Optical Zeno Gate: Bounds for Fault Tolerant Operation
Optical Zeno Gate: Bounds for Fault Tolerant Operation
Patrick M. Leung∗ and Timothy C. Ralph
Centre for Quantum Computer Technology, Department of Physics,
University of Queensland, Brisbane 4072, Australia
(Dated: September 8, 2021)
In principle the Zeno effect controlled-sign gate of Franson et al’s (PRA 70, 062302, 2004) is a
deterministic two-qubit optical gate. However, when realistic values of photon loss are considered its
fidelity is significantly reduced. Here we consider the use of measurement based quantum processing
techniques to enhance the operation of the Zeno gate. With the help of quantum teleportation, we
show that it is possible to achieve a Zeno CNOT gate (GC-Zeno gate) that gives (near) unit fidelity
and moderate probability of success of 0.76 with a one-photon to two-photon transmission ratio
κ = 104. We include some mode-mismatch effects and estimate the bounds on the mode overlap
and κ for which fault tolerant operation would be possible.
PACS numbers: 03.67.Lx, 42.50.-p
I. INTRODUCTION
Photons are a natural choice for making qubits be-
cause the quantum information encoded can have a long
decoherence time and is easy to manipulate and mea-
sure. Also, photonic qubits are the only type of qubits
that are feasible for long distance quantum communica-
tion. Quantum information processing requires universal
two-qubit entangling gates. Knill et al [1] showed that
it is theoretically possible to do scalable quantum com-
puting with linear optics by using measurement induced
interactions to perform the two-qubit gates. However, de-
spite a continuous effort in reducing the resource require-
ment [2, 3, 4, 5], the resource overhead is still high for
linear optical quantum computing. Franson et al[6] pro-
posed the use of two-photon absorption non-linearity and
exploiting the quantum Zeno effect to implement a con-
trol sign (CZ) gate that requires much less resources than
linear optics schemes. However, the problem with the
Zeno gate is that photon loss affects the performance of
the Zeno gate significantly and the single photon to two-
photon loss ratio requirement is very stringent. One solu-
tion couldbe to combine measurement induced quantum
processing with the Zeno gate. Previously we have shown
that when using the Zeno gate for qubit fusion [8], state
distillation [9] with post-selection can boost the gate fi-
delity to unity and that for less stringent absorption ra-
tios the gate has an advantage in success probability over
linear optics gates for fusing clusters of qubits. These
clusters of qubits can then be used for cluster state quan-
tum computing [10]. In addition, Myers and Gilchrist [7]
have shown that the performance of the Zeno gate may
be enhanced by using error correction codes such as re-
dundancy and parity encoding.
Here we design a high fidelity Zeno CNOT gate
suitable for circuit-based quantum computing. Although
with the current estimate of the photon loss ratio, only
a poor fidelity Zeno gate is directly achievable, here we
show that it is possible to use two of these Zeno gates
to do Bell measurements and implement a Gottesman-
Chuang[11] teleportation type of CNOT gate (GC-Zeno
gate) that, like the fusion gate, gives high fidelity via
state distillation and moderate success probability via
partially off-line state preparation. We include the effect
of mode-mismatch and detector efficiency on the scheme
and estimate lower bounds on the parameter which in
principle allow fault tolerant operation.
The paper is arranged in the following way. The in-
troduction continues with a subsection on the Zeno CZ
gate, which describes the scheme and modelling of the
gate and give descriptions on the modelling parameters
that are also used for modelling the GC-Zeno CZ gate.
In section II, we discuss the GC-Zeno gate and the effect
of mode-mismatch and detector efficiency on the gate. In
section III, we give estimates of the lower bounds on the
photon loss ratio and mode-matching, followed by a sub-
section on the advantage in using state distillation. We
conclude and summarize in section IV.
Ia. Zeno CZ Gate
Franson et al’s control sign gate scheme consists of a
pair of optical fibers weakly evanescently coupled and
doped with two-photon absorbing atoms. The purpose of
the two-photon absorbers is to suppress the occurrence
of two photon state components in the two fibre modes
via the Quantum Zeno effect. This allows the state to
remain in the computational basis. After a length of
fibre corresponding to a complete swap of the two fibre
modes, a π phase difference is produced between the |11〉
term and the other three basis terms. After swapping
the fibre modes by simply crossing them, a CZ gate
is achieved. The gate becomes near deterministic and
performs a near unitary operation when the Quantum
Zeno effect is strong and photon loss is insignificant.
However, with current technology, the strength of the
http://arxiv.org/abs/0704.1069v1
Quantum Zeno effect is a few orders of magnitude below
what is required, and thus the Zeno gate has significant
photon loss.
In [8], the gate is modelled as a succession of n weak
beamsplitters followed by two-photon absorbers as shown
in Fig. 1. As n → ∞ the model tends to the continu-
ous coupling limit envisaged for the physical realization.
The gate operates on the single-rail encoding for which
|0〉L = |0〉 and |1〉L = |1〉 with the kets representing pho-
ton Fock states. Fig. 2 shows how the single rail CZ can
be converted into a dual rail [13] CZ with logical encoding
|0〉L = |H〉 and |1〉L = |V 〉. Let L be the total length of n
absorbers. Also, let γ1 = exp(
) and γ2 = exp(
the probability of single photon and two-photon trans-
mission respectively for one absorber. Here the param-
eter λ = χL, where L is the length of the absorber and
χ is the corresponding proportionality constant related
to the absorption cross section. Furthermore, κ specifies
the relative strength of the two transmissions and relates
them by γ2 = γ
1 . This CZ gate does the following oper-
ation:
|00〉 → |00〉
|01〉 → γn/2
|10〉 → γn/2
|11〉 → −γn
τ |11〉+ f(|02〉, |20〉) (1)
where the new expression for τ is given by:
τn,λ =
(gn,λ +
dn,λ√
2dn,λ − hn,λ)
+(gn,λ −
dn,λ√
2dn,λ + hn,λ)
dn,λ =
(1 + cos
)(1 + γ2) + 2
γ2(cos(
)− 3)
gn,λ = (cos
γ2 + 1)
hn,λ = 2(cos
γ2 − 1) (2)
The explicit form of the |02〉, |20〉 state components
are suppressed, as they lie outside the computational
basis and so do not explicitly contribute to the fidelity.
FIG. 1: Construction of our CZ gate.
FIG. 2: CZ gate in dual rail implementation.
From equation 1, it is clear that the amplitude of the
four computational states are unequal and this lowers the
gate fidelity. With the current best estimate of κ = 104,
the unherald fidelity is only 0.94. If the gate is used in
a measurement based strategy then state distillation can
be used and the fidelity of the gate can be improved by
trading off some success probability. Figure 3 shows the
distillated Zeno CZ gate circuit. The τ gate is simply
an interferometer consisting of two 50-50 beam splitters
with a two-photon absorber in each arm, which gives op-
eration: |00〉 → |00〉, |01〉 →
|01〉, |10〉 →
|10〉,
and |11〉 → γ′
τ |11〉. Here γ′
= τ1/κ is the single photon
transmission coefficient of the absorber. The distillation
beam splitters labelled 1 to 3 have transmission coeffi-
cient
τ respectively. With these dis-
tillations in place, the operation of the distillated Zeno
CZ gate is:
|00〉 → γn
τ |00〉
|01〉 → γn1 γ
1τ |01〉
|10〉 → γn
τ |10〉
|11〉 → −γn
τ |11〉+ f(|02〉, |20〉) (3)
After measuring the output (detectors at output are
not shown in fig.3) and treating the photon bunching
terms (|02〉, |20〉) and the terms with photon loss as fail-
ures (excluded from Eq. 3 for clarity), renormalising the
states gives unit heralded fidelity independent of λ and
probability of success Ps = γ
τ2 = e−2λ/κτ2+2/κ.
FIG. 3: Schematic of distillated Zeno CZ gate.
II. ZENO GATE USING GOTTESMAN-CHUANG
SCHEME
As argued above, state distillation can improve the
fidelity of the Zeno gate to unity by trading off success
probability. However, the output of the distillated Zeno
FIG. 4: Schematic of GC-Zeno gate. The state |χ〉 is
((|00〉 + |11〉)|00〉 + (|01〉 + |10〉)|11〉)/2
gate contains terms outside the computational basis
due to photon loss and photon bunching. Hence if we
want the gate to have unit fidelity, it is necessary to
measure the output and exclude these failure terms by
post-selection. However, such post-selection means that
the gate can no longer be used directly as a CNOT gate
for circuit-based quantum computing.
Gottesman and Chuang [11] showed the viabil-
ity in using state teleportation and single qubit
operations to implement a CNOT gate. The
scheme requires the four qubit entangled state
|χ〉 = ((|00〉+ |11〉)|00〉+ (|01〉+ |10〉)|11〉)/2. Preparing
the entangled state requires a CZ operation, which can
be done off-line with linear optics with high fidelity. Bell
measurements are made between the input qubits and
the first and last qubits of |χ〉. The measurement results
are fed forward for some single qubit corrections such
that the circuit gives a proper CNOT operation. Here
we propose using such scheme, as shown in figure 4, to
implement a GC-Zeno CNOT gate with high fidelity.
Since this gate includes state distillation, post-selection
and off-line state preparation, the gate has unit fidelity
(under perfect mode-matching) and moderate success
probability. Figure 5 plots the probability of success
against the one-photon to two-photon transmission ratio
κ. It shows that with κ = 104 (current best estimate),
the probability of success is about 0.76, which is better
than the break even point of 0.25 for the linear optics
version of this gate [1].
IIa. Effect of Mode-Mismatch
From source preparation to gate operation to detec-
tion, mode-mismatch is an unavoidable issue in optical
quantum computing that causes unlocated errors which
lowers the fidelity of the device[15]. Fortunately, with
the help of quantum error correction, a certain amount of
unlocated error rate, including but not limited to mode-
mismatch errors, can be tolerated. A reliable quantum
FIG. 5: Plot of probability of success versus log(κ) (in base
10) for GC-Zeno gate. Note that the success probability is not
one at κ = 108, but that the curve asymptotically approaches
one for very large κ. Result is per two input qubits. Detector
inefficiency is taken into account in accord with Dawson et
al’s bound.
gate must therefore have unlocated error rates below this
threshold.
The dominant source of mode-mismatch error in the
GC-Zeno gate is from the CZ gate and τ gate, where
two-photon interaction occurs. Here we follow Rohde et
al’s [14] analysis to examine the effect of such error. We
take the simplest model in which the mode-mismatch is
present between the photons entering the gate but re-
main constant through the gate. In this case, the mode-
mismatch in two-photon interaction can be analysed as
having two-photons fail to interact with some probabil-
ity. This allows us to write the operations for the CZ
gate as follow, where 0 < Γ < 1 quantifies the over-
lap of the two wavepackets. Γ2 is the probability that
the two photons successfully interacted and Γ = 0 for
completely mode-mismatched and Γ = 1 for completely
mode-matched. The bar in the |1̄1〉 term indicates mode-
mismatched component of the state.
|00〉 → |00〉
|01〉 →
|10〉 →
|11〉 → −Γγn1 τ |11〉+
1− Γ2γn1 |1̄1〉
+f(|02〉, |20〉) (4)
And similarly for the operations of τ gate:
|00〉 → |00〉
|01〉 →
|10〉 →
|11〉 → Γγ
τ |11〉+
1− Γ2γ
|1̄1〉
+f(|02〉, |20〉) (5)
With the equations for the CZ and τ gate[16], and
given a normalized input state (α|00〉 + β|01〉 + δ|10〉 +
ǫ|11〉), we can obtain analytical expression for the fidelity
F (per qubit) and success probability Ps (per qubit) of
the GC-Zeno gate as follow. Equation 6 and 7 show
that both the fidelity and success probability are state
dependent due to mode-mismatch. The worst case of
fidelity occurs when the input state is the equal super-
position state (|00〉+ |01〉+ |10〉+ |11〉)/2 (i.e. α = δ =
β = ǫ = 1/2) and the worst case of success probability
occurs when the input state is the pure state |11〉 (i.e.
α = β = δ = 0 and ǫ = 1).
α∗A1 + β
∗A2 + δ
∗A3 + ǫ
|A1|2 + |A2|2 + |A3|2 + |A4|2
e−2λ/κτ2/κ
2(1 + e−λ/κτ2+1/κ)
|A1|2 + |A2|2 + |A3|2 + |A4|2 (7)
where a1 = (τ+τΓ+
1− Γ2), a2 = (τ−τΓ+
1− Γ2),
a3 = (τ − τΓ−
1− Γ2), a4 = (τ + τΓ−
1− Γ2), and
A1 = αa
1 + βa1a2 + δa1a2 + ǫa
2, A2 = αa1a3 + βa2a3 +
δa1a4 + ǫa2a4, A3 = αa1a3 + βa1a4 + δa2a3 + ǫa2a4,
A4 = αa
3 + βa3a4 + δa3a4 + ǫa
IIb. Effect of Detector Efficiency
In practice, even for the most advanced photon detec-
tor, detector inefficiency is always present. The effect of
this noise is to reduce the probability of success of the
gate but not the fidelity because the errors are locatable.
III. ESTIMATE OF BOUNDS FOR FAULT
TOLERANCE
We now wish to estimate lower bounds on the mode-
matching, Γ, and photon loss ratio, κ, that will still allow
fault-tolerant operation. We allow a small amount of
detector inefficiency but assume all other parameters are
ideal. To make this estimate we directly use the bounds
obtained by Dawson et al [12] for a deterministic error
correction protocol. For this protocol, they numerically
derived one bound using the 7-qubit Steane code and
another bound using the 23-qubit Golay code.
In order to use the Dawson et al’s bounds we need
to identify the unlocated and located error rates for our
gate. In general, the unlocated error rate is less than
1 − F but here we take it to be 1 − F because in our
analysis, γ is almost 1, which means the other terms in-
volved are very small. The located error rate is simply
1 − Ps (both F and Ps are per qubit). Using these re-
lationships, we convert each of the bounds into a fidelity
versus success probability bound. For a gate built with
two-photon absorbers that have a certain single-photon
to two-photon transmission ratio κ, we can find an op-
timal λ (i.e. choosing an optimal absorber length) that
FIG. 6: Lower bounds of amount of mode-matching Γ
required for a fault tolerant GC-Zeno gate versus single-
photon to two-photon transmission ratio κ. The bounds are
derived from Dawson et al’s [12] results on deterministic
error correction protocol. The top and bottom curves are
for the 7-qubit Steane code and the 23-qubit Golay code
respectively. Above the curves are the regions where the
amount of mode-mismatch is tolerable. Here we have used
the worst case input.
gives a maximum success probability. Hence, by match-
ing the success probability with the bound, we can deter-
mine the corresponding fidelity threshold and therefore
find the least amount of mode-matching required for fault
tolerant gate operation. We note that the error model
used by Dawson et al is specific to their optical cluster
state architecture and will differ in detail from the appro-
priate error model for the GC-Zeno gate. Nonetheless we
assume that a comparison based on the total error rates
will give a good estimate of the bounds.
Figure 6 shows the lower bounds on the mode-
matching parameter Γ for a gate with a certain κ. Since
the fidelity and success probability are state dependent
due to mode-mismatch, in that figure, we have plotted
for the case of worst fidelity input state (i.e. the equal
superposition state). The top and bottom curves are
best fit curves for using the 7-qubit Steane code and the
23-qubit Golay code respectively. The curves show that
highly mode-matched photons are essential for robust
gate operation. With the worst fidelity input state,
(|00〉 + |01〉 + |10〉 + |11〉)/2, for the Steane code, the
lowest Γ required for fault tolerant operation is about
0.998, where κ = 106, and for the Golay code, the lowest
Γ required is about 0.996, where κ = 5 × 105. With
the worst success probability input state, |11〉, for the
Steane code, the lowest Γ required for fault tolerant
operation is about 0.995, where κ = 106, and for the
Golay code, the lowest Γ required is about 0.989, where
κ = 5 × 105. Figure 6 also shows that under (near)
perfect mode-matching, the required κ can be as low as
approximately 6000 for the Steane code and 2000 for the
Golay code. Two-photon absorbers with such κ values
may be achievable with the best of current technology.
IIIa. Advantage of Using State Distillation
State distillation allows us to trade off some success
probability against fidelity for the GC-Zeno gate, or in
other words, reducing the unlocated error rate by hav-
ing a larger located error rate. Since the determinis-
tic error correction protocol can tolerate both unlocated
and located errors, therefore we should ask whether state
distillation is truly advantageous? We can answer this
question by comparing two GC-Zeno gates in the case of
perfect mode matching, where one has complete distil-
lation and the other has no distillation. For the case of
complete distillation, the deterministic error correction
protocol with the 7-qubit Steane code can tolerate errors
of a GC-Zeno gate with κ = 6100, and with the 23-qubit
Golay code, it can tolerate errors of a GC-Zeno gate with
κ = 2100. For state distillation to be advantageous un-
der the same protocol, these values of κ must be smaller
than the values of κ for the case of no distillation[17].
For the case of no distillation, the fidelity and suc-
cess probability of the gate becomes state dependent. In
the parameters space of interest, the input that gives the
worst fidelity is (|00〉 + |01〉 + |10〉 − |11〉)/2. With this
input state, we find that for the protocol using the 7-
qubit Steane code and no distillation, the critical κ is
12000. Similarly, for the protocol using the 23-qubit Go-
lay code and no distillation, the critical κ is 4300. With
an arbitrary amount of distillation, the value of κ lies
between the limit of no distillation and full distillation
cases. Hence it is evident that state distillation is advan-
tageous. Also, it should be noted it is better to have only
located error, which is the case when there is full distilla-
tion, than have both located and unlocated errors, which
is the case when no or some distillation is utilized.
IV. CONCLUSION
In this paper, we have shown that it is possible to
build a high fidelity Zeno CNOT gate with two distil-
lated Zeno gates implemented in the Gottesman-Chuang
teleportation CNOT scheme. For one-photon to two-
photon transmission ratio κ = 104 (current best esti-
mate), the gate has a success probability of 0.76 under
perfect mode-matching. When including measurement
noise that equals one-tenth of the gate’s noise, and the
effect of mode-mismatch in the CZ and τ gate, we find
that with the deterministic error correction protocol us-
ing the 7-qubit Steane code, the lowest Γ required for
fault tolerant gate operation is 0.998, where κ = 106.
For using the 23-qubit Golay code, the lowest Γ required
is 0.996, where κ = 5 × 105. Hence, the requirement on
mode-matching is stringent for a fault tolerant GC-Zeno
gate.
∗ Electronic address: pmleung@physics.uq.edu.au
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[15] Note that mode mismatch in CZ and τ gate causes unlo-
cated error that lowers the fidelity, in which we find that
it cannot be improved with state distillation.
[16] Due to mode-mismatch, the τ gate is less effective in two-
photon distillation. It is true that we can increase the
two-photon absorption strength in the τ gate to make
up for the inefficiency. However, here we assume that we
do not know the mode-matching parameter Γ of the in-
put wavepackets, and therefore this adjustment cannot
be made. In addition, increasing the two-photon distilla-
tion will increase single-photon loss as well, which lowers
the probability of success.
[17] An ideal Zeno gate requires strong quantum Zeno effect
and strong quantum Zeno effect corresponds to a large
κ value. However such large non-linearity is difficult to
engineer. Hence it is desirable to have a gate that works
with modest κ.
mailto:pmleung@physics.uq.edu.au
