state, as we prove in Proposition II.1 below. Specifi-cally, we will prove that application of the sequence

ECl

N _{⬅ E共N−1,N兲}

¯ E共1,2兲 共5兲

transforms an arbitrary disentangled state of the N-partite standard basis into a cluster state. The definition of linear cluster states for N qubits is as follows: the cluster state is the state resulting when applying the Ising interaction Z共n,n+1兲共␪兲=exp共−i␪/ 4兲共1−␴共n兲_z 兲共1−␴_z共n+1兲兲 with ␪=␲ 共the so-called z-phase gate兲 to each neighbor in a N-qubit chain prepared in the state 丢i=1

N

1 /

冑

2共兩0典+兩1典兲. The z-phase gate can be generated in quantum dots as Z共n,n+1兲 = R_z共n兲共␲/ 2兲R_z共n+1兲共−␲/ 2兲

冑

U共n,n+1兲_SWAPR_z共n兲共␲兲

冑

U_SWAP共n,n+1兲 共see 关37兴兲, where we omit the overall phase factor共as in the rest of this paper兲. In order to generate a cluster state, one thus has to apply this z-phase gate to each pair of qubits in the state 1 /

冑

2共兩0典+兩1典兲=Ry共␲/ 2兲兩0典. With these observations, we are

now ready to prove the following. Proposition II 1. The two sequences

ZN⬅ Z共N−1,N兲¯ Z共1,2兲Ry共N兲

冉

␲ 2

冊

¯ Ry 共1兲

冉

␲ 2

冊

共6a兲 and ECl N _{⬅ E共N−1,N兲} ¯ E共1,2兲 共6b兲

are locally equivalent.

Proof. We start by rewriting the sequence ZN as

ZN= Z共N−1,N兲Ry共N兲

冉

␲ 2

冊

¯ Z 共2,3兲_R y 共3兲

冉

␲ 2

冊

Z 共1,2兲_R y 共2兲

冉

␲ 2

冊

Ry 共1兲

冉

␲ 2

冊

共7a兲 =Z˜共N−1,N兲Z˜共N−2,N−1兲¯ Z˜共1,2兲Ry共1兲

冉

␲ 2

冊

, 共7b兲

which consists of N − 1 applications of the operator Z˜共n,n+1兲 ⬅Z共n,n+1兲_R

y

共n+1兲_{共␲/ 2兲, plus an additional rotation R}

y

共1兲_{共␲/ 2兲.} We now write Z˜共n,n+1兲 in terms of E共n,n+1兲关Eq. 共4兲兴, with the goal to express ZNas ZN⬅LECl

N

, where L is a product of local operations. To this end, we use the identity

Z ˜共1,2兲_R y 共1兲

冉

␲ 2

冊

= Ry 共1兲

冉

␲ 2

冊

Rx 共1兲

冉

␲ 2

冊

Ry 共2兲

冉

␲ 2

冊

Rx 共2兲

冉

₋ ␲ 2

冊

E 共1,2兲_, 共8兲 and substitute Eq. 共8兲 into Eq. 共7b兲. The two rotations on qubit 1 commute with the sequence to the left of them, and can thus be absorbed into the local operation L, leaving Z˜共2,3兲 acting on R_y共2兲共␲/ 2兲R_x共2兲共−␲/ 2兲. By rewriting Z ˜共n,n+1兲_R y 共n兲

冉

␲ 2

冊

Rx 共n兲

冉

₋␲ 2

冊

= Ry 共n兲

冉

␲ 2

冊

Ry 共n+1兲

冉

␲ 2

冊

⫻Rx共n+1兲

冉

− ␲ 2

冊

E 共n,n+1兲_{, 共9兲} repetitive substitution of Eq.共9兲 into Eq. 共7b兲 for increasing

n and using commutation relations to reorder the resulting sequence such that rotations are shifted to the left of all

冑

USWAPoperations, we find

ZN= LE共N−1,N兲¯ E共1,2兲= LECl N , 共10兲 with L = Ry共N兲

冉

␲ 2

冊

Rx 共N兲

冉

₋␲ 2

冊

Ry 共N−1兲

冉

␲ 2

冊

¯ Ry 共2兲

冉

␲ 2

冊

Ry 共1兲

冉

␲ 2

冊

⫻Rx共1兲

冉

␲ 2

冊

. 共11兲

Since L does not change the entanglement class of the state that has been generated by ECl

N

, we have thus proven that application of the transformation EN_{to any state of the}

stan-dard basis leads to a cluster state. 䊏

Inspecting the sequence共6b兲, we see that 2共N−1兲

冑

USWAP operations and 共N−1兲 rotations are needed to generate a N-partite cluster state, whereas when using previously pro-posed implementations of the z-phase gate Z共n,n+1兲 关40兴 a total of共4N−3兲 rotations are required. The sequences given by Eqs. 共6a兲 and 共6b兲 transform the standard basis into a basis of cluster states. Note that in order to transform the ground state 兩0¯0典 into a cluster state, the sequence E共N−1,N兲¯E共2,3兲

冑

U_SWAP共1,2兲 R_x共1兲共␲兲, which contains one

冑

USWAP operation less than Eq. 共6b兲, is sufficient, since

冑

U_SWAP共1,2兲 R_x共1兲共␲兲 关see Eq. 共3兲兴 already maximally entangles the first two qubits.

A special property of the z-phase gate, being a diagonal matrix, is that the Z共n,n+1兲 matrices commute for different n. As a result, the transformation 共6a兲 and 共6b兲 from the stan-dard basis to the cluster basis can be done in two steps: first, all the even-numbered qubits are simultaneously entangled to their共odd兲 neighbor to the right, and then the same is done for the odd-numbered qubits. In the next proposition, we show that the same commutation relation applies for the 共nondiagonal兲 E共n,n+1兲 _{matrices in Eq.共5兲.}

Proposition II 2. The two sequences

Z共N−1,N兲Z共N−3,N−2兲¯ Z共3,4兲Z共1,2兲Z共N−2,N−1兲¯ Z共2,3兲 ⫻Ry共N兲

冉

␲ 2

冊

¯ Ry 共1兲

冉

␲ 2

冊

共12a兲 and E共N−1,N兲E共N−3,N−2兲¯ E共1,2兲E共N−2,N−1兲¯ E共2,3兲 共12b兲 are locally equivalent.

Proof. Since the Z共n,n+1兲matrices commute for different n, the sequence共12a兲 is equivalent to the right-hand side of Eq. 共6a兲 and hence by Proposition II.1 to the sequence 共6b兲. It now remains to be shown that 共6b兲 is equivalent to 共12b兲. This directly follows from the fact that the matrices E共n,n+1兲 and E共n+1,n+2兲关Eq. 共4兲兴 commute

Propositions II 1 and II 2 imply that the sequences 共6a兲 and共12a兲, which consist only of the entangling operations E共n,n+1兲, create a state of the cluster class. Can this be done with fewer operations? To answer this question, consider first three qubits: we have already seen that the first

冑

U_SWAP op-eration in the sequence共6b兲 can be omitted when entangling the ground state, and if we start in an appropriate excited state also the first rotation is not needed. That leaves the sequence

冑

U_SWAP共1,2兲 R_x共1兲

冑

U_SWAP共1,2兲

冑

U_SWAP共2,3兲 . It is straightforward to check, e.g., by calculating the tangle␶关42兴 of the resulting entangled state, that this is the shortest sequence of

冑

U_SWAP operations and single qubit rotations that creates a tripartite cluster state: if any of the four operations is omitted, ␶= 0 and the resulting state is no longer a cluster state. General-izing to an arbitrary number of qubits, we note that omitting any operation in the sequence共6b兲 or 共12b兲 leads to a state which is not maximally connected in the sense of Ref. 关9兴, and therefore cannot be a cluster state.

B. GHZ states

In this section we show how the disentangled N-qubit state 兩00¯典 can be transformed into a N-qubit GHZ state using single-qubit rotations and

冑

USWAP-operations. We start with the observation that GHZ states are generated by suc-cessive application of the controlled-NOT共CNOT兲 gate UCNOT: Observation 1. Starting from the disentangled N-qubit state 兩00¯0典, the N-partite GHZ state 兩GHZN典=兩00¯0典

+兩11¯1典 共disregarding normalization兲 is generated by N − 2 applications of the UCNOT,

兩GHZN典 =

冉

兿

n=N−1 2 UCNOT共n,n+1兲

冊

Ry共1兲

冉

− ␲ 2

冊

Rx 共1兲

冉

␲ 2

冊

⫻Rx共2兲

冉

␲ 2

冊

冑

USWAP 共1,2兲 _R y 共1兲_{共␲兲兩00 ¯ 0典, 共14兲} where U_CNOT共n,n+1兲 denotes a UCNOToperation with the nth qubit as control bit, and the共n+1兲th qubit as target bit.

Note that the order in the product in Eq. 共14兲, starting with the highest n = N − 1, is essential. In Eq.共14兲, the opera-tion R_y共1兲

共

−␲₂

兲

R_x共1兲

共

␲₂

兲

R_x共2兲

共

␲₂

兲

冑

U_SWAP共1,2兲 R_y共1兲共␲兲 on the first two qubits yields the Bell state共1/

冑

2兲共兩00典+兩11典兲 and each suc-cessive UCNOTgate entangles one more qubit to this super-position, resulting in the N partite GHZ state兩GHZN典. Using

single-spin rotations and

冑

USWAPoperations only, the short-est sequence of operations required to implement the UCNOT gate is given by关41兴 U_CNOT共n,n+1兲⬅ Ry共n兲

冉

− ␲ 2

冊

Rx 共n兲

冉

₋␲ 2

冊

Rx 共n+1兲

冉

␲ 2

冊

冑

USWAP 共n,n+1兲 ⫻Rx共n兲共␲兲

冑

USWAP共n,n+1兲Ry共n兲

冉

␲ 2

冊

. 共15兲

We now substitute Eq.共15兲 into Eq. 共14兲. By moving all single-qubit rotations that commute with the sequence of op-erations to the left of them in front of all 冑SWAPoperations and defining U_CNOT共n,n+1兲
⬅

冑

U_SWAP共n,n+1兲Rx共n兲共␲兲

冑

USWAP共n,n+1兲Ry共n兲

冉

␲ 2

冊

Rx 共n兲

冉

␲ 2

冊

, 共16兲 Eq.共14兲 becomes 兩GHZN典 = L˜

冉

兿

n=N−1 2 U_CNOT共n,n+1兲

冊

冑

_U SWAP 共1,2兲 _R y 共1兲_{共␲兲兩00 ¯ 0典,} Nⱖ 3, 共17兲

where L˜ consists of single-qubit rotations. We see from Eq. 共17兲 that a total of 共2N−3兲

冑

U_SWAP operations and a mini-mum of 共3N−5兲 single-qubit rotations are needed to trans-form the separable state兩00¯典 into an N-partite entangled state in the GHZ class. Compared to cluster states 共see the previous section兲, the generation of a GHZ state thus requires 共2N−4兲 more single-qubit rotations. In practice, implemen-tation of the sequence共17兲 can be done in the most efficient way by applying two

冑

USWAP operations simultaneously in each step. This can be achieved by starting with qubit num-ber m⬅N/2 共for N even, or m⬅共N+1兲/2 for N odd兲 in the middle of the chain and reordering the sequence共17兲 as 共for N even兲:

兿

j=N−1 m+1

共U
UCNOT共N−j+1,N−j兲
兲CNOT共j,j+1兲

冑

USWAP共m,m+1兲Ry共m兲共␲兲兩00 ¯ 0典.

共18兲 The two U
operations in between the brackets in Eq.CNOT 共18兲 can be performed simultaneously. An analogous expres-sion as Eq.共18兲 applies if N is odd.

Using the same line of reasoning as in the previous sub-section and the fact that GHZ states are also maximally con-nected关9兴, one can directly show that the number of

冑

USWAP operations and the number of rotations over␲in Eq.共17兲 is minimal. Although we have no formal proof for this, we suspect that the total number of rotations in Eq.共17兲 is mini-mal.

C. W states

A W state for N qubits is a multipartite entangled state that can be written in the form

兩WN典 = 1

冑

N共兩10 ¯ 0典 + 兩01 ¯ 0典 + ¯ + 兩00 ¯ 1典 N terms 兲. 共19兲 In this section we show how兩WN典 can be generated using

the least possible number of single-qubit rotations and

qubits such that each qubit is excited with a fraction 1 / N. Rotating qubit 1 and then applying a共USWAP兲␣interaction to qubits 1 and 2 results in the fraction of the first excitation being cos共␣/ 2兲; so in order to have the first qubit excited with a fraction 1 / N, ␣ has to be chosen as ␣ = 2 arccos共

冑

1 / N兲. By applying the same reasoning to all the following qubits, we find

兩WN典 = J共N−1,N兲共␮N−1兲 ¯ J共1,2兲共␮1兲Ry共1兲共␲兲兩00 ¯ 0典, 共20兲 with ␮n= 2 ␲arccos

冉

冑

1 N − n + 1

冊

,

where J共n,n+1兲共␮兲 denotes a 共USWAP兲␮ gate on the qubits n and n + 1.

For implementation of the sequence共20兲 we can apply the same trick as for the GHZ states by starting from the middle qubit in the two opposite directions along the chain and per-forming two

冑

U_SWAPoperations simultaneously. To find the correct interaction angles we then need to distinguish be-tween even and odd number of qubits N. For the even case, we rotate the qubit m = N / 2, apply an interaction J共m,m+1兲共1/2兲, and then proceed as with two independent strings of length m. For the odd case, we rotate the qubit m =共N+1兲/2, apply an interaction J共m,m+1兲共␮兲 with ␮ = 2 arccos共

冑

m / N兲, and then proceed as with two independent strings of length m共to the left兲 and m−1 共to the right兲. Thus the required number of operations to transform the state 兩00¯0典 into a N-partite W state consists of 共N−1兲 共USWAP兲␮_{-operations and one rotation.}

III. FIDELITY

In the previous sections we have assumed perfect control of the single- and two-qubit operations, i.e., we assumed that all the pulses were perfectly timed. However, for a physical implementation it is important to estimate the effect of im-perfections in the 共USWAP兲␣ gate operations and in single-qubit rotations on the intended final entangled states. In this section, we provide such an estimate for the generation of cluster states 关Eq. 共5兲兴, assuming that the control of each

冑

U_SWAP-operation is off by a small parameter ⑀, and simi-larly for each rotation by a small parameter␦, i.e., we replace

冑

USWAP→共USWAP兲共1/2+⑀兲 and R共␲兲→R共␲+␦兲, where 兩⑀兩1/2 and 兩␦兩␲. As measure for the effect of the inac-curacies⑀and␦ we use the fidelity F关43兴, which describes the overlap between the intended 共“perfect”兲 state 兩␾N典 and

the generated共in the presence of the inaccuracies兲 state 兩␾˜N典.

For pure states, F is defined as:

F⬅

冑

具␾N兩␾˜N典具␾N兩␾N典, 共21兲

where兩␾˜N典 and 兩␾N典 are both normalized.

In the following, we calculate F as a function of N for the cluster state

兩␾N典 = E共N−1,N兲¯ E共1,2兲兩00¯典, 共22兲

up to second order in ⑀ and ␦. We start by expanding 共USWAP兲1/2+⑀_{关Eq. 共2兲兴 and R}

x共␲+␦兲 关Eq. 共1兲兴 up to second order in␦ and⑀: 共USWAP兲1/2+⑀_{= e}−i␲/4共1/2+⑀兲

冤

ei␲/2共1/2+⑀兲 0 0 0 0 cos

关

␲₂

共

1₂+⑀

兲兴

i sin

关

␲₂

共

1₂+⑀

兲兴

0 0 i sin

关

␲₂

共

1₂+⑀

兲兴

cos

关

␲₂

共

1₂+⑀

兲兴

0 0 0 0 ei␲/2共1/2+⑀兲

冥

, 共23兲 Rx关␲+␦兴 =

冤

cos

共

␲+⑀₂

兲

0 − i sin

共

␲+⑀₂

兲

0 0 cos

共

␲+⑀₂

兲

0 − i sin

共

␲+⑀₂

兲

− i sin

共

␲+⑀₂

兲

0 cos

共

␲+⑀₂

兲

0 0 − i sin

共

␲+⑀₂

兲

0 cos

共

␲+⑀₂

兲

冥

. 共24兲

sin

冉

␲ 2 + ␦ 2

冊

= sin

冉

␲ 2

冊

cos

冉

␦ 2

冊

+ cos

冉

␲ 2

冊

sin

冉

␦ 2

冊

= 1 − B2 2 + O共B 3_{兲, 共25d兲} cos

冉

␲ 2 + ␦ 2

冊

= cos

冉

␲ 2

冊

cos

冉

␦ 2

冊

− sin

冉

␲ 2

冊

sin

冉

␦ 2

冊

= − B + O共B 3_{兲. 共25e兲}

Here c⬅exp共i␲/ 4兲 is a constant, A⬅␲⑀/ 2, and B⬅␦/ 2. Using Eqs.共23兲, 共24兲, and 共25a兲–共25e兲, we construct the entangling operation E˜ ⬅共USWAP兲1/2+⑀R_x共1兲共␲+␦兲共USWAP兲1/2+⑀up to second order in the parameters⑀and␦:

E ˜ =

冤

− c共1 + 2iA兲B _冑1₂

关

1 +共1 + i兲A − 共1 − i兲A2₋B2

2

兴

¯ 1 冑2

关

1 +共1 + i兲A − 共1 − i兲A 2₋B2 2

兴

− 2c3AB ¯ −i 冑2

关

1 −共1 − i兲A − 共1 + i兲A 2₋B2 2

兴

− cB ¯

0 _冑−i₂

关

1 −共1 − i兲A − 共1 + i兲A2₋B2

2

兴

¯

. . . −i_冑2

关

1 −共1 − i兲A − 共1 + i兲A2₋B2

2

兴

0

. . . − cB 冑−i2

关

1 −共1 − i兲A − 共1 + i兲A

2₋B2 2

兴

. . . − 2c3_AB 1 冑2

关

1 +共1 + i兲A − 共1 − i兲A 2₋B2 2

兴

. . . _冑1₂

关

1 +共1 + i兲A − 共1 − i兲A2₋B2 2

兴

− c共1 + 2iA兲B

冥

. 共26兲

The cluster state兩␾˜N典 is defined as

兩␾˜_{N典 = E˜}共N−1,N兲_{¯ E}˜共1,2兲_{兩0 ¯ 0典. 共27兲} The order of the entangling operations E˜共n,n+1兲in Eq.共27兲 has to be the same as for the intended state兩␾N典 in Eq. 共22兲,

since different ordering generates different states共although in the same entanglement class兲. We now calculate 兩␾N典 and

兩␾˜_{N典 and from these the fidelity F 关Eq. 共21兲兴 for an increasing} number of qubits. We then find关44兴

F =

冑

1 −共N − 1兲A2−5N − 9

2 B

2_{, Nⱖ 3, 共28兲} for 共N−1兲A2−共5N−9兲B2/ 2ⱕ1. We have verified that Eq. 共28兲 is valid up to N=10, and suspect that it is true for all values of N. We remark that the fidelity F increases as the square root of N, the number of qubits. Figure1共a兲shows the fidelity as a function of the inaccuracies ⑀ and ␦ for the case of three qubits. Numerical evaluation shows remarkably high fidelities even for systems with many qubits, e.g., F共N=10, ⑀= 0.05, ␦= 0.1兲=0.95. This suggests that the proposed sequences for the cluster states enable generation of many-qubit entangled states with high fidelity. Although the fidelity of generating, e.g., N-partite GHZ states 共not shown兲 are lower because there are more single-qubit rota-tions required to transform the N-qubit ground state into a GHZ state, they are also within reach of experimental imple-mentation, as we discuss in the next section.

IV. DISCUSSION

In this section we briefly discuss the experimental feasi-bility of generating multipartite entangled states of electron spins in quantum dots. As demonstrated experimentally, the duration of a

冑

USWAP operation of two electron spins is ⬃180 ps 关35兴, and a spin rotation over␲/ 2 requires⬃27 ns 关34兴. As a rough estimate, we then find that the sequence Eq.

共12b兲 to implement a N-partite cluster state using simulta-neous application of

冑

USWAPoperations to all pairs of qubits requires ⬃2⫻50=100 ns for any N 共since the linear array of qubits can be entangled in just two steps: first each of the even-numbered qubits to their right neighbor, and then the same for the odd-numbered qubits兲. The time required to implement the N-partite GHZ state共17兲 depends on the

num-1.00 0.96 0.98 F 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 ε δ F N=3 N=4 N=5 N=10N=9 N=8 N=7 N=6 0.98 0.96 0.90 0.92 0.94 0.02 0.04 0.06 0.08 ε,2δ (b) (a)

FIG. 1. 共Color online兲 共a兲 Three-dimensional 共3D兲 plot of the fidelity F, Eq.共28兲, for N=3 as a function of the two parameters⑀

ber of qubits and amounts to ⬃共N−1兲₂ ⫻100 ns 关using the ordering given in Eq.共18兲兴. The limiting time for the imple-mentation of these sequences of operations is the decoher-ence time T2, which has not yet been measured for a single spin. Rabi oscillations of a single electron spin 关34兴 have been seen for more than 1 ␮s, indicating a decoherence time T₂ⲏ1 ␮s. Based on this estimate for T₂, the generation of a N-partite cluster state and a N-partite W state thus seems feasible for any N in a time shorter than T2, whereas the generation of GHZ states should be possible for up to⬃10 qubits关45兴.

To conclude, we have calculated general sequences to generate genuine N-partite entangled states in various en-tanglement classes starting from a separable N-qubit state in the computational basis and using the least possible number of single-qubit rotations and two-qubit exchange interac-tions. For all entangled states that we considered 共cluster states, GHZ states and W states兲 we find that the total num-ber of operations required to generate these states scales lin-early with the number of qubits N. The generation of N-partite W states requires the least amount of operations, namely共N−1兲

冑

USWAPoperations and one rotation. They are followed by the N-partite cluster states that require a

mini-mum of共2N−3兲 exchange interactions and 共N−1兲 rotations and the GHZ states that also require a minimum of共2N−3兲 exchange interactions and共3N−5兲 single-qubit rotations. We also calculated the fidelity F for the generation of N-partite cluster states in the presence of imperfect single-qubit rota-tions and

冑

USWAP operations, and find that F decreases as F⬃

冑

1 −␮N +␯ as the number of qubits grows, with ␮, ␯ ⬎0, and␮N −␯ⱕ1.

Our results can be implemented for electron spins in quantum dots 关46兴, for which Heisenberg exchange is the naturally available two-qubit interaction in the system. We estimate that our proposed scheme for the generation of mul-tipartite entangled states is feasible for at least ten qubits within current experimental accuracy. Finally, we emphasize that the approach used in this paper can be used for any kind of two-qubit entangling interaction and provides an analyti-cal scheme to analyti-calculate the implementation of multipartite entangled states for any type of qubit.

ACKNOWLEDGMENTS

This research was supported by the Netherlands Organi-sation for Scientific Research共NWO兲.

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关44兴 For N=2 the fidelity is given by F=

冑

1 − A2_{− B}2_.

关45兴 We note in this context that proposals have been presented to replace the共slow兲 rotations by more of the 共fast兲 interactions, but they require ancilla qubits and increase the number of re-quired qubits by a factor of 3; see D. P. DiVincenzo et al., Nature共London兲 408, 339 共2000兲.