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# A robust W-state encoding for linear quantum optics

Madhav Krishnan Vijayan<EMAIL_ADDRESS>Centre for Quantum Software &
Information (UTS:QSI), University of Technology Sydney, Sydney NSW, Australia
Austin P. Lund  [ Centre for Quantum Computation & Communications Technology,
School of Mathematics & Physics, The University of Queensland, St Lucia QLD,
Australia  Peter P. Rohde  [ http://www.peterrohde.org Centre for Quantum
Software & Information (UTS:QSI), University of Technology Sydney, Sydney NSW,
Australia

###### Abstract

Error-detection and correction are necessary prerequisites for any scalable
quantum computing architecture. Given the inevitability of unwanted physical
noise in quantum systems and the propensity for errors to spread as
computations proceed, computational outcomes can become substantially
corrupted. This observation applies regardless of the choice of physical
implementation. In the context of photonic quantum information processing,
there has recently been much interest in passive linear optics quantum
computing, which includes boson-sampling, as this model eliminates the highly-
challenging requirements for feed-forward via fast, active control. That is,
these systems are passive by definition. In usual scenarios, error detection
and correction techniques are inherently active, making them incompatible with
this model, arousing suspicion that physical error processes may be an
insurmountable obstacle. Here we explore a photonic error-detection technique,
based on W-state encoding of photonic qubits, which is entirely passive, based
on post-selection, and compatible with these near-term photonic architectures
of interest. We show that this W-state redundant encoding techniques enables
the suppression of dephasing noise on photonic qubits via simple fan-out style
operations, implemented by optical Fourier transform networks, which can be
readily realised today. The protocol effectively maps dephasing noise into
heralding failures, with zero failure probability in the ideal no-noise limit.
We present our scheme in the context of a single photonic qubit passing
through a noisy communication or quantum memory channel, which has not been
generalised to the more general context of full quantum computation.

###### Contents

  1. 1 Introduction
  2. 2 Conceptual basis – Redundancy & entanglement robustness
    1. 2.1 GHZ states
    2. 2.2 Cluster states
    3. 2.3 W-states
  3. 3 Protocol
  4. 4 Error heralding
    1. 4.1 Heralding probability
      1. 4.1.1 Absence heralding
      2. 4.1.2 Presence heralding
        1. 4.2 Heralded fidelity
          1. 4.2.1 Absence heralding
          2. 4.2.2 Presence heralding
            1. 4.3 Probability and fidelity plots
              1. 5 Single-qubit unitary operations
                1. 6 Discussion
                  1. 6.1 Pros and cons
                  2. 6.2 Economic justification
                  3. 6.3 Robustness against different noise models
                  4. 6.4 Compatability with no-go theorem
                    1. 7 Comparison with other schemes
                      1. 8 Conclusion

##  1 Introduction

Within quantum information processing systems, the ability to detect errors is
an absolute prerequisite for the road towards fault-tolerance. In the standard
approach to fault-tolerant quantum computing, one first constructs error-
detection circuits, upon which we build error-correction capabilities, finally
revisiting the construction to ensure error transversality, facilitating
recursive nesting of the protocol to suppress error rates Shor [1996],
Preskill [1998], Nielsen and Chuang [2000]. In the absence of the initial
error detection stage, such a construction for mitigating errors cannot
function.

The standard framework when considering quantum error-correction is in the
context of universal quantum computation Knill et al. [2001]. Given that it is
universal, multiple levels of error correcting codes can be implemented. In
general this requires large, but sub-exponential, resource overheads each with
sub-threshold error rates. Although such constructions are essential for
realising the full potential of quantum computing, it remains a distant
target. Hence there is currently a pursuit to find utility for achievable
near-term devices with post-classical capabilities, even if not universal
Harrow and Montanaro [2017], Lund et al. [2017]. This has lead to the
alternative target where universality is discarded as a requirement and the
sole purpose is demonstrating some form of quantum computational advantage
with pragmatically reasonable resources. Two examples for such paradigms whose
quantum power is proven by links to widely presumed structures in
computational complexity theory are IQP (Instantaneous Quantum Polynomial),
and boson-sampling devices, both examples of so-called sampling problems. IQP
is the class of sampling problems consisting of commuting gates acting on
qubits that are prepared and measured in superposition basis relative to that
of the eigenbasis of the commuting gates Shepherd and Bremner [2009], Bremner
et al. [2011]. Boson-sampling is the set of problems that can be constructed
from the preparation and measurement of individual bosons subject to evolution
via passive linear interferometers Aaronson and Arkhipov [2011, 2014].

In the development of classical hardness arguments for these restricted
models, the consideration of errors under a trace-norm induced distance has
been of prime importance. Sampling from these distributions with bounded error
(which is actually an input to the problem definition) is called approximate
sampling. The arguments for the classical computational hardness of
approximate sampling do not utilize any form of extra resources to deal with
errors as the main purpose of these models was the minimisation of quantum
resources. Therefore, in standard error analysis for restricted modes the
objective has been to find the scaling relationships between trace-norm
distance and the parameters defined within the error process (e.g. loss rates,
mode overlaps, unitary noise, etc.) Rohde and Ralph [2012], Aaronson and Brod
[2016], Shchesnovich [2014], Tichy [2015], Leverrier and García-Patrón [2015],
Kalai and Kindler [2014], Arkhipov [2015], Rahimi-Keshari et al. [2016].
However, some quantum resources come cheaper than others within these models.
In particular, additional modes prepared with vacuum states within the boson-
sampling paradigm are considered to have much lower cost than additional modes
prepared with single photons. However, given that this model is passive, one
may suspect that it is not possible to perform any kind of error correction
without leaving the constraints of the model, and hence dealing with errors
defaults back to the requirements associated with universal models.

Marshman _et al._ Marshman et al. [2018] have shown that, for boson-sampling,
it is possible to detect the presence of random phase errors without leaving
the paradigm and that the conditional state on detecting the error has a lower
error than would otherwise be the case. This was done using a redundant
encoding of the passive linear interferometer with a particular network chosen
for encoding and decoding of input single photons. The presence of the photon
within a particular mode was used as the error detection mechanism. Devices
requiring higher photon numbers could be accommodated by parallel combinations
of single photon interferometers. This is distinct from the considerations of
Arkhipov [2015] for errors within unitary networks as there it was assumed
that there was no redundancy utilizing additional resources.

In this paper, we extend this result by considering single photon encoding
that involve W-state path entanglement encoding of photonic qubits encoded in
dual-rail form. These states can be generated from single photons through
passive linear interferometers, and resemble a generalisation of an optical
fan-out operation, having desirable properties for error correction such as
the maintenance of path entanglement when single systems are lost. The
expansion in mode number can be conceptually related to conventional error-
correction schemes based on redundancy, such as Shor’s original 3-qubit code
Shor [1995]. We show that this encoding yields an improvement on local phase
shift errors Rohde and Ralph [2006] much like that of the previous work but
also show that photon loss is the constraining factor in the heralded fidelity
for this localised noise model. We also show that this performance is
independent of the type of distribution underlying the random phase errors
provided that the errors acting on different modes are independent (i.e., no
correlated errors), identical (all modes are treated the same) and the
characteristic function for the distribution is well defined. Under these
conditions any level of encoding will improve fidelity when conditioned on
detecting no error and with a large enough encoding we can fully mitigate the
dephasing error.

To present our results we will first discuss different classes of multipartite
entangled states and elaborate on why W-states are a good candidate for
encoding and define the W-basis in section 2. In section 3 we introduce our
W-state based encoding using only linear optics and single-photon inputs and
describe how to post-select to filter our error. In this section we will also
describe the linear optics error model that we will consider. Then in section
4 we compute the success probability of the protocol to succeed and compute
the fidelity of the output logical state with the input logical state and show
that the performance improves as the level of encoding increases. We discuss
how to implement qubit gates on the logical qubit while in the W-state
encoding in section 5. Finally and we will discuss some implications of this
work and finally make some concluding remarks in section 6.

Importantly, we present an elementary economic argument in Sec. 6.2 for the
merit of our scheme from the perspective of engineering economics that,
regardless of the state of precision engineering, this approach is likely to
be economically justified is some regimes, complimentary to investment into
improved precision engineering. This simple observation is based on the
intuitive notion that investment into enhanced engineering precision scales
exponentially with precision requirements, whereas redundancy scales roughly
linearly, in economic overhead, from which the inevitability arises of there
being a crossover point between how resources should be allocated to maximise
economic efficiency.

##  2 Conceptual basis – Redundancy & entanglement robustness

An inherent feature of any kind of multi-qubit entangled state is that, by
virtue of its entanglement, loss or decoherence of a single constituent qubit
diminishes its degree of entanglement, similarly reducing its purity (or
conversely, increasing its collective entropy). Some entangled states are more
robust than others in this respect and, as discussed below, the W-states are a
quintessential example of entangled states with this robustness property. Note
that the resultant state following a partial trace operation upon a qubit
(equivalent to loss when using single photon encoding) is independent of
anything done only to traced out qubits prior to the partial tracing
operation. Therefore considering loss via partial trace is completely
sufficient to understand the worst-case degradation of an entangled state
under any kind of local noise process.

###  2.1 GHZ states

The worst-case scenario is the GHZ state, a maximally-entangled $n$-qubit
state of the form,

$\displaystyle\ket{\mathrm{GHZ}_{n}}=\frac{1}{\sqrt{2}}(\ket{0}^{\otimes
n}+\ket{1}^{\otimes n}),$ (1)

whereby all qubits are collectively perfectly correlated. That is, measurement
of any one (in the computational $Z$-basis), reveals the equivalent
measurement outcome of all others. However, this directly implies that losing
access to this information similarly implies loss of knowledge of the others.
Loss or dephasing directly correspond to such loss of information. For this
reason, dephasing a single qubit, or losing it outright, implies complete
decoherence of the entire $n$-qubit state. Specifically, partial tracing out a
single qubit from a GHZ state leaves behind the hopelessly mixed stated,

$\displaystyle\operatorname{Tr}_{i}(\ket{\mathrm{GHZ}_{n}}\bra{\mathrm{GHZ}_{n}})$
$\displaystyle=\frac{1}{2}\ket{0}^{\otimes n-1}\bra{0}^{\otimes n-1}$
$\displaystyle+\frac{1}{2}\ket{1}^{\otimes n-1}\bra{1}^{\otimes n-1},$ (2)

where the partial trace is performed upon any qubit $i$.

###  2.2 Cluster states

Cluster (or graph) states Raussendorf and Briegel [2001], Raussendorf et al.
[2003] are a highly useful class of states, enabling universal quantum
computation using the measurement based model for quantum computing (MBQC).
Despite being more computationally useful than GHZ states, they are far less
entangled, and hence far more robust against localised noise processes. For
example, by measuring out the immediate neighbours of a lost qubit from within
a graph state, a reduced, yet perfect graph state is recovered, given by the
sub-graph of the original graph, with the neighbourhood of the lost qubit
removed.

###  2.3 W-states

An especially robust (and so far not particularly useful) class of entangled
states are the W-states Zeilinger et al. [1997], Dür et al. [2000], given by
the equal superposition of a single excitation across $n$-sites. In qubit form
this can be expressed,

$\displaystyle\ket{W_{n}}$
$\displaystyle=\frac{1}{\sqrt{n}}(\ket{1,0,0,\dots}+\ket{0,1,0,0,\dots}$
$\displaystyle+\ket{0,0,1,0,\dots}+\ket{0,0,0,1,\dots}+\dots).$ (3)

Alternately, this can be expressed in terms of creation or excitation
operators, $\hat{a}^{\dagger}_{i}$ for the $i$th site,

$\displaystyle\ket{W_{n}}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\hat{a}^{\dagger}_{i}\ket{\Omega},$
(4)

where $\ket{\Omega}$ is the collective ground or optical vacuum state. The
latter representation is the one we will focus on here, given its direct
applicability to photonic encoding.

These states exhibit complete permutational symmetry under qubit interchange.
That is, the state is invariant under any permutation $\hat{\pi}\in S_{n}$ in
the symmetric group,

$\displaystyle\hat{\pi}\ket{W_{n}}=\ket{W_{n}}.$ (5)

Tracing out a single qubit from a W-state yields,

$\displaystyle\operatorname{Tr}_{i}(\ket{W_{n}}\bra{W_{n}})$
$\displaystyle=\frac{n-1}{n}\ket{W_{n-1}}\bra{W_{n-1}}$
$\displaystyle+\frac{1}{n}\ket{0}^{\otimes n-1}\bra{0}^{\otimes n-1}.$ (6)

That is, upon loss of a single qubit, with probability $p=(n-1)/n$ it simply
undergoes a reduction in its level of encoding to a $\ket{W_{n-1}}$ state,
preserving its W-type structure entirely, otherwise collapsing to the
$\ket{0}^{\otimes n-1}$ state. This implies that that for large $n$, W-states
are extremely robust (indeed almost invariant) against single-qubit loss. As
discussed earlier, this directly implies similar single-qubit robustness
against other noise channels.

Note that atomic ensemble qubits Duan et al. [2001] are a direct alternate
physical manifestation of W-type encoding, whereby an ensemble (or cloud) of
collectively-addressed atomic qubits undergo collective excitation,
mathematically of the form given in Eq. (4). This approach to realising
physical qubits has attracted much attention, especially as good candidates
for quantum memories, given their notably high coherence lifetimes, often at
room temperature, which can be intuitively associated with the described
robustness of their underlying W-type entanglement structure – if a few atoms
go missing from the cloud, little is lost.

The $n$-qubit W-state can be easily generalised to an entire orthonormal
W-basis, by appropriately manipulating the phase relationships within the $n$
terms in the superposition. One way in which to choose these phases is by
taking the elements from the Quantum Fourier Transform (QFT) matrix, or
generalised Hadamard matrices, both of which have equal $1/\sqrt{n}$
amplitudes across all matrix entries, with phase relationships ensuring
orthonormality.

These different phase relationships do not change the earlier observations
about the states’ robustness against local noise. This immediately leads to
the intuition, that by choosing such a W-basis for encoding logical qubits,
the encoded logical qubit must inherit via linearity these same robustness
characteristics. This makes them a direct candidate for optical encoding,
given that photonic implementation of QFT mappings may be implemented via
passive linear optics, in the absence of any active control, and is realisable
with today’s technology integrated wave-guide architectures across a large
number of modes.

##  3 Protocol

Figure 1: Photonic W-state error-correction and -detection protocol. Encoding
of a single dual-rail photonic qubit proceeds via a Quantum Fourier Transform
($\hat{QFT}$), which maps the 2-mode encoding across a larger number of
redundant modes. The independent dephasing noise channel is denoted by
$\mathcal{E}$. Decoding proceeds via the inverse Quantum Fourier Transform
($\hat{QFT}^{\dagger}$). Post-selection upon detecting the single photon
within the desired 2 output modes defining the single qubit, projects the
logical state into one with reduced noise action.

The error-detection and correction protocol is shown in Fig. 1. Consider $N$
optical modes, the first two of which contain a single photon state, defining
a dual-rail-encoded photonic qubit. This qubit can defines a logical qubit,

$\displaystyle\ket{L}$ $\displaystyle=\alpha\ket{0}_{L}+\beta\ket{1}_{L}$
$\displaystyle=(\alpha\hat{a}^{\dagger}_{0}+\beta\hat{a}^{\dagger}_{1})\ket{\Omega},$
(7)

where $\ket{\Omega}$ is the $N$ mode vacuum state. To W-encode the logical
qubit we pass the $N$ modes through a linear optical network implementing the
$N$-mode quantum Fourier transform,

$\displaystyle\hat{a}_{i}^{\dagger}\to\hat{W}^{\dagger}_{i}=\sum_{j=0}^{N-1}\hat{\mathrm{Q}}_{ij}\hat{a}^{\dagger}_{j},$
(8)

where,

$\displaystyle\hat{\mathrm{Q}}_{jk}=\dfrac{\omega_{N}^{jk}}{\sqrt{N}};\hskip
28.45274ptj,k\in\\{0,1,\dots,N-1\\},$ (9)

are matrix elements of the $N$-dimensional quantum Fourier transformation
operator $\hat{\rm{Q}}$ with $\omega_{N}=e^{\frac{2\pi i}{N}}$. This
transforms the logical qubit to the encoded qubit,

$\displaystyle\ket{W}=(\alpha\hat{W}^{\dagger}_{0}+\beta\hat{W}^{\dagger}_{1})\ket{\Omega},$
(10)

which represents the same state of quantum information, but in expanded form.
Next the W-encoded state passes through a noisy channel that independently
adds random phases to each optical mode,

$\displaystyle\hat{a}^{\dagger}_{j}\to e^{i\theta_{j}}\hat{a}^{\dagger}_{j},$
(11)

where the $\theta_{j}$ represent random variables, whose distribution is
considered arbitrary at this point, that form a vector $\vec{\theta}$
describing the phases applied to each mode. The state $\ket{W^{\vec{\theta}}}$
denotes the W-encoded state following application of the phase noise channel.
We now apply the decoding operation (the inverse QFT operation), and the first
two output modes represent the decoded logical state, $\hat{\rho}_{L}$.
Because of the noise in the channel we are not guaranteed to observe the
photon strictly within the first two modes. Thus we post-select and treat
cases where photons are found in the other modes as heralding a failure. The
intuition is then that for the heralded success cases the phase noise errors
would have been filtered out.

The fidelity of the decoded state compared to the input logical qubit
$\ket{L}$ is

$\displaystyle F_{N}=\bra{L}\hat{\rho}_{L}\ket{L},$ (12)

where $\ket{L}$ is implicitly a function of the superposition parameters
$\alpha$ and $\beta$. Note that the overlap between two states is invariant
under common unitary operations. As the encoding and decoding operations are
unitary it suffices to consider the fidelity of the W-encoded state,

$\displaystyle F_{N}$
$\displaystyle=\braket{W}{W^{\vec{\theta}}}\braket{W^{\vec{\theta}}}{W},$ (13)

where $F_{N}$ is used here to show that the fidelity will depend on the number
of modes used for the encoding $N$. Eq. (13) assumes knowledge of the phase
errors in each mode as represented by $\vec{\theta}$ but these are of course
unknown. However we can model them as independent random variables acting on
each mode separately according to some arbitrary distribution $p$,

$\displaystyle p(\vec{\theta})=\prod_{j=0}^{N-1}p_{i}(\theta_{j}),$ (14)

where $p_{j}$ is the distribution for mode $j$. The encoded state after
application of the error channel on average is given by,

$\displaystyle\hat{\rho}_{W}=\int
p(\vec{\theta})\ket{W^{\vec{\theta}}}\bra{W^{\vec{\theta}}}\,d\vec{\theta}.$
(15)

The fidelity between the output and input of the error channel is given by,

$\displaystyle F_{N}$ $\displaystyle=\bra{W}\left(\int
p(\vec{\theta})|W^{\vec{\theta}}\rangle\langle
W^{\vec{\theta}}|d\vec{\theta}\right)\ket{W}.$ (16)

As with all quantum operations, the noise channel is a linear map on the state
space. Let the channel map be denoted by $\mathcal{L}_{\vec{\theta}}$, then we
have for the encoded qubit state,

$\displaystyle\ket{W^{\vec{\theta}}}$
$\displaystyle=\mathcal{L}_{\vec{\theta}}(\ket{W})=\alpha\mathcal{L}_{\vec{\theta}}(\ket{W_{0}})+\beta\mathcal{L}_{\vec{\theta}}(\ket{W_{1}})$
$\displaystyle=\alpha\ket{W_{0}^{\vec{\theta}}}+\beta\ket{W_{1}^{\vec{\theta}}},$
(17)

where $\ket{W_{k}}=\hat{W}_{k}^{\dagger}\ket{\Omega}$, following the
definition in Eq. (8),

$\displaystyle\ket{W_{k}^{\vec{\theta}}}=\sum_{j}^{N-1}\hat{\mathrm{Q}}_{kj}e^{i\theta_{j}}a^{\dagger}_{j}\ket{\Omega}.$
(18)

These equations can now be used to compute $\hat{\rho}_{W}$,

$\displaystyle\hat{\rho}_{W}$ $\displaystyle=\int
p(\vec{\theta})\left(|W^{\vec{\theta}}\rangle\langle
W^{\vec{\theta}}|\right)d\vec{\theta}$ $\displaystyle=\int
p(\vec{\theta})\left(\sum\limits_{i,j=0}^{1}c_{i}c^{*}_{j}|W_{i}^{\vec{\theta}}\rangle\langle
W_{j}^{\vec{\theta}}|\right)d\vec{\theta},$ (19)

where $c_{0}=\alpha$ and $c_{1}=\beta$. Substituting the definition of
$\ket{W_{i}^{\vec{\theta}}}$ from Eq. (18) and defining,

$\displaystyle
k_{m,n}=(\alpha+\beta\omega_{N}^{m})(\alpha+\beta\omega_{N}^{n})^{*},$ (20)

we obtain,

$\displaystyle\hat{\rho}_{W}=\frac{1}{N}\int
p(\vec{\theta})\begin{pmatrix}k_{0,0}&k_{01}e^{i(\theta_{0}-\theta_{1})}&k_{0,2}e^{i(\theta_{0}-\theta_{2})}&\dots&k_{0,(N-1)}e^{i(\theta_{0}-\theta_{N-1})}\\\
k_{1,0}e^{i(\theta_{1}-\theta_{0})}&k_{1,1}&k_{12}e^{i(\theta_{1}-\theta_{2})}&\dots\\\
k_{2,0}e^{i(\theta_{2}-\theta_{0})}&k_{2,1}e^{i(\theta_{2}-\theta_{1})}&k_{2,2}&\dots\\\
\vdots&&&\ddots\\\
k_{(N-1),0}e^{i(\theta_{N-1}-\theta_{0})}&&&&k_{(N-1),(N-1)}\end{pmatrix}d\vec{\theta},$
(21)

where the matrix within the integral represents the state after the noise
channel in the photon number basis. The characteristic function of a
probability distribution $p(x)$ is defined as,

$\displaystyle\phi_{p(x)}(z)=\int\limits_{-\infty}^{\infty}p(x)e^{ixz}dx.$
(22)

If we assume all $\theta_{j}$ are identically and independently distributed as
$p(\theta)$ then we have,

$\displaystyle\lambda=\int
p(\vec{\theta})e^{i(\theta_{j}-\theta_{k})}d\vec{\theta}=|\phi_{p(\theta)}(1)|^{2},$
(23)

whenever the indices $j$ and $k$ are different. Thus,

$\displaystyle\hat{\rho}_{W}$
$\displaystyle=\frac{\lambda}{N}\sum_{i,j=0}^{N-1}(1-\delta_{ij})k_{i,j}|i\rangle\langle
j|+\frac{1}{N}\sum_{i}^{N-1}k_{i,i}|i\rangle\langle i|$
$\displaystyle=\lambda\left(|W\rangle\langle W|-\Delta(|W\rangle\langle
W|)\right)+\Delta(|W\rangle\langle W|)$ $\displaystyle=\lambda|W\rangle\langle
W|+(1-\lambda)\Delta(|W\rangle\langle W|),$ (24)

where we have used the fact that,

$\displaystyle\sum\limits_{i,j=0}^{N-1}\frac{k_{i,j}}{N}|i\rangle\langle
j|=|W\rangle\langle W|,$ (25)

and $\Delta$ is the completely dephasing map in the photon number basis
defined as,

$\displaystyle\Delta(\hat{\rho})=\sum_{i}\braket{i}{\hat{\rho}}{i}|i\rangle\langle
i|.$ (26)

We can see from Eq. (24) that the our error channel is essentially a dephasing
channel with dephasing parameter $\lambda$. To analyze our protocol further we
will choose a particular error model by assuming that the phase error in each
mode is distributed as a Gaussian with mean $\mu$ and variance
$\delta^{2}$111This assumption allows for values of $\theta$ that are larger
than single multiples of $2\pi$, but the theory used here does not need to be
changed to incorporate this. The operations used in defining the phase shift
channel are periodic and hence having larger value of phase does not
invalidate this description. However, it does mean that there is no unique
probability density function for any given distribution on the range
$[0,2\pi)$.,

$\displaystyle
p(\theta)=\frac{1}{\sqrt{2\pi}\delta}e^{-\frac{(\theta-\mu)^{2}}{2\delta^{2}}}.$
(27)

This is a natural choice when we do not have any knowledge about the nature of
the processes that generates the errors beyond that many underlying random
distributions average to give a final contribution to the error (à la the
central limit theorem). The characteristic function of a normal distribution
is given by,

$\displaystyle\phi_{p(\theta)}(z)=e^{-\frac{\delta^{2}z^{2}}{2}+i\mu z}.$ (28)

This leads to $\lambda=e^{-\delta^{2}}$ and,

$\displaystyle\hat{\rho}_{W}=e^{-\delta^{2}}|W\rangle\langle
W|+(1-e^{-\delta^{2}})\Delta\left(|W\rangle\langle W|\right).$ (29)

We can interpret the error channel as performing the identity with probability
$e^{-\delta^{2}}$ and applying the Fock basis dephasing operator with
probability $(1-e^{-\delta^{2}})$. This channel is thus a dephasing channel
with probability of no error occurring $p=e^{-\delta^{2}}$. In practical
terms, the variance $\delta^{2}$ of the phase error will depend on the
physical implementation of the quantum channel. For fibre-optic cables we
would generally expect the variance to increase with the length of the cable
$L$ or equivalently the propagation time of the photon in cable $t_{p}=L/v$,
where $v$ is the propagation velocity of the photon in the fibre. If we model
the variance as increasing linearly with propagation time, i.e,

$\displaystyle\delta^{2}=\frac{t_{p}}{T_{2}},$ (30)

where $T_{2}$ is a constant defining a characteristic time for the dephasing
channel, we can write down our error channel in the standard dephasing channel
notation as,

$\displaystyle\hat{\rho}_{W}$
$\displaystyle=\mathcal{E}^{\text{dephasing}}_{t_{p}}(|W\rangle\langle W|)$
$\displaystyle=e^{-t_{p}/T_{2}}|W\rangle\langle
W|+(1-e^{-t_{p}/T_{2}})\Delta(|W\rangle\langle W|).$ (31)

##  4 Error heralding

In implementing the protocol as described in Fig. 1, we can perform the post-
selection in two different ways:

  * •

Presence heralding: Success is assumed based upon the detection of exactly one
photon between the output modes 0 and 1, which define the logical qubit space.
Note that in the absence of quantum non-demolition measurements, this is
necessarily destructive, limiting its applicability. The heralding operator is
effectively the projector

$\displaystyle\hat{\Pi}_{\mathrm{presence}}=\hat{a}_{0}^{\dagger}\ket{\Omega}\bra{\Omega}\hat{a}_{0}+\hat{a}_{1}^{\dagger}\ket{\Omega}\bra{\Omega}\hat{a}_{1}.$
(32)

  * •

Absence heralding: Success is inferred via the detection of no photons in any
of the remaining modes outside the logical qubit space. This is non-
destructive on the logical qubit, broadening its utility. However, photon loss
contributes to the occurrence of this signature, implying higher error rates
on the remaining logical qubit. The heralding operator is equivalently a
projection given by

$\displaystyle\hat{\Pi}_{\mathrm{absence}}=\mathbb{\hat{I}}-\sum_{i=2}^{N-1}\hat{a}_{i}^{\dagger}\ket{\Omega}\bra{\Omega}\hat{a}_{i}$

###  4.1 Heralding probability

####  4.1.1 Absence heralding

We define the absence heralded probability $P_{H_{a}}$ as the probability that
no photons are detected in modes 2 to ($N-1$) and $\hat{\rho}_{\mathrm{out}}$
to be the $N$-mode state of the system at the end of the protocol before the
final measurement. Assuming a uniform loss model parameterized by $\eta$, for
our choice of input states the probability of detecting the photon in mode $m$
is

$\displaystyle(1-\eta)\cdot\braket{\Omega}{\hat{a}_{m}\hat{\rho}_{\mathrm{out}}\hat{a}_{m}^{\dagger}}{\Omega},$
(33)

where $\ket{\Omega}$ is the global vacuum state. Using this expression for the
probability of detection under loss we can see that,

$\displaystyle P_{H_{a}}$ $\displaystyle=\text{Pr}(\text{Photon is in modes 0
or 1})$ $\displaystyle+\text{Pr}(\text{Photon is in modes 2-($N-1$)})$
$\displaystyle\times\text{Pr}(\text{Loss in modes 2-($N-1$)}),$
$\displaystyle=\sum_{i=0}^{1}\braket{\Omega}{\hat{a}_{i}\hat{\rho}_{\mathrm{out}}\hat{a}_{i}^{\dagger}}{\Omega}+\eta\left(1-\sum_{i=0}^{1}\braket{\Omega}{\hat{a}_{i}\hat{\rho}_{\mathrm{out}}\hat{a}_{i}^{\dagger}}{\Omega}\right),$
$\displaystyle=\eta+(1-\eta)\left(\sum_{i=0}^{1}\braket{\Omega}{\hat{a}_{i}\hat{\rho}_{\mathrm{out}}\hat{a}_{i}^{\dagger}}{\Omega}\right),$
$\displaystyle=\eta+(1-\eta)\left(\sum_{i=0}^{1}\braket{\Omega}{\hat{a}_{i}{\rm\hat{Q}}^{\dagger}\hat{\rho}_{W}{\rm\hat{Q}}\hat{a}_{i}^{\dagger}}{\Omega}\right),$
$\displaystyle=\eta+(1-\eta)\left(\sum_{i=0}^{1}\braket{W_{i}}{\hat{\rho}_{W}}{W_{i}}\right).$
(34)

Using Eq. (24) we can write,

$\displaystyle\braket{W_{k}}{\hat{\rho}_{W}}{W_{k}}$
$\displaystyle=\lambda\left|\braket{W_{k}}{W}\right|^{2}$
$\displaystyle+(1-\lambda)\braket{W_{k}}{\Delta(|W\rangle\langle W|)}{W_{k}}.$
(35)

The first term on the R.H.S. of Eq. (4.1.1) is simply,

$\displaystyle\left|\braket{W_{0}}{W}\right|^{2}$
$\displaystyle=|\alpha|^{2},$
$\displaystyle\left|\braket{W_{1}}{W}\right|^{2}$ $\displaystyle=|\beta|^{2},$
(36)

which are the values of $k$ that preserve the encoded qubit. The second term
can be calculated as,

$\displaystyle\braket{W_{k}}{\Delta(|W\rangle\langle W|)}{W_{k}}=$
$\displaystyle\frac{1}{N^{2}}\sum_{m,n=0}^{1}\sum_{j,q,p=0}^{N-1}\omega_{N}^{(m-n)j+(q-p)k}c_{m}c_{n}^{*}\delta_{pj}\delta_{jq},$
$\displaystyle=\frac{1}{N^{2}}\sum_{m,n=0}^{1}\sum_{j=0}^{N-1}\omega_{N}^{(m-n)j}c_{m}c_{n}^{*}$
$\displaystyle=\frac{1}{N}\sum_{m,n=0}^{1}\delta_{mn}c_{m}c_{n}^{*}=\frac{1}{N},$
(37)

where in the first equality we have used the fact that,

$\displaystyle\ket{W_{k}}=\sum_{q=0}^{N-1}\hat{\rm{Q}}_{kq}\hat{a}_{q}^{\dagger}\ket{\Omega},$
(38)

and

$\displaystyle\Delta(|W\rangle\langle
W|)=\sum\limits_{m,n=0}^{1}c_{m}c_{n}^{*}\sum_{j=0}^{N-1}\omega_{N}^{(m-n)j}\hat{a}_{j}^{\dagger}|\Omega\rangle\langle\Omega|\hat{a}_{j}.$
(39)

This implies that the photon in the error state is equally spread over all the
modes after decoding. If the state contains an error, the heralding will
detect it with a probability of $\frac{N-2}{N}$ and miss it with probability
$\frac{2}{N}$. So there will be a linear advantage in error detection with the
number of modes. Substituting these results in Eq. (34) we get,

$\displaystyle P_{H_{a}}$
$\displaystyle=\eta+(1-\eta)\left[\lambda|\alpha|^{2}+\frac{1}{N}(1-\lambda)\right]$
$\displaystyle+(1-\eta)\left[\lambda|\beta|^{2}+\frac{1}{N}(1-\lambda)\right]$
$\displaystyle=\eta+(1-\eta)\left[\lambda+\frac{2}{N}(1-\lambda)\right].$ (40)

If we assume the Gaussian error model in Eq. (27) this will reduce to,

$\displaystyle
P_{H_{a}}=\eta+(1-\eta)\left[e^{-\delta^{2}}+\frac{2}{N}(1-e^{-\delta^{2}})\right].$
(41)

As the number of modes $N$ increases the heralded probability will decrease,
this is because the probability of the error state being in the modes 1 and 2
is inversely proportional to $N$. As we connected the phase error variance to
a $T_{2}$ time via Eq. (30), we can also reparameterize the loss probability
as $\eta=1-e^{-tp/T_{1}}$. In terms of the $T_{1}$ and $T_{2}$ parameters and
propagation time $t_{p}$, the absenence heralded probability can be written
as,

$\displaystyle P_{H_{a}}$ $\displaystyle=(1-e^{-t_{p}/T_{1}})$
$\displaystyle+e^{-t_{p}/T_{1}}\left[e^{-t_{p}/T_{2}}+\frac{2}{N}(1-e^{-t_{p}/T_{2}})\right].$
(42)

####  4.1.2 Presence heralding

The presence heralded case is the case where we post-select on there being no
photon loss. The presence heralded fidelity is the probability of getting a
photon in modes 1 and 2 and this can be easily seen to be,

$\displaystyle P_{H_{p}}=P_{H_{a}}-\eta.$ (43)

###  4.2 Heralded fidelity

####  4.2.1 Absence heralding

The absence heralded state is the state in the output modes 0 and 1 when no
photons are detected in the modes modes 2 – $(N-1)$. This can happen in two
mutually exclusive ways; either the photon is lost and there is no photon in
any mode or there is no loss and our negative measurement of modes 2 – $(N-1)$
projects the quantum state into the subspace spanned by
$a_{0}^{\dagger}\ket{\Omega}$ and $a_{1}^{\dagger}\ket{\Omega}$. So, the
absence heralded state is given by,

$\displaystyle\hat{\rho}_{H_{a}}=(1-\eta)\frac{\hat{\Pi}_{0,1}\hat{\rho}_{\mathrm{out}}\hat{\Pi}_{0,1}}{\operatorname{Tr}(\hat{\Pi}_{0,1}\hat{\rho}_{\mathrm{out}})}+\eta|\Omega\rangle\langle\Omega|,$
(44)

where,

$\displaystyle\hat{\Pi}_{0,1}=a_{0}^{\dagger}|\Omega\rangle\langle\Omega|a_{0}+a_{1}^{\dagger}|\Omega\rangle\langle\Omega|a_{1},$
(45)

is the projector on to the subspace of modes 1 and 2. But we observe that,

$\displaystyle\operatorname{Tr}(\hat{\Pi}_{0,1}\hat{\rho}_{\mathrm{out}})$
$\displaystyle=\braket{\Omega}{\hat{a}_{0}\hat{\rho}_{\mathrm{out}}\hat{a}_{0}^{\dagger}}{\Omega}+\braket{\Omega}{\hat{a}_{1}\hat{\rho}_{\mathrm{out}}\hat{a}_{1}^{\dagger}}{\Omega}$
$\displaystyle=\braket{\Omega}{\hat{a}_{0}\hat{\rm{Q}}^{\dagger}\hat{\rm{Q}}\hat{\rho}_{\mathrm{out}}\hat{\rm{Q}}^{\dagger}\hat{\rm{Q}}\hat{a}_{0}^{\dagger}}{\Omega}$
$\displaystyle+\braket{\Omega}{\hat{a}_{1}\hat{\rm{Q}}^{\dagger}\hat{\rm{Q}}\hat{\rho}_{\mathrm{out}}\hat{\rm{Q}}^{\dagger}\hat{\rm{Q}}\hat{a}_{1}^{\dagger}}{\Omega}$
$\displaystyle=\braket{W_{0}}{\hat{\rho}_{W}}{W_{0}}+\braket{W_{1}}{\hat{\rho}_{W}}{W_{1}}$
$\displaystyle=\lambda+\frac{2}{N}(1-\lambda).$ (46)

The fidelity of the heralded state with our logical input state is given by,

$\displaystyle F_{H_{a}}$ $\displaystyle=\braket{L}{\hat{\rho}_{H_{a}}}{L}$
$\displaystyle=\braket{L}{{\rm\hat{Q}}^{\dagger}{\rm\hat{Q}}\hat{\rho}_{H_{a}}{\rm\hat{Q}}^{\dagger}{\rm\hat{Q}}}{L}$
$\displaystyle=(1-\eta)\frac{\braket{W}{\hat{\rho}_{W}}{W}}{\operatorname{Tr}(\hat{\Pi}_{W_{0},W_{1}}\rho_{W})}$
$\displaystyle=\frac{1-\eta}{\lambda+\frac{2}{N}(1-\lambda)}\cdot
F(\ket{W},\hat{\rho}_{W}),$ (47)

where,

$\displaystyle\hat{\Pi}_{W_{0},W_{1}}=|W_{0}\rangle\langle
W_{0}|+|W_{1}\rangle\langle W_{1}|,$ (48)

and in the third equality we have used the fact that
$\hat{\Pi}_{0,1}|\Omega\rangle\langle\Omega|\hat{\Pi}_{0,1}=0$. We can
interpret Eq. (47) as saying that heralding improves the output fidelity of
our protocol by a factor of,

$\displaystyle\frac{1-\eta}{\lambda+\frac{2}{N}(1-\lambda)}.$ (49)

We have,

$\displaystyle
F(\ket{W},\hat{\rho}_{W})=\lambda+(1-\lambda)\braket{W}{\Delta(|W\rangle\langle
W|)}{W}.$ (50)

Notice that,

$\displaystyle\braket{W}{\Delta(|W\rangle\langle W|)}{W}$
$\displaystyle=\operatorname{Tr}(|W\rangle\langle W|\Delta(|W\rangle\langle
W|))$ $\displaystyle=\sum_{i=0}^{N-1}([|W\rangle\langle W|]_{ii})^{2},$ (51)

where, $[|W\rangle\langle W|]_{ii}$ are diagonal elements of the the state
$|W\rangle\langle W|$ in the computational basis. We know that,

$\displaystyle[|W\rangle\langle W|]_{ii}=$
$\displaystyle\braket{\Omega}{\hat{a}_{i}\left(\sum_{m,n=0}^{1}c_{m}c_{n}^{*}\sum_{k,j=0}^{N-1}\hat{\rm{Q}}_{mk}\hat{a}_{k}^{\dagger}|\Omega\rangle\langle\Omega|\hat{a}_{j}\hat{\rm{Q}}^{*}_{nj}\right)\hat{a}_{i}^{\dagger}}{\Omega}$
$\displaystyle=\frac{1}{N}\sum_{m,n=0}^{1}\sum_{k,j=0}^{N-1}c_{m}c_{n}^{*}\omega_{N}^{mk-
nj}\delta_{ik}\delta_{ji}$
$\displaystyle=\frac{1}{N}\sum_{m,n=0}^{1}c_{m}c_{n}^{*}\omega_{N}^{(m-n)i}.$
(52)

Using the above expression we obtain,

$\displaystyle\braket{W}{\Delta(|W\rangle\langle W|)}{W}$
$\displaystyle=\sum_{i=0}^{N-1}\left(\frac{1}{N}\sum_{m,n=0}^{1}c_{m}c_{n}^{*}\omega_{N}^{(m-n)i}\right)\left(\frac{1}{N}\sum_{p,q=0}^{1}c_{p}c_{q}^{*}\omega_{N}^{(p-q)i}\right)$
$\displaystyle=\frac{1}{N^{2}}\sum_{i=0}^{1}\sum_{\begin{subarray}{c}m,n,\\\
p,q\end{subarray}=0}^{1}c_{m}c_{p}c_{n}^{*}c_{q}^{*}\omega_{N}^{(m-n+p-q)i}$
$\displaystyle=\frac{1}{N}\sum_{\begin{subarray}{c}m,n,\\\
p,q\end{subarray}=0}^{1}c_{m}c_{p}c_{n}^{*}c_{q}^{*}\delta_{m-n+p,q}$
$\displaystyle=\frac{1}{N}\sum_{\begin{subarray}{c}m,n,p=0\\\ m-n+p\geq
0\end{subarray}}^{1}c_{m}c_{p}c_{n}^{*}c_{m-n+p}^{*}$
$\displaystyle=\frac{|\alpha|^{4}+|\beta|^{4}+4|\alpha|^{2}|\beta|^{2}}{N}$
$\displaystyle=\frac{1+2|\alpha|^{2}|\beta|^{2}}{N}.$ (53)

Therefore,

$\displaystyle
F_{H_{a}}=(1-\eta)\cdot\frac{\lambda+(1-\lambda)\frac{1+2|\alpha|^{2}|\beta|^{2}}{N}}{\lambda+\frac{2}{N}(1-\lambda)}.$
(54)

In terms of Bloch variables $\theta$ and $\phi$ where
$\alpha=\cos\frac{\theta}{2}$, $\beta=e^{i\phi}\sin\frac{\theta}{2}$ as, it
can be seen that,

$\displaystyle
F_{H_{a}}=(1-\eta)\cdot\frac{e^{-\delta^{2}}+(1-e^{-\delta^{2}})(\frac{2+\sin^{2}\theta}{2N})}{e^{-\delta^{2}}+\frac{2}{N}(1-e^{-\delta^{2}})}.$
(55)

In the limit of large $N$ the heralded fidelity will approach $(1-\eta)$. This
implies that the only error will be from photon loss. In terms of the
dephasing and amplitude damping channel parameters $T_{2}$ and $T_{1}$ and a
propagation time $t_{p}$ this can be written as,

$\displaystyle
F_{H_{a}}=e^{-t_{p}/T_{1}}\cdot\frac{e^{-t_{p}/T_{2}}+(1-e^{-t_{p}/T_{2}})(\frac{2+\sin^{2}\theta}{2N})}{e^{-t_{p}/T_{2}}+\frac{2}{N}(1-e^{-t_{p}/T_{2}})}.$
(56)

####  4.2.2 Presence heralding

In the presence heralded case, we are post selecting the case where there is
no photon loss in the system, so the post measurement state in this scenario
will be,

$\displaystyle\hat{\rho}_{H_{p}}=\frac{\hat{\Pi}_{0,1}\hat{\rho}_{\mathrm{out}}\hat{\Pi}_{0,1}}{\operatorname{Tr}(\hat{\Pi}_{0,1}\hat{\rho}_{\mathrm{out}})}$
(57)

this just improves the fidelity by a factor of $(1-\eta)$ giving, the fidelity
as,

$\displaystyle F_{H_{p}}=\frac{F_{H_{a}}}{1-\eta}.$ (58)

The heralded probability is plotted in Fig. 2 as a function of $\delta$ and
$T_{2}$ with fixed values of $\eta$ and $T_{1}$ respectively. From these plots
it is evident that the heralded fidelity improves with the number of modes
$N$. The choice of the parameters values $T_{1}$ and $\eta$ do not influence
the ordering of these plots.

Heralding type | Probability | Fidelity
---|---|---
Presence | $P_{H_{p}}=(1-\eta)[\lambda+\frac{2}{N}(1-\lambda)]$ | $F_{H_{p}}=\frac{N\lambda+(1-\lambda)(1+2|\alpha\beta|^{2})}{(N-2)\lambda+2}$
Presence ($N\to\infty$) | $P_{H_{p}}=(1-\eta)\lambda$ | $F_{H_{p}}=1$
Absence | $\eta+P_{H_{p}}$ | $(1-\eta)F_{H_{p}}$
Table 1: Heralding probabilities and post-selected logical qubit fidelities of
a single photon qubit under the W-state encoding protocol, according to the
two different modes of post-selection operation. Note that here
$\lambda=|\phi_{p(\theta)}(1)|^{2}$ as defined in equation (23).

###  4.3 Probability and fidelity plots

The heralding probability and associated post-selected state fidelities are
shown as a function of the channel parameters in Fig. 2, and the respective
analytic expressions in Tab. 1. We note that while we are specifically
plotting for a Guassian noise model, the qualitative features can be expected
to be the same for any i.i.d. error model. This is because depolarizing
parameter $\lambda$ is related to the characteristic function
$\phi_{p(\theta)}(z)$ through equation (23). A function and it’s Fourier
transform will have their variances inversely related like quadrature
variances so even if the exact expressions for the fidelity and heralding
probabilities might vary we can expect the qualitative behaviour to remain the
same and the average map to be a depolarizing channel.

Figure 2: Analytic heralded error-correction results for the absence heralding
technique. Results for presence heralding are given by simple transformations
of these results (shift by $\eta$ for $P_{H}$, and scale by $(1-\eta)$ for
$F_{H}$). (left) Heralding probability and (right) post-selected fidelity,
parameterized in terms of loss-rate, and dephasing in terms of (top) phase-
variance $\delta$, (bottom) $T_{2}$-time for $t_{p}=1$.

##  5 Single-qubit unitary operations

Once in the encoded basis, can we directly perform single-qubit unitary
operations, without the rigmarole of decoding and encoding? The answer is yes.

Consider the single qubit operation,

$\displaystyle\ket{\psi_{\mathrm{out}}}_{L}=\hat{U}\ket{\psi_{\mathrm{in}}}_{L},$
(59)

in the logical qubit basis. In the encoded photonic basis, this can be
expressed,

$\displaystyle\ket{\psi_{\mathrm{out}}}_{L}$
$\displaystyle=\hat{Q}^{\dagger}\hat{Q}[\hat{U}\oplus\mathbb{\hat{I}}_{N-2}]\hat{Q}^{\dagger}\hat{Q}\ket{\psi_{\mathrm{in}}}_{L},$
(60)

where we have inserted the identity operation $\mathbb{\hat{I}}_{N-2}$ on the
ancillary input photonic modes, and
$\mathbb{\hat{I}}=\hat{Q}^{\dagger}\hat{Q}$. This yields the equivalent
redundantly-encoded photonic unitary operation (i.e between encoding and
decoding),

$\displaystyle\ket{\psi_{\mathrm{out}}}_{\mathrm{enc}}=\tilde{U}\ket{\psi_{\mathrm{in}}}_{\mathrm{enc}},$
(61)

where,

$\displaystyle\tilde{U}=\hat{Q}[\hat{U}\oplus\mathbb{\hat{I}}_{N-2}]\hat{Q}^{\dagger},$
(62)

is the redundantly-encoded equivalent of the logical 2-qubit operation,
obtained by conjugating with $\hat{Q}$.

##  6 Discussion

###  6.1 Pros and cons

Our scheme has the following advantages:

    1. 1.

It can be implemented in a number of quantum memory architectures such as
atomic ensembles, optical cavities and delay lines.

    2. 2.

Any independent uncorrelated phase noise can be corrected for with sufficient
levels of encoding. The physical source of these phase errors will depend on
the particular architecture. For example, in a delay line temporal mismatch
between photon arrival times will manifest as a phase error. If this is
influenced by thermal fluctuations, it will be manifested as a dephasing error
reflecting our theoretical calculations of the post error state. In an optical
cavity array, the source of the phase error could be decay rate mismatch
between cavities. All these cases are consistent with our model.

    3. 3.

Normally to correct phase mismatch one could either thermally or mechanically
isolate the system or use a high intensity source to periodically measure
phase errors and actively correct it using feedback. Our scheme mitigates
this.

    4. 4.

Because we only need passive linear optics without feed-forward, this is quite
scalable with present-day technology, notably integrated photonic waveguide
chips.

    5. 5.

Robustness against mode loss. Standard QECs require the use of entangling
gates such as CNOTs and the code state themselves can be highly entangled such
as the GHZ states. These states however are not robust against loss in the
sense of a partial trace operation while the W-state encoding will robust
against such loss and increasingly so with higher levels of encoding.

The disadvantages of our scheme are:

    1. 1.

Inability to correct correlated phase fluctuations (e.g a uniform phase shift
across all redundant memory cells).

    2. 2.

A multiplier in production cost and resource overhead, determined by the
degree of redundancy.

###  6.2 Economic justification

There is a strong economic argument for the merit of our scheme, regardless of
the state of engineering.

The main economic overhead associated with the protocol is the substitution of
a single quantum memory with a bank of $N$ identical ones, a roughly linear
cost overhead. However, via this trade-off, dephasing processes inherent
within them can be asymptotically suppressed, enabling the construction of a
quantum memory bank that overcomes the fidelity bounds of a single one.

Given that the engineering and production cost of a single quantum memory unit
can increase exponentially with inverse infidelity, an $N$-fold cost overhead
is expected to be a more economically efficient mechanism for noise-
suppression in the regime of very high fidelity targets.

The net cost of the memory bank scales roughly linearly with $N$, whereas the
cost of a single cell within it grows exponentially with fidelity. The
economically optimal configuration is determined by the minimum of this cost
trade-off curve over $N$, for a given target fidelity $F_{\mathrm{target}}$,

$\displaystyle C_{\mathrm{net}}(F_{\mathrm{target}})\approx N\cdot
C_{\mathrm{unit}}(F_{\mathrm{unit}}),$ (63)

where $C_{\mathrm{unit}}(F_{\mathrm{unit}})$ is the engineering cost of a
single memory cell with fidelity $F_{\mathrm{unit}}$, and the relationship
between the target and unit fidelity follows from the respective heralded
fidelity given in Tab. 1, related by $N$. The crossover point, at which it
becomes economically efficient to begin utilising our encoding, occurs when
manufacturing a single memory cell with the target fidelity matches that of a
redundant bank of cells with lower unit cost Duan et al. [2001],

$\displaystyle N\cdot C_{\mathrm{unit}}(F_{\mathrm{unit}})\approx
C_{\mathrm{unit}}(F_{\mathrm{target}}).$ (64)

###  6.3 Robustness against different noise models

Whilst we have used a Gaussian noise model for detailed analysis in Section 3,
this is by no means an absolute requirement on many of the results we present.
As mentioned in Section 4 C, the property that determines the output state is
the characteristic function for the random variables in the noise model for
the unitary errors evaluated at $z=1$. Due to the nature of the characteristic
function, this value will be well defined in virtually all possible
distributions, even ones that do not have a well defined moment generating
function. The exact details of specific properties of the scheme will change
under different distributions (e.g. the $T_{1}$ and $T_{2}$ decoherence
factors identified here won’t be well defined in general), but the analysis
from the point of view of the encoded state will be essentially the same as
what we have presented here.

###  6.4 Compatability with no-go theorem

It might seem that this protocol permits the distillation of entanglement that
is present within the input state using only operations that maintain the
Gaussian form of a Gaussian state, which has been proven impossible by a no-go
theorem Eisert et al. [2002]. The conditions for the no-go theorem do not
apply here as the final heralding measurement, which heralds success on a
(Gaussian) vacuum state, is overall not a Gaussian measurement under our
proposal to measure in the Fock basis. The input Fock states are also non-
Gaussian. This is similar to how the no-go Gaussian distillation theorem is
avoided in current bosonic entanglement distillation schemes Browne et al.
[2003], Ralph and Lund [2009], Xiang et al. [2010], Walk et al. [2013].

##  7 Comparison with other schemes

The idea of error filtration in a passive linear optic network has been
explored in Gisin et al. [2005], Li et al. [2007], Jiang et al. [2017].
Broadly these schemes transmit a photon through a linear optical network such
that some measurement outcomes will indicate an uncorrupted state in some
output. We have formalized this intuition by giving an explicit code space and
showed how it is robust against mode loss and i.i.d. dephasing noise.

Other schemes have explored the use of probabilistic gates to protect against
transmission loss such as Ewert and van Loock [2017] where optical Bell
measurements are used along with a parity encoding. However the encoding
states used for these schemes are highly entangled GHZ-like states. These
states cannot be deterministically prepared using passive linear optics
without introducing active feed-forward and additional photons to accommodate
the higher level of encoding — making such schemes highly impractical using
present-day technology.

It is important to clarify the distinction between this protocol, which can be
regarded as a form of error correction, and the more general concept of fault-
tolerance where gate errors are accommodated for.

Here we have assumed that our encoding and decoding operations are ideal, and
all the dephasing errors are associated with the channel between them.
Furthermore, we are not considering full quantum computations, but rather the
storage or communication of just a single photonic qubit.

While future work might consider the effects of errors in the encoding and
decoding errors in this protocol, the presented analysis is nonetheless
reasonably well justified in most practical circumstances.

Current linear optics technology, both using discrete elements or in
integrated wave-guides, has become extremely mature and precise, enabling
passive linear networks to be implemented with very high degrees of fidelity.

On the other hand, photonic qubits communicated over long-distance links, via
any medium, or which are held in quantum memories by coupling them to non-
optical physical systems, are far more likely to contribute to these noise
processes.

A further distinction between our scheme and conventional error correction
schemes, is that we don’t rely on any notion of code concatenation to
asymptotically improve error thresholds. Instead, we directly expand our level
of encoding by increasing the number of optical modes in the fan-out operation
implemented by the QFT encoding operation.

Unlike most well-known codes whereby error syndrome measurements are used to
apply feed-forward corrections to encoded qubits, this protocol does not rely
on any form of active correction via syndrome extraction. Rather, dephasing
noise is effectively mapped to non-determinism, such that upon success the
effective dephasing rate has been reduced.

The final important distinction between this scheme and conventional QEC
schemes, is that we do not create our encoded state via the introduction of
additional qubits (i.e photons), but via the the introduction of additional
optical modes, where the number of photons is preserved.

Owing to these conceptual differences compared to more familiar QEC and fault-
tolerance techniques, our scheme as presented is especially suited to the
context of photonic quantum communication or storage via coupling into quantum
memories.

##  8 Conclusion

We have proposed a passive linear optics encoding, using W-states which have
the property of being strongly robust against entanglement degradation from
qubit loss. This encoding was shown to be robust against any dephasing error
modelled as an uncorrelated independent and identically distributed dephasing
process on each subsystem. We showed that the effective error probability is
inversely related to the level of encoding $N$, vanishing in the large $N$
limit. The loss rate upper-bounds the fidelity and success probability, but
its effect does not scale with $N$, given that uniform losses can be commute
through passive linear optics systems.

The protocol is naturally suited to optical quantum memories (e.g via atomic
ensembles, cavities, or delay lines), where the dominant error processes are
independent dephasing and loss. Single-qubit operations are readily
implementable within the encoded basis using conjugated, passive, linear
optics operations.

We argued that for high-fidelity quantum memories, utilising this technique
complimentary to improving engineering precision, has merit from an economic
perspective, given the only linear overhead in cost associated with simple
redundancy, versus the far greater cost of improving engineering precision.

## Acknowledgements

This work was supported by the Australian Research Council Centre of
Excellence for Engineered Quantum Systems (Grant No. CE170100009), and
Australian Research Council Centre of Excellence for Quantum Computation &
Communication Technology (Grant No. CE170100012). Peter Rohde is funded by an
ARC Future Fellowship (project FT160100397). Austin Lund would also like to
acknowledge fruitful discussions with Joshua Combes and Nicholas Menicucci in
relation to this work. Madhav Krishnan Vijayan would like to thank Alexis Shaw
for an insightful discussion on the physical source of errors.

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