Loss Tolerant Optical Qubits

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by  T. C. Ralph

Loss Tolerant Optical Qubits
T. C. Ralph, A. J. F. Hayes, Alexei Gilchrist
Centre for Quantum Computer Technology, Department of Physics,
University of Queensland, QLD 4072, Brisbane, Australia. [email protected]
(Dated: February 1, 2008)
We present a linear optics quantum computation schemethat employsa new encoding approach
that incrementally adds qubitsand is tolerant to photon loss errors. The scheme employs a circuit
modelbutusestechniquesfrom clusterstatecomputation andachievescomparable resourceusage.
To illustrate ourtechniqueswe describe a quantum memory which is fault tolerant to photon loss.
PACSnumbers:
5
Quantum logic gates can be built using linear op- operates at the redundant-encoding level to protect in-
0
0 tics, photon detection and ancillary resources in a scal- formation from photon loss.
2 able manner, as shown by Knill, Laflamme and Milburn Physical encoding: At the first tier are the ba-
n (KLM) [1]. A number of experimental efforts are cur- sic physical states that we will use to construct qubits,
a rently focused on testing the building blocks of linear these will be the polarisation states of a photon so that
J optical quantum computing (LOQC) [2, 3, 4]. However, 0 H and 1 V . The advantage of this choice
1 optimism for large scale quantum computation based on i|nio≡pt|ics,i is th|ati w≡e|cain perform any single physical-
3
LOQChasbeentemperedbythemajoroverheadsinher- qubit unitary deterministically with passive linear opti-

entintheKLMschemeandthehighdetectorandsource cal elements. Of course gates between different physi-
1
v efficiencies apparently required [1]. cal qubits become difficult and in LOQC these are non-
4 An alternative approach to implementing LOQC was deterministic.
8 proposedbyNielsen[5]andfurtherdevelopedbyBrowne Parity encoding: at the second tier of encoding are
1
and Rudolph [6] (see also [7] for related work). This parity qubits encoded across many physical qubits. We
1
approach combines the model of cluster-state quantum shall use the notation ψ (n) to mean the logical state
0
| i
5 computation [8] with the non-deterministic gates pre- ψ of a qubit, which is parity encoded across n physical
| i
0 sentedbyKLM,andachievesaverysignificantreduction qubits. In this notation the physical qubits are the first
h/ in the overheads. The fault tolerance of the scheme has level,andwewilloftendropthe superscriptforthis level
p also been studied [9]. as was done above.
- In this paper we present a new approach to LOQC Specifically, the parity encoding is given by
t
n based on an incremental parity encoding [10]. Our
a method combines ideas from both the KLM and the 0 (n) (+ ⊗n+ ⊗n)/√2
u | i ≡ | i |−i
q cluster-stateapproaches. Parityencodingwasusedinthe 1 (n) (+ ⊗n ⊗n)/√2, (1)
| i ≡ | i −|−i
: originalKLMproposaltoprotectagainstbothteleporter
v
failures (i.e. the non-determinism of the gates)and pho- where = (0 1 )/√2. The main feature of this
i
X ton loss. By using parity encoding but re-encoding in- encodin|±giis tha|tia±co|miputational basis measurement of
r crementally (instead of by concatenation) we can obtain anyoneofthephysicalqubitswillnotdestroythelogical
a
the reduction in overheads characteristic of the cluster state,butratherwillreducethelevelofencodingbyone.
stateapproachwhilstretainingthephotonlosstolerance There are two operations which are easily performed
of KLM. on parity encoded states, one is a rotation by an arbi-
In particular we will describe a quantum memory trary amount around the x axis of the Bloch sphere (ie
which is fault tolerant [11] to photon loss. Though our X = cos(θ/2)I +isin(θ/2)X) [16], which can be per-
θ
techniques for detecting and correcting loss are them- formedbyapplyingthatoperationto anyofthe physical
selves themselves also subject to loss, above a particular qubits;andtheotherisaZ operation,whichcanbe per-
threshold efficiency the effect of loss can be negated to formed by applying Z to all the physical qubits (since
arbitrary accuracy. A previous description of an optical the odd-parity states will acquire an overall phase flip).
quantum memory based on errorcorrectiondid not con- Akeyoperationwewilluseis thepartialBellstatemea-
sider fault tolerance [12]. Although we specifically only surement [13, 14]. This consists of mixing two physical
consider memory, our construction is compatible with qubits on a polarising beam splitter followed by mea-
gate operations and thus can form a template for fault surement in the diagonal-antidiagonal basis. A success-
tolerant quantum computation with respect to photon ful event occurs when a photon is counted at each out
loss. We will deal with qubits in three different tiers of put of the beamsplitter. An unsuccessful event occurs
encoding: physical encoding, parity encoding andredun- when both photons appear at one of the outputs. When
dant encoding. Theapplicationwedescribeinthisletter successful it projects onto the Bell states 00 + 11 and
| i | i
2
00 11 . When unsuccessful it projects onto the sep- madeupofqn“new”photonsintroducedbytheresource
| i−| i
arable states 01 and 10 , thus measuring the qubits in and n k of the “old” photons that made up the par-
| i | i −
the computational basis. This operation can be used to ity qubit. By measuring the old photons in the compu-
add n physical qubits to a parity encoded state using a tational basis and making a bit flip (on all new parity
resourceof 0 (n+2). Wewillrefertothisastype-IIfusion qubits if needed) we obtain the expected encoded state
| i
(f ) following the nomenclature of Brown and Rudolph (Eq.3). The previous universal set of gates can be used
II
[6]. We will discuss the production of the required re- at this highest level of encoding also.
source states shortly. The result of type-II fusion is
Loss tolerant qubit memory: A schematic of the
ψ (m+n) (success) memory circuit for the example of a 2 qubit redundancy
fII|ψi(m)|0i(n+2) →(cid:26) ψ (|m−i1) 0 (n+1) (failure) (2) code is shown in Fig.1. The basic idea is as follows. The
| i | i logical qubit is held in memory for some time as shown,
When successful (with probability 1/2), the length of during which a photon may be lost. The logical qubit is
the parity qubit is extended by n. A phase flip correc- thentakenoutofmemoryandoneofitsconstituentpar-
tion may be necessary depending on the outcome of the ity qubits, P2, is sent into the encoder described above.
Bell-measurement. If unsuccessful a physical qubit is re- Theencoderperformstwotasks: (i)itaddsanotherlevel
moved from the parity encoded state, and the resource of redundancy encoding to the logical qubit and; (ii) it
state is left in the state 0 (n+1) (which may be recy- makes a quantum non-demolition measurement of the
| i
cled). This encoding procedure is equivalent to a gam- photon number of P2, which determines if a photon has
bling game wherewe either lose one levelofencoding,or been lost, without determining the logical value of the
gain n depending on the toss of a coin. Clearly if n 2 qubit. Fig.1(a) shows the procedure if no photons are
≥
this is a winning game. found to have been lost. The state straight after the en-
The remaining gates in order to achieve a universal coder is: ψ = α0 (n) 0 (n) 0 (n) +β 1 (n) 1 (n) 1 (n)
gate set (a Z and a cnot gate) can be efficiently per- The other|pairLity qu|biit1, P|1ii2s n|oiw3measu|reid1 in| it2he|diia3g-
90
formed on the parity encoded states by making use of onal basis 0 (n) 1 (n). This disentangles it from the
| i ±| i
the encoder above and will be described elsewhere [15]. other parity qubits which are returned to the state of
Theresourceoverheadforperforminggatesinthiswayis Eq.3 by the possible application of a phase-flip (depen-
approximately equal to the best quoted for cluster state dent on the outcome of the measurement on P1). They
encoding [6]. are returned to memory as shown.
Redundant encoding: The parity encoding has two
Fig.1(b) shows the procedure when the encoder finds
purposes. Firstly the non-deterministic gates which we
a photon missing in P2. Now the encoded state may
will employ, fail by measuring the qubit in the computa-
have suffered a bit flip and we may have the state:
tional basis. Hence this code enables recovery from gate (n) (n) (n) (n) (n) (n)
ψ = α0 1 1 +β 1 0 0 However,
failures. Secondly, loss of a photon can be considered | iL | i1 | i2 | i3 | i1 | i2 | i2
recovery is possible by now measuring the modes pro-
a computational basis measurement in which we did not
duced by the encoder in the diagonal basis. This dis-
findouttheanswer. Thusuponlossofaphotonweknow
entangles P1 from the other parity qubits without dis-
that the remaining state is at worst a bit flipped version
turbing its logical value. Importantly the correct P1 is
of the original. The final level of encoding is a redun-
obtained regardless of whether a bit flip occurredto P2,
dancy code which enables recovery from this possibility
though again a phase flip on P1 may be required de-
of a bit flip. Thus at the highest level our logical qubits
pendent on the outcome of the diagonal basis measure-
are given by:
ments. Finally, P1 is put through an encoder, sent back
ψ =α0 (n) 0 (n).....0 (n)+β 1 (n) 1 (n).....1 (n) (3) to memory and the sequence is repeated. This will cor-
| iL | i1 | i2 | iq | i1 | i2 | iq rect photon loss errors in which up to a single photon is
We can create an “encoder” gate that correctly en- lost per sequence. Higher levels of loss can be tolerated
codes a parity qubit by simply fusing a more com- by increasing the size of the redundancy code placed in
plicated resource state onto the parity qubit, namely memory and generalizing the protocol. For example a 3
0 0 (n) 0 (n).....0 (n) + 1 1 (n) 1 (n).....1 (n). We at- qubitcodecouldbekeptinmemoryand3qubitencoders
t|eim|pit1 t|ypie2-II fu|siioqn of|thi|isi1res|oui2rce on|tioq the parity used. Then two loss events could be tolerated with re-
qubit, ψ (n), repeating till successful (on average twice) covery from the third qubit. We will describe how the
giving|thie (phase flip corrected) result various operations required for the memory circuit can
beachievedusingonlylinearoptics,feedforwardandBell
α[0 (n−k) 0 (n).....0 (n)+ 1 (n−k) 1 (n)....1 (n)]+ state resources.
| i | i1 | iq | i | i1 | iq
β[1 (n−k) 0 (n).....0 (n)+ 0 (n−k) 1 (n).....1 (n)] (4) Threshold: Firstly consider the effect of photon loss
| i | i1 | iq | i | i1 | iq intheencoder. Ifalosseventoccursinthefusionprocess,
where 0 < k < n 1 is the number of unsuccessful at- that is, only one photon is detected when a fusion is
−
tempts made before fusion was achieved. This state is attempted, then the process is aborted. The presence of
3
takesinto accountfailureto decouple using the new par-
ity qubits also (measuring the components in diagonal
basis). That leaves the probability that a photon loss
occurs in the encoding of one parity qubit but that we
successfully disentangle it from the other parity qubit in
the redundancy code: P =1 P P .
Qf Qs ff
− −
Wecannowcalculatethethresholdforthememorycir-
cuit. Therearetwowaysinwhichthecircuitcansucceed.
FIG. 1: A schematic of memory circuit. First, one of the parity qubits can be encoded without
photon loss and then successfully disentangled from the
other. ThiswilloccurwithprobabilityP [1 (1 η )n].
Qs 1
− −
Secondly,aparityqubitcansufferphotonlossbutbesuc-
the redundancy code allows the following recovery. One
cessfully disentangled, where-upon another parity qubit
oftheremainingoldphotonsismeasuredinthediagonal
is successfully re-encoded. This will occur with proba-
basis. This disentangles the entire parity qubit onwhich
bility P P . Thus the probability of one successful
Qf Qs
theencoderwasattemptedfromtheotherparityqubitas
sequence of the memory circuit for q parity qubits is:
described earlier. If fusion is successful but a loss occurs
whilst measuring the old photons in the computational q−1
basis then measurement of any one of the remaining old P = Pj P [1 (1 η )n]q−1−j (7)
E Qf Qs − − 1
physical qubits (or indeed one from eachof the new pair Xj=0
ofencodedparityqubits)willdisentangletheotherparity
qubit which can then be re-encoded. Although for fixed n, limq→∞PE = 0 and for fixed q,
Theprobabilitythataparityqubitwillbesuccessfully limn→∞PE = 0 numerical investigations indicate that
it’s still possible to find n and q so that P approaches
encoded, without photon loss, is given by: E
one.
n−1 The optimal q can be found from d P =0 and using
PQs = (21η1η2)iη1n−i (5) this value numerically we find thatdPqEEapproaches one
Xi=1 for increasing n provided the threshold η >0.82 is satis-
fied. Forefficienciesaboveabout0.96apolynomialover-
where the size of the original parity qubit is n and
head in the code size results in an exponential decrease
the probability of detecting an old photon is given by
in the failure probability (1 P ). For lower efficien-
η1 = ηdηsηm, for a detector efficiency of ηd, a photon − E
cies the overheadis exponential. In figure 2 we show the
source efficiency of η and a memory efficiency of η .
s m
behaviour of P for optimal q as a function of η and n.
The probability of successfully detecting a new photon E
is given by η = η η . The photon source efficiency ap-
2 d s 20 nn
pears in the detection efficiency of an old photon be- 1155
causeaphotonmayhavebeenmissingfromthe resource 1100
state used in the previous encoding sequence. In ‘read- 55
ing’ these probabilities it pays to keep in mind that the 1
fusionprocesswillsucceedorfailwithprobabilityη1η2/2 0.75
and detect a photon loss with probability 1 η1η2. 0.5P
−
Now let us consider the case of complete (unrecover- 0.25
able) failure. This will occur if there is a sequence of fu- 0
sionfailures and photon loss events whichresult in allof 11
00..99
theparityqubitcomponentphotonsbeinglostwithouta 00..88
successfuldisentanglingoperationbeingcarriedout. The 0.7 Η
probability of this occurring is given by:
FIG. 2: PE for optimal q.
n−1
1
P = ( η η )j−1(1 η η )(1 η )n−j
ff 1 2 1 2 1
Xj=1 2 − − Resources: We now discuss the creation of the re-
source states used to implement our memory circuit and
n−2 n−2−j
1
+R ( η η )j+1 ηk(1 η )n−1−j−k hence the overheads needed. To this end we introduce a
Xj=0 2 1 2 Xk=0 1 − 1 secondoperation,thesinglerailpartialBellmeasurement
[1]. This is achieved by mixing one of the polarization
1
+( η1η2)n−1(1 η1) (6) modes from each of 2 physical qubits on a beamsplit-
2 −
ter and counting photons at the outputs. A successful
Where R = q q (1 η )kn[1 (1 η )n]q−k and event occurs when one and only one photon is counted,
k=1 k − 2 − − 2
P (cid:0) (cid:1)
4
otherwise it is unsuccessful. When successful it projects the productionofa 0 (3) onto whichtwo 0 (8) are fused
| i | i
onto single -railBellstates in whicha logicalzero is rep- using f . A simple recycling strategy leads to a cost
II
resented by the vacuum and a logical one by a single of approximately 1690 (2). This is not necessarily opti-
| i
photon state. In terms of dual rail qubits its effect is mal. Increasing the redundancy in the encoder resource
to project onto the states (H HH + V VV )/√2 or requires only a linear overhead, i.e. the resource state
| ih | | ih |
(H HH V VV )/√2whensuccessful,andmeasures for a q-fold redundancy encoder costs approximately
| ih |−| ih |
each qubit in the computational basis when it fails. We (q 1) 1690 (2). Increasingn similarly carriesa linear
− × | i
will refer to this operation as type-I fusion, (f ) [6]. overhead.
I
We willtake asourbasicresourcethe Bellstate 0 (2). Conclusion In this paper we have introduced optical
| i
Non-deterministic sources for such states are currently qubits with fault tolerance to loss under linear optical
available and considerable effort is being made to cre- manipulations. We numerically determine the threshold
ate deterministic, or at least heralded sources of these foranopticalmemorybasedonthesequbitstobe82%ef-
states. To create the state 0 (3), two 0 (2) can be fused ficiency. That is, in principle, for efficiencies higher than
| i | i
together using the fI gate. When successful, the 0 (3) this threshold, it is possible to find a suitable encoding
| i
state is produced, when unsuccessful, both Bell states such that the probability of a successful sequence of the
are destroyed. Since fI functions with a probability of quantum memory is arbitrarily close to 1. If we restrict
1/2,onaveragetwoattemptsarenecessary,soonaverage ourselves to two parity qubits each encoded across five
each 0 (3) consumes 40 (2). physical qubits, and ask only when our quantum mem-
| i | i
Once there is a supply of 0 (3) states, either fI or fII oryworkswithhigherprobabilitythanapassivememory,
| i
can be used to further build up the resource state via thenthe answeristhatthe efficienciesofthe sourcesand
detectors must exceed 96%.
0 (m+n−1) (success)
(H⊗H)fIH|0i(n)|0i(m) →(cid:26)| i (failure) The parity encoding we use was first introduced by
− KLM,howeverbyusingincrementalencodingtechniques
(8)
and the fusion technique we dramatically reduce the re-
andEq.2. Usingf withHadamardgateshastheadvan- sourceusage andincrease the thresholdoverthe original
I
tage of losing only a single qubit from the input states, scheme. Although we have only specifically discussed
butthedisadvantageofcompletelydestroyingtheencod- a quantum memory the techniques can be generalized
inginbothinputstatesintheeventoffailure. Usingf to include gate operations. We expect a number of the
II
to join the input states is at the expense of losing two techniques described here could also be useful in optical
of the initial qubits. There are two advantages to using quantuminformationprocessingwithnon-linearitiesand
f — firstly, in the case of failure, we do not destroy other quantum information platforms.
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the encoding so-far produced, just reduce this encoding We wouldlike to acknowledgehelpful discussions with
by one and; secondly, the operation is “fail-safe” in that Bill Munro and Stefan Scheel.
a detection loss event is immediately recognizable as a
failure (as 2 photons will not be counted) in contrast to
f where photon loss can lead to a false positive.
I
We can avoid the problem of the f failure mode in
I [1] E. Knill, R. Laflamme, and G. Milburn, Nature 409, 46
the following way. If f gives a false positive it means
I
(2001).
that the mode exiting the fusion gate does not contain
[2] T. B. Pittman, M. J. Fitch, B. C. Jacobs, and J. D.
a photon. Thus our supply of 0 (3) states each have
Franson, Phys.Rev.A p. 032316 (2003).
| i
one “suspect” mode which may be vacuum. We now [3] J. L. O‘Brien, G. J. Pryde, A. G. White, T. C. Ralph,
simply fuse two 0 (3) with fI to form a 0 (5) using the and D.Branning, Naturep. 264 (2003).
| i | i
suspect modes as the fusion point. We now are able to [4] S. Gasparoni, J.-W. Pan, P. Walther, T. Rudolph, and
produce a supply of 0 (5) states which again have one A. Zeilinger, Phys. Rev.Lett. p.020504 (2004).
suspect mode each. F|inially we use f to fuse two 0 (5) [5] M.A.Nielsen,Phys.Rev.Lett.p.040503(2004), quant-
II
| i ph/0402005.
toproducea 0 (8),onceagainusingthesuspectmodesas
| i [6] D.E.BrowneandT.Rudolph(2004),quant-ph/0405157.
the fusionpoint. Thisfinalfusioncannotgiveapositive [7] N. Yoran and B. Reznik, Phys. Rev. Lett. 91, 037903
if a photon had been lost in either of the previous fusion (2003).
events. In this way we can reliably produce the resource [8] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86,
state, 0 (8), regardless of detection efficiency. Of course 5188 (2001).
missing| iphotons due to finite source efficiency can still [9] M. A. Nielsen and C. M. Dawson (2004), quant-
ph/0405134.
occur and are accounted for by η .
s
[10] A. J. F. Hayes, A. Gilchrist, C. R. Myers, and T. C.
Using this approach and recycling f failures car-
II Ralph, J. Opt. B 6, 533 (2004).
ries an average cost of approximately 440 (2) per 0 (8),
[11] M. A. Nielsen and I. L. Chuang, Quantum Computation
| i | i
where we have assumed high detection and source effi- andQuantumInformation (CambridgeUniversityPress,
ciencies. Producing the encoder resource requires first Cambridge, 2000).
5
[12] R. M. Gingrich, P. Kok, H. Lee, F. Vatan, and J. P. ration.
Dowling, Phys.Rev.Lett. 91, 217901 (2003). [16] X, Y, and Z are the usual Pauli operators and an angle
[13] H.Weinfurter, Europhys.Lett. 25, 559 (1994). subscript denotes a rotation about that axis, analogous
[14] S.L. Braunstein and A. Mann, Phys.Rev. A 51, R1727 to Xθ definedin thetext.
(1995).
[15] A. Gilchrist, A. J. F. Hayes, and T. C. Ralph, in prepa-