Controlled-NOT for multiparticle qubits and topological quantum computation based on parity measurements

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by  Oded Zilberberg

Controlled-NOT for multiparticle qubits and topological quantum computation based
on parity measurements
Oded Zilberberg, Bernd Braunecker, and Daniel Loss
Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
(Dated: February 1, 2008)
We discuss a measurement-based implementation of a controlled-NOT (CNOT) quantum gate.
Such a gate has recently been discussed for free electron qubits. Here we extend this scheme for
8 qubits encoded in product states of two (or more) spins-1/2 or in equivalent systems. The key to
0 suchanextensionistofindafeasible qubit-paritymeter. Wepresentageneralschemeforreducing
0 this qubit-parity meter to a local spin-parity measurement performed on two spins, one from each
2 qubit. Twopossible realizations ofamultiparticleCNOTgatearefurtherdiscussed: electron spins
indoublequantumdotsinthesinglet-tripletencoding,andν =5/2Isingnon-Abeliananyonsusing
n
topological quantumcomputation braiding operations and nontopological charge measurements.
a
J
PACSnumbers: 03.67.Lx,73.21.La,05.30.Pr,85.35.Be
5
2

I. INTRODUCTION Many qubit-encoding schemes, however, encode a qubit
]
l in two states from a Hilbert space larger than the two-
l
a Single-quantum-bit(qubit)operationsandatwo-qubit dimensional spin-1/2 Hilbert space, specifically, from a
h gatethatgeneratesentanglementaresufficientforuniver- productHilbertspaceoftwo(ormore)two-levelsystems.
s- sal quantum computation [1]. One such two-qubit gate An example is the singlet-triplet (S−T0) encoding [13].
e is the controlled-NOT (CNOT) which flips the state of Forsimplicity,werefertothecomposingparticlesofthis
m a target qubit if the control qubit is in the logical 1 kind of qubit as spins-1/2, yet we emphasize that they
. state. A physical implementation of the CNOT ga|tei canhavevariousphysicalorigins. Suchencodingschemes
t
a typically requires a control of the interaction between result from system-dependent constraints, for instance,
m the qubits, e.g. for spin qubits see [2, 3, 4]. However, seeking a less noisy physical system as in the case of
- introducing interactionsbetweenqubits inevitably intro- electron spins in double quantum dots [13], or due to
d duces additional decoherence sources and is not possi- topological constraints in the case of ν = 5/2 Ising-type
n
ble in some quantum computation proposalssuch as, for anyons,wheretwoquasiparticlesformatwo-levelsystem
o
c instance, in linear-optics quantum computation due to equivalent to a spin-1/2 [14].
[ the fact that photons interact in a negligible way. How- In this paper we discuss a qubit-parity measurement-

ever, Knill, Laflamme, and Milburn (KLM) have shown basedimplementationofaCNOTgateforsuchmultipar-
2
that measurements rather than interactions can provide ticle qubits. The implementation is a direct extension of
v
2 the means to implement a CNOT gate on photons us- the schemes proposed in [5, 10]. The key to such an ex-
6 ing nonunitary operations [5]. Shortly thereafter, addi- tension is to find a feasible qubit-parity measurement.
0 tionalmeasurement-basedapproachesfor quantumcom- Wepresentageneralschemetoreducethismeasurement
1 putation were proposed [6, 7]. to a local spin-parity measurement of a representative
8. The KLMmodel indeed servedas a stepping stone for spin from each qubit. For concreteness, we specifically
0 coherent quantum information processing, but was re- discuss qubits based on the S T0 basis and present for
−
7 strictedtotheunderlyingphysicalsystem,relyingonthe this case a proof of the linearity of the measurement-
0 bosonicpropertiesofphotons. Attemptstodesignasim- based CNOT gate operation. The linearity proof is re-
:
v ilar implementation for fermionic systems encountered quired due to the nonlinear nature of the measurement-
i some difficulties in the form of a no-go theorem [8, 9], based implementation of the gate and can be used simi-
X
which showed that for fermions, single-electron Hamil- larlyforthecaseofRef. [10]. We alsoproposeapossible
ar tonians and single-spin measurements are simulated effi- realization of such a S −T0 CNOT on double quantum
cientlybyclassicalmeans. Thisno-gotheorem,however, dots using a recently proposed spin-parity meter [11].
was sidestepped recently in a work by Beenakker et al. Forν =5/2Ising-typeanyons,ameterequivalenttoa
[10]. Bytakingadvantageoftheadditionalchargedegree spin-parity meter involves measuring the charge of four
offreedomofanelectron,atwo-spinparitymeasurement quasiparticles. Such measurements have been recently
was proposed. Using this parity meter, a measurement- proposed [15, 16, 17, 18, 19] and first steps toward their
based CNOT gate for free “flying” electrons was de- implementationhavebeenpresented[20,21]. InRef. [14]
signed. Followingthisresult,implementationsofaparity this type of parity measurement was invoked alongside
gate for spin [11] and charge qubits [12] have been pro- topologicalbraidingoperationstoimplementatwo-qubit
posed. entangling gate. We use this parity meter to construct
The setup in Ref. [10] was proposed for qubits en- the measurement-based CNOT gate for this system. A
coded in the spin states of free electrons, i.e. the elec- comparison to the scheme of Ref. [14] shows the follow-
tronspinup(down)isinterpretedasalogical1(0)state. ing differences: The present scheme requires only local
2
braiding between the anyonscomposing a qubit but also
a Hˆ Hˆ c
additionalanyonsfor anancilla and anadditionalparity 1 1
ˆ
measurement. The scheme in Ref. [14] is thus more effi- P
s
cientinanyonresourcesandusesoneparitymeasurement
less, but it requires long-ranged anyon braiding opera- b Hˆ Hˆ d
2 2
tions between qubits, which will be experimentally chal-
lenging.
The paper is structured as follows: In Sec. II we Figure 1: A gate that uses a spin-parity measurement to
present the scheme for the qubit-parity meter using a measure theparity of S−T0 qubits. A pair of S−T0 qubits
entersthe gate in arms a and b. Each of the qubitsis ro-
representative spin-parity measurement. The scheme is
tated by a Hadamard gate Hˆ. The spin parity of the left
presentedfullyfortheS T0qubitencoding. Wethenex-
− spins (see Eq. (1)) from each qubit is then measured in the
tendtheresultofRef. [10]andpresentthemeasurement-
based CNOT setup using the S−T0 qubit-parity meter. aPˆnsdbtohxe. pTahreityquobfittshearsepirnostaitseedqubiavcaklebnytHtoadthaempaardritgyatoefsthe
In Sec. III we discuss possible implementations of the
qubits.
CNOT scheme, focusing on two physically entirely dif-
ferent systems: double quantum dots and ν =5/2 Ising-
type anyons. In the Appendix we prove the linearity of
operation
the measurement-based CNOT gate.
Pˆ =Hˆ1Hˆ2PˆsHˆ1Hˆ2, (2)
II. SCHEME FOR QUBIT-PARITY where Hˆ1,Hˆ2 are the Hadamard gates operating on
MEASUREMENT AND CNOT qubits 1 and2, andPˆs measuresthe spin paritybetween
the twoleftspinsofqubits1and2. Asketchofthis gate
is shown in Fig. 1.
Inordertoextendthemeasurement-basedCNOTgate
As an example for the operation of Pˆ, let ψ =
proposed in Ref. [10] to a multiparticle qubit encoding, | i
one must find a way to measure the qubit parity of two |T0i1 ⊗ (α|T0i2 + β|Si2) be a two-qubit state. Once
rotated by Hadamard gates the state becomes ψ˜ =
such qubits. We propose a general scheme in which the
| i
(α +β ). Measuringthespinparityofthe
qubits are rotated to “witness” states such that a rep- 1 2 2
|↑↓i ⊗ |↑↓i |↓↑i
resentative spin parity measurement demonstrates their left spins in each qubit results in ψ˜ 1 = 1 2
qubit parity. We illustrate this scheme on a specific two- if even spin parity is measured (p|s{=}i1) a|n↑d↓i ψ˜⊗0|↑↓i=
spin singlet-triplet qubit encoding where two selected ifoddspinparityismeasured(p =| 0{).}iRo-
1 2 s
|↑↓i ⊗|↓↑i
Bell states serve as the qubit’s logical state, i.e. 0 = tatingthequbits byHadamardgatesagainresultsinthe
| i
T0 = ( + )/√2, 1 = S = ( )/√2. projected qubit states with a qubit parity equivalent to
| i |↑↓i |↓↑i | i | i |↑↓i−|↓↑i
This is an encoding scheme used for electron spins in the measured spin parity.
double-quantum-dot setups [13, 22, 23, 24, 25, 26, 27]. The factthatHadamardgatesrotateto witness states
We show that a spin-parity measurementis sufficient for and back in this S T0 encoding results from the fact
−
a S T0 parity meter and detail the CNOT implemen- that the computational states are a superposition of the
−
tation. two product spin states , with equal ampli-
{|↑↓i |↓↑i}
In order to demonstrate the equivalence between spin tudes. Thus, for an x-aligned single-spin qubit encoding
parity and S T0 qubit parity, we rotate the qubit = ( )/1/√2 the same qubit parity routine as
− |±i |↑i±|↓i
states to witness states over which a spin parity mea- shown in Fig. 1 is valid. The difference between the
surement will make the distinction of qubit parity. An above setup and a parity setup for other types of encod-
important building block in this scheme is the single- ingliesinthesingle-qubitrotationthatrotatesthequbits
qubit Hadamard gate Hˆ. Applied to the computational to the witness states and back, i.e. the Hadamard gates
basis states 0 , 1 , it has the matrix representation in the entrance and exit of the parity gate are replaced
√12(11 −11), an{|diit|yiie}lds for the S−T0 encoding bdyitidoniffaelrlye,notnreotmatuisotnsnoftoertohtahtersoemnceoodfinsgucshchreomtaetsio.nAs dto-
Hˆ T0 = , witness states might require the qubit to leave the com-
| i |↑↓i putational subspace. For example, if one uses a unitary
Hˆ S = . (1) two-spinrotationthat maps S , T0 , the
| i |↓↑i | i→|↓↓i | i→|↑↑i
target states are outside the qubit encoding subspace.
Therefore,theleftspinintheright-handsideofEq. (1) However, they can still be used as witness states for the
canserveasawitnessfortheoriginaltwo-spinstate. For qubit-parity measurement. In this case it is, of course,
example, if the left spin is in the state, the original requiredthatthoseadditionalstatesareenergeticallyac-
|↑i
prerotated state was a T0 . Hence, the spin parity of cessible from the computational subspace.
| i
the left spins of two rotated S T0 qubits indicates the We can now extend the result of Ref. [10], where
qubit parity. If Pˆ is a spin-par−ity gate (as used in Refs. spinparityisusedtoimplementaCNOToperationona
s
[10, 28]) we obtain a S T0 qubit-parity gate from the single-spinqubit, by using the S T0 parity in the same
− −
3
cin Hˆ Hˆ σˆc cout III. POSSIBLE IMPLEMENTATIONS
ˆ
P
ain=|T0i s,1 Hˆ maeaosuutred We present here two possible implementations of the
Pˆ CNOTgatefortwotypesofsystemsthathavebeenpro-
s,2
tin σˆt tout posed for quantum computation. In the first part we
discuss how it may be realized on double-quantum-dot
qubits. In the second part we consider an implemen-
Figure 2: Measurement-based CNOT gate for S−T0 qubits. tation for non-Abelian Ising-type anyons that have been
The boxes represent spin-parity measurements of the left proposedtoexistaselementaryexcitationsinafractional
spins (see Eq. (1)) of each qubit. Three Hadamard gates quantum Hall system with filling factor ν =5/2.
rotate the qubitsenteringand leaving thefirst box. The in-
put of the CNOTgate consists of control and target qubits
plus an ancilla which is prepared in the |T0i state. The an- A. Double quantum dots
cilla is measured at theoutput in a |Si or |T0i state. The
outcome of thismeasurement plus thetwo measured spin
Since the introduction of electron spins in quantum
parities determinewhich operators σˆc,σˆt one has to apply
on thecontrol and target qubits,respectively, in order to dots (QDs) as a platform for quantum information pro-
complete the CNOToperation: Weapply on thecontrol cessing[2],therehasbeenmuchresearchinthisdirection.
qubit σˆc =σˆz if p2 =0 and σˆc =1 if p2 =1. For thetarget Severalproposalsspecifically focus onaS T0 qubiten-
−
qubit,σˆt = σˆx if p1 = 1 and theancilla is measured in the coding where two electrons in neighboringQDs form the
|Si state, or if p1 =0 and theancilla is measured in the |T0i S and T0 states. Possible implementations of single-
state. Otherwise, σˆt =1. See[10]. q|uibitope|ratiionsaswellasaCNOTgatebasedoncontrol
ofthedesignandtheinteractionsinthesystemhavebeen
discussedinthelastfewyears[13, 22,23,24,25, 26,27].
InFig. 4weshowthatthemeasurement-basedCNOT
way for a S T0 CNOT implementation. The result- gate can be realized in such systems as well. The spin
−
ing gate is shown in Fig. 2. The gate can be seen as parity can be measured using a recently proposed spin-
a Hadamard-rotated version of the gate from Ref. [10] parity meter [11]. This meter, however,is local and can-
that operates on two-spin qubits instead of free flying not measure spins in distant QDs, i.e. if we label the
electron qubits. In addition, the gate has the following electrons by 1,2 (first qubit) and 3,4 (second qubit), the
advantages over the gate from Ref. [10]: (1) The an- required witness parity of spins 1 and 3 cannot be mea-
cilla is prepared in a pure computational state instead sured. From Eq. (1) we see, however, that measuring
ofasuperpositionofcomputationalstates,and(2)fewer even parity between spins 2 and 3 is the same as mea-
Hadamard operations are required. suring odd parity between spins 1 and 3, and vice versa.
Upon this reinterpretation, the CNOT gate remains un-
The parity and ancilla measurements (see Fig. 2)
changed. With the suggested geometric arrangement of
are projective nonlinear operations. Each measurement
QDs shown in Fig. 4, it may further be possible that a
projects the state onto one of two possible outcome
single spin-parity meter, coupling alternately to the left
states. In Fig. 3 we present the “calculation tree” of
or right QD of the ancilla, is sufficient for the operation.
the CNOT gate where the three consecutive measure-
ments lead to eight possible outcome states. With the
last tuning step of the gate, however, we obtain a single
deterministic result, i.e. all branches have the outcome
B. Ising-type anyons
(up to a global phase),
Topological quantum computation (TQC) [29, 30, 31]
|ψic⊗|ψit →αγ|T0ic|T0it+αδ|T0ic|Sit+ ptiroonpiossedsonaescbhyetmoepoinlowgihcaiclhopcoehraetrieonntsqpuearnfoturmmedcoomnpnuotna--
βγ|Sic|Sit+βδ|Sic|T0it, (3) Abelian anyons. A physical system that may serve as
a platform for TQC is the two-dimensional electron gas
inthefractionalquantumHallregime. Atfillingfraction
where |ψic = α|T0ic+β|Sic and |ψit = γ|T0it+δ|Sit ν =5/2,localizedelementaryexcitations(quasiparticles)
are the control and target input states, respectively.
are proposed to have non-Abelian anyon statistics and
Hence Eq. (3) describes the operation of a CNOT gate
are dubbed Ising anyons [32, 33, 34].
on S T0 qubits.
− Twosuchquasiparticlesformatwo-levelsystemequiv-
In the Appendix we follow the calculation tree in Fig. alenttoaspin-1/2. However,inRef. [14]itisshownthat
3 when the gate in Fig. 2 is applied to an arbitrary two- due to topological superselection rules the qubit is en-
qubit state. The result yields Eq. (3) and proves that coded in two product states 0 = 0,0 , 1 = 1,1 from
| i | i | i | i
the gateisindeedaCNOTgateandthatitsoperationis the Hilbert space formed by four quasiparticles. Thus,
linear. this system forms a Hilbert space equivalent to that of a
4
Hˆc Pˆ1 Hˆc Hˆa Pˆ2 Mˆa σˆc σˆt
z=1 |ψ1,1,1i
|ψ1,1,zi
˜ p2=1 z=0 |ψ1,1,0i
|ψ1,p2,zi |ψ1,p2,zi
p2=0 z=1 |ψ1,0,1i
|ψ1,0,zi
p1=1 z=0 |ψ1,0,0i
˜
|ψii |ψi |ψfi
p1=0 z=1 |ψ0,1,1i
|ψ0,1,zi
p2=1 z=0 |ψ0,1,0i
˜
|ψ0,p2,zi |ψ0,p2,zi p2=0 z=1 |ψ0,0,1i
|ψ0,0,zi
z=0 |ψ0,0,0i
Figure 3: Calculation tree of themeasurement-based CNOT gate in Fig. 2. The computation splits in accordance with the
parity and ancilla measurements. Dueto theHadamard gate rotations, the measurements donot destroy the initial state and
each path of the computation has thesame probability of occurring. Weobtain eight possible result states which we denote
as |ψp1,p2,zi. In the Appendixwe follow theexecution of thegate and show that theresults in each of thecalculation arms
indeed merge intoa single result |ψ i which is equalto theresult of theCNOT operation.
f
product Hilbert space of two spins-1/2 [37]. In addition, iHˆ = eiπ4σˆzeiπ4σˆxeiπ4σˆz, the operation shown in Fig. 5
itisshowninRef. [14]thatinordertoimplementuniver- performs a Hadamard rotation on such a qubit (up to a
salquantumcomputationonthis system,nontopological global phase) [35, 36].
parity-like measurements are required. Such measure-
ments may be carried out by an interferometric device
Figure6presentsthemeasurement-basedCNOToper-
recently proposed in [15, 16, 17, 18, 19] and first steps
ationonthis system. The paritymeasurementofanyons
have been taken toward its implementation [20, 21]. We
3,4,5,6 gives the parity of the control and ancilla qubits.
refer to these references for more details.
Theparitymeasurementofanyons7,8,9,10givesthepar-
The measurement-based CNOT scheme can be imple-
ity of the ancilla and target qubits.
mented over this system as well. The parity meter here
acts directly on the computational states so that no ro-
tationpriorto the paritymeasurementis required. If we Upon comparing the CNOT operation in Fig. 6 to
labelthe anyonsformingthe firstqubit1,2,3,4andthose the two-qubit gate proposed in Ref. [14], we see that
of the second qubit 5,6,7,8, the parity of two qubits can the present measurement-based scheme requires an an-
be measured by an interferometer which measures the cilla and an additional parity measurement that are not
charge of the four adjacent anyons 3,4,5,6. This mea- needed in [14]. This is due to the fact that the present
surement is equivalent to the spin-parity measurement scheme does not take advantageof the underlying anyon
of two neighboring spins, one from each qubit, as dis- system statistics. Using the anyon statistics as in Ref.
cussed in Sec. IIIA. The required Hadamard rotations [14], however, requires braiding of distant anyons be-
bythemeasurement-basedCNOTscheme[10]canbeim- tween qubits, while in the present scheme all braiding
plemented using topologicalbraiding of the Ising anyons operationsarestrictlylocal,i.e. werequireonlybraiding
[35, 36]. If we consider the qubit formed by the anyons operations of nearest-neighbor quasiparticles. Further-
1,2,3,4,thebraidingofanyons1,2resultsinaeiπ4σˆz qubit more, we braid the anyons only within the qubit they
rotation and braiding of 2,3 results in eiπ4σˆx [14]. Since define.
5
Control Ancilla Target
1 2 3 4 5 6 7 8 9 10 11 12
ˆ
P
1
T
im
ˆ e
P
2
Figure 4: Double-quantum-dotimplementation setup for the
measurement-based CNOTgate. A dot with an electron in
it is represented by an emptycircle containing a filled circle.
The ancilla dots are situated next to a spin-paritymeter
ˆ
M
(proposed in Ref. [11]). In order to measure theparity of a
theancilla and control qubits,P1, thespin parity of the
right electron spin of thecontrol and theleft electron spin
of the ancilla is measured. The parity of theancilla and Figure 6: Measurement-based CNOT gate implemented on
target qubits, P2, is measured by thespin parity of theright ν =5/2 Ising anyon qubits. The control, ancilla, and target
electron spin of the ancilla and theleft electron spin of the qubitsare shown from left to right, e.g. thecontrol qubitis
target. represented by anyons1,2,3,4. The representative “spin”-
parity measurements are shown by the Pˆ boxesand thean-
cilla measurement bythe box at thebottom. The braiding
1 2 3 4
between the measurements represents Hadamard rotations
on the qubits.
T
iHˆ = im S −T0 setup to provide a proof of the linearity of the
e gate (see Fig. 3), which is required as the CNOT is im-
plemented by nonunitary operations. As an illustration,
wepresentedtwopossibleimplementationsoftheCNOT
gate. We have proposed a possible setup for the S T0
−
encoding(seeFig. 4). Forν =5/2Ising-typeanyons,the
CNOTgatecanbeimplementedwithbraidingoperations
and the parity meter proposed in [14] (see Fig. 6). In
Figure 5: Hadamard gate using braiding of ν = 5/2 Ising
contrast to a similar gate described in [14], the present
anyons[35, 36]. The gate adds a global π/2 phase which
can beignored. CNOT requires one more parity measurement and the
additional ancilla. But all braiding operations remain
strictly local and confined within the individual qubits.
Bothschemes have their strengths but it is yet unknown
IV. CONCLUSION
which is a more efficient route for implementation.
We have presented a general scheme to measure the
qubit parity of two multiparticle qubits via a represen-
tative spin-parity measurement in some rotated state.
Using this qubit-parity meter we have extended the
Acknowledgments
measurement-based CNOT setup proposed in Ref. [10]
to additional encoding schemes. As an example, we dis-
cussed the S T0 qubit encoding case in detail. In this WethankW.A.Coish,S.Bravyi,L.Chirolli,D.Stepa-
−
encoding, as shown in Fig. 2, the rotations used by the nenko, and D. Zumbu¨hl for useful discussions. Financial
qubit-paritymeterledtoaslightlysimplerrotatedsetup supportbytheNCCRNanoscienceandtheSwissNSFis
ofthe CNOTgateascomparedto[10]. We alsousedthe acknowledged.
6
Appendix A: PROOF OF LINEARITY Measuring the spin parity of the left spins of the an-
cillaandtargetqubitsagainsplitsthe resultsetintotwo
To prove the linearity of the measurement-based possible branches:
CNOT gate shown in Fig. 2, we follow the gate exe-
αγ
ctaurtgioentqtuhbatitsisapreorttarkaeynediniintiaFlilgy.to3bwehiennatrhbeitcroanrytrsotlaatensd: |ψ1,1,zi=√2[|T0ic|↑↓ia|↑↓it+|T0ic|↓↑ia|↓↑it]+
αδ
|ci=α|0i+β|1i=α|T0i+β|Si (A1) √2[|T0ic|↑↓ia|↑↓it−|T0ic|↓↑ia|↓↑it]+
t =γ 0 +δ 1 =γ T0 +δ S (A2) βγ
| i | i | i | i | i
[S S ]+
The initialstate ofthe inputqubits plus the ancillais: √2 | ic|↑↓ia|↑↓it−| ic|↓↑ia|↓↑it
βδ
[S + S ], (A12)
|ψii=|ci⊗|ai⊗|ti. (A3) √2 | ic|↑↓ia|↑↓it | ic|↓↑ia|↓↑it
The calculation splits in accordance with the parity
and ancilla measurements. We obtain eight optional re-
αγ
s3u.ltWsetaptersovwehtihchatwtehedernesouteltsasin|ψepa1c,ph2,ozfi tahseseceanlcuinlatFiiogn. |ψ1,0,zi=√2[|T0ic|↑↓ia|↓↑it+|T0ic|↓↑ia|↑↓it]−
armsfinallymergeintoasingleresult ψ whichisequal αδ
f
to the result of the CNOT operation.| i √2[|T0ic|↑↓ia|↓↑it−|T0ic|↓↑ia|↑↓it]+
Atthefirststep,thecontrolispassedthroughaS T0 βγ
−
Hadamard gate, resulting in [S S ]
√2 | ic|↑↓ia|↓↑it−| ic|↓↑ia|↑↓it −
|ψ˜i= h√α2(|T0ic+|Sic)+ √β2(|T0ic−|Sic)i⊗ √βδ2[|Sic|↑↓ia|↓↑it+|Sic|↓↑ia|↑↓it], (A13)
a t . (A4)
| i⊗| i
The firstparitymeasurementis performedonthe con-
αγ
trolandancillaqubits. To presentthe resultofthe spin-
paritymeasurement,wefirstwritethecontrolandancilla |ψ0,1,zi=√2[|T0ic|↑↓ia|↑↓it−|T0ic|↓↑ia|↓↑it]+
in the product spin basis , , αδ
{|↑↓i |↓↑i} √2[|T0ic|↑↓ia|↑↓it+|T0ic|↓↑ia|↓↑it]+
+
ψ˜ =[α +β ] |↑↓ia |↓↑ia t . (A5) βγ
| i |↑↓ic |↓↑ic ⊗ √2 ⊗| i [S + S ]+
√2 | ic|↑↓ia|↑↓it | ic|↓↑ia|↓↑it
Measurementof the spin parity of the left spins of the βδ
[S S ], (A14)
control and ancilla qubits has two possible outcomes: √2 | ic|↑↓ia|↑↓it−| ic|↓↑ia|↓↑it
|ψ˜1,p2,zi=[α|↑↓ic|↑↓ia+β|↓↑ic|↓↑ia)]⊗|ti, (A6)
|ψ˜0,p2,zi=[α|↑↓ic|↓↑ia+β|↓↑ic|↑↓ia]⊗|ti. (A7) |ψ0,0,zi=√αγ2[|T0ic|↑↓ia|↓↑it−|T0ic|↓↑ia|↑↓it]−
The ancilla and control qubits are then rotated by
Hadamard gates: αδ
√2[|T0ic|↑↓ia|↓↑it+|T0ic|↓↑ia|↑↓it]+
|ψ1,p2,zi=[α|T0ic|T0ia+β|Sic|Sia]⊗|ti, (A8) βγ
[S + S ]
|ψ0,p2,zi=[α|T0ic|Sia+β|Sic|T0ia]⊗|ti. (A9) √2 | ic|↑↓ia|↓↑it | ic|↓↑ia|↑↓it −
βδ
Now the ancilla and target qubits enter a spin-parity
[S S ]. (A15)
measurement. Once more we write their states in the √2 | ic|↑↓ia|↓↑it−| ic|↓↑ia|↑↓it
product spin basis:
The ancilla and target qubits can be rewritten in the
+
|ψ1,p2,zi=(cid:20)α|T0ic |↑↓ia√2|↓↑ia +β|Sic |↑↓ia√−2|↓↑ia(cid:21)⊗ S−T0 basis using the following relations:
+ 1
γ|↑↓it |↓↑it +δ|↑↓it−|↓↑it , (A10) = [T0 + S ]
(cid:20) √2 √2 (cid:21) |↑↓i √2 | i | i
+ 1
|ψ0,p2,zi=(cid:20)α|T0ic |↑↓ia√−2|↓↑ia +β|Sic |↑↓ia√2|↓↑ia(cid:21)⊗ |↓↑i= √2[|T0i−|Si] (A16)
+
γ|↑↓it |↓↑it +δ|↑↓it−|↓↑it . (A11)
(cid:20) √2 √2 (cid:21) Substituting Eq. (A16) into Eqs. (A12)-(A15) and
7
rewriting the expressions we obtain: |ψ1,1,0i=αγ|T0ic|T0it+αδ|T0ic|Sit+
|ψ1,1,zi=√αγ2[|T0ic|T0ia|T0it+|T0ic|Sia|Sit]+ βγ|Sic|Sit+βδ|Sic|T0it, (A22)
αδ
√2[|T0ic|Sia|T0it+|T0ic|T0ia|Sit]+ |ψ1,0,1i=−αγ|T0ic|Sit−αδ|T0ic|T0it+
√βγ2[|Sic|Sia|T0it+|Sic|T0ia|Sit]+ βγ|Sic|T0it+βδ|Sic|Sit, (A23)
βδ
√2[|Sic|T0ia|T0it+|Sic|Sia|Sit], (A17) |ψ1,0,0i=αγ|T0ic|T0it+αδ|T0ic|Sit−
βγ|Sic|Sit−βδ|Sic|T0it, (A24)
|ψ1,0,zi=√αγ2[|T0ic|T0ia|T0it−|T0ic|Sia|Sit]+ |ψ0,1,1i=αβγγ||ST0iicc|S|Ti0ti+t+βδα|δS|iTc0|iTc0|iSti,t+ (A25)
αδ
√2[|T0ic|T0ia|Sit−|T0ic|Sia|T0it]+
βγ
√2[|Sic|Sia|T0it−|Sic|T0ia|Sit]+ |ψ0,1,0i=αγ|T0ic|Sit+αδ|T0ic|T0it+
βδ βγ|Sic|T0it+βδ|Sic|Sit, (A26)
√2[|Sic|Sia|Sit−|Sic|T0ia|T0it], (A18)
αγ |ψ0,0,1i=αγ|T0ic|T0it+αδ|T0ic|Sit−
|ψ0,1,zi=√2[|T0ic|Sia|T0it+|T0ic|T0ia|Sit]+ βγ|Sic|Sit−βδ|Sic|T0it, (A27)
αδ
√2[|T0ic|T0ia|T0it+|T0ic|Sia|Sit]+
βγ |ψ0,0,0i=−αγ|T0ic|Sit−αδ|T0ic|T0it+
√2[|Sic|T0ia|T0it+|Sic|Sia|Sit]+ βγ|Sic|T0it+βδ|Sic|Sit. (A28)
βδ
√2[|Sic|Sia|T0it+|Sic|T0ia|Sit], (A19) Applying the gates σˆc,σˆt on the control and target
qubits in accordance with the results of the parity and
ancillameasurements(asshowninFig. 2)givesthesame
αγ
|ψ0,0,zi=√2[|T0ic|Sia|T0it−|T0ic|T0ia|Sit]+ spthaatsee)in: all eight computation branches (up to a global
αδ
√2[|T0ic|Sia|Sit−|T0ic|T0ia|T0it]+ |ψfi= αγ|T0ic|T0it+αδ|T0ic|Sit+
βγ
√2[|Sic|T0ia|T0it−|Sic|Sia|Sit]+ βγ|Sic|Sit+βδ|Sic|T0it. (A29)
βδ
√2[|Sic|T0ia|Sit−|Sic|Sia|T0it]. (A20) ofUthnedCerNtOheTqugabtite:encodingthisstateisindeedtheresult
Measuring the ancilla in a singlet or triplet results in
the final eight states: |ψfi= αγ|0ic|0it+αδ|0ic|1it+
βγ 1 1 +βδ 1 0 . (A30)
|ψ1,1,1i=αγ|T0ic|Sit+αδ|T0ic|T0it+ | ic| it | ic| it
βγ|Sic|T0it+βδ|Sic|Sit, (A21) This proves the linearity of the gate.
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