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Fault Tolerance in Parity-State Linear Optical Quantum Computing
A. J. F. Hayes,1, ∗ H. L. Haselgrove,2 Alexei Gilchrist,3 and T. C. Ralph1
1Centre for Quantum Computer Technology and Physics Department,
University of Queensland, QLD 4072, Brisbane, Australia.
2C3I Division, Defence Science and Technology Organisation, Canberra, 2600, Australia.
3Physics Department, Macquarie University, Sydney, NSW 2109, Australia.
(Dated: November 4, 2009)
We use a combination of analytical and numerical techniques to calculate the noise threshold
and resource requirements for a linear optical quantum computing scheme based on parity-state
encoding. Parity-state encoding is used at the lowest level of code concatenation in order to efficiently
correct errors arising from the inherent nondeterminism of two-qubit linear-optical gates. When
combined with teleported error-correction (using either a Steane or Golay code) at higher levels of
concatenation, the parity-state scheme is found to achieve a saving of approximately three orders
of magnitude in resources when compared to a previous scheme, at a cost of a somewhat reduced
noise threshold.
PACS numbers: 42.50Dv
I. INTRODUCTION
It was shown by Knill, Laflame, and Milburn (KLM)
[1] that, in principle, scalable optical quantum com-
puting could be achieved using only passive linear ele-
ments, single-photon sources, measurement and feedfor-
ward. Non-deterministic gates were developed that failed
through the accidental measurement of qubit value. Par-
ity state error codes were developed to protect against
such accidental measurement. KLM showed that, by
concatenating the parity codes, gate failures could be re-
duced to arbitrarily small levels, thus justifying the claim
of scalibility. However, because of the massive complex-
ity of the scheme, it has not been practical to couple it to
a higher level error correction protocol capable of correct-
ing environmental errors, and hence evaluate its resource
requirements and fault tolerant threshold.
A major simplification of the KLM circuit approach
was achieved by the introduction of incremental parity
codes [2] and fusion gate techniques [3, 4]. We refer to
this modification of KLM as parity state quantum com-
puting. These techniques reduce the complexity of the
scheme sufficiently that it becomes possible to make a
full fault tolerant analysis, thus completing the original
KLM program.
In this paper, we derive the resource usage and er-
ror thresholds achievable when a concatenated error-
correcting code such as the Steane code [5] is used to
to handle environmental and residual gate errors in the
parity-state optical quantum computing scheme. This
type of analysis has previously been done for cluster state
[6] and coherent state [7] schemes. It is important to es-
tablish these thresholds for the parity-state implementa-
tion both as a target for technological development, and
for comparison with the other proposals. Our results
∗Electronic address: ahayes@physics.uq.edu.au
show that the parity-state protocol may offer a useful
trade-off between the higher resource usage of cluster-
state schemes and the lower noise threshold of coherent
state schemes.
Significant progress has been made in optical quantum
computing experiments in the last decade [8].In particu-
lar the basic principles of optical parity state production
and their ability to correct Z-measurement errors has
been demonstrated [9, 10, 11]. These promising experi-
ments indicate that effective use of such error-correcting
codes in future designs is viable.
The layout of the paper as follows. Section II gives a
brief review and introduction of parity encoding and the
operations that may be done on parity-encoded states.
Section III describes the physical noise model that we
consider, and gives expressions for the effective noise
rates on various parity-encoded operations. In Section IV
these error-rate expressions are used as the basis of sim-
ulations of higher levels of encoding. The results of the
simulations are presented in the form of noise-threshold
curves. Finally, Section V considers the resources re-
quired by this scheme, and provides a comparison with
the thresholds and resource requirements of some other
schemes for fault-tolerant optical quantum computation.
II. UNIVERSAL GATE SET
This section describes the states and operations used
in the two lowest levels of encoding in our scheme: the
physical qubits in the dual-rail nondeterministic linear-
optical architecture, and the first level of logical qubits
which use parity-state encoding. These designs lead to
the central focus of this paper, a discussion of the effects
of noise on the parity-state encoding and the effects of
higher levels of encoding (fault-tolerant Steane and Golay
encoding), which is covered in Sections III–V.

2A. Physical Encoding and Operations
At the physical level, qubits in our scheme are encoded
and manipulated according to the techniques of nonde-
terministic dual-rail linear optics. Although other im-
plementations, such as spatial encoding, are possible, we
will explicitly consider polarisation qubits, encoded in
the horizontal and vertical polarisation modes of single
photons. This entails a series of physical and technologi-
cal assumptions including: that single-photon states can
be produced on-demand in a desired mode, such modes
can be made to interact using mode-matched linear op-
tics, modes can be stored in a quantum memory, and, the
number of photons in a mode can be measured and the re-
sults used in the fast control of optical switching. Single-
qubit operations in this scheme are relatively straightfor-
ward, but two-qubit gates are inherently nondeterminis-
tic even in the absence of noise.
As in reference [3], our scheme utilises two particu-
larly simple nondeterministic two-qubit gates, the so-
called type-I (fI) and type-II (fII) fusion gates (Fig 1). A
type-II fusion gate performs a two-qubit destructive mea-
surement in the basis {|00〉+ |11〉, |00〉 − |11〉, |01〉, |10〉}.
The first two outcomes (corresponding to maximally-
entangled basis elements) are considered “successful”,
whereas the second two outcomes are considered to be
failures of the gate. A type-I fusion gate is a partial
Bell measurement on two qubits. Two outcomes, con-
sidered successful, project the input state into the space
spanned by {|00〉, |11〉} and outputs a single qubit ac-
cording to the operator |0〉〈00| ± |1〉〈11|. Again, there
are two failure outcomes which correspond to a destruc-
tive measurement in the basis {|01〉, |10〉}.
For the type of input states we will consider, both the
type-I and type-II fusion gates are successful with 50%
probability (in the absence of noise).
B. Parity Encoding
A length-n parity code encodes one logical qubit into
n physical qubits. The logical basis states of the code,
denoted |0〉(n) and |1〉(n), are defined to be:
|0〉(n) ≡ (|+〉⊗n + |−〉⊗n)/
√
2
|1〉(n) ≡ (|+〉⊗n − |−〉⊗n)/
√
2, (1)
where |±〉 = (|0〉± |1〉)/
√
2. Note that |0〉(n) is the equal
superposition of all even-parity n-bit strings, and |1〉(n) is
the equal superposition of all odd-parity strings. A useful
property of the parity code basis states is that they have
a simple expansion in terms of smaller code states:
|0〉(n) = (|0〉(n−j)|0〉(j) + |1〉(n−j)|1〉(j))/
√
2, (2)
|1〉(n) = (|1〉(n−j)|0〉(j) + |0〉(n−j)|1〉(j))/
√
2, (3)
where 1 ≤ j ≤ n − 1. For j = 1 this expansion shows
that a computational basis measurement (such as occurs
FIG. 1: a) The type-I fusion gate. b) The type-II fu-
sion gate. Here the gates are shown being performed on
two polarization-encoded photonic input qubits, Q1 and Q2.
These input qubits are typically part of larger entangled
states. The gates are constructed from photon-counting de-
tectors, polarising beam-splitters, and half-wave plates, where
the half-wave plates act as Hadamard operations on the po-
larization qubits.
with a failed fusion gate) of one of the physical qubits will
not destroy a parity-encoded qubit, but will only reduce
the length of encoding by one (and possibly introduce
a known logical Pauli X operation, depending on the
measurement outcome).
C. Generating Parity States
The production of parity-encoded states is a necessary
procedure both for the preparation of encoded qubits as
sources, and as part of the resource generation required
by some of the non-deterministic logical gates in our uni-
versal gate set for parity-encoded qubits.
The state |0〉(n) is locally equivalent to a star shaped
cluster state (by a Hadamard operation applied to the
central node of the star). Consequently, given a supply
of Bell states (|0〉(2)), the resource |0〉(n) can be built up
using essentially the same techniques as used in [3] to
build up star-shaped cluster states.
Parity states |0〉(n) and |0〉(m) can be fused using the
fI gate as follows:
HfI(H⊗H)|0〉(n)|0〉(m) →
{
|0〉(m+n−1) (success)
|+〉⊗n−1|−〉⊗m−1 (failure)
(4)
(where we have omitted other possible locally equivalent
outcomes for both success and failure). The operator
HfI(H⊗H) should be understood as a Hadamard gate
acting on one of the physical qubits from each of the
encoded states, followed by the fI gate applied to the

3same pair, followed by a Hadamard gate applied to the
output of fI in the case of success. Alternatively the fII
gate can be used to carry out fusion, as follows:
fII|0〉(n)|0〉(m) →
{
|0〉(m+n−2) (success)
|0〉(m−1)|0〉(n−1) (failure) (5)
(again, some additional locally-equivalent outcomes have
been omitted).
The first alternative, where fI is used with Hadamard
gates to perform fusion, has the advantage of losing only
a single physical qubit from the input states, but the
disadvantage of completely destroying the entanglement
in both input states in the event of failure. In the second
case, fII is used to join the input states at the expense of
losing two of the initial physical qubits. There are three
advantages to the second scheme — in the case of failure
we do not destroy the entanglement of the input states,
just reduce their encoding by one; we do not need photon
number discriminating detectors to operate fII ; and it
is failsafe with respect to loss, in the sense that a lost
photon will not cause a failed fII gate to appear to have
operated successfully.
Thus, to create the state |0〉(3), two |0〉(2) are fused
together using the fI gate. Since fI functions with a
probability of 1/2, on average two attempts are necessary
so on average each |0〉(3) consumes 4|0〉(2). Once there is
a supply of |0〉(3) states, either fI or fII can be used to
progressively build up larger resource states.
D. Simple Single-Qubit Gates
For the parity code, encoded single-qubit unitaries can
be divided into those which have a particulary simple de-
terministic implementation, and those which have a more
complicated nondeterministic implementation involving
the consumption of resource states.
We can deterministically perform encoded versions of
any of the gates in the set {Xθ, Z}. Here, Xθ refers to an
arbitrary rotation about the X-axis of the Bloch sphere,
Xθ = cos(θ/2)I+ i sin(θ/2)X . An encoded Xθ operation
is achieved by applying Xθ to just one physical qubit
in the code state. The encoded Z gate is achieved by
applying a Z gate transversally to all physical qubits in
the code state.
E. Z90 Gate
To make our set of encoded single-qubit gates univer-
sal, we add the Z90 operation. Similar to the notation
introduced in the previous subsection, Z90 refers to a
rotation by 90 degrees around the Z-axis of the Bloch
sphere.
The logical Z90 operation is based on the process of re-
encoding. Re-encoding can be understood by considering
the following generalization of Equation 5:
fII|Ψ〉(n)|0〉(m) →
{
|Ψ〉(m+n−2) (success)
|Ψ〉(m−1)|0〉(n−1) (failure) , (6)
where |Ψ〉(n) ≡ α|0〉(n) + β|1〉(n) is an arbitrary parity-
encoded input state, and like Equation 5 we have omitted
other locally-equivalent outcomes.
To re-encode a logical qubit |Ψ〉(n), a type-II fusion
gate is first performed between the logical qubit and an-
other ancillary parity state |0〉(n+1). Then, each of the
remaining n− 1 qubits that belonged to the original en-
coded input state are measured in the computational ba-
sis, leaving the new ancilla qubits in the same state as
the original input, |Ψ〉(n). A logical X operation may be
required as a correction depending on the total parity of
the measurements made.
A slight modification of this procedure yields the en-
coded Z90 gate. Let
|Ψ〉(n) = α(|0〉(n−1)|0〉I + |1〉(n−1)|1〉I) + (7)
β(|0〉(n−1)|1〉I + |1〉(n−1)|0〉I)
be the logical qubit on which we wish to perform an en-
coded Z90 operation. One of the component physical
qubits (here denoted by the subscript I) is chosen to rep-
resent the input for the type-II fusion in the re-encoding
procedure. To achieve an encoded Z90 gate on |Ψ〉(n), we
simply apply the un-encoded Z90 gate to qubit I, then
carry out the re-encoding procedure detailed above. The
final logical state following a successful fusion is Z90|Ψ〉.
In this case, a correction corresponding to a logical Y
operation may need to be applied to the output state
depending on the parity of the measurements made. In
the event that the fusion gate fails, the size of the input
state is reduced, and the operation may be re-attempted
if there are enough remaining qubits. If all qubits in the
input state are depleted, then the logical gate is consid-
ered to have failed.
To implement a logical Hadamard operation in this
gate set we use the decomposition H = X90Z90X90.
Since operations of the form Xθ are relatively easy to
perform, the logical Hadamard is essentially equivalent
to the Z90 gate in terms of difficulty, time taken, and
error emergence.
F. XX ′90 Gate
We define two maximally-entangling two-qubit gates
XX90 and XX ′90 as follows:
XX90 ≡
1√
2
(I1I2 − iX1X2), (8)
XX ′90 ≡ (X−90 ⊗X−90)XX90. (9)
The relationship between the XX90, XX ′90 and controlled
not (cnot) gates is shown in Figure 2 in circuit form.

4FIG. 2: The relationship between theXX90 gate and the cnot
gate. The combined operation inside the dotted area defines
the XX ′90 gate.
FIG. 3: The circuit used to create the resource state |RXX〉
used in implementing the XX ′90 gate. The circuit can be mod-
ified by treating the top and bottom qubits as length-2 parity
encoded qubits in order to improve the nondeterministic be-
haviour of the gate.
XX90 and XX ′90 have the useful property that the parity-
encoded versions of these gates can be achieved by ap-
plying just one copy of the unencoded gate to a pair of
physical qubits (where one physical qubit is selected from
each of the encoded input blocks).
An unencoded XX ′90 can be achieved nondeterminis-
tically as follows. First, a four-qubit resource state is
created using the circuit shown in Figure 3. If the three
fusion gates are successful, the resulting state is
|RXX〉 =
1
2[|++〉(|00〉+|11〉)+| − −〉(|01〉+|10〉)]. (10)
Next, two fII gates are applied between the input qubits
and the resource state in the following manner: a fII gate
is applied between a qubit of the first input and the first
qubit of the resource state, and a fII gate is applied be-
tween a qubit of the second input and the fourth qubit of
the resource state. If both are successful, the remaining
two qubits will be in the state XX ′90|ψ〉, where |ψ〉 was
the input state (subject to possible known Pauli correc-
tions).
For parity-encoded inputs the procedure is almost
identical, except that one has more opportunities to at-
tempt the fII gates between the inputs and the resource
state. If both of the fII gates fail, the corresponding en-
coded input qubits will be reduced in size by one and the
gate can be re-attempted. Note that if one of the fII gates
fails and the other succeeds, then one of the input states
will be reduced in size by one but the other input effec-
tively retains its size, if we consider one of the remaining
qubits in the resource state to now belong to the partic-
ular encoded input that corresponds to the fusion gate
that succeeded.
The procedure described above provides an encoded
XX ′90 gate which on average decreases the size of its
inputs by one. A simple modification can be made to
the resource-state generation circuit so that the result-
ing gate will instead on average preserve the size of its
inputs. The modification involves treating the first and
fourth qubits in Figure 3 as length-2 parity qubits in-
stead of physical qubits, meaning that the resource state
is now a state of 6 physical qubits. We will make use
of this version of the XX ′90 gate in the remainder of the
paper.
III. ANALYSIS OF ERRORS
A. Error Modelling
The theory of error correction in general requires that
some assumptions be made concerning the nature of er-
rors that may occur in a system. The set of possible
errors that are considered, and their probability, form
an error model on which the conclusions of a theory are
based. In developing error correction methods, the aim
is to choose error models which closely match the physi-
cal reality. Typically, this involves focussing on the most
common types of error found in the corresponding exper-
imental systems. However, it is not always possible to
make the error model a detailed fit to the requirements
of a particular system, especially when the technology is
still in development.
In optical quantum computing systems with single
photons, the greatest source of error is normally pho-
ton loss. This can occur in many different ways, includ-
ing tunnelling, detection failure, imperfect coupling and
unreliable sources. Another important type of error in
optical systems is dephasing. It is expected that dephas-
ing errors will typically be less frequent than loss errors
in future optical computer components, but still likely to
have a significant effect in any large-scale quantum cir-
cuit. These errors can also be caused by imperfect cou-
pling, or indeed any misalignment or flaw in the optical
elements that can affect the polarization of the photon.
We use a simple error model for the noise on physical
qubits. Each gate operation is divided into timesteps,
with a single timestep being roughly the time required to
make a measurement or set of measurements and perform
feedforward based on the results. Each physical qubit is
considered to experience loss at a rate of γ per qubit per

5timestep, and Pauli errors at a rate of η per qubit per
timestep. The Pauli error is selected randomly from the
set {X , Y , Z} with equal probability. This corresponds
to a depolarization error of rate 43η, and to a marginal
probability of a bit-flip of 23η. Depolarization errors are a
generic way of representing the effects of dephasing noise,
as well as other errors which act locally on each qubit and
do not cause leakage from the qubit state space.
The physical error model described above is used as the
basis for estimating the effective error rates on encoded
qubits at higher levels of encoding. The relationship be-
tween the levels of encoding are summarised in Figure 4.
At each encoding level, the aim is to estimate the rates
of two different error types: located errors, which are
those errors which are heralded (in a way analogous to
the failure of fusion gates at the physical encoding level),
and unlocated errors, which are Pauli errors that are not
directly heralded and must be found indirectly via syn-
drome measurement. The remainder of this section is de-
voted to deriving expressions approximating the effective
error rates of the various parity-encoded operations, i.e.,
operations at the first level of encoding above the physical
level. Then in the following section, these expressions are
used as the error model for simulations of concatenated
fault-tolerant teleported error correction (telecorrection),
using Steane and Golay coding.
In deriving expressions for the effective error rates at
level 1, we use approximations which are correct only to
first order in the physical error rate. In particular, we find
the probability that a qubit remains error-free after mul-
tiple timesteps by taking the product of the individual
probabilities of an error not occurring for each timestep.
In general, this is correct only to first order, since in re-
ality multiple errors of the same type occurring on the
same qubit can cancel. For bit- or sign-flip errors of the
type included in our error model, an even number of er-
rors has no overall effect on the qubit. However, since we
are only considering very small values of physical error
rate, and since each parity-encoded gate consists of only
a relatively small number of optical components, these
higher order error terms are insignificant.
It is important to note that this model assigns the same
error probability to a timestep regardless of whether a
qubit was involved in operations during that time. Hence
this model takes into account errors that arise while a
qubit is kept in memory, and assumes that such error
rates are similar to those for a qubit actively involved in
computation. This aspect of the model may be unduly
pessimistic, but we have chosen to consider the worst case
in this regard.
Having defined the error model, it is also necessary
to consider how these errors propagate through the el-
ements of an optical system. First, it should be noted
that measurement in the Z basis renders any X errors
on that qubit irrelevant (likewise for Z errors and X-
basis measurements). Measurement also serves to locate
loss errors. Due to the regular measurements that occur
in our protocol (either directly or via a fusion gate), it
can be seen that loss errors will always be quickly trans-
formed into located errors.
The properties of parity state encoding are such that
any X errors on physical qubits immediately become X
errors on the logical, parity-encoded qubit. Hence the
probability of a logical X error will depend simply upon
the rate of such errors at the physical level, the number of
qubits and the duration of the computation. Z errors on
individual photonic qubits do not automatically become
logical, however if a photonic qubit on which a Z error
has occurred is used as an input for a fusion gate, the
error is then applied to all the qubits forming the parity
state of which the input qubit was a member. We will use
ηX = ηZ = 23η to denote the marginal error probabilities
of X and Z errors (where Y is considered to be “both
an X and a Z error”). This notation allows one at a
glance to see which physical error type contributes to
a particular logical error rate in the expressions of the
following subsections.
B. Size of the Parity Code
There is a trade-off in the located error rate for the
non-deterministic gates between the errors due to gate
failure and those due to loss (Figure 5). This occurs
as each additional qubit in the encoding increases the
possibility of a loss occurring. However, a certain level
of encoding is necessary in order to reduce the proba-
bility of gate failure. We can optimise the size of the
parity states by calculating the located error rates for
the non-deterministic operations examining how these er-
rors vary with code size. Fortunately, the optimal parity
code size to minimize the trade-off is found to be similar
for both non-deterministic operations, with a seven qubit
code proving to offer the best threshold. All error rate
calculations will therefore assume a code of this size.
C. Source Production
We assume that copies of the state |0〉(7) are prepared
as needed by massively parallel production in order to
have them ready when required by the circuit. We begin
with Bell pairs in the state (|00〉 + |11〉)/
√
2, and link
these by means of type-I and -II fusion gates. For a parity
state of size 7, 7 Bell pairs and 6 fusion gates are required
to build the resource. As each gate has a 50% failure rate,
parallel production of each resource requires an average
of 7× 26 Bell pairs, and takes 3 timesteps. The resulting
state has an unlocated X error rate of 1−(1−ηX)41 and a
located error rate of 1−(1−γ)21. (There are 41 locations
in the optical circuit which can contribute errors to the
output state. For 20 of these locations, photon loss will
be immediately heralded, and so we post-select on no loss
occurring at those locations).

6FIG. 4: The teleported error-corrector (telecorrector) circuit shown at different scales. This diagram demonstrates the manner
in which logical operations are broken down into a series of operations at the lower layers of encoding. Box 1 shows the circuit
for the production of the resource required for one round of telecorrection. Box 2 demonstrates how a cnot gate on the
Steane-encoded qubits can be performed using 7 XX ′90 gates and 14 Hadamards at the parity encoding layer. Box 3 shows the
process of using resource preparation plus teleportation to implement an XX ′90 gate, and Box 4 provides the circuit used to
prepare the required resource.
D. Z90 Error Rates
This gate occurs in two steps: the first step is the at-
tempt to fuse the encoded state with a resource state, and
the second step is measuring the remaining component
qubits from the original state once a successful fusion has
been performed. The probability of a failure at the first
step is:
1−
7
∑
j=1
2−j(1− γ) 12 j(1+j) (11)
which combines the possibility of a loss during fusion with
the fusion failure rate to get the total located error rate
during the fusion attempts. To cover the possibility of
loss occurring on any of the other component qubits, we
include the factor
(1− γ)3j+(1+j)(7−j) (12)
in the sum, which depends on the number of qubits re-
maining and the time they have been in memory.
Hence the Z90 gate has a combined located error rate
per parity qubit of
PLE = 1−
7
∑
j=1
2−j(1−γ) 12 j(1+j)(1−γ)3j+(1+j)(7−j) . (13)
For unlocated errors, the average rate can be found by
combining the error rates for memory and for producing
an ancillary parity state, due to the re-encoding process
used to implement the gate. The average time required to
implement this gate would be 2 timesteps. However, for

7FIG. 5: The probability of success for the XX90 gate as a
function of the photon loss rate (1− η) and code size n.
all error types, it is assumed that the average time spent
is 4 timesteps (this corresponds to the average time re-
quired for the slowest gate, XX ′90). This is done so that
all encoded operation types can be treated as taking an
equal amount of time. By counting the number of loca-
tions that may contribute to logical errors on the output,
we obtain overall effective rates of unlocated X and Z er-
rors of 1− (1− ηX)69 and 1− (1− ηZ)10 respectively, for
the Z90 gate.
E. XX90 Error Rates
The propagation of errors through this gate is fairly
simple: a Z-error on either input qubit will cause an X-
error on the opposite qubit. X-errors on an input qubit
propagate to the output without having an effect on the
other qubit.
As described in Section II F, the gate is achieved by
performing fusion gates between qubits from the encoded
input state and a 6-qubit resource state. As we are in-
terested in an error rate per parity qubit, we model the
success or failure of the fusion gates applied to an in-
put parity qubit using a random walk on a semi-infinite
1-dimensional lattice with 1 absorbing boundary at −n
[12]. Here the lattice represents the size of the parity
state, with failure occurring either due to a loss or when
all component qubits are measured due to repeated tele-
portation failures.
The number of paths that reach the boundary at time
t is
N(n, t) = t!
( t+n2 !)(
t−n
2 !)
for (t− n) mod 2 = 0 (14)
N(n, t) = 0 for (t− n) mod 2 = 1
where n is the initial size of the parity code. To incor-
porate the possibility of a successful gate operation, we
include a scaling factor in the number of paths:
Nscaled(n, t) = 2−
t−n
2 N(n, t). (15)
The number of first passage paths for this type of walk
is
F (n, t) = nt Nscaled(n, t). (16)
Therefore, total probability of success per parity qubit in
the absence of loss is:
PS = 1−
∞
∑
t=n
F (n, t)
2t . (17)
For a parity qubit of size 7, this evaluates to PS = 0.9763.
As the average size of the state and the rate of loss per
timestep is constant, the average loss per parity qubit
during this operation can be simplified to:
PL = 1− (1− γ)t(n+3). (18)
The average time taken for this operation is 4 timesteps.
Thus, the approximate located error rate per parity-
encoded input is 1 − 0.9763(1− γ)40. The unlocated X
and Z error rates, obtained by counting error locations in
the average-time case, are 1−(1−ηX)28 and 1−(1−ηZ)4
respectively.
F. Memory and Measurement
The memory or identity operation on an encoded qubit
simply entails keeping the constituent physical qubits in
memory while other operations are performed. We treat
the encoded memory operation as taking 4 time steps,
equal to the average time taken to perform the slowest
encoded gate, XX ′90. This yields approximate rates of X ,
Z, and located errors of 1 − (1 − ηX)28 , 1 − (1 − ηZ)4,
and 1− (1− γ)28 respectively.
An encoded computational-basis measurement in-
volves measuring each physical qubit in the computa-
tional basis and finding the parity of the measurement
results. Thus, the rates of located and unlocated errors in
the measurement outcome are 1−(1−γ)7 and 1−(1−ηX)7
respectively.
IV. FAULT TOLERANCE THRESHOLD
Having analysed the emergence of logical errors at the
parity code layer, we used numerical simulation to cal-
culate the error rates at higher levels of concatenation,
and from this, the value of the noise threshold for optical
parity-state quantum computation.
The simulations were performed using similar tech-
niques to those described in Sec V.D of [6]. In par-
ticular, we simulate one level of the telecorrector pro-
tocol, for both the 7-qubit (Steane) and 23-qubit (Go-
lay) codes. Our simulation differs from [6] in the noise

80 0.5 1 1.5 2
x 10−3
0
0.5
1
1.5
2
2.5
x 10−5
Located error rate, γ
Un
lo
ca
te
d
er
ro
r p
ar
am
et
er
, η
FIG. 6: The threshold curve for optical parity-state comput-
ing using the 7-qubit Steane code. The region below the solid
curve represents the set of error rates which can be tolerated
by the scheme.
0 0.5 1 1.5 2 2.5 3 3.5
x 10−3
0
1
2
3
4
5
6
7
x 10−5
Located error rate, γ
Un
lo
ca
te
d
er
ro
r p
ar
am
et
er
, η
FIG. 7: The threshold curve for optical parity-state comput-
ing using the 23-qubit Golay code.
model and gate set used. The telecorrector circuit in
[6] uses the following gate set: preparation of |0〉 and
|+〉 states, cnot, cphase, and X-basis measurement.
We converted this circuit to our gate set in the fol-
lowing way. First the circuit was expressed solely in
terms of cphase gates, Hadamard gates, |+〉 creation
and X-basis measurements, by making appropriate sub-
stitutions of each cnot gate, |0〉 creation, and compu-
tational basis measurements in the circuit, and simplify-
ing the resulting circuit where possible. Then a simple
change of basis |0〉 ↔ |+〉, |1〉 ↔ |−〉 yielded a circuit in
our gate set (XX ′90 gates, Hadamard gates, |0〉 creation,
and computational-basis measurement).
We carried out a series of Monte Carlo simulations for
a range of values of the physical error rates (γ, η), in
each case measuring the resulting rate of unlocated and
located errors at the next highest level of encoding. For a
particular choice of the physical noise rates, Pauli errors
(both unlocated and located) are introduced by each gate
with a probability that is governed by the noise model
derived in the previous section. In the case of unlocated
errors,X and Z errors are introduced independently. For
FIG. 8: Process for generating a 6 qubit parity state, begin-
ning with 6 Bell states and performing two rounds of fusion
gates.
the XX ′90 gate, errors are introduced independently on
each of the two output qubits.
The results of the simulations were used to characterise
the mapping of error rates from the physical level to the
second level of encoding (i.e., the parity encoding plus
one level of telecorrection encoding) by way of a poly-
nomial fit to the measured values. Our characterization
of the mapping for all levels of encoding above this was
obtained by simply using the particular polynomial that
was obtained in [6]. Thus, by applying the appropriate
sequence of polynomials to a particular setting (γ, η), we
are able to estimate effective error rates at any level of
encoding.
The threshold curve is defined to be the curve in the
γ− η plane below which effective error rates tend to zero
for many levels of encoding. Our calculated threshold
curves are shown in Figures 6-7 (the small circles show
the values of (γ, η) for which a Monte Carlo simulation
was run). Like previous schemes for optical quantum
computing, the results demonstrate a trade-off between
a tolerance of photon loss and depolarization errors. It
is worth noting that the code is always required to deal
with some probability of located errors due to the non-
deterministic gates and the finite size of the underlying
parity encoding. These errors are taken into account
when calculating the threshold for the loss and depolar-
izing rates plotted here.
V. RESOURCES
For a useful comparison with other schemes for error-
correcting quantum computing, it is necessary to also
consider the resources that would be required to imple-
ment this scheme. As noted previously, the method con-
sidered here for the creation of resources involves par-
allel production of many copies of a resource to ensure
it is available on demand. This is a simple approach
that leads to a higher cost in terms of entangled photon
generation, and avoids more complicated resource-saving
techniques such as storing previously prepared resources
and “recycling” entangled states from unsuccessful at-
tempts. Such techniques tend to require more intricate
design and photon-switching, as well as potentially in-
troducing more errors due to the longer photon storage

9time.
We calculate the resources required in terms of the
number of Bell states used. This a useful unit to consider
as it is a common starting point for building entangled
states across many optical schemes. It is also handy for
comparisons with current optical quantum computation
experiments as most use the entangled output of optical
parametric down-conversion systems as photon sources.
As no recycling is used, the resource production has
a 50% chance of failing for every fusion gate performed,
meaning that the number of parallel attempts required
doubles with each fusion gate. This exponential growth
can be tolerated as it only occurs on a small scale – in
our case none of the resource states need to be larger
than 8 qubits. To estimate the resources we calculate
the average number of Bell states required to produce
the appropriate resources using this parallel production
method and implement a parity-encoded operation.
Much of this resource production simply involves the
creation of ancillary parity states, which is done by link-
ing the initial Bell states together using type-I and -II
fusion gates (Figure 8). State preparation at the parity
layer requires a 7-qubit parity state, which would take
an average of 448 Bell pairs to produce. Re-encoding,
and hence the Z90 operation, would require an 8-qubit
parity state for each attempt, and an average of two at-
tempts to implement the operation. Hence we estimate
the resources for the Z90 operation at 2048 Bell pairs on
average. The circuit for the preparation of the XX ′90 re-
source was shown previously in Figure 3. This resource
requires an average of 128 Bell pairs to produce.
As a basis for comparison, we will consider the require-
ments for producing the telecorrector state needed for
one round of correction. Using the gate costs described
above, it can be calculated that the state requires ap-
proximately 177670 Bell pairs to generate. In Table I
resource usage and threshold values are compared be-
tween parity-state and cluster-state schemes for optical
quantum computing. Due to its greater difference from
the other two schemes, we did not include in the table
a scheme for quantum computing using “cat” states [7].
Cat states are in general more difficult to generate than
Bell states, so this makes a direct comparison of resource
usage difficult. We note that on average 103 cat states
are needed to create a telecorrector state in that scheme.
The loss threshold for the cat states is 2 × 10−4, and
a threshold for depolarization was not specified in that
work.
Thus, our scheme for fault-tolerant parity-state quan-
tum computing gives a resource usage figure three orders
of magnitude smaller than that for an equivalent clus-
ter state circuit [6] (and, in some sense, three orders of
magnitude larger than the requirements for the coherent
state version [7]). However threshold is poorer than that
of the cluster state protocol, but better (with respect to
loss) than the scheme using cat states.
Loss Depolarization
Scheme Threshold Threshold Resources
Cluster states 4 × 10−3 8 × 10−5 1.3 × 108 a
Parity states 2 × 10−3 2.4 × 10−5 1.8 × 105
aReferece [6] quotes an incorrect resource usage number, the one
cited here is the corrected value.
TABLE I: A comparison of thresholds and resources for linear
optics quantum computing error correction schemes using a
7-qubit Steane code. Here the resources are those required
for the first level of telecorrection.
VI. CONCLUSIONS
We have shown that an error-correcting system based
on parity encoding falls in-between other schemes in both
threshold and resource requirements. The parity scheme
has a higher error threshold than that found for coher-
ent states, but also significantly larger resource require-
ments. Conversely, it is two orders of magnitude cheaper
in resources than a cluster-state implementation, but also
has lower thresholds for both located and unlocated error
rates. Together, these results suggest a necessary trade-
off between resources and achievable threshold, which in-
dicates that the preferred encoding method in any exper-
imental attempt to demonstrate optical quantum error
correction will depend on the capabilities and limitations
of the physical system and its components.
It is worth noting the general principle demonstrated
here, that concatenation can be used to tailor error cor-
rection to suit the relative error rates. In this case, some
parity encoding is used to reduce the rate of failures due
to non-deterministic gates to a level at which the remain-
ing errors can be handled by the general error correction
code. However, the same principle could be applied to
other errors. For example, a system with a very high loss
rate in comparison with other errors could use concate-
nation with a dedicated loss-correction code [13] to allow
a higher threshold for loss at the cost of lower thresholds
for other errors.
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