quantum simulation of pairing models on an nmr quantum computer

5
Physics Letters A 352 (2006) 304–308 www.elsevier.com/locate/pla Quantum simulation of pairing models on an NMR quantum computer An Min Wang , Xiaodong Yang Department of Modern Physics and Institute for Theoretical Physics, University of Science and Technology of China, Hefei 230026, PR China Received 20 April 2005; received in revised form 28 November 2005; accepted 2 December 2005 Available online 9 December 2005 Communicated by P.R. Holland Abstract We give out a scheme of quantum simulation of pairing models on an NMR quantum computer. We show that it can obtain the spectrum of paring model in principle, and present a simple example. Similarly, it is possible to discuss the quantum simulation to obtain more differences of energy levels. 2005 Elsevier B.V. All rights reserved. PACS: 03.67.-a; 74.20.Fg; 76.60.-k Keywords: Quantum simulation; Pairing model; NMR quantum computer Simulating a real physical system by a quantum computer (QC) was originally conjectured by Feynman [1]. It has at- tracted a lot of physicists working on it, for example, Lloyd confirmed the idea of quantum simulating in a two-state array [2], and Somaroo et al. presented a general scheme for the quan- tum simulation [3]. NMR quantum computer is one of successful realizations of quantum computers so far [4,5]. Quantum simulations of a four- level truncated oscillator [3], a three-spin effective Hamiltonian [6] and the migration of excitation in an eight-state quantum system [7] on it have been researched. Recently, Wu et al. [8] re- ported an NMR experimental scheme performing a polynomial- time simulation of pairing models. More and more works in this aspect come forth continually [9,10]. In this Letter, we give out a scheme of quantum simulation of pairing models on an NMR quantum computer. Two main features of our scheme are: (1) the choice of an appropriate initial state which can be eas- ily prepared in experiment; (2) the use of the second (discrete) Fourier transform which can give the spectrum of paring model in principle. We concretely discuss the cases in the concerned subspaces of pairing model and then, as an example, present a * Corresponding author. E-mail address: [email protected] (A.M. Wang). URL: http://qtg.ustc.edu.cn. simple initial state to get the gap of the two lowest energy lev- els in a given subspace. Similarly, it is possible to discuss the quantum simulation to get more differences of energy levels. In order to understand what is the theoretical foundation of our scheme and whether our scheme can arrive at the needed pre- cision, we also carry out some relevant research which can be seen in our preprints [11,12]. In addition, after proposing our theoretical scheme, we presented its simple experimental im- plementation [13]. As is well known, the spin-analogy of paring model Hamil- tonian has the form [8,14] (1) H p = N m=1 m 2 σ (m) z V 2 N m<l =1 ( σ (m) x σ (l) x + σ (m) y σ (l) y ) . Without loss of generality, a general working initial state in the quantum simulation can be written as (2) |ψ ini = i a i v i p = i,j a i b ij |j = i,j a i b ij φ j nmr , where |v i p = j b ij |j are the H p ’s eigenvectors with the corresponding eigenvalues E i p , and |j are the computational bases. Here, we have used the fact that the eigenvectors |φ j nmr of H nmr with eigenvalues E j nmr are just |j since H nmr in the 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.12.012

Upload: an-min-wang

Post on 29-Jun-2016

213 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Quantum simulation of pairing models on an NMR quantum computer

Physics Letters A 352 (2006) 304–308

www.elsevier.com/locate/pla

Quantum simulation of pairing models on an NMR quantum computer

An Min Wang ∗, Xiaodong Yang

Department of Modern Physics and Institute for Theoretical Physics, University of Science and Technology of China, Hefei 230026, PR China

Received 20 April 2005; received in revised form 28 November 2005; accepted 2 December 2005

Available online 9 December 2005

Communicated by P.R. Holland

Abstract

We give out a scheme of quantum simulation of pairing models on an NMR quantum computer. We show that it can obtain the spectrum ofparing model in principle, and present a simple example. Similarly, it is possible to discuss the quantum simulation to obtain more differences ofenergy levels. 2005 Elsevier B.V. All rights reserved.

PACS: 03.67.-a; 74.20.Fg; 76.60.-k

Keywords: Quantum simulation; Pairing model; NMR quantum computer

Simulating a real physical system by a quantum computer(QC) was originally conjectured by Feynman [1]. It has at-tracted a lot of physicists working on it, for example, Lloydconfirmed the idea of quantum simulating in a two-state array[2], and Somaroo et al. presented a general scheme for the quan-tum simulation [3].

NMR quantum computer is one of successful realizations ofquantum computers so far [4,5]. Quantum simulations of a four-level truncated oscillator [3], a three-spin effective Hamiltonian[6] and the migration of excitation in an eight-state quantumsystem [7] on it have been researched. Recently, Wu et al. [8] re-ported an NMR experimental scheme performing a polynomial-time simulation of pairing models. More and more works inthis aspect come forth continually [9,10]. In this Letter, we giveout a scheme of quantum simulation of pairing models on anNMR quantum computer. Two main features of our scheme are:(1) the choice of an appropriate initial state which can be eas-ily prepared in experiment; (2) the use of the second (discrete)Fourier transform which can give the spectrum of paring modelin principle. We concretely discuss the cases in the concernedsubspaces of pairing model and then, as an example, present a

* Corresponding author.E-mail address: [email protected] (A.M. Wang).URL: http://qtg.ustc.edu.cn.

0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2005.12.012

simple initial state to get the gap of the two lowest energy lev-els in a given subspace. Similarly, it is possible to discuss thequantum simulation to get more differences of energy levels.In order to understand what is the theoretical foundation of ourscheme and whether our scheme can arrive at the needed pre-cision, we also carry out some relevant research which can beseen in our preprints [11,12]. In addition, after proposing ourtheoretical scheme, we presented its simple experimental im-plementation [13].

As is well known, the spin-analogy of paring model Hamil-tonian has the form [8,14]

(1)Hp =N∑

m=1

εm

2σ (m)

z − V

2

N∑m<l=1

(σ (m)

x σ (l)x + σ (m)

y σ (l)y

).

Without loss of generality, a general working initial state inthe quantum simulation can be written as

(2)|ψini〉 =∑

i

ai

∣∣vip

⟩ = ∑i,j

aibij |j 〉 =∑i,j

aibij

∣∣φjnmr

⟩,

where |vip〉 = ∑

j bij |j 〉 are the Hp’s eigenvectors with the

corresponding eigenvalues Eip , and |j 〉 are the computational

bases. Here, we have used the fact that the eigenvectors |φjnmr〉

of Hnmr with eigenvalues Ejnmr are just |j 〉 since Hnmr in the

Page 2: Quantum simulation of pairing models on an NMR quantum computer

A.M. Wang, X. Yang / Physics Letters A 352 (2006) 304–308 305

laboratory is diagonal:

(3)Hnmr = 1

2

(N∑

i=1

ωi0σ

iz +

N∑i,j=1,i<j

πJσ izσ

jz

).

Considering the following evolution

(4)e−iHpτ/h|ψini〉 =∑

i

aie−iEi

pτ/h∣∣vi

p

⟩,

one obtains the density matrix at time τ

ρ(τ) =∑i,i′

∑j,j ′

aibii′a∗j b∗

jj ′

(5)× exp{−i

(Ei

p − Ejp

)τ/h

}∣∣φi′nmr

⟩⟨φ

j ′nmr

∣∣.Recalling the procedure of the NMR measurement, one can

write the NMR frequency spectrum as

(6)SNMR(ω) ∝ f t

[Tr

(e−iHnmrt/hρf (τ )eiHnmrt/h

N∑k=1

σ+k

)],

where σ±i = (σ i

x ± iσ iy)/2, ρf (τ) is the density matrix to be

measured and f t is the Fourier transform applied on the NMRmeasured signal—free induction decay (FID) from time do-main to frequency domain. It must be emphasized that theFourier transform is applied to the NMR measure time t ratherthan to the evolution time τ . Moreover, note that the FID alsoincludes the information on the system evolution. This impliesthat we can extract the information of the system Hamiltonianwith a second Fourier transform for the evolution time τ .

Now, substituting ρ(τ) in Eq. (5) into the above equation,we have

SNMR(ω) ∝∑i,i′

∑j,j ′

aibii′a∗j b∗

jj ′ exp{−i

(Ei

p − Ejp

)τ/h

}

(7)× δ(Ei′

nmr − Ej ′nmr

)Tr

(|i′〉〈j ′|

N∑k=1

σ+k

).

For an appearing peak in NMR frequency spectrum, for ex-ample Eα

nmr −Eβnmr, we can obtain its corresponding difference

of energy levels (the values of α and β) based on the NMRHamiltonian and the parameters of the sample. Then, we carryout the second Fourier transform and are able to obtain the Hp’sfrequency spectrum

(8)Sp(Ep) ∝∑i,j

aibiαa∗j b∗

jβδ(Ei

p − Ejp

),

where we have used the fact that Tr(|i′〉〈j ′|∑Nk=1 σ+

k ) is a sum-mation of a series of discrete delta functions in Eq. (11), whichis proved in the following. Thus, summing over i′ and j ′ willlead to Eq. (7), proportional to the summation of a series ofterms with form aibiαa∗

j b∗jβ exp{−i(Ei

p − Ejp)τ/h}δ(Eα

nmr −E

βnmr). Choosing an existing pair of the subscripts α and β

(meaning there is the peak), we can obtain this equation by thesecond Fourier transform.

In experiment, the second (discrete) Fourier transform canbe applied from the data that are collected by measuring the

areas (heights) of peaks Eαnmr − E

βnmr at a series of evolution

times τi with Hp . As soon as the Eip − E

jp are obtained and

the lowest energy level is known, the spectrum of Hp is justobtained in principle.

However, there are still three difficulties we have to face:(1) which state is an appropriate working initial state for ourpurpose; (2) which Eα

nmr − Eβnmr peaks can appear in NMR fre-

quency spectrum for a given initial state; (3) what i and j valuescorrespond to the Ei

p − Ejp peaks in Hp’s frequency spectrum.

In order to solve them, we need some mathematical andphysical preparations.

Firstly, we should derive the obvious expression ofTr(|i′〉〈j ′|∑N

k=1 σ+k ). Denoting by S

(N)n a subspace with n spin-

up states |0〉, we can write that S(N)spin = S

(N)0 ⊕ S

(N)1 ⊕ S

(N)2 ⊕

· · · ⊕ S(N)N , that is, the spin space can be divided into the direct

summation of different subspaces which correspond to the dif-ferent numbers of spin-up states. It is clear that these subspacescontain the following basis:

(9a)S(N)0 = {∣∣2N

⟩}, S

(N)N = {|1〉},

S(N)n =

{∣∣S(N)i1i2...in

⟩ =∣∣∣∣∣2N −

n∑a=1

2(N−ia)

⟩,

(9b)ia �= ib, ia = 1,2, . . . ,N

}.

Further, based on the relations σ+|0〉 = 0, σ+|1〉 = |0〉,σ−|0〉 = |1〉 and σ−|1〉 = 0, we have that

(10a)N∑

k=1

σ−k |1〉 =

N∑k=1

∣∣2(N−k) + 1⟩,

(10b)N∑

k=1

σ−k

∣∣s(1)i1

⟩ = ∣∣2N⟩,

N∑k=1

σ−k

∣∣2N⟩ = 0,

(10c)N∑

k=1

σ−k

∣∣s(N)i1i2...in

⟩ = ∑k=i1,...,in

∣∣∣∣∣2N −n∑

a=1

2(N−ia) + 2N−k

⟩.

In terms of these equations, it is easy to get

Tr

(|i〉〈j |

N∑k=1

σ+k

)

=N∑

k=1

δi,2(N−k)+1δj1 + δi,2N

N∑k=1

δj,2N−2(N−k)

(11)+∑

k=i1,i2,...,in

δi,j+2(N−k)δj,2N−∑na=1 2(N−ia ) ,

where in the last term: n with possible values 2,3, . . . ,N − 1,ia �= ib and ia = 1,2, . . . ,N for any a = 1,2, . . . , n.

Secondly, we need to analyze the structure of the eigen-vectors of Hp . From Eq. (1) and the relations (σ

(m)x σ

(l)x +

σ(m)y σ

(l)y ) = (σ

(m)x + iσ (m)

y )(σ(l)x − iσ (l)

y ) (m �= l), there followsthat |1〉 and |2N 〉 must be Hp’s two eigenvectors respectivelycorresponding to the maximum and the minimum eigenvalues,

Page 3: Quantum simulation of pairing models on an NMR quantum computer

306 A.M. Wang, X. Yang / Physics Letters A 352 (2006) 304–308

which are denoted respectively by |v1p〉 and |v2N

p 〉. Moreover,

if the arbitrary basis |s(N)i1...in

〉 belongs to S(N)n , then Hp|s(N)

i1...in〉

also belongs to S(N)n because σ+ and σ− appear either in pairs

or do not appear in the various terms of Hp . This implies that

〈s(N)i1...im

|Hp|s(N)

i′1...i′n〉 = 0 (m �= n;m,n = 1,2, . . . ,N −1). There-

fore

(12)H(N)p = H

(N)sub0 ⊕ H

(N)sub1 ⊕ H

(N)sub2 ⊕ · · · ⊕ H

(N)subN.

So we can denote the other eigenvectors of Hp as |vi(n)

p 〉 =∑j (n) bi(n)j (n) |j (n)〉 ∈ S

(N)n with the corresponding eigenvalues

Ei(n)

p , and i(n), j (n) take only the sequence number of spin bases

belonging to the subspace S(N)n , for example i(1)(k) = 2N −

2(N−k) (k = 1,2, . . . ,N). In other words, |i(n)〉, |j (n)〉 ∈ S(N)n .

Making use of the above conclusions, we can solve the men-tioned three difficulties one by one.

Let us start with solving the first difficulty. If our purposeis only to obtain the spectrum Ei(1)

p and Ei(N−1)

p in the sub-space 1 and subspace N − 1 respectively, we should choosesuch working initial state that it does not include any |vi(m)

p 〉(m = 2,3, . . . ,N − 2). Obviously, the appearing NMR fre-quency spectrum will be simplified as at most 2N peaks:

SNMR(ω)

∝N∑k

∑i,j

′[aibi,2(N−k)+1a

∗j b∗

j1 exp

{− i

h

(Ei

p − Ejp

}

× δ(E2(N−k)+1

nmr − E1nmr

)+ aibi,2N a∗

j b∗j,2N−2(N−k) exp

{− i

h

(Ei

p − Ejp

}

(13)× δ(E2N

nmr − E2N−2(N−k)

nmr

)],

where in the summation∑′

i,j , i, j take only over 1,2(N−k)+1,

2N − 2(N−k),2N (k = 1,2, . . . ,N). In this simplified case, theNMR spectrum includes only the differences of energy levels inthe subspaces 0 and 1, N −1 and N . In particular, if the working

initial state is taken as c|vi(1)

p 〉 + d|vj(1)

p 〉, then Ei(1)

p − Ej(1)

p canbe obtained from the second Fourier transform, unless no peakappears in the NMR frequency spectrum.

Next, we solve the second difficulty, that is, how to guar-antee the needed peaks in the NMR frequency spectrum. Thekey matter is that we have to take an appropriate workinginitial state including the kets |1〉, |2(N−k) + 1〉(∈ S

(N)(N−1))

or(/and) |2N − 2(N−k)〉(∈ S(N)1 ), |2N 〉, that is, |ψini〉 = a1|1〉 +∑

i(N−1) ai(N−1) |vi(N−1)

p 〉 or |ψini〉 = a2N |2N 〉 + ∑i(1) ai(1) |vi(1)

p 〉.Respectively for the two cases we have

SNMR(ω)

∝N∑k

∑i(N−1)

ai(N−1)bi(N−1),2(N−k)+1a∗1b∗

11

(14)× exp

{− i (

Ei(N−1)

p − E1p

}δ(E2(N−k)+1

nmr − E1nmr

),

h

SNMR(ω) ∝N∑k

∑j (1)

a2N b2N ,2N a∗j (1)b

∗j (1),2N−2(N−k)

(15)

× exp

{− i

h

(E2N

p − Ej(1)

p

}δ(E2N

nmr − Ej(1)

nmr).

Since ai(N−1)bi(N−1),2(N−k)+1a∗1b∗

11 or a2N b2N ,2N a∗j (1) ×

b∗j (1),2N−2(N−k) are not all zero, NMR spectrum must have some

peaks appearing.The last difficulty is easy to be solved. In fact, setting

|ψini〉 = a2N |2N 〉 + a2N−2(N−K) |v2N−2(N−K)

p 〉, we can read

E2N−2(N−K)

p − E2N

p from Hp’s spectrum and then obtain

E2N−2(N−K)

p in terms of E2N

p = −∑Nm=1 εm/2. If setting

|ψini〉 = a2N |2N 〉 + a2N−2(N−k1) |v2N−2(N−k1)

p 〉 + a2N−2(N−k2) ×|v2N−2(N−k2)

p 〉 (k1 �= k2), we can read E2N−2(N−k1)

p − E2N

p

and E2N−22N −(N−k2)

p − E2N

p , and then obtain E2N−2(N−k1)

p −E2N−2(N−k2)

p (set E2N−2(N−k1)

p � E2N−2(N−k2)

p if k2 > k1). There-fore, in principle, we always can find the energy levels and theirdifferences in the subspace 1 of Hp . Likewise, one ought to beable to obtain the other energy levels in the other subspaces bythe different choices of the working initial states.

The preparation of the desired initial state can start with thepseudo-pure state in the NMR experiment. The complexity ofpreparing for the pseudo-pure state grows exponentially withthe system size, i.e., the qubit number, though there are someof ingenious methods which can reduce the complexity intopolynomial size. Based on the pseudo-pure state, we can pro-duce the desired initial state by a series of Hadamard gates, ordividing the desired initial state into results corresponding tothe multi-experiment and sum up the results together. Thus, thecomplexity or the efficiency of preparing for the desired initialstate in the experiment depends mostly on the complexity of thepreparing the pseudo-pure state. This implies that our choicemethod of the initial state is not suitable for a “very large” N

beyond the power of an NMR quantum computer. In fact, theNMR quantum computer is not scalable using the presentedtechnology. For a “very large” N , not only the preparation ofinitial state but also the other aspects pose some difficulties.

It must be emphasized that it is interesting to find whatis the physical meaning of E2N−2(N−k1)

p − E2N−2(N−k2)

p in the-ory and what is an appropriate working initial state to obtainE2N−2(N−k1)

p − E2N−2(N−k2)

p in experiment. Actually, by the di-rect diagonalization of Fock space for spin-analogy of pair-ing model [11] and the previous numerical calculation [12],we have found that the relation when N is large enough, is(ξ2

2N−2+ ∆2)1/2 − (ξ2

2N−1+ ∆2)1/2 ≈ E2N−2

p − E2N−1p , where

E2N−2p − E2N−1

p is the difference of the two lowest energylevels in the subspace 1 of Hp , ∆ is the solution of the en-ergy gap equation [15]. Therefore, the physical meaning ofE2N−2(N−k1)

p − E2N−2(N−k2)

p in theory is understood.Now we can determine the working initial state which can

obtain E2N−2(N−k1)

p − E2N−2(N−k2)

p by our scheme of quantumsimulation of pairing models on an NMR quantum computer.

Page 4: Quantum simulation of pairing models on an NMR quantum computer

A.M. Wang, X. Yang / Physics Letters A 352 (2006) 304–308 307

As an example, we consider the case of k2 = 1, k1 = 2. It iseasy to prove that

Hp|W 〉 = −[

1

2

N∑m=1

εm + (N − 1)V

]|W 〉

(16)+ 1√N

M∑m=1

εm

∣∣∣11 . . .1 0︸︷︷︸m

1 . . .1⟩,

Hp|uij 〉 = −[

1

2

N∑m=1

εm − εj − V

]|uij 〉

(17)− (εj − εi)1√2

∣∣∣11 . . .1 0︸︷︷︸i

1 . . .1⟩,

where we have defined

(18a)|W 〉 = 1√N

N∑i=1

∣∣∣11 . . .1 0︸︷︷︸i

1 . . .1⟩,

(18b)

|uij 〉 = 1√2

(∣∣∣11 . . .1 0︸︷︷︸i

1 . . .1⟩−

∣∣∣11 . . .1 0︸︷︷︸j

1 . . .1⟩)

.

Here, i �= j and i, j = 1,2, . . . ,N . Obviously, |uij 〉 are notcompletely independent of each other. In practice, we can fixi = 1 and j = 2,3, . . . ,N , and obtain (N − 1) linearly inde-pendent |uij 〉. Note that when all εm are equal, the so-calledanti-W state |W 〉 corresponds to the lowest energy level in thesubspace 1. When εm varies with the value of m and εj+1 − εj

is a very small positive parameter, we know from the perturba-tion theory that the lowest energy level is still related to |W 〉,and the second lowest energy ought to relate to |u12〉. Thus, in

order to make the working initial state to contain |v(2N−1)p 〉 and

|v(2N−2)p 〉 definitely, we should take it as

(19)∣∣ψ(0)

ini

⟩ = 1√3

(∣∣2N⟩ + |W 〉 + |u12〉

).

Thus, the final Hp’s frequency spectrum includes consequently

the peaks E2N−1p − E2N

p and E2N−2p − E2N

p . Obviously, such aninitial state is simple and can be prepared in the NMR exper-iment for a suitable N within the power of an NMR quantumcomputer. It should be pointed out that within N -qubits that anNMR quantum computer can deal with, it is of significance toseek for other principles or conditions to simplify the choice ofour initial state in order to advance the efficiency. That is, weshould partially know the structure of involved eigenvectors,and then drop the independent bases and decrease the probabil-ities (the norm of coefficients) of some bases appearing, whichis being studied in the results of our previous works [11].

In the summary, our scheme of quantum simulation of par-ing models on an NMR quantum computer has four steps: (1)prepare the initial state based on the quantum simulation pur-pose, for example (19), in order to obtain E2N−2

p − E2N−1p ; (2)

use NMR pulse series to implement evolution e−iHpτ/h [8,13]at a series of times τi ; (3) carry out NMR measure (the first

Fourier transform) to collect data which are a series of ampli-tudes corresponding to the evolution time τi by Hp; (4) makethe second (discrete) Fourier transform for the collecting dataand then obtain Hp’s frequency spectrum. Recently, we havefinished the experiment of quantum simulation for the simplestsystem of two qubits. More experimental detail is presented inour another paper [13].

Comparing with the known scheme [8], where the workinginitial states obviously are prepared by the process of a quasi-adiabatical evolution, our scheme does not need such a processand chooses directly an appropriate working initial state whichcan be more easily prepared in experiment. In terms of featureof NMR measure and the second (discrete) Fourier transform,our scheme can obtain the spectrum of paring model in princi-ple. It is different from the scheme in Ref. [8] where the secondFourier transform was not used. In addition, because we chosean appropriate working state, the rotation step is not needed.However, it must be pointed out that for every time evolution,we use the same series of pulses as in Ref. [8]. These fea-tures of our proposal prove that the simulating paring modelon an NMR quantum computer is actually feasible and reallycomplete. Moreover, we concretely discuss the case in the sub-spaces 1 and N − 1 of pairing model and then, as an example,give out a simple initial state to get the gap of the two lowestenergy levels in the subspace 1. Similarly, it is possible to dis-cuss the quantum simulation to get more differences of energylevels. This work is in progress.

Acknowledgements

We particularly thank Jiangfeng Du for his valuable sugges-tions and indispensable support in the experimental implemen-tation of our scheme, and we are grateful to Feng Xu, XiaoSan Ma, Wan Qing Niu, Ning Bo Zhao, Hao You, Ren GuiZhu and Xiao Qiang Su for helpful discussions. This workwas founded by the National Fundamental Research Programof China with grant No. 2001CB309310, partially supported bythe National Natural Science Foundation of China under GrantNo. 60573008.

References

[1] R.P. Feynman, Int. J. Theor. Phys. 21 (1982) 467.[2] S. Lloyd, Science 273 (1996) 1073.[3] S. Somaroo, C.H. Tseng, T.F. Havel, R. Laflamme, D.G. Cory, Phys. Rev.

Lett. 82 (1999) 5381.[4] D.G. Cory, R. Laflamme, E. Knill, L. Viola, T.F. Havel, N. Boulant, G.

Boutis, E. Fortunato, S. Lloyd, R. Martinez, C. Negrevergne, M. Pravia,Y. Sharf, G. Teklemariam, Y.S. Weinstein, W.H. Zurek, in: Experimen-tal Proposals for Quantum Computation, Special Issue, Fortschr. Phys. 48(2000) 875.

[5] J.A. Jones, Prog. Nucl. Magn. Reson. Spectrosc. 38 (2001) 325.[6] C.H. Tseng, S. Somaroo, Y. Sharf, E. Knill, R. Laflamme, T.F. Havel, D.G.

Cory, Phys. Rev. A 61 (2000) 012302.[7] A.K. Khitrin, B.M. Fung, Phys. Rev. A 64 (2001) 032306.[8] L.-A. Wu, M.S. Byrd, D.A. Lidar, Phys. Rev. Lett. 89 (2002) 057904;

J. Dukelsky, J.M. Román, G. Sierra, Phys. Rev. Lett. 90 (2003) 249803;L.-A. Wu, M.S. Byrd, D.A. Lidar, Phys. Rev. Lett. 90 (2003) 249804.

[9] R. Somma, G. Ortiz, J.E. Gubernatis, E. Knill, R. Laflamme, Phys. Rev.A 65 (2002) 042323.

Page 5: Quantum simulation of pairing models on an NMR quantum computer

308 A.M. Wang, X. Yang / Physics Letters A 352 (2006) 304–308

[10] C. Negrevergne, R. Somma, G. Oritz, E. Knill, R. Laflamme, quant-ph/0410106.

[11] A.M. Wang, cond-mat/0502222;A.M. Wang, R.G. Zhu, cond-mat/0505499.

[12] X. Feng, A.M. Wang, X. Yang, X. Ma, H. You, Commun. Theor. Phys. 44(2005) 171–176.

[13] X. Yang, A.M. Wang, F. Xu, J. Du, quant-ph/0410143.[14] P.B. Allen, in: R.G. Lerner, G.L. Trigg (Eds.), Concise Encyclopedia of

Solid State Physics, Addison–Wesley, Reading, MA, 1983, p. 266.[15] P.L. Taylor, O. Henonen, A Quantum Approach to Condensed Matter

Physics, Cambridge Univ. Press, Cambridge, 2002.