Input states for quantum gates

5 Pages·0.16 MB·English

by  A. Gilchrist

Input states for quantum gates
A. Gilchrist,1,∗ W.J. Munro,2 and A.G. White1
1Center for Quantum Computer Technology and Department of Physics,
University of Queensland, QLD 4072, Brisbane, Australia.
2Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol, BS34 8QZ, United Kingdom.
(Dated: February 1, 2008)
We examine three possible implementations of non-deterministic linear optical cnot gates with
a view to an in-principle demonstration in the near future. To this end we consider demonstrating
the gates using currently available sources such as spontaneous parametric down conversion and
coherent states, and current detectors only able to distinguish between zero or many photons. The
demonstration is possible in the co-incidence basis and the errors introduced by the non-optimal
inputstates and detectors are analysed.
3
0 PACSnumbers: 03.67.Lx,42.50.-p
0
2
n INTRODUCTION selective detectors(detectorsonlyabletoresolvezeroand
a multiple photons). The aim is to identify a scheme that
J Optics is a natural candidate for implementing a va- allows a scalable cnot implementation to be initially
2 riety of quantum information protocols. Photons make examined with current sources and detectors, and into
2
beguiling qubits: at optical frequencies the qubits are which we can easily incorporate single photon sources

1 largely decoupled from the environment and so experi- and selective detectors as they become available.
v ence little decoherence, and single qubit gates are easily Typicallythegatesinvolvefourphotonswiththequbit
2 realised via passive optical elements. Some protocols, states are encoded in the polarisation state of the con-
1 notably quantum computation, also require two-qubit trol and target modes c and t, and the cnot operation
1
gates. Until recently this was regarded as optically in- is implemented with the aid of some ancillary modes a,
1
0 feasible, since the requirednonlinear interaction is much betc. We willconsiderstartingwiththe controlandtar-
3 greater than that available with extant materials. How- getmodes eachin a generalsuperposition(we couldalso
0 ever, it is now widely recognisedthat the necessary non- consider initially entangled states though these may be
h/ linearity can be realised non-deterministically via mea- more difficult experimentally)
p surement, and that deterministic gates can be achieved
t- by combining such non-deterministic gates and telepor- |ψinict =(Ahcˆ†h+Avcˆ†v)(Bhtˆ†h+Bvtˆ†v)|0i (1)
n tation [1].
a with A 2+ A 2 = B 2+ B 2 = 1, and where cˆ†
u There are a number of proposals for implementing a | h| | v| | h| | v| h,v
q non-deterministic cnot gate with linear optics and pho- andtˆ†h,v arebosoniccreationoperatorsformodech,v and
: todetectors [1, 2, 3, 4, 5, 6]. The proposals require de- t etc. In the interest of brevity we will use the nota-
v h,v
i terministic, or heralded, single photon sources, and/or tion above where we write the state in terms of creation
X selective detectors, that can distinguish with very high operators acting on the vacuum state.
r efficiency between zero, one and multiple photons. Cur- The modesarefirstentangledwitha linearopticsnet-
a
rent commercial optical sources and detectors fall well work Ucnot comprised of beamsplitters , phase shifters,
short of these capabilities. Although there are a num- waveplates, and polarising beam splitters. Finally the
berofactiveresearchprogramsaimedatproducingboth gate is conditioned on detecting the ancillary modes in
efficient selective detectors [7, 8], and deterministic pho- someappropriatestate,whichleavesthestateofthecon-
ton sources [9, 10, 11], nonselective avalanche photodi- trol and target modes as if a cnot had been applied.
odes, spontaneous parametric downconversion (SPDC) The key simplification for our purposes is to detect
and coherent states remain the best accessible labora- in the ‘coincidence basis’ — where we detect the out-
tory options. While we couldside-step the single photon put of the ancillary modes and also of the target and
sourceproblembyusinganSPDCsourceconditionedon control modes and postselect out those events that do
the detection of a photon in one arm if we had selec- not simultaneously register a photon in all four modes.
tive detectors, demonstrating a four-photon cnot gate The advantage of this configuration is that now we can
without quantum memory would be frustratingly slow. use non-selective detectors, since if we get a “click” on
In this paper we examine three proposals which allow all four detectors we’ve accounted for all the photons
acnottobe implementednon-destructivelyonthe con- in the system. This is a much less stringent require-
trolandtargetmodes,toascertainunderwhatconditions ment on the detectors and in particular can be fulfilled
it is possible to demonstrate and characterise the gates by existing avalanche photodiodes. We model the non-
operationusing SPDC sources,coherent states and non- selective detectors with a positive-operator-valued mea-
2
sure(POVM),withthe POVMelementsassociatedwith and cos2θ is the reflectivity. The angle choices for
detecting no photons or photons (one or more) simply the gate are given by θ = cos−1 5 3√2 and θ =
1 2
∞ −
being Π0 =|0ih0| and Πm = n=1|nihn| respectively. cos−1 (3 √2)/7; c and t are thpe control and target
The output state of a typeP-I SPDC can be described q −
modes and a, b, v and v are independent ancillary
as 1 2
modes. The gate is conditioned on detecting a single
λ = λ(00 +λ11 +λ2 22 + ) (2) photon in the modes a and b and detecting no photons
| i M | i | i | i ···
∞ (λaˆ†ˆb†)n2 in the modes v1 and v2.
= 0 (3)
Mλ n! | i Consider the case where both the control, target and
nX=0 2 ancillaryphotonsaresuppliedbytwoindependentSPDC
(even) sources. Theinputstateis λ ǫ whichcanbewritten
| ict| iab
where =(1 λ2)−1 andthesumisoverevennwhere as a sum over total photon number
λ
M −
n is the number of photons in each term.
Nowsupposethatourinputstatetotheopticalcircuit ∞
is some initial pure state |ψini, and that after passing |φini = MλMǫ X Qˆn|0i (9)
throughthelinearopticalelementsweareleftinthestate n=0
ψout =Ucnot ψin . The probabilitythat we geta count (even)
| i | i
simultaneously in modes c, t, a and b with non-selective n
detectors is Qˆ = 2 ǫmλn2−m (aˆ†ˆb†)m(cˆ†tˆ†)n2−m
n m!(n m)!
P = ψ Π(c) Π(t) Π(a) Π(b) ψ (4) mX=0 2 −
h out| m ⊗ m ⊗ m ⊗ m | outi
For the ideal case where we had single photon inputs to
The control and target horizontal and vertical polarisa-
the gate, we will label this probability as P . We can
1 tionmodesaretheneachmixedonabeamsplittersothat
now introduce the “single photon visibility” as a figure
we achieve the input state (1) for those modes.
of merit for how close the gate operates to the ideal:
Since we are postselecting on getting a ‘click’ at four
1 s e
= − +1 (5) detectorsthenthetermswithn<4willalwaysgetpost-
V 2(cid:18)s+e (cid:19) selected out. Similarly, the terms with n > 4 will get
wheresistheproductoftheprobabilityofobtainingthe postselected out if we used selective detectors otherwise
single photon terms from the source, with P the proba- they represent error terms. In the latter case, so long as
1
bilityofthegatefunctioning. The“error”e=max(s P) ǫ,λ 1 these terms will be small. For the case were
− ≪
whereP istheactualprobabilityofobtainingacounton n=4, three input terms contribute:
the detectors. The maximisation is over all qubit input
sthtaetevsistiobitlihtye gisatcel.osHeetnocezeifrot,heifetrhroerntooitsaellyisdsommailnlatthees ψ(4) =(λǫaˆ†ˆb†cˆ†tˆ†+ λ2cˆ†2tˆ†2+ ǫ2aˆ†2ˆb†2)0 (10)
| in i 2! 2! | i
visibility is close to one. As a guide a visibility of 0.8
correspondstoanerrora quarterofthe sizeofthe single
Whilethefirstofthesetermsisequivalenttohavingfour
photon “signal” s.
initialFockstates,theremainingtwotermshavethepos-
sibilityofsurvivingthepostselectioncriteriaandskewing
SIMPLIFIED KLM CNOT the statistics observed. Fortunately these last two terms
leadtooutputtermswhichall getpostselectedoutinthe
Intheoriginallyproposednon-deterministiccnotgate coincidencebasis(e.g. twophotonsinthecontrolmode).
Thismeansthatwithselectivedetectorswecouldinprin-
[1]thenonlinearsignshiftelementswereinterferometric:
ciple postselect out all terms that do not correspond to
these elements can be replaced by sequential beamsplit-
ters to make a simplified cnot gate [2], one example of single photon inputs from the output statistics. With
non-selective detectors the error terms will scale at least
which is
ˆ ˆ π ˆ π ˆ ˆ asλ3 inamplitude (duetothen>4terms)sothe figure
Usklm = Bthtv(4)Bcvth(4)Bbth(θ2)Bacv(θ2) ofmeritwillscalewithλ(takingǫ=λ)asV ∼1/(1+λ2)
ˆ π ˆ ˆ π ˆ and λ is typically very small.
B ( )B (θ )B ( )B (θ ) (6)
cvth 4 thv2 1 thtv 4 v1cv 1 NowconsiderthesituationwhereaSPDCsuppliesthe
where Bˆ represents a beam splitter with the following two photons for the control and target modes and weak
ab
action coherentstatesareusedfor the ancillarymodes. The in-
Bˆab(θ)aˆBˆ†ab(θ) = aˆcosθ+ˆbsinθ (7) pthuetcsrteaatteioisntohpeenra|φtoinris=for|λt,hαe,cβoihewrhenertestaˆa†teasn.dAˆbf†tewrirlelabre-
Bˆ (θ)ˆbBˆ† (θ) = aˆsinθ ˆbcosθ (8) ranging the state as a primary sum over photon number
ab ab −
3
we get
(a)
∞ n n−p(λcˆ†tˆ†)p2 (αaˆ†)q (βˆb†)n−p−q
|φini= p! q! (n p q)!|0i 1
nX=0 pX=0 Xq=0 2 − − 10−3 V
0.9
(even) 10−4
ǫ
(11) 0.8
10−5 1100−−44
Again, terms with n<4 will get postselected out and 1100−−55
1100−−66 λ
terms with n > 4 will be weak error terms. The ex- 10−6 1100−−77
1100−−8
trafreedomfromtwoindependentcoherentstatesmeans
that now there will be nine terms with n = 4 and only (b)
one of these is equivalent to using single photon inputs.
The terms were a single coherentstate supplies allthe 1
photons always gets postselected out. By setting β =iα 10−3 V
0.9
the two terms where a single coherent state supplies two 10−4
photons and the paramp supplies two will cancel each ǫ 0.8
10−5 1100−−44
other due to the symmetry in the circuit. Finally the 1100−−55
1100−−66 λ
term where the paramp supplies all the photons is post- 10−6 1100−−77
1100−−8
selected out as before. This means that we will still get
errors arising from the input terms:
FIG.1: Thesinglephotonvisibility withnon-selectivedetec-
iα4(aˆ†3ˆb† aˆ†ˆb†3)0 (12) tors as a function of the strengths of the SPDC sources. (a)
6 − | i the entangled ancilla gate, (b) the Knill gate. In both cases
the input state was truncated at six photon terms, and the
Note that these do not depend onthe input state that is
maximisation of the error was performed numerically.
encoded on the control and target modes and by setting
α λ we can scale away these terms relative the single
≪
photon terms. Unfortunately this means that we cannot Consider that the entangled pair in modes a and b
beat the photon collection rate that could be achieved are provided by two type-I parametric downconverting
using two independent SPDC sources. crystalssandwichedtogether. We’llfixtherelativephase
It should be noted that all the observations made for to get a particular Bell pair for the two photon term:
the simplified KLM cnot also hold for the full KLM
cnot in the coincidence basis. However from the per- |ǫ2i = M2ǫ(|00i+ǫ|11i+···)(|00i+ǫ|11i+···)
spective of an initial demonstration of the gate the sim- = M2ǫ[···+ǫ(|0011i+|1100i)+···] (14)
plified version is more desirable. In the following two
wherethemodesarea ,b ,a ,andb respectively. Such
sections we will compare these results against two other h h v v
sources have been previously built and provide a rel-
implementations of optical cnot gates.
atively bright source of polarisation entangled photons
[12, 13]. We can write this source succinctly as
ENTANGLED ANCILLA CNOT
∞
ǫ = 2 Lˆ 0 (15)
| 2i Mǫ n| i
In a recent paper, Pittman, Jacobs and Franson [6] nX=0
proposed using entangled ancilla to further simplify im- (even)
tpdhliesepmnoewsnaetlacataninoneimnotpfaltnehgmeleecdnntseotdattt.ehCe|φocinns=oidt(eaˆrbhetˆbhtwhate+ewnaˆemvhˆboavdv)ee/s√act2a|o0nuidr, Lˆn = mXn/=20ǫn2(aˆ†hmˆb†h!()n2m(−aˆm†vˆb)†v!)n2−m (16)
t by first applying the unitary
With another independent paramp, λ , supplying the
Uˆ =Pˆ Pˆ Wˆ Wˆ Wˆ Pˆ Wˆ Wˆ Pˆ (13) photonsforthecontrolandtargetmode|s,itheinputstate
ent bd ae a t b bt t b ac
becomes
ˆ
where W represents a half-wave plate on mode a and
Ptˆhaebeiffseactpatohlaartisainhg→beaamh, sbphli→tterbhin, amvo→desbav,aannddbbwvi→th |φini≡M2ǫMλ nX∞=0 qX=n 0 Lˆqλn−2q(ncˆ−2†qtˆ!†)n−2q (17)
a . Finally the resulting state is then conditioned on
v (even) (even)
detecting a single photon in modes a and b. The raw
successprobabilityofthis gateis 1/16whichrisesto 1/4 where we will encode the qubits in the polarisationstate
if fast feed-forward and correction is used. of the control and target modes, as in (1).
4
Again all terms with n < 4 will get postselected out. do not get postselected out leading to inherent errors in
There are six terms with n = 4 of which two terms rep- the statistics we will observe. Notice however that all
resents our single photon input terms, the rest are error these terms will be proportional to λ2 so again by mak-
terms due to the sources. With non-selective detectors ing λ ǫ we can scale these terms away with selective
≪
terms with n>4 will also contribute to the error. detectors at the expense of the count rate. With non-
The fourphotontermsin the outputstate thatdonot selective detectorsthere willagainbe anoptimum λ, see
get postselected out are figure 1 (b), which is very similar to the previous gate.
out = 1 λaˆ†ˆb†(A B ǫcˆ†tˆ† +A B ǫcˆ†tˆ†
| i 2√2 v h v v v v v h
CONCLUSION
+A [A B2λ A B2λ+B ǫ]cˆ†tˆ†
h v h − v v v h v
+A [A B2λ A B2λ+B ǫ]cˆ†tˆ†)0 (18) We have examined three possible implementations for
h v h − v v h h h | i linear optics cnot gates with a view to experimentally
and by making λ ǫ we can recover the single photon demonstratingtheiroperationinthenearfuture. Incon-
termsandtheactio≪nofthecnotwithselectivedetectors. sidering demonstrating the gates with SPDC and coher-
This of course means that the count rate with this gate ent state sources and non-selective detectors there is a
wouldbeconsiderablylessthanwiththesimplifiedKLM clearadvantagetothesimplifiedKLMcnotgate,where
gate. With non-selective detectors, if we make λ too the inherent symmetries in the gate allow the use of two
small the error due to the six photon input terms will independent SPDC sources to supply the control, tar-
dominate, so there is an optimum λ for a given ǫ see get and ancillary photons, with errors from the use of
figure 1 (a). non-Fock states making little contribution. The other
There does not appear to be a way of using two co- twoimplementationssufferfromerrorsintroducedbythe
herent states to replace one of the SPDC sources. If we non-Fock state inputs which cannot be postselected out.
replace either the control or target mode then it is hard Whilethesituationmaybemitigatedsomewhatbyusing
to see how the 02 and 20 terms could cancel as with a weak SPDC source this would occur at the expense of
the simplified K|LMi cno|tisince these terms will have the countrateofvalideventsthatmaybe collectedfrom
factors that depend on the encoded qubit. Similarly re- the gate.
placingthesourceofentangledphotonswouldthenmean The conclusion we arrive at is that an experimen-
wewouldhavetoentanglethesinglephotoncomponents tal program focusing on the simplified KLM cnot gate
which is difficult. would then allow immediate characterisationof the gate
with current sources and detectors, with the operation
of the gate in a non-destructive fashion becoming pos-
KNILL CNOT
sible when single photon sources and selective detectors
become available.
A recent numerical search for optical gates by Knill
We would like to acknowledge support from the the
yieldedacnotgate[14]whichoperateswithaprobabil-
AustralianResearchCouncilandthe US Army Research
ity of 2/27 and is described by the following unitary,
Office. AG was supported by the New Zealand Founda-
ˆ ˆ π ˆ ˆ ˆ tion for Research, Science and Technology under grant
U = B ( )B (θ )B (θ )B (θ )
Knill tvth 4 ab 3 cvtv 2 tvb 1 UQSL0001. WJM acknowledges support for the EU
ˆ ˆ π ˆ project RAMBOQ. We would also like to thank Michael
B (θ )B ( )F (π) (19)
cva 1 tvth 4 a Nielsen, Jennifer Dodd, Nathan Langford, Tim Ralph
and Gerard Milburn for helpful discussions.
ˆ
where F (θ) is a phaseshift of θ on mode a and the re-
a
flectivities are given by θ = cos−1 1/3, θ = θ and
1 2 1
−
p
θ = cos−1 1/2+1/√6. The gate requires two ancil-
3
q
lary modes a and b initially in Fock states to be finally ∗ Electronic address: [email protected]
detected also in single Fock states. [1] E. Knill, R. Laflamme, and G. Milburn, Nature 409, 46
Consider the case where both the control, target and (2001).
ancillaryphotonsaresuppliedbytwoindependentSPDC [2] T. C. Ralph, A. G. White, W. J. Munro, and G. J. Mil-
sources. The input state is given by (9) with the usual burn,Phys. Rev.A 65, 012314 (2002).
[3] H.F.HofmannandS.Takeuchi,Phys.Rev.A66,024308
qubit encoding as in equation (1). We will again get the
(2002).
three terms (10) possibly contributing to the error for
[4] T.C.Ralph,N.K.Langford,T.B.Bell,andA.G.White,
n=4. The last term again leads to output terms which Phys. Rev.A 65, 062324 (2002).
all get postselected out in the coincidence basis. Unfor- [5] K. Sanaka, K. Kawahara, and T. Kuga,
tunately the output terms produced by the second term quant-ph/0108001,2001.
5
[6] T.Pittman,B.C.Jacobs,andJ.D.Franson,Phys.Rev. [11] A. Beveratos et al.,Eur. Phys. D 18, 191 (2002).
A 64, 062311 (2001). [12] P. G. Kwiat et al.,Phys. Rev.A 60, R773 (1999).
[7] A.Imamo¨glu, Phys. Rev.Lett. 89, 163602 (2002). [13] A.G.White,D.F.V.James, P.H.Eberhard,andP.G.
[8] D.F.JamesandP.G.Kwiat,Phys.Rev.Lett.89,183601 Kwiat, Phys. Rev.Lett. 83, 3103 (1999).
(2002). [14] E. Knill, quant-ph/0110144, 2001.
[9] P.Michler et al.,Science 290, 2282 (2000).
[10] M. Pelton et al., quant-ph/0208054, 2002.