Quantum information processing via a lossy bus

0.19 MB·English

by  S. D. Barrett

Quantum information processing via a lossy bus
S. D. Barrett1,2,3,∗ and G. J. Milburn2
1Hewlett-Packard Laboratories, Filton Road Stoke Gifford, Bristol BS34 8QZ, United Kingdom
2Centre for Quantum Computer Technology, University of Queensland, St Lucia, Queensland 4072, Australia
3Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, United Kingdom
(Dated: February 1, 2008)
Wedescribeamethodtoperformtwoqubitmeasurementsandlogicoperationsonpairsofqubits
which each interact with a harmonic oscillator degree of freedom (the bus), but do not directly
interact with one another. Ourscheme uses only weak interactions between the qubitand the bus,
homodynemeasurements,andsinglequbitoperations. Incontrasttoearlierschemes,thetechnique
presented here is extremely robust to photon loss in the bus mode, and can function with high
fidelity even when therate of photon loss is comparable to the strength of the qubit-buscoupling.
7
PACSnumbers: 03.67.Lx,42.50.Pq,85.25.Cp,78.67.Hc
0
0
2
The practicalimplementationofquantuminformation qubits and a bus which is a single mode of the optical

n processing (QIP) devices is one of the central aims of field, though the scheme could be modified to work for
a
quantuminformationscience. Largescalequantumcom- other systems with a suitable qubit-bus coupling Hamil-
J
puters can dramatically outperform classical computers tonian. In this scheme, the bus mode is initially pre-
7 for certain problems [1], while few qubit devices would pared in a coherent state α = e−|α|2/2 αn n /√n!,
2 allow other useful QIP applications, including quantum where n denotesthe n’th|exicitedstate oPftnhe o|sciillator.
| i
v repeaters [2] and cryptography [3]. Subsequently the bus interacts with the qubit degrees
6 Of particular interest are bus mediated QIP architec- of freedom via appropriately chosen cross-Kerr interac-
0 tures,inwhichcoherentquantumoperationsonmultiple tions. Finally, by performing a homodyne measurement
2
qubits are implemented indirectly, via a continuous de- onthebusdegreeoffreedom,thebusisdisentangledfrom
7
greeoffreedom(thebus)suchasaquantizedelectromag- the qubits, and the entanglement is transferred to the
0
6 netic field mode which interacts with each of the qubits. qubits. In the original proposal [10], this technique was
0 Sucharchitectureshaveanumberofpotentialadvantages used to implement two-qubit entangling measurements
/ for practical implementations. Firstly, in some systems, onthe qubits, whichare sufficientfor universalquantum
h
engineering a coherent interaction between qubits and a computation [11]. A modification of the scheme also al-
p
- bus mode can be easier than directly coupling pairs of lows the implementation of a CNOT gate [12]. Other
t qubits. Secondly,architecturesinvolvingabusmodecan schemes for implementing gates using a bus mode have
n
a bemorereadilyscaledtolargenumbersofqubits. When also been proposed [13, 14].
u a new qubit is added to the system, only it’s interaction However, these schemes have a substantial drawback
q withthebusmodeneedstobecalibratedandcontrolled. with regard to practical implementations. At various
:
v Bus mediated operations can then be implemented on stages in the protocol, the bus mode is in a superposi-
Xi any pair of qubits, rather than being restricted to (say) tionofcoherentstates,|αeiθqi, withdistinct phases{θq}
nearestneighbourqubits. Finally, in many casesthe bus whichcorrespondtodifferentstatesofthequbits. Super-
r can be propagated long distances, which can be useful positionsofthisformareratherfragileinthepresenceof
a
for distributed applications such as quantum repeaters. dissipation (e.g. photon loss) of the bus degree of free-
Recently, additional interest in bus-mediated QIP dom. Hence, losing even a small number of quanta from
has been generated by several experiments which have thebuscanleadtocompletedecoherenceofthecombined
demonstrated the coherent coupling between qubits and qubit-bus state, and destroy any entanglement between
continuous degrees of freedom. These experiments in- the qubits. A careful analysis shows that the scheme in-
clude the strong coupling of a superconducting charge troducedin [10], andthose derivedfrom it [12], can only
qubits to a single electromagnetic field mode [4], the work in the regime of extremely low dissipation defined
coupling of trapped ion qubits to a collective vibrational by κ/χ χτ, where κ is the loss rate for quanta in
≪
mode[5]andtheobservationofstrongcouplinginvarious the bus mode, χ is the interaction strength between the
quantum dot-cavityandatom-cavitysystems [6, 7, 8, 9]. qubits and the bus, and τ is the corresponding interac-
Previously, one of us introduced a scheme for imple- tion time. Since, in most realistic implementations, the
menting two-qubit operations via a bus mode [10]. This busmodewillbesubjecttodissipation,thisrequirement
was described in terms of an all-optical implementation, can be a significant restriction.
usingaweakcross-Kerrinteractionbetweenthephotonic In this Letter, we describe a method for bus mediated
quantum information processing which is robust to dis-
sipation of the bus mode. The physical resources used
for this scheme are similar to those used in the original
∗Electronicaddress: [email protected] scheme [10], namely preparation of a coherent state of
2
the bus mode, conditional phase shifts of the bus and path along the x axis. Since β > β , the final state
34 12
−
homodyne measurements. In addition, the new scheme of the bus, for the even parity states, is displaced by a
alsomakesuseofunconditionaldisplacements ofthe bus small distance δ α θ2. By repeating this procedure
i
≈
mode; such displacements are straightforward to imple- N times, the relative displacement of the bus mode for
mentinmostsystems,e.g. bydrivingthebusmodewith the odd and even parity qubit inputs can be made suffi-
aclassicalfieldresonantwiththe oscillatorfrequency. In cientlylargethatthetwocasescanbedistinguishedbya
particular, we show how, with these resources, a near- homodyne measurement of the x quadrature of the bus
−
deterministic, two qubit projective parity measurement mode. Finally, if the measurement outcome indicates an
(i.e. a measurement of the operator Z Z ) can be im- odd parity state, a fixed rotation about the z axis is
1 2
−
plemented. Such a measurement, when augmented with applied to one of the qubits to correct for an unwanted
single qubit operations,is universalforquantum compu- phase accumulation during the protocol.
tation, either using the CNOT construction of Ref. [15] As we show below,by making θ sufficiently small, this
or by constructing cluster states [16, 17]. The interac- procedurecanbemadeextremelytoleranttophotonloss.
tion betweenthe qubits andthe bus canbe controlledin Heuristically, this works because, for a given input state
sucha waythat the scheme is robustto lossesinthe bus parity, the bus mode is never in a superposition of co-
mode even in the regime κ > χ, where the strength of herentstates with substantially differentcomplex ampli-
the qubit-businteractionis weakcomparedthe lossrate. tudes. Thus, while a moderate amount of photon loss
reduces the total amplitude of the bus state, it does not
destroythecoherenceofthesuperposedstateswithinthe
relevant parity subspaces.
To analyze this scheme in more detail, we consider a
pair of qubits coupled to a harmonic oscillator (the bus)
via the interaction picture Hamiltonian (~=1)
χ (t) χ (t)
H (t)= 1 Z a†a+ 2 Z a†a+iε(t)(a† a). (1)
I 1 2
2 2 −
Here, χ (t) denotes the strength ofthe coupling between
i
qubitiandthe busmode, Z isthe Pauliz operatorfor
i
−
the ith qubit, a is a lowering operator for the bus, and
ε(t)denotesthestrengthoftheexternaldrivingfieldact-
ingonthe bus,whichis resonantwiththebareoscillator
frequency [21]. We assume the phase and amplitude of
this driving term can be accurately controlled. The first
twotermsinEq. (1)generatetheconditionalphaseshifts
FIG. 1: The photon loss tolerant scheme. (a) The ‘circuit’ of the oscillator, while the third term generates the un-
thatimplementsthescheme. (b)Aphasespaceillustrationof conditional displacements of the oscillator. This Hamil-
a single round of the protocol. In the absence of dissipation, tonian can be realized in a variety of settings, including
we would have θ′ = −θ, β12 = −2αicosθ and β34 = 2αi. thedispersiveregimeofcavityquantumelectrodynamics
Dissipationleadstoaslightdeformationofthebus’trajectory.
(CQED)[22](whichcanberealizedinavarietyofatomic
and solid-state systems, and can also be simulated in an
Our scheme for implementing parity measurements is iontrapsystem[5]),andinanall-opticalsetting,viathe
illustratedinFig. 1. Thebusispreparedinthecoherent cross-Kerreffect [10].
state αi , with αi real. The bus then interacts with the A single round of the scheme can be implemented
| i
qubits via repeated applications of the circuit shown in by a piecewise constant time dependence for χ (t) and
i
Fig. 1(a). The Uj(θ) operation is a conditional phase ε(t). We take χ1(t) = χ2(t) = χ0, 0, χ0, 0 , and
shift operation, which acts on qubit j and the bus mode ε(t) = 0, ε , 0, ε for−the interv{als 0 <−t < t}, t <
0 0 1 1
as 0 α 0 αeiθ and 1 α 1 αe−iθ (through- t < t ,{t −< t < t ,}t < t < t . Th{is has the effect
out| tih|isiL→ett|eri,| theistates|air|eiw→ritt|eni|withithe ket for of imp2lem2enting th3e co3nsecutive4p}airs of U ( θ) opera-
j
the bus mode at the right). D(β) =eβa†−β∗a is the dis- tions [shown in Fig. 1(a)] simultaneously, wh±ich is con-
placement operator acting on the bus mode. The effect ceptually equivalent to performing them consecutively,
of a single round of the protocol can be understood by since the Hamiltonian terms that generate these oper-
considering the phase space diagram in Fig. 1(b). For ations commute with one another. We also define the
the 2-qubit input states 01 and 10 , the bus mode fol- time intervals τ =t t corresponding to the length
i i i−1
lows the indicated loops|iniphase| spiace. θ, θ′, β and of each step of the evo−lution. The results obtained using
12
β are chosen such that the bus returns its initial state, this choice for χ (t) and ε(t) will be qualitatively valid
34 i
α , at the end of each round of the protocol. For the for other choices, too. For example, in a particular im-
i
| i
input states 00 and 11 , the consecutive U (θ) opera- plementation, it may be easier to have a fixed, constant
j
| i | i
tions cancel each other out, and the bus mode follows a qubit-buscoupling,andtoperiodicallyapplyverystrong,
3
short driving pulses (with ε χ ) to the bus mode. e−2κτ3) + α2 κ (e−2(κ−iχ)τ1 1), with α = β +
| |≫| i| 2κ−iχ − 2 12
We describe dissipation using a standardapproachfor e−κ(τ1+τ2)eiχτ1α . The real part of the exponent
i
the CQED setting: the damping from the oscillatormay
Γ Γ Φ Γ corresponds to a dephasing er-
01 12 12 23
bemodeledasleakagefromthecorrespondingcavitymir- − − − −
ror (see below), while the imaginary part leads to a de-
rorat a rate 2κ. Within the Born,Markov,and rotating
terministic phase rotation which is ultimately corrected
waveapproximations,suchlossescanbedescribedbythe
by applying the single qubit unitary U [see Fig. 1 (a)],
z
non-unitary term in a Lindblad master equation [18]:
after the homodyne measurement stage of the protocol.
The timescales τ , τ and τ are chosen as follows. τ
ρ˙(t)= i[H (t),ρ(t)]+2κ [a]ρ(t), (2) 2 3 4 2
− I D andτ3 canbefixedrequiringthat,forsymmetry,attime
t , the bus mode associated with the input state Ψ
where ρ(t) is the density matrix of the combined system 3 | o±i
is in the state α = α . This leads to the approximate
AofρAbo†th (qρuAb†itAs+anAd†Athρe)/o2sc[2il3la].toBrelmowod,ew,easnodlveDE[Aq].ρ(2=) valuesτ2 ≈(2−κ32τ12−−χi2τ12)αi/ǫ0 andτ3 ≈τ1(1−2κτ1).
− τ is chosen such that α = α , i.e. that the bus mode
forasingleroundoftheprotocol,andthenusetheresults 4 o i
associated with the input state Ψ is returned to it’s
todetermineboththetotalnumberofroundsrequiredto | o±i
initial state at the end of each round of the protocol.
distinguishevenparitystatesfromoddparitystates,and
Using these values of τ , it is possible to obtain
the total dephasing error at the end of these N rounds. i
approximate expressions for the total dephasing error
ThesolutionofEq.(2)canbesimplifiedbynotingthat,
at the very end of the protocol. After N rounds,
since the aim of the scheme is to perform a parity mea-
the state corresponding to an initial state Ψ is
surement, only dephasing errors within the correspond- | o+i
ing parity subspaces are of concern. Thus we are only ρ(oN) = h(1 p)Ψ˜o+ Ψ˜o+ +pΨ˜o− Ψ˜o− i αi αi ,
− | ih | | ih | ⊗ | ih |
interested in solutions of Eq. (2) for initial states of the where p = (1 e−NRe(Γ01+Γ12+Γ23))/2 is the probability
xforamnd(xy|0a0rie+aryb|1it1ria)r|αyiiamanpdlit(uxd|e1s0.i+Wye|0t1aik)e|αtihi,ewwhoerrset ofadephasing−error,and|Ψ˜o±icorrespondsto|Ψo±i,up
tothesinglequbitcorrectionU discussedabove. Thefi-
case, x = y = 1/√2, and so the resulting errors should z
(N)
nalstatecorrespondingtoaninitialstate Ψ isρ =
be viewed as an upper bound. The solution can be fur- | e+i e
thersimplifiedbyprojectingρ(t)ontothecomputational |Ψe+ihΨe+|⊗|α(eN)ihα(eN)|, where α(eN) ≈αi+Nαiχ2τ12.
basisofthequbits. Thisresultsinasetofuncoupleddif- Writing the expression for the dephasing error in
wfehreenretiai,ljequa0t0io,n0s1,f1o0r,t1h1e,owphericahtocrasnρbi|ej(sto)lv=edhiin|ρd(etp)|ejni-, t4eNrmαs2iκχof2τ1τ31/3g+iv4esNαth3ieκχa2pτ12p/roǫ0x.imTahtee fierxsptretsesrimoninpth≈is
dently. Th∈es{e equations a}re analogous to those describ- expression originates from decoherence on the ‘rotation’
ing the single-qubit, driven Jaynes-Cummings model in partsoftheevolution(i.e. t0 t1 andt2 t3),whereas
→ →
the dispersive limit, and thus can be solved exactly us- the second term originates from the first ‘displacement’
ing techniques similar to those used in Refs. [19, 20]. part (t1 t2). Note that no decoherence results from
→
We omit the details of this calculation, but give the full theseconddisplacementstage(t3 t4),asthebusmode
→
analytic results here. is not in a superposition state during these parts of the
For an input state Ψ α , where Ψ = (00 evolution.
e+ i e±
11 )/√2, the state af|ter ai|sinigle round| of tihe pr|otoic±ol The total number of required rounds of the protocol,
|(i.ei. at time t4), is given by |Ψe+i|αei, with N, is fixed by the requirement that |α(eN)i and |αii are
distinguishable by a X homodyne measurement at the
α =β +e−κ(τ3+τ4)(β +e−κ(τ1+τ2)α ), endoftheprotocol. For−coherentstates,thisimpliesthat
e 34 12 i
α(N) α =A,whereA&1isaconstantthatdependson
where β = (ǫ /κ)[1 e−κτ2], β =(ǫ /κ)[1 e−κτ4], e − i
12 0 34 0 therequirederrorindistinguishingthestates Ψ from
− − − o±
and the time intervals τi = ti ti−1 correspond to the thestates Ψ . ThisrequirementgivesN =|A/αiχ2τ2.
length of each step of the evolu−tion. | e±i 0 1
Thus the total error is well approximated by
For an input state Ψ α , where Ψ = (01
o+ i o±
| i| i | i | i±
10 )/√2, the state after a single round is 4Aα κτ 4Aα2κ
| i p i 1 + i . (4)
≈ 3 ǫ
0
1
[(01 01 + 10 10) α α +...
2 | ih | | ih | ⊗| oih o| The firstterm in this expressioncanbe made arbitrarily
e−Γ01−Γ12−Φ12−Γ23 01 10 α α∗ +h.c. , (3) smallbydecreasingthe valueofκτ1. Thesecondtermin
(cid:0) | ih |⊗| oih o| (cid:1)(cid:3) thisexpressionisproportionaltheratioκ/ǫ . Thuserrors
0
where can be minimized provided the strength of the classical
driving field can be made large compared to the pho-
α =β +e−iχτ3e−κ(τ3+τ4)(β +e−κ(τ1+τ2)eiχτ1α ), ton loss rate. An important point to note is that both
o 34 12 i
terms are independent of the ratio κ/χ, and therefore
and Γ01 = α2i[(1 − e−2κτ1) + κ−κiχ(e−2(κ−iχ)τ1 − the scheme is robust to losses in the bus mode even in
1)], Γ = α2e−2κτ1(1 e−2κτ2)(1 e2iχτ1), Φ = the regime κ > χ, where the strength of the qubit-bus
2iIm(α12β e−κ(iτ2+τ1)eiχτ1−), and Γ − = α 21(21 interaction is weak compared to the loss rate. The ap-
i 12 23 2
| | −
4
proximate error, together with the exact values for the Nσ2(α ) N[σ2(β ) + σ2(β ) + β2 sin2θσ2(θ)]
same values of the parameters τ , are plotted in Fig. 2. 2Nσ2(β0 )≈+4α Aσ21(2θ), where3σ42(...)1d2enotes the var≈i-
i 12 0
ance in each parameter. Thus the noise will not signif-
icantly affect the outcome of the X-homodyne measure-
10−1 (a) 106 (b) ment provided σ(β12) . N−21 and σ(θ) . (4α0A)−21. If
−2 4 these conditions cannotbe met then the scheme canstill
10 10
be made to work by increasing the number of rounds,
p N
10−3 102 since the peak separation grows linearly in N, while the
1
peak width grows only as N2. Another potentially im-
−4 0
10 10
portant source of noise is single qubit errors. These can
−4 −3 −2 −1 −3 −2 −1
10 10 10 10 10 10 10
potentially build up during the implementation of the
τ1 p
two-qubit operation, leading to an effective error rate
that scales with N. Thus our scheme is particularly
FIG. 2: (a) Exact (curves) and approximate (points) values
suited to systems in which the single qubit decoherence
of the final dephasing error against τ1. Other parameters:
A = 1.5, αi = 2, κ = 0.05, χ = 1, ǫ0 = 102,103,104, (top rate is small compared to the bus damping rate. Fortu-
tobottom). (b)Overhead,N,requiredtoreachagivenerror nately,therearemanysystems(suchastrappedions[5])
rate, p, for A=1.5, αi =2, κ=0.05, χ=1, ǫ0 =105. where the single qubit error rate is much longer than all
other timescales.
The price to be paid for this robustness is an increase In conclusion, we have proposed a method for imple-
in the overheadcost of the scheme: the first term in Eq. menting two-qubit operations via a quantum bus, which
(4) can be reduced by reducing τ , but as τ is reduced, is extremely robust to dissipation of the bus degree of
1 1
the required number of rounds of the protocol required freedom. Our method implements a paritymeasurement
increases. If p is dominated by the first term, then the of the two-qubit system, which, when augmented with
requirednumberofroundsisN (16A3α /9)(κ/χ)2/p2. single qubit operations,is sufficient for universalcompu-
i
Thus the overhead cost grows qu≈adratically with the re- tation. The residual errors in our scheme can be made
quired reduction in the errors (see Fig. (2)). negligible even when the strength of the qubit-bus inter-
In view of this overhead, it is worth briefly consider- action is weak compared to the loss rate. We anticipate
ing the effect of errorsin the scheme. Stochastic noise in that this robustness will permit bus-mediated QIP in a
the control pulses used to implement the protocol tend varietyofphysicalsetups,includingsuperconductingsys-
to broaden final state of the bus mode, making the odd tems, trapped ions or atoms, and all-optical systems.
and even parity outcomes harder to distinguish by the Acknowledgments: SDB was supported by the eu, the
X homodyne measurement. Treating the noise in each cqct, and the epsrc. SDB thanks Aggie Branczyk,
−
pulseasindependentGaussianprocesses,thefinaluncer- Alexei Gilchrist, Pieter Kok, Bill Munro, Tim Spiller,
tainty in the real part of α(N) is found to be σ2(α(N))= and Tom Stace for useful conversations.
0 0
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[21] Note Eq. (1) is defined in an interaction picture relative cavity coupling strengths, and ∆i are the qubit-cavity
tothebareHamiltonianoftheuncoupledqubitsandbus, detunings.
H0=Hq1+Hq2+Hbus,whereHqi=−E2iZi andHbus= [23] Equation(2)describeserrorsonlyduetodampingofthe
ω0a†a, with Ei the qubit energy splittings, and ω0 the oscillator mode; in general, there will also be decoher-
bareoscillator frequency.In arriving at Eq. (1), we have ence processes acting directly on the qubit degrees of
madearotatingwaveapproximationforthedrivingterm, freedom. However, such processes are typically indepen-
which is valid provided ω0 ≫ ε(t), and provided ε(t) dentofthoseduetodampingandcanbedealtwithusing
changes slowly on timescales of order ω−1. standardfaulttolerancetechniques,andsoweomitthem
0
[22] InaCQEDimplementation,additionalmulti-qubitterms from this analysis.
canappearintheHamiltonianduetohigherorderinter-
actionsviathebus.Thesecanbeavoidedbyengineering