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PHYSICAL REVIEW A JULY 1999VOLUME 60, NUMBER 1
Qubit-qubit interaction in quantum computers. II. Adder algorithm with diagonaland off-diagonal interactions
Julio Gea-BanaclocheDepartment of Physics, University of Arkansas, Fayetteville, Arkansas 72701
~Received 23 December 1998!
The effect of off-diagonal interactions between quantum bits is studied in a simple quantum computeralgorithm designed to calculate the sum of twon-bit numbers. Scaling laws are derived numerically for thedependence of the error on the numbern of bits, the strength of the interaction, and the calculation time. Thecase of diagonal interactions, previously studied in the context of the quantum Fourier transform algorithm, isalso considered. No substantial differences are found between the off-diagonal and diagonal forms of theinteraction: in both cases, the errors accumulate almost coherently, despite the fact that the system’s freeevolution is constantly being interrupted by the logical gates, which scramble the coefficients of the wavefunction. Some comments on the possibility of using ‘‘field-insensitive’’ atomic states in ion-chain quantumcomputers are also given.@S1050-2947~99!04907-0#
PACS number~s!: 03.67.Lx, 89.70.1c, 89.80.1h
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I. INTRODUCTION
Unwanted interactions between the quantum bits, or ‘‘qbits,’’ that make up a quantum computer@1#, could be, inprinciple, as damaging as interactions with the environm@2#. In a recent paper@3# I have considered, as an explicexample of this, the influence of magnetic dipole-dipoleteractions on a chain of ions@4# when it is used as a quantumcomputer running the quantum Fourier transform~QFT! al-gorithm @5#. Although magnetic dipole interactions betweatoms are typically very weak, their cumulative effect canlarge for atoms separated by distances of the order of amicrons. This is because, as seen in@3#, the error introducedappears to accumulate with time in an almost coherent faion: that is, the magnitude of the error~phase error, in thecase studied in@3#! grows linearly with time, in spite of thefact that the coherent evolution of the system is being cstantly interrupted from the outside by the logical gatesplied to it.
The results in@3#, however, were obtained under somrestricted assumptions and hence leave open a few quesFirst, the qubit-qubit interaction was treated as being dianal in the computational basis, which means that only pherrors were considered, and it is not immediately obviowhether ‘‘bit’’ errors ~as would be induced by off-diagonaterms! would also have the property of quasicoherent acmulation. Also, it is not immediately obvious what wouhappen if one were to use for the computational basiscalled ‘‘field-free,’’ or ‘‘field-insensitive’’ states, that isstates in which the diagonal elements of the dipointeraction operator are zero~or can be set to zero afterenormalization!. Finally, the QFT, although an essential paof Shor’s factorization algorithm, is not the most timconsuming part~in fact, it takes a negligible amount of timein the limit of many qubitsN, since it requires of the order oN2 operations, whereas the full algorithm requires of torder of N3 operations!, and is also a somewhat atypicexample of a quantum computing algorithm, in that it donot involve anyCNOT or Toffoli gates. Thus, there is som
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possibility that the results of@3# ~regarding, for instance, thescaling of the error with the number of operations! may notbe broadly applicable to other quantum computing algrithms.
This paper addresses all of the issues raised in the preing paragraph. The effect of interaction between qubitsexplored in a simple adder algorithm, typical of the sortbuilding block necessary for doing arithmetic on a quantcomputer. Diagonal and off-diagonal forms of the interactiare considered, and the possibility of using ‘‘fieldinsensitive’’ states is explicitly addressed. Although sframed, for the most part, in terms of chains of ions amagnetic-dipole interactions, the forms used for the intertion Hamiltonian here are sufficiently general that manythe conclusions of this paper would probably hold, with suable modifications, for more general kinds of quantum coputing hardware, wherever unwanted qubit-qubit interactiomight play an important role.
The paper is organized as follows: Section II deals wthe magnetic dipole-dipole interaction in a chain of atomand introduces the main simplifications and assumptiused to derive the simple model forms of the interactHamiltonian to be used in the remainder of the paper. Stion III describes briefly the adder algorithm and the procdure used for the numerical calculation, and presents thesults, including numerically calculated scaling laws. FinalSec. IV attempts to give an idea of the magnitude ofeffect for a chain of ions, by showing the slowdown thwould result from the requirement to keep the ions suciently far apart to prevent unwanted interactions. This lsection reprises arguments presented earlier this year in@6#,although some of the conclusions~or, rather, the assumptions! of @6# need to be modified in ways discussed here.
II. MAGNETIC DIPOLE INTERACTIONIN AN ION CHAIN
The first proposal@4# to build a quantum computer out oa chain of cold ions in an rf trap envisioned using the groustate and a metastable excited state as the two compu
185 ©1999 The American Physical Society
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186 PRA 60JULIO GEA-BANACLOCHE
states for the qubit. Typically, the excited state wouldreached via an electric quadrupole transition; electric ocpule transitions have also been considered.
It is not immediately obvious, but it is generally true, thin these systems one would also end up with a differmagnetic moment for the ion in each of the two computstates~henceforth referred to asu0&, for the ground state, anu1&!. Consider, for instance, a system such as Ba1, whereu0&is anS1/2 state andu1& is a D5/2 state. If the ground state hamJ521/2, then exciting it with1 circularly polarized radia-tion would result in a state withmJ51/2, of the form
u1&5A35 u l 52, mL50&umS5 1
2 &1A25 u l 52,
mL51&umS52 12 &. ~1!
The magnetic moment operator associated with the valeelectron is
m52mB
\~L12S!, ~2!
wheremB is the Bohr magneton. It is easy to see that thxandy components ofm have zero matrix elements betweethe statesu0& and u1&, because of the orthogonality of thel52 andl 50 states; also they have zero expectation valubecause whenever the spin part does not vanish the orpart vanishes and vice versa. These results are quite genHowever, thez component ofm, while still having zero off-diagonal matrix elements, does have the distinct expectavalues
^0umzu0&5mB , ~3a!
^1umzu1&52mB@ 35 1 2
5 ~121!#52 35 mB . ~3b!
@It is straightforward to verify that other choices for the fiepolarization and the initial and final states in Eq.~1! lead tosimilar results.#
As a result of this, two neighboring ions will ‘‘see’’ adifferent magnetic dipole-dipole interaction when they aretheir ground states than when they are in their excited stor in a combination thereof. The magnetic dipole-dipoHamiltonian has the general form
Hi j 5m0
4pr i j3 @mi•mj23~mi•n!~mj•n!#, ~4!
wheren is a unit vector pointing from the first dipole to thsecond. Since, for these systems, all components ofm otherthan mz vanish, the interaction Hamiltonian~4! can be re-written as
Hi j 5m0
4pr i j3 ~123 cos2 u!m izm jz , ~5!
whereu is the angle that the vectorn makes with the quantization axisz. This shows that the interaction energy vaishes for the special choice cos2 u51/3 (u554.7°), whichcorresponds to a setting where each dipole is perpendicto the magnetic field of the other one at its location. To dirthe laser beam that interacts with the ions at this angle to
e-
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ce
s,italral.
on
es
-
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axis of the chain, however, would probably not be a vegood idea since it would result in a large coupling to tranverse vibrational degrees of freedom; typically one expecuin Eq. ~5! to be very small.
Equation ~5!, therefore, leads to an interaction Hamtonian that in the basis$u100&, u01&, u10&, u11&% of the internalstates of the two ionsi,j will have a matrix like the follow-ing:
Hi j 5S a2
000
0ab00
00
ab0
000b2D , ~6!
with
a5F m0
4pr i j3 ~12cos2 u!G1/2
^0umzu0&, ~7a!
b5F m0
4pr i j3 ~12cos2 u!G1/2
^1umzu1&. ~7b!
This is a nonadditive kind of interaction energy@that is,E01Þ(E001E11)/2], which, as explained in@3#, will, in gen-eral, result in entanglement of the joint wave function of ttwo ions. It is convenient to separate Eq.~6! into the additiveand nonadditive parts,
Hi j 5S a2 0 0 0
0a21b2
20 0
0 0a21b2
20
0 0 0 b2
D2 1
2 ~a2b!2S 0000
0100
0010
0000D . ~8!
The first matrix in Eq.~8! can be removed by going to ainteraction picture in which the stateu0& has the additionalenergya2/2 and the stateu1& has the energyb2/2. The differ-ence (b22a2)/2 is an effectivedetuning for the u0&→u1&transition; although it may be significant as a source ofhomogeneous broadening along the ion chain for extremlong chains and calculation times, it will not be discussfurther in this paper.
Equations~7! show that, in general,a andb depend oniandj through the spacingr i j . The procedure used to splitHi jas in Eq.~8! can be applied to all pairs of interacting ionsturn; e.g., renormalize the energies of ions 1 and 2, theand 3, then 1 and 3, and so forth. In the end, the effecenergy of the lower state of thei th ion is shifted by anamount
~DE0! i5(j Þ i
ai j2
25
m0
8p~12cos2 u!^0umzu0&2(
j Þ iS 1
di j3 D
~9a!
ise
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PRA 60 187QUBIT-QUBIT INTERACTION IN . . . . II. . . .
and the upper state by an amount
~DE1! i5(j Þ i
bi j2
25
m0
8p~12cos2 u!^1umzu1&2(
j Þ iS 1
di j3 D ,
~9b!
and, in the interaction picture implied by Eq.~9!, the totalinteraction Hamiltonian is simply
H int5(j . i
Hi j 52 12 (
j . i~ai j 2bi j !
2~ u0&^0u i ^ u1&^1u j
1u1&^1u i ^ u0&^0u j !. ~10!
A special case of this interaction Hamiltonian was studieddetail in @3#, namely, the simple case where only neareneighbor interactions are considered, and the inhomogenalong the chain~the dependence ofr i ,i 11 on i! is ignored. Inthe model of @3#, the quantityd5(a2b)2/2\ was intro-duced, and it was shown that the computation of the quanFourier transform was seriously degraded if the proddN1/2T was of the order of 1, whereN is the total number ofions andT the total computation time. The reason for tfactor of N1/2 is that the standard deviation of the possibconfiguration energies of the chain, as given by the Hamtonian~10!, scales with the number of ions asN1/2, as shownin @3#.
By Eqs.~2! and ~7!, we may estimated as
d;m0mB
2
4p\d3 ;0.1
d3 s21, ~11!
where the last expression assumesd is in microns, a scalecharacteristic of the traps considered, for example, in@7#.This immediately shows that the effect is not necessasmall, since the times necessary to factorize numbers ofew as ten bits on a quantum computer have been estimto be of the order of seconds@2#, even without error correction protocols.
In @3# I studied only the errors induced by the second tein the Hamiltonian~8!, which are phase errors that accumlate coherently, occasionally interrupted by the action o‘‘gate’’ operation on a qubit. Clearly, however, the interation ~4! can lead to other types of errors. Withm given byEq. ~2!, leakage of the qubit out of the computing manifocan happen in a number of ways, asLx , Ly , Sx , andSy allcan change the spin or angular-momentum state appeariEq. ~1!.
Other proposed ion-chain systems may exhibit a magndipole-dipole interaction, which leads also to bit errors in tcomputational basis. This is the case, in general, whenstatesu0& andu1& are hyperfine sublevels or Zeeman sublevof the ground state. For these systems, thex and y compo-nents of the magnetic dipole moment operator do, in genehave nonvanishing matrix elements between the statesu0&andu1&, and the Hamiltonian~4! is, therefore, not diagonal inthe $u00&, u01&, u10&, u11&% basis. This is pretty much selfevident if the ground state is anS state and hyperfine interaction is negligible, since in that case the statesu0& and u1&differ only in the electron spin state~u11
2& or u212&!. In the
presence of strong hyperfine interaction, this is still gener
nt-ity
mt
l-
yased
-a
in
icehes
l,
ly
true, since the electron spin state will still be typically dferent in the statesu0& and u1&.
For definiteness, consider the quantum logic gate demstrated by Monroeet al. @8# using the hyperfine splitting othe ground state of9Be1. The stateu0& is uF52,mF52& andthe stateu1& is uF51,mF51&. The nuclear spin isI 5 3
2 , andthe two states can be rewritten in terms ofmI andmS as
u0&5umI532 , mS5 1
2 &,
~12!
u1&51
2umI5
12 , mS5 1
2 &2)
2umI5
32 , mS52 1
2 &.
The magnetic dipole moment operator is now
m522mB
\S, ~13!
neglecting the nuclear magnetic field, which is much weaover interatomic distances. Then the Hamiltonian~4! has thematrix form~assuming the quantization axisz to be along thechain, for simplicity!
Hi j 52m0mB
2
4pr i j3 S 2
000
021230
023210
00012
D ~14!
in the $u00&, u01&, u10&, u11&% basis. The diagonal part of Eq~14! is still of the form ~6!, but now there are additionaoff-diagonal elements, basically because the couplingtransform the stateu01& into u10&, or vice versa. These arpotential bit errors, and the rate at which they may occuessentially the same as for the phase errors discussed eand in@3#, since the coupling is of the same order of magtude.
Interestingly, one also gets off-diagonal elements fromcertain kind of so-called ‘‘field-insensitive state,’’ namelthose in which the expectation values of all componentsthe magnetic dipole moment are zero. For instance, fonucleus with a spin 1/2, one could choose for the statesu0&and u1& the singlet state and themF50 component of thetriplet state, respectively:
u0&51
&~ umI5
12 , mS5 1
2 &2umI52 12 , mS52 1
2 &),
~15!
u1&51
&~ umI5
12 , mS5 1
2 &1umI52 12 , mS52 1
2 &).
It is easy to verify that for these states^m&50, so that, tolowest order, they are insensitive to perturbation~of the formB•m) by any stray external magnetic fieldsB; hence thename ‘‘field-insensitive states.’’ It is clear, however, froEq. ~15! that, while the electronic operatorsSx andSy haveno matrix elements between any combination ofu0& and u1&,due to the orthogonality of the nuclear spin states, the optor Sz has the effectSzu0&5(\/2)u1&, andSzu1&5(\/2)u0&.
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188 PRA 60JULIO GEA-BANACLOCHE
Therefore, the interaction Hamiltonian~4! in the computa-tional basis$u00&, u01&, u10&, u11&% has the fully off-diagonalform
Hi j 52m0mB
2
4pr i j3 S 0
001
0010
0100
1000D . ~16!
This means that in these systems the magnetic dipole-diinteraction between neighboring atoms could, in fact, indtransitions such asu00&→u11& or u01&→u10&. The first typeof transition would in general require a significant amountenergy~of the order of the hyperfine splitting!, which meansthat it would not be likely to play an important role~i.e., leadto secular terms! over the time scales associated with tlogical gate operations, which should be at least sevtimes the reciprocal of theu0&→u1& transition frequency. Ihave, therefore, neglected such terms in what follows.
On the other hand, the transitionu01&→u10& has an en-ergy penalty only of the order of the inhomogeneous broening along the chain. In all likelihood, this would besmall that the states could, for practical purposes, begarded as degenerate, over time scales long compared tduration of a logical gate. This is how such off-diagonterms will be treated in this paper. One can, of course, esion quantum computing systems where the statesu10& andu01& are chosen to have a resolvable energy difference; thso for the NMR systems and would almost certainly becase for most solid-state implementations, where the difence in transition frequencies becomesthe way to addressindividual qubits that are not spatially resolvable. For susystems one would have to reconsider the role playedthese off-diagonal terms~in the NMR systems, for instancethey are actually used to implement conditional logic btween qubits!. For the ion-chain systems, however, andany other systems with small inhomogeneous broadenthese off-diagonal terms would most likely be simplysource of error~specifically, bit errors! and that is how theyare studied here.
The states of the form~15! are not, however, the mosgeneral kind of field-insensitive state. In a very comprehsive study of the technical and fundamental problems facthe ion-chain quantum computers, Winelandet al. @9# haveproposed using states that become field insensitive~to firstorder! in a strong external field. Such states have been ufor atomic clocks, as in@10#, where the external field is largenough to break the hyperfine coupling in the ground stat9Be1, so that one could then choose the following computional basis:
u0&5umI52 32 , mS5 1
2 &,~17!
u1&5umI52 12 , mS5 1
2 &.
The difference between the states~17! and all the ones considered previously in this section is that the electronic statthe same in bothu0& and u1&: all the information about thecomputational state is contained in thenuclear spin. Natu-rally, this renders the magnetic dipole-dipole interactiontween the valence electrons irrelevant. Only the interac
lee
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hy
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-g
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is
-n
between the nuclear spins can now disrupt the quantum cputer, and this is much weaker, as was pointed out aboverelevant quantity is the nuclear magneton, which is so1800 times weaker than the Bohr magneton@11#. Nonethe-less, except for the scaling, all the above considerationsapply, and the nuclear spin interaction can still be modealong the same lines as the electronic spin interaction, oreplacingme by mp in mB @for the case~17!, the relevantinteraction operator is clearly diagonal#. Accordingly, it willbe shown in Sec. IV that even this extremely weak intertion might still be unacceptably large for long computatioin some cases~specifically, for the center-of-mass–bus quatum computers!.
In summary, depending on the choice of a computatiobasis the magnetic dipole-dipole interaction may lead toteraction operators that are fully diagonal in the computional basis and that correspond to pure phase errors, sucEq. ~8!; or to operators that are fully off-diagonal and thcorrespond to bit errors~specifically, two simultaneous biflips!, such as Eq.~16!; or to operators with both diagonaand off-diagonal terms, such as Eq.~14!. In what follows, Ishall deal only with purely diagonal interactions and pureoff-diagonal interactions, and assume, which seems reaable, that things do not change much, qualitatively, whboth diagonal and off-diagonal elements are involved. I shalso restrict myself, for simplicity, to nearest-neighbor inteactions, and neglect very off-resonant transition terms sasu00&→u11&, as explained above. The two model Hamiltnians that result are, therefore,
HD52\d(i
~ u01&^01u1u10&^10u! i , ~18a!
for the diagonal interactions, and
HOD5\d(i
~ u01&^10u1u10&^01u! i , ~18b!
for the off-diagonal interactions. The angular frequencydthat characterizes the strength of the interaction is oforder of magnitude given by Eq.~11! @except for the choiceof states~17!, where it is smaller by a factor (me /mp)2]. It isunderstood that a ket such asu01& in the i th term in this sumrepresents the state of two neighboring atoms, of the fou0& i u1& i 11 , and the sum runs fromi 51 to i 5Natoms21. TheHamiltonian~18a! was the one used in@3#. In the followingsection the effects of interactions described by Eqs.~18a! and~18b! on the operation of a simple adder algorithm are stied numerically.
III. NUMERICAL RESULTS
The procedure followed to study the effect of the interations ~18! on the quantum computer is analogous to the oused in @3#. Assume each gate takes a timetg to be per-formed. After each gate has acted on a pair of qubits~thereare no single-qubit gates in this algorithm!, evolve the stateof the system using the evolution operator
U5e2 iH tg /\, ~19!
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PRA 60 189QUBIT-QUBIT INTERACTION IN . . . . II. . . .
whereH is one of the operators~18!. Apply the next gate inthe algorithm, then evolve the state again using Eq.~19!, andso on. At the end of the algorithm, compare the stateresults,uCout&, with the ideal result of the unperturbed caculation, uC ideal&. The quantity used for this comparisonthe fidelityF:
F5 z^C idealuCout& z2. ~20!
The algorithm used for the calculation is an implemention of a simple adder due to Vedralet al. @12#. ~Note thatFig. 3 of @12# is slightly confusing, and is better understoowhen turned upside down.! The computer consists of threregisters,a, b, andc, of n, n11, andn bits, respectively. Asequence of controlled-NOT~CNOT! and Toffoli gates (4n22 Toffoli gates and 4n CNOT gates! performs the additionof ann-bit number in registera to ann-bit number in registerb, and stores the result in registerb. The auxiliary registercis used to store the ‘‘carries’’ in the course of the compution: it starts in theu0& state and is reset to zero at the endthe calculation.
The fidelity observed in any run depends~deterministi-cally, of course! on the numbers to be added. The followinprocedure has been used to determine an average fidStart with registera in a coherent superposition of all thpossible numbers from 0 to 2n21, and registerb in a statecorresponding to a singlen-bit numberb. Carry out the cal-culation and compute the fidelity for this value ofb, F(b);do this for every possible value ofb separately and thenaverage the results:
Fav[1
2n (b50
2n21
F~b!. ~21!
I believe that starting the computer with the registera in asuperposition of all possible states makes for a fair test, smost practical applications of quantum computers wouldsomething just like that, namely, perform an operation ‘parallel’’ on all the possible values of a register.
Since the operator~18a! is diagonal in the computationabasis, the calculation of the time evolution given by Eq.~19!requires no special effort. The off-diagonal operator~18b! is,however, more complicated. As written there it representsoff-diagonal~though extremely sparse! matrix of dimension23n11323n11, which needs to be diagonalized and applito the state vector at every step. Even forn as low as 5, onecan see thatU will be a 65 536365 536 matrix, with some 4billion elements. On a personal workstation, such as theused for these calculations, storage alone of such a mbecomes essentially impossible.~This is, of course, an indi-rect demonstration of the potential of quantum computerssimulating the evolution of quantum systems, since it bodown to the realization that even a simple array of 16 twlevel systems, with a general coupling, already is intractaby most conventional computers.!
What I have done, then, is to reduce the (3n11)-qubitproblem, formally, to twon- and one (n11)-qubit problems,by introducing the artifical assumption that thea, b, and cregisters do not interact with each other. This is a very smmodification of the physical situation: if one imagines tregisters lying one after the other along the chain, it amou
at
-
-f
ity.
ceo
n
nerix
rs-le
ll
ts
to neglecting the interaction between the last qubit of regisa and the first one of registerb, and the last one ofb and thefirst one ofc. It cannot be much worse than the assumptalready made of keeping only nearest-neighbor interactioand, in fact, as will be seen in a moment, the calculation wthis assumption yields results qualitatively identical to tcalculation with the diagonal Hamiltonian~18a!, for whichno such assumption has been made.
Figures 1 and 2 show the result of the calculations, inform of a plot of the average fidelity versus the interactistrengthd, for different values ofn; Fig. 1 is for the diagonalinteraction~18a! and Fig. 2 is for the off-diagonal interactio~18b!. Note how, when the variabled is scaled as shown, thresults seem to approach an asymptotic curve independen. This scaling was suggested by the results in@3#, whichseemed to indicate that the degradation of the resultsproportional to the total spread in energy of the eigenvalof the interaction Hamiltonian and to the total calculatitime @which is here 2(4n21)tg]. For the spread~standarddeviation! in energy of the eigenvalues of Eqs.~18a! and~18b!, I find
FIG. 1. Average fidelity versus scaled interaction strength foralgorithm designed to add twon-bit numbers in the presence odiagonal interactions between qubits~phase errors!. Solid curve,n54; dashed curve,n55; dotted curve,n56.
FIG. 2. As in Fig. 1, but with purely off-diagonal interactionbetween qubits~bit errors!. Solid curve,n53; dashed curve,n54; dotted curve,n55.
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190 PRA 60JULIO GEA-BANACLOCHE
DED5 12 \d~N21!1/2 ~22!
for the diagonal interaction~18a! and
DEOD5\dS N21
2 D 1/2
~23!
for the off-diagonal interaction~18b!, whereN is the totalnumber of qubits. For the diagonal interaction, I just useN53n11 in Eq. ~22!. For the off-diagonal interaction, however, since I have broken up the chain into three sepapieces withn, n11, andn qubits, respectively, the overaHamiltonianHa^ Hb^ Hc has an energy dispersion
DEOD8 5\dS n21
21
n
21
n21
2 D 1/2
5\dS 3n22
2 D 1/2
,
~24!
and this is the scaling used for the horizontal axis in FigThe good overlap of the curves forn54, 5, and 6 in Fig.
1 suggests that the scaling used here ford captures suffi-ciently well the essential dependence of the fidelity onn forlarge n. A Gaussian fits all the curves quite well, and I etrapolate a dependence
[email protected]~3n!~dNgatestg!2#, ~25!
whereNgates58n22 is the total number of gates, andtg isthe time needed for each gate. More meaningfully perhain terms of the total number of qubitsN53n11 and thetotal calculation timeT5Ngatestg , one has
Fav.e20.16N~dT!2. ~26!
For the off-diagonal interaction, Fig. 2, the calculatiofor n56 are too long to accumulate enough points, butcurves forn53, 4, and 5 overlap quite well and again sugest that the essential dependence onn is included in thescaling ofd. This time I estimate
[email protected]~3n22!~dNgatestg!2#
.e20.23N~dT!2. ~27!
The results~26! and ~27! are of the same form as thresults reported in@3# for the quantum Fourier transform agorithm; it would appear, therefore, that the basic depdence on the total calculation time and interaction strengtthe same in all cases. The details~numerical factors!, on theother hand, vary somewhat, depending on the algorithmthe form of the interaction; note, for instance, that tslightly larger numerical factor in Eq.~27! does not fullyreflect the fact that the spread in energy for the off-diagoHamiltonian, Eq.~24!, is actually& times larger than theone for the diagonal Hamiltonian~22!.
Apart from this small correction, however, it seems thareally makes no substantial difference whether the intetion between the qubits is diagonal in the computationalsis ~phase errors! or off-diagonal~bit errors!. This is, at leastto me, a somewhat surprising result; in a previously plished paper, I conjectured that bit errors would be meffectively prevented from accumulating in the course ofcomputation because of the constant interruptions due to
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action of the logical gates~a sort of quantum Zeno effect!.This simulation, instead, indicates that, while there may bsmall correcting factor involved, the asymptotic dependeof the overall error on the number of gates~i.e., on the totalcomputation time! is the same in both cases, and correspoto a quasicoherent accumulation of error with time. A veremarkable consequence of this is that at least one kin‘‘field-insensitive’’ state—namely, the kind that would bcoupled, in a fully off-diagonal way, by the magnetic dipoldipole interaction, as explained in the previous sectionwould actually be as error prone as ordinary field-sensitstates.
It is interesting to compare the results of this simulationa much simpler one, namely, a calculation of the surviprobability of the initial state of the system of qubits asfunction of time and ofn, in the presence of interaction. Thsurvival probability could be calculated numerically alonthe same lines as above, only applying no gates at all toquantum computer, which means that, in fact, it can be cculated for any given value oft in one single step. The quasicoherent accumulation of error found earlier suggeststhe basic scaling of the survival probability will be the samas that of the fidelity when gates are applied. One may rethat, in fact, Garg’s@2# early investigation of the effects ointeractions~specifically, in that case, zero-point fluctuatioof the ions about their equilibrium positions, and the corsponding changing electric fields! was explicitly based on acalculation of the survival probability.
For the diagonal interaction, if one assumes as earliethis section that the computer is prepared in the state
uC0&5S 1
2n/2 (a50
2n21
ua& D ub&u0&, ~28!
then the magnitude of the coefficients in the coherent suposition does not change with time: they just accumuldifferent phase shifts. Clearly, also, the stateub& does notmake a difference~the high bit ofub&, which is the only onethat interacts withua&, is fixed in the u0& state!. One cancalculate the survival probability from
P5 z^C0uU~ t !uC0& z2, ~29!
where U(t) is the time-evolution operator. For the initiastate~28! and the diagonal interaction, this reduces to
P5U 1
2n (a50
2n21
e2 iEa8t/\U2
, ~30!
where Ea8 is the energy of the configuration of ionsua&^ u0&. It is straightforward to evaluate Eq.~30! using theapproximation, valid for largen, that theEa8 are distributed ina Gaussian manner, with a standard deviation only slighdifferent from Eq.~22! ~n, rather thanN21, should be usedbecause of the interaction with the first qubit ofub&!. Theresult is
P5e2n~dt !2/4, ~31!
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PRA 60 191QUBIT-QUBIT INTERACTION IN . . . . II. . . .
which indeed shows basically the same scaling as Eq.~26!.Numerical calculations yield very good agreement with E~31!, even for the relatively small values ofn consideredhere.
For the off-diagonal interaction, I have not attemptedderive an analytical expression for the survival probabilibut only done the numerical calculations. This time the itial stateub& matters, so again I average over all the initstates. On the other hand, since no calculation is beingried out, then qubits in theuc& register play no role@the stateu0& is an eigenstate of~18b!#. When the scaling is modified ina trivial manner to account for this, the calculations shthat the average survival probability is very well appromated by
Pav5e20.37~2n21!~dt !2. ~32!
Again, this is essentially the same scaling as Eq.~27!, exceptfor a numerical factor in the exponent. Hence, the quasherent accumulation of error during the operation ofquantum computer has the important consequence thascaling of the fidelity can be predicted, except for numerifactors of the order of unity, from the scaling of the survivprobability, which is much easier to calculate numerica~and sometimes even analytically!. The consequences thathese results might have for quantum computing are explofurther in the next section.
IV. DISCUSSION
According to the simulations in the preceding sectionseems that, in order to keep the error due to qubit-qubitteraction sufficiently small in the course of a calculation, owould require
dTN1/2,1, ~33!
whered is the interaction strength,T the total computationtime, andN the number of interacting qubits. It seems,first, that one could always make the interaction strengtdsufficiently small merely by increasing the separationtween qubits. For most configurations, however, this woalso decrease the strength of the desired coupling betwqubits~some kind of coupling will always be needed in ordto carry out conditional logic on the qubits!. The result willtypically be a slower computer.
In Ref. @6# I have illustrated this point for an ion-chaiquantum computer, for two different kinds of ‘‘buses~mechanisms to transfer the information between qubits!: acenter-of-mass bus, and a cavity photon bus. For referensummarize the results below. Note that my assumption in@6#that bit errors would scale differently from phase errorspears to be unfounded, by the results of the precedingtion, and therefore the ‘‘phase errors’’ formulas derived@6# should apply in all cases, except when the bit errorvolves a nondegenerate transition~that is, when there is anenergy penalty involved in the bit flip that is greater th\/tg); this nondegenerate case is treated briefly in@6#.
In the center-of-mass–bus quantum computer, the sling down takes place because the frequencyvx of the center-of-mass motion of the ions in the trap decreases as the sration between ionsd increases. Since the pulses needed
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perform conditional logic gates have to be longer than 1/vx ,increasingd leads to a largertg ~cf. also @7#!. As shown in@6#, the combination of the constraint~33! @with the estimate~11! for d# and the requirementtg. f /vx ~wheref is a suit-able ‘‘safety factor’’! leads to a requirement, for the minmum calculation timeT,
T.1
2 S A
32Z2DN2.18Ngates2 mp\
c2mef 2. ~34!
This involves only fundamental constants and atomic masand numbers, as well as the safety factorf .1. It may betranslated into a sort of figure of merit as follows: assuthat the factor involvingA andZ is of the order of unity, andthat the computational task is the factorization of anL bitnumber using Shor’s algorithm, which I will take to requiof the order of 200L3 gates~no error correction of any sort isassumed!. Then the lower limit~34! on T scales as
T. f 2L8.1831.5310212s. ~35!
This is an extremely long time for any reasonable-size nuber; for instance, if one choosesf 5103 one finds it wouldtake 900yearsto factor a 100-bit number.~What constitutesan acceptable value forf depends on the number of gates thneed to be executed, since the probability of error per gwill scale as 1/f 2.)
A word may be said at this point about the possibilityusing the field-insensitive states of the kind~17!, with mostof the information stored in the nuclear spin. Going over tderivation of Eq.~34!, one finds that to address the effectthe nuclear spin–nuclear spin interaction it is necessaryreplaceme by mp , which results in a lower limit forT some1800 times smaller than Eq.~35!. Remarkably, even this maprove too long; the 900 years of the previous example wobecome only half a year, but this is worse than conventiocomputers can do today; and increasing the number of bita really challenging size, for instance, toL5400, would in-crease the calculation time by a factor 48.18584 000.
For the cavity-photon bus@13#, increasing the distancebetween the ions also slows down the computer becausecavity becomes larger and hence the coupling to the camode, characterized by the ‘‘one-photon Rabi frequency’g,becomes weaker; since, as in the previous case, one neehave tg.1/g, this again leads to a slower computer.~Im-plicit in this discussion is the assumption that one harather improbable arrangement in which an optical cavitybuilt around an ion trap, with the axis of the cavity along taxis of the trap, so that the total length of the cavity is of torder of l 5Nd, where d is, again, the distance betweeneighboring ions.! The estimate shown in@6# was based onthe most favorable case in which the atomic electric-dipmoment~matrix element for the transition in question! is ofthe order of magnitude ofea0 ~e is the electron charge, ana0 is the Bohr radius!, and the mode volume of the opensided cavity is only limited by diffraction, hence of the ordof l l 2. Then the atom-cavity coupling may be estimated
g51
\ea0A \v
2e0V. ~36!
Then Eqs.~33! and ~11! lead to the constraint
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192 PRA 60JULIO GEA-BANACLOCHE
T.N7/4Ngates3/2 f 3/2l3/2a5/4A me
64p3\c, ~37!
wherel is the resonant optical wavelength of the cavity,athe fine-structure constant, andf is the safety factor,tg5 f /g. For the factorization of anL-bit number, as beforeand assumingl;1027 m, one gets
T. f 3/2L25/434310213s, ~38!
which scales much more favorably than Eq.~35!, but stillleads to a factorization time of seven years for a 400number, if f is chosen to have a reasonable value for tcase, such asf >103. With the nuclear-spin-based fieldinsensitive states~17!, however, these estimates are agreduced by the ratiome /mp , which in this case leads to thmuch more attractive estimate of less than two days to faa 400-bit number.
These results suggest that the center-of-mass–bus qtum computer is likely to be too slow for any large-scapractical applications, unless the problem of qubit-qubitteraction is addressed in some other way not considered hwhereas cavity-photon bus quantum computers might pobly work, if the very large couplings implied by Eq.~36! canbe realized and the information is stored in the nuclear sas in@10#. There are, however, formidable technical difficuties involved in building a linear trap around, and along taxis of, a high finesse optical cavity, and alternative scheare being eagerly pursued by several groups.
One of these schemes is an intriguing proposal by Chmanet al. @14#, which consists of two parallel chains of netral atoms trapped in a standing-wave e.m. field, perhapsor hundreds of microns apart, so thatd would be truly neg-ligible. An atom from one chain could only interact at angiven time with an atom on the other chain. To do this,atoms are slid along the chain into and out of an opticavity whose axis is at right angles to the chains’ axis; owhen the atoms are inside the cavity do they interact, ancourse the cavity can be very small~which will result in alarge couplingg! since it only needs to accomodate two aoms at a time. This is a very clever scheme whose ultimspeed and potential for scaling has yet to be determined
Somewhat similar to the above are proposals for ‘‘distruted computing’’@15#, consisting of many nodes with relatively few qubits, connected via a photonic channel that ctransfer quantum entanglement from one node to anotAgain, having only relatively few qubits in close proximittends to minimize the impact of direct, unwanted interactiobetween qubits, of the sort discussed here.
Leaving aside these distributed systems, it is my bethat the simulations presented here are based on a sufficigeneral set of Hamiltonians and a sufficiently basic alrithm to be relevant to other quantum computing systemsyet in the planning stages, such as, for instance, solid-ssystems. One would only need to identify the main relevsort of unwanted interaction between nearby qubits and emate its characteristic strengthd, and how it scales with thedistance between qubits, in order to be able to use the represented here to set possible limits on the computer’stential performance. If necessary, for more detailed simu
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Lastly, a few words need to be said about error correctiwhich has been ignored up to this point. For the specideterministic kind of error considered here, one might posbly envisionad hocerror correction schemes, perhaps alothe lines of some of the methods used in nuclear magnresonance to, in effect, ‘‘turn on’’ or off various interactioterms in the Hamiltonian. A concern in any practical schewould have to be, of course, that such methods, probabased on sequences of pulses, should not introducelarger errors than the extremely small ones they wouldtrying to compensate for.
Alternatively, there are also the by now standard ercorrection methods based on encoding the logical 0 and 1superpositions of states of many physical qubits, and caing out periodic measurements of ancillary qubits to extran error syndrome and perform error correction if necessIt was claimed in@3# that these methods might not necessily work on interaction-induced errors, where the phase erfor instance, accumulated at each step depends in princon the states of all the physical qubits in the computerturns out, however, that these methods might nonethelesused to correct for errors at least up to some desired leveaccuracy; that is, they could correct for errors generatedterms up to some given order in the expansion of the evotion operator~19! in powers of the interaction strength.
Consider, for definiteness, only the diagonal Hamilton~18a!, although this would apply to the off-diagonal form~18b! as well. For timest short enough for (dN1/2t)2 to benegligible, the action of the time-evolution operator e(2iH intt/\) of Eq. ~19! is approximately given by
e2 iH intt/\512 iH intt
\1¯
511 idt
22 i
dt
2 (j .1
s izs jz1¯ . ~39!
Here sz is the operatoru1&^1u2u0&^0u. Thus, u0&^0u5(sz21)/2, u1&^1u5(sz11)/2, and the last equality in Eq.~39!follows immediately from~18a!.
Error-correcting codes are specifically designed to hanerrors such as those that result when an operator likes iz actson the state of the system. A code is characterized, amother things, by how many such errors it may correct in opass. Codes with ‘‘distance’’d can correct (d21)/2 errors.~The ‘‘distance’’ is the minimum number of digits in whicany of the code words used to encode the logical 1 sdiffer from any of the code words used to encode the logi0.! The first few codes presented by Steane, Calderbank,Shor and others all hadd53, and could therefore correctsingle error, but codes that could correct many more errexist as well; they just require more physical qubits to ecode each logical qubit. Steane@16# has recently considerea code with 23 qubits and distance 7, which could therefcorrect up to three simultaneous errors in as many qubthis would be enough to eliminate the error associated wthe terms shown explicitly in Eq.~39!.
It is important to notice that a superposition operatorthe forms1z1s2z . . . acting on the state of the system st
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PRA 60 193QUBIT-QUBIT INTERACTION IN . . . . II. . . .
representsone single error, or, more precisely, a coheresuperposition of possible single-qubit errors; measuringerror syndrome will project the system onto the approprisingle-qubit error space, and then the appropriate correcaction may be taken. The superposition in Eq.~39! is a su-perposition of two-qubit errors and requires a code of mmum distance 5 to be corrected fully. However, even a cof distance 3 could correct for part of the errors, namethose in which the physical qubitsi and j are in differentlogical blocks, since then one only has to deal withoneerrorper logical block, which can be corrected even if the qubiin fact entangled with something else.
Assume that an@@n,1,d## code is used, whered is thedistance and the 1 and then refer to the use ofn physicalqubits to encode one logical qubit. Steane has considcodes@@7, 1, 3##, @@23, 1, 7##, @@55, 1, 11##, and@@87, 1, 15##.As mentioned above, the code of distance 3 can correcrors associated with the first-order term in the expansion~39!when the qubits involved are in different ‘‘blocks’’~a blockrefers to the set of physical qubits associated with a logqubit!, but not when they are in the same block. The codedistance 7 could correct all errors associated with the fiorder term in the expansion~39! and all those associatewith the second-order term~products of four error operators!,except, again, when all four errors happen in the same blThe code of distance 11 can correct the first- and seco
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.
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order errors, and the third-order~six consecutive error operators!, except when all six errors happen in the same bloand so on. In general, the first term in the expansion~39! thata code of distanced will notbe able to correct will be the onof orderp[(d11)/4, which involves 2p5(d11)/2 simul-taneous errors in the same logical block ofn physical qubits.
In principle, therefore, quantum-error correction methomight be used to bring the probability of error per gate tosmall a level as desired. With independent errors, this shobe enough to ensure the possibility of fault-tolerant comtation @17#, but the errors discussed here accumulate morless coherently, which means that the constraints or threolds for fault-tolerant computation may need to be modifieThis may be a point deserving further study. In any evenshould be noted that realistic, large-scale error correcwould require, among other things, a large degree of palelism, which means a very different architecture, and prably very different ‘‘hardware,’’ from those explicitly considered here.
ACKNOWLEDGMENTS
This research has been supported by the National SecAgency and the National Science Foundation. I am grateto C. Monroe for useful conversations.
e-
ofld,offgy
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@1# See, for example, the recent review article by A. Steane, RProg. Phys.61, 117 ~1998!.
@2# These have been studied in many cases for many diffesystems. For ion-chain systems without error correction,the work by M. B. Plenio and P. L. Knight, Phys. Rev. A53,2986 ~1996!; Proc. R. Soc. London, Ser. A453, 2017~1997!;see also A. Garg, Phys. Rev. Lett.77, 964 ~1996!, where the‘‘environment’’ is the zero-point oscillations of the ion chain
@3# J. Gea-Banacloche, Phys. Rev. A57, R1 ~1998!.@4# J. I. Cirac and P. Zoller, Phys. Rev. Lett.74, 4091~1995!.@5# For a description of the QFT, see, for instance, A. Ekert and
Jozsa, Rev. Mod. Phys.68, 733 ~1996!.@6# J. Gea-Banacloche, inPhotonic Quantum Computing II, edited
by Steven P. Hotaling and Andrew R. Pirich, special issueProc. SPIE3385, 64 ~1998!.
@7# R. J. Hughes, D. F. V. James, E. H. Knill, R. Laflamme, andG. Petschek, Phys. Rev. Lett.77, 3240~1996!.
@8# C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and DJ. Wineland, Phys. Rev. Lett.75, 4714~1995!.
@9# D. Wineland, C. Monroe, W. Itano, D. Leibfried, B. King, anD. Meekhof, J. Res. Natl. Inst. Stand. Technol.103, 259~1998!; see also D. Wineland, C. Monroe, W. Itano, B. KinD. Leibfried, D. Meekhof, C. Myatt, and C. Wood, FortschPhys.46, 363 ~1998!.
p.
nte
.
f
.
@10# J. J. Bollinger, J. D. Prestage, W. M. Itano, and D. J. Winland, Phys. Rev. Lett.54, 1000~1985!.
@11# Actually, the form given here is only the asymptotic formthese states for infinitely strong applied field. For a finite fiethe hyperfine interaction would result in a small amountcoupling to~mixing with! other states with different values omS ; in fact, it is this coupling that must cancel the enerdependence on the nuclear spin for the special value ofB atwhich near field independence is achieved. However, it flows from this that the magnitude of the coupling~the ampli-tude of other states in the mix! must be as small as the ratio othe nuclear to the electronic dipole, and thus the effectvalue ofd in the direct electron-electron magnetic interactiwould also be correspondingly reduced.
@12# V. Vedral, A. Barenco, and A. Ekert, Phys. Rev. A54, 147~1996!.
@13# T. Pellizzari, S. A. Gardiner, J. I. Cirac, and P. Zoller, PhyRev. Lett.75, 3788~1995!.
@14# M. Chapman, L. You, and T. A. B. Kennedy~unpublished!.@15# J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Phys. Re
Lett. 78, 3221~1997!.@16# A. M. Steane, Fortschr. Phys.46, 443 ~1998!.@17# See, e.g., E. Knill, R. Laflamme, and W. H. Zurek, Scien
279, 342 ~1998!, and references therein.
