# Flying-cat parity checks for quantum error correction

Z. M. McIntyre<EMAIL_ADDRESS>W. A. Coish
<EMAIL_ADDRESS>Department of Physics, McGill University, 3600 rue
University, Montreal, QC, H3A 2T8, Canada

###### Abstract

Long range, multi-qubit parity checks have applications in both quantum error
correction and measurement-based entanglement generation. Such parity checks
could be performed using qubit-state-dependent phase shifts on propagating
pulses of light described by coherent states $\ket{\alpha}$ of the
electromagnetic field. We consider “flying-cat” parity checks based on an
entangling operation that is quantum non-demolition (QND) for Schrödinger’s
cat states $\ket{\alpha}\pm\ket{-\alpha}$. This operation encodes parity
information in the phase of maximally distinguishable coherent states
$\ket{\pm\alpha}$, which can be read out using a phase-sensitive measurement
of the electromagnetic field. In contrast to many implementations, where
single-qubit errors and measurement errors can be treated as independent,
photon loss during flying-cat parity checks introduces errors on physical
qubits at a rate that is anti-correlated with the probability for measurement
errors. We analyze this trade-off for three-qubit parity checks, which are a
requirement for universal fault-tolerant quantum computing with the subsystem
surface code. We further show how a six-qubit entangled “tetrahedron” state
can be prepared using these three-qubit parity checks. The tetrahedron state
can be used as a resource for controlled quantum teleportation of a two-qubit
state, or as a source of shared randomness with potential applications in
three-party quantum key distribution.

##  I Introduction

The ability to scale up quantum computers is crucial for the long-term goal of
performing computations fault-tolerantly. Many current proposals for scaling
up take a modular approach where groups of qubits at the nodes of a network
are linked by quantum photonic interconnects [1, 2, 3]. Recent experiments
have shown significant progress, demonstrating entanglement distribution
across nodes that are separated by tens of metres [4, 5, 6]. In a distributed
architecture, the ability to perform long-range (inter-node) parity checks
would also provide a clear path towards implementing quantum error correction
[7, 8].

Many quantum error correcting codes, including the surface code [9, 10, 11],
fall within a class of stabilizer codes [12] known as Calderbank-Shor-Steane
(CSS) codes [13, 14]. In CSS codes, errors can be detected by measuring the
parities of groups of qubits, given by the eigenvalues of stabilizer operators
of the form $X^{\otimes n}$ and $Z^{\otimes m}$, where $X$ and $Z$ are
Pauli-$X$ and Pauli-$Z$ operators, respectively. Here, $n$ and $m$ are
integers giving the weight of the stabilizer, equal to the number of qubits on
which the operator acts nontrivially. A parity check can be performed using a
sequence of entangling gates applied between an ancilla qubit and each of
several code qubits. The ancilla qubit can then be measured to infer the
parity of the code qubits.

There are many strategies that can be used to implement two-qubit gates at
long range. Given a source of distributed entanglement, these gates can be
performed by teleportation [15, 16, 17, 18]. Alternatively, long-range two-
qubit gates can be realized in solid state systems by coupling qubits via the
real or virtual excitations of superconducting [19, 20, 21, 22, 23],
ferromagnetic [24, 25], or normal-metal [26] elements.

Rather than mapping parity onto an ancilla qubit, parity information can more
broadly be extracted by measuring any observable—not necessarily corresponding
to a qubit observable—whose outcome depends on parity. In a circuit quantum
electrodynamics architecture, for instance, this could be a shift in the
resonance frequency of a microwave resonator dispersively coupled to multiple
code qubits [27, 28, 29, 30, 31, 32].

In an architecture involving photonic interconnects, a natural way to connect
qubits is using pulses of light. An inter-node parity check could then be
performed via qubit-state-selective phase shifts $\alpha\rightarrow
e^{i\phi}\alpha$ on a light pulse prepared in a coherent state $\ket{\alpha}$.
Such state-selective phase shifts have been analyzed theoretically as a means
of implementing quantum gates [33, 34]. In addition, small conditional phase
shifts ($\phi<\pi$) generated from weak light-matter interactions have been
theoretically considered as a resource for parity measurements [35, 36]. More
recently, large phase shifts ($\phi\simeq\pi$) have been realized
experimentally in both the optical [37, 38] and microwave [39, 40, 41, 42]
domains. These experiments demonstrate the enabling technology for parity
checks based on “flying-cat” states given by superpositions of macroscopically
distinct coherent states,
$\ket{\mathrm{C}_{\pm}}\propto\ket{\alpha}\pm\ket{-\alpha}$, associated with
propagating pulses of optical or microwave light. In ancilla-based parity
checks (involving gates on a common ancilla qubit), measurement errors and
gate errors typically have independent physical origins and thus occur with
independent probabilities. This is not the case for flying-cat parity checks,
where these two error sources are correlated with the average number of
photons in the coherent state, $\lvert\alpha\rvert^{2}$. In particular, there
is a trade-off between the rate of measurement errors (errors made in
discriminating $\ket{\pm\alpha}$) and the rate of errors on code qubits due to
backaction arising from photon loss. While the coherent-state
distinguishability is controlled by a finite overlap
$\langle\alpha|-\alpha\rangle$ that decreases with increasing $|\alpha|^{2}$,
the rate of photon loss increases with increasing $|\alpha|^{2}$.

In this paper, we quantify the performance of flying-cat parity checks in the
presence of photon loss. To do this, we derive a superoperator describing the
backaction of photon loss on code qubits during a weight-3 flying-cat parity
check. From this superoperator, error patterns on code qubits together with
their associated error probabilities can be identified. This knowledge can
then be used to optimize $\alpha$ to minimize the total error probability.
High quality weight-3 parity checks have applications in both universal
quantum computing with surface codes [43] and entanglement distribution in
multi-node quantum networks. As a short-term demonstration of these ideas, we
further show that weight-3 Pauli-$X$ and Pauli-$Z$ parity checks can be used
to prepare a particular six-qubit entangled state. This state can be used for
three-party controlled quantum teleportation of an arbitrary two-qubit state.
It also provides a natural testbed for flying-cat parity checks in early
implementations, en route to the more ambitious goal of performing fault-
tolerant quantum computation.

The layout of this paper is as follows. In Sec. II, we describe how qubit-
state-selective phase shifts imprint parity information onto propagating light
pulses. In Sec. III, we discuss the aforementioned trade-off between
measurement errors and errors due to photon loss. Finally, in Sec. IV, we
illustrate how flying-cat parity checks can be used to generate entanglement
in a multi-node network. In particular, we show how they can be used to
prepare a six-qubit entangled state that can be used as a resource for three-
party controlled quantum teleportation.

##  II Coherent states as parity meters

A parity measurement of two or more physical qubits can be performed by
allowing a propagating light pulse in coherent state $\ket{\alpha}$ to
interact with each qubit according to the entangling operation described by

$\displaystyle\begin{aligned}
&\ket{0,\pm\alpha}\rightarrow\ket{0,\pm\alpha},\\\
&\ket{1,\pm\alpha}\rightarrow\ket{1,\mp\alpha},\end{aligned}$ (1)

where $\ket{0}$ and $\ket{1}$ are the qubit Pauli-$Z$ eigenstates [with
$Z\ket{s}=(-1)^{s}\ket{s}$ for $s=0,1$]. This operation can be realized by
coupling a three-level system with states $\ket{1},\ket{0}$, and $\ket{e}$ to
a single quantized cavity mode, such that the $\ket{0}\leftrightarrow\ket{e}$
transition is resonantly coupled to the cavity mode [33, 34] (Fig. 1). An
input coherent state $\ket{\alpha}$ occupying a propagating spatiotemporal
mode (light pulse) that is resonant with the decoupled cavity frequency
$\omega_{\mathrm{c}}$ will then acquire a different phase upon reflection from
the cavity, depending on the qubit state. When the qubit is in state
$\ket{1}$, the cavity frequency is unaffected by the presence of the qubit,
and the input pulse is reflected with a $\pi$ phase shift [44]. However, when
the qubit is in state $\ket{0}$, hybridization of the qubit and cavity
suppresses the cavity density of states at $\omega_{\mathrm{c}}$, and the
input pulse is reflected with no phase shift. This is of course not the only
way of realizing the interaction given in Eq. (1). Another way to realize this
scenario would be to introduce a large dispersive coupling that shifts the
cavity resonance frequency by more than a linewidth, conditioned on the qubit
state.

Figure 1: A qubit-state-selective phase shift on an incoming coherent state
$\ket{\alpha}$ occupying some propagating quasimode of the electromagnetic
field can be obtained by resonantly coupling a cavity mode having frequency
$\omega_{\mathrm{c}}$ to the transition $\ket{0}\leftrightarrow\ket{e}$
involving qubit state $\ket{0}$ and some auxiliary level $\ket{e}$ [33, 34,
40, 38, 41].

After interacting with $n$ qubits in state $\ket{s_{1}s_{2}\dots s_{n}}$
($s_{i}\in\\{0,1\\}$), $\ket{\alpha}$ will have acquired a total phase shift
$(-1)^{s}=e^{is\pi}$, where $s=\sum_{i}s_{i}\>(\mathrm{mod}\>2)$ is the
$Z$-basis parity of the $n$ qubits:

$\ket{s_{1}s_{2}\dots s_{n}}\ket{\alpha}\rightarrow\ket{s_{1}s_{2}\dots
s_{n}}\ket{e^{is\pi}\alpha}.$ (2)

The use of such an entangling interaction for continuous stabilization of a
Bell state was analyzed theoretically in Ref. [45] and demonstrated
experimentally (with phase shifts $<\pi$ due to a small dispersive coupling)
with superconducting qubits in Ref. [46].

Switching from $Z$-basis to $X$-basis parity checks simply requires that
Hadamards be applied to the qubits both before and after the entangling
operation described in Eq. (1):

$\displaystyle\begin{aligned}
&\ket{+,\pm\alpha}\overset{H\;}{\rightarrow}\ket{0,\pm\alpha}\rightarrow\ket{0,\pm\alpha}\overset{H\;}{\rightarrow}\ket{+,\pm\alpha},\\\
&\ket{-,\pm\alpha}\overset{H\;}{\rightarrow}\ket{1,\pm\alpha}\rightarrow\ket{1,\mp\alpha}\overset{H\;}{\rightarrow}\ket{-,\mp\alpha},\end{aligned}$
(3)

where here, $X\ket{\pm}=\pm\ket{\pm}$. Hence, for both $X$\- and $Z$-basis
parity checks, the parity of the code qubits can be inferred by measuring the
phase of the coherent-state amplitude.

###  II.1 Weight-3 checks are enough

Many quantum error correcting codes are formulated as stabilizer codes [12]
defined by a stabilizer group $\mathcal{S}$. The associated code space $C$ is
then given by the span of the vectors $\ket{\psi}$ stabilized by
$\mathcal{S}$, i.e., satisfying $S_{j}\ket{\psi}=\ket{\psi}$ for all
stabilizers $S_{j}\in\mathcal{S}$. In the usual surface code [11], for
instance, correctable errors can be detected by measuring weight-4 stabilizers
of the form $X^{\otimes 4}$ and $Z^{\otimes 4}$.

Figure 2: (a) The unit cell of the subsystem surface code [43]. Physical
qubits represented by solid black dots are located on the edges and vertices
of a square lattice. (b) Gauge operators: Each stabilizer can be represented
as the product of two weight-3 gauge operators. Gauge operators do not all
mutually commute, but they do commute with all stabilizers and therefore
preserve the code space. (c) Stabilizers of the subsystem surface code: Each
unit cell hosts two stabilizers, one of the form $Z^{\otimes 6}$ (left) and
one of the form $X^{\otimes 6}$ (right). (d) A light pulse is prepared in a
coherent state $\ket{\alpha}$ and interacts sequentially with three qubits.
The interaction of the light pulse with each qubit results in the entangling
operation of Eq. (1). Photon loss in transit is modeled by a series of beam
splitters having reflectivities $\eta_{i}$ ($i=1,2,3$). The parity of the code
qubits is inferred by performing a phase-sensitive measurement of the coherent
state amplitude. This phase-sensitive measurement could be achieved through
homodyne detection by interfering the light pulse with a reference signal and
measuring the resulting intensity.

A surface code can alternatively be operated as a subsystem code [43] [Fig.
2(a-c)]. A subsystem code [47, 48, 49] is defined by a non-Abelian gauge group
$\mathcal{G}$ together with an Abelian stabilizer subgroup
$\mathcal{S}\subseteq\mathcal{G}$. This construction allows the code space $C$
stabilized by $\mathcal{S}$ to be partitioned into subsystems, $C=A\otimes B$.
Here, the subsystems $A$ and $B$ are defined by the action of gauge operators
$g\in\mathcal{G}$, which may act nontrivially only on subsystem $B$:
$g\ket{\psi_{A}}\otimes\ket{\psi_{B}}=\ket{\psi_{A}}\otimes\ket{\psi_{B}^{\prime}}$
for $\ket{\psi_{A}}\in A,\,\ket{\psi_{B}},\ket{\psi_{B}^{\prime}}\in B$.
Errors acting on the codespace are then defined only up to an equivalence
relation involving $\mathcal{G}$. In particular, error operators $E$ and
$E^{\prime}$ are equivalent in their action on encoded logical information (in
subsystem $A$) if $E^{\prime}=Eg$ for some $g\in\mathcal{G}$. Much of the
appeal of subsystem codes comes from the fact that stabilizer eigenvalues can
be inferred from measurements of lower-weight gauge operators. For the
subsystem surface code [43] [Fig. 2(a-c)], for instance, the eigenvalues of
the weight-6 stabilizers $X^{\otimes 6}$ can be inferred by multiplying
together the measurement outcomes of two weight-3 gauge operators of the form
$X^{\otimes 3}$. Correctable errors in a surface code can therefore be
detected using weight-3 parity checks only (as opposed to the usual weight-4
parity checks), with a fault-tolerance threshold of order 1% [43, 50]. With
this long-term application in mind, we consider error sources—photon loss and
finite coherent-state distinguishability—that are specific to weight-3 flying-
cat parity checks [Fig. 2(d)] and show how to minimize their total impact.

An important advantage of the subsystem surface code is that errors on ancilla
qubits used for syndrome readout propagate onto at most one other qubit,
modulo gauge operators [43]. The extra degree of freedom provided by the gauge
group therefore eliminates the horizontal hook errors considered in Ref. [51]
for the surface code, wherein ancilla-qubit errors propagate onto several
qubits.

##  III Trade-off between measurement errors and errors due to photon loss

Parity checks performed using coherent states exhibit an unavoidable trade-off
between error sources depending on the average number of photons per pulse,
$\lvert\alpha\rvert^{2}$. Due to the non-orthogonality of $\ket{\pm\alpha}$,
larger field amplitudes $\alpha$ yield better measurement fidelities. However,
they also produce a stronger backaction on the code qubits in the presence of
photon loss. In this section, we consider these two competing error sources:
measurement errors and errors due to photon loss.

In a real-world implementation, there may be a number of other error sources
affecting the quality of the operation required for parity checks [Eq. (1)].
For example, the qubit-state-dependent conditional phase introduced may not be
precisely $\pi$, phase noise or thermal light may corrupt the prepared
coherent states, and photodetectors may suffer from amplifier noise, dark
counts, etc. In this paper, we assume that these implementation-dependent
imperfections lead to a background error rate that can be controlled, leaving
the effects of photon loss and imperfect coherent-state distinguishability as
the dominant error sources. These effects are likely to be common to any
implementation.

###  III.1 Photon-loss-induced backaction

In this subsection, we evaluate the backaction of photon loss on the state of
the three code qubits involved in the parity check. Photon loss during a
flying-cat parity check can lead to two-qubit correlated errors on the code
qubits (horizontal hook errors, Fig. 3). As is the case for parity
measurements via noisy ancilla qubits [43], these horizontal hook errors are
gauge-equivalent to single-qubit Pauli errors, and they can therefore be
corrected as if they were single-qubit Pauli errors. This key advantage of the
subsystem surface code is therefore retained for flying-cat parity checks.

Figure 3: Horizontal hook error $Z_{2}Z_{3}$ due to photon loss during a
parity check.

To perform a weight-3 parity check, a coherent state $\ket{\alpha}$ will
interact with three qubits. In what follows, it will be helpful to define the
even and odd ($Z$-basis) parity subspaces given by

$\displaystyle\begin{aligned}
&\mathcal{H}_{+}=\mathrm{span}\\{\ket{000},\ket{110},\ket{011},\ket{101}\\},\\\
&\mathcal{H}_{-}=\mathrm{span}\\{\ket{001},\ket{010},\ket{100},\ket{111}\\}.\end{aligned}$
(4)

We write the state $\ket{\Xi}$ of the three qubits prior to the parity check
as

$\ket{\Xi}=c_{+}\ket{\Xi_{+}}+c_{-}\ket{\Xi_{-}},$ (5)

where $\ket{\Xi_{\pm}}\in\mathcal{H}_{\pm}$, and where
$\sum_{\sigma=\pm}\lvert c_{\sigma}\rvert^{2}=1$. In the absence of photon
loss, the states $\ket{\Xi_{\pm}}$ are preserved by Eq. (1):
$\ket{\Xi}\ket{\alpha}\rightarrow
c_{+}\ket{\Xi_{+}}\ket{\alpha}+c_{-}\ket{\Xi_{-}}\ket{-\alpha}$. To perform an
$X$-basis parity check, Hadamards are simply applied before/after the
$Z$-basis parity check, as described in Eq. (3). In that case, we take
$\ket{\Xi}$ to describe the state of the three qubits after the first set of
Hadamards.

In the presence of photon loss, the parity check has a backaction on the code
qubits that leads to the introduction of errors beyond the background error
rate. We model the effect of photon loss to the environment via a beam
splitter-type interaction [52]

$R(\eta)=e^{\mathrm{arcsin}\sqrt{\eta}(a_{\ell}^{\dagger}a-\mathrm{h.c.})},$
(6)

where here, $a$ is an annihilation operator defined such that
$a\ket{\alpha}=\alpha\ket{\alpha}$, $a_{\ell}$ is associated with an
environmental loss mode into which photons may be scattered, and $\eta$ is the
beam splitter reflectivity. Under this operation,
$R^{\dagger}(\eta)aR(\eta)=\sqrt{1-\eta}a+\sqrt{\eta}a_{\ell}$. The amplitude
$\alpha$ of $\ket{\alpha}$ is therefore reduced according to

$R(\eta)\ket{\alpha}\ket{0}_{\ell}=\ket{\sqrt{1-\eta}\alpha}\ket{\sqrt{\eta}\alpha}_{\ell},$
(7)

where here, $\ket{\;}_{\ell}$ is the state of the loss mode. Although
coherent-state amplitude may be lost to many different environmental modes in
principle, these modes can be treated as a single effective mode since the
state of the environment will eventually be traced over. To account for the
effect of loss at different points in the parity-check operation, we imagine
that the light pulse undergoes such a beam splitter interaction $R(\eta_{i})$,
$i=1,2,3$, after every entangling operation [Fig. 2(d)]. Accounting for loss
in this manner, the initial state
$\ket{s_{1}s_{2}s_{3}}\ket{\alpha}\ket{0,0,0}_{\ell}$ of the qubits, coherent
state, and environment (here, $\ket{0,0,0}_{\ell}$ denotes the vacuum state of
the three loss modes) evolves into

$\displaystyle\ket{s_{1}s_{2}s_{3}}\ket{(-1)^{s}\bar{\alpha}}\ket{(-1)^{s_{1}}\alpha_{1},(-1)^{\mu}\alpha_{2},(-1)^{s}\alpha_{3}}_{\ell},$
(8)

where $s=s_{1}+s_{2}+s_{3}\>(\mathrm{mod}\>2)$ is the $Z$-basis parity of
$\ket{s_{1}s_{2}s_{3}}$, $\mu=s_{1}+s_{2}\>(\mathrm{mod}\>2)$, and where

$\displaystyle\begin{aligned}
&\bar{\alpha}=\alpha\prod_{i=1}^{3}(1-\eta_{i})^{1/2},\\\
&\alpha_{i}=\alpha\eta_{i}^{1/2}\prod_{j<i}(1-\eta_{i})^{1/2}.\end{aligned}$
(9)

The joint state $\rho$ of the qubits and electromagnetic field following the
entangling operations (but prior to measurement) can then be obtained by
tracing over the loss modes:

$\rho=\sum_{\sigma,\sigma^{\prime}=\pm}\rho_{\sigma,\sigma^{\prime}}\outerproduct{\sigma\bar{\alpha}}{\sigma^{\prime}\bar{\alpha}},$
(10)

where $\rho_{\pm,\pm}\in\mathcal{H}_{\pm}$. For
$\ket{\Xi_{+}}=\sum_{i=0,3,5,6}c_{i}\ket{i}$, where here, $i$ is the base-10
value of $s_{1}s_{2}s_{3}$ in binary, $\rho_{+,+}$ takes the following form in
the basis
$\\{\ket{0}=\ket{000},\ket{3}=\ket{011},\ket{5}=\ket{101},\ket{6}=\ket{110}\\}$:

$\frac{\rho_{+,+}}{\lvert c_{+}\rvert^{2}}=\begin{pmatrix}\lvert
c_{0}\rvert^{2}&c_{0}^{*}c_{3}e^{-2\lvert\alpha_{2}\rvert^{2}}&c_{0}^{*}c_{5}e^{-2(\lvert\alpha_{1}\rvert^{2}+\lvert\alpha_{2}\rvert^{2})}&c_{0}^{*}c_{6}e^{-2\lvert\alpha_{1}\rvert^{2}}\\\
c_{3}^{*}c_{0}e^{-2\lvert\alpha_{2}\rvert^{2}}&\lvert
c_{3}\rvert^{2}&c_{3}^{*}c_{5}e^{-2\lvert\alpha_{1}\rvert^{2}}&c_{3}^{*}c_{6}e^{-2(\lvert\alpha_{1}\rvert^{2}+\lvert\alpha_{2}\rvert^{2})}\\\
c_{5}c_{0}^{*}e^{-2(\lvert\alpha_{1}\rvert^{2}+\lvert\alpha_{2}\rvert^{2})}&c_{5}^{*}c_{3}e^{-2\lvert\alpha_{1}\rvert^{2}}&\lvert
c_{5}\rvert^{2}&c_{5}^{*}c_{6}e^{-2\lvert\alpha_{2}\rvert^{2}}\\\
c_{6}^{*}c_{0}e^{-2\lvert\alpha_{1}\rvert^{2}}&c_{6}^{*}c_{3}e^{-2(\lvert\alpha_{1}\rvert^{2}+\lvert\alpha_{2}\rvert^{2})}&c_{6}^{*}c_{5}e^{-2\lvert\alpha_{2}\rvert^{2}}&\lvert
c_{6}\rvert^{2}\end{pmatrix}.$ (11)

The block $\rho_{-,-}/\lvert c_{-}\rvert^{2}$ takes the same form as Eq. (11)
under the mapping $(0,3,5,6)\mapsto(1,2,4,7)$. The off-diagonal blocks
$\rho_{\pm,\mp}\propto c_{\pm}c_{\mp}^{*}$ in Eq. (10) appear as a result of
considering an initial state $\ket{\Xi}$ [Eq. (5)] of mixed parity and will be
suppressed exponentially $\sim e^{-2\lvert\bar{\alpha}\rvert^{2}}$ in the
final post-measurement state.

In the absence of photon loss ($\eta_{j}=0$ for $j=1,2,3$),
$\rho_{\sigma,\sigma^{\prime}}=c_{\sigma}c_{\sigma^{\prime}}^{*}\outerproduct{\Xi_{\sigma}}{\Xi_{\sigma^{\prime}}}$
and $\bar{\alpha}=\alpha$. With $\eta_{j}\neq 0$, however,
$\rho_{\sigma,\sigma^{\prime}}\neq
c_{\sigma}c_{\sigma^{\prime}}^{*}\outerproduct{\Xi_{\sigma}}{\Xi_{\sigma^{\prime}}}$
due to the photon-loss-induced backaction of the parity check. Hence, while
the interaction with the electromagnetic field remains quantum non-demolition
(QND) in the parity of $\ket{s_{1}s_{2}s_{3}}$ in the presence of photon loss,
it is no longer QND in $\ket{\Xi_{\pm}}$ as desired. It may be verified that
the backaction leading to this non-QND character can be described by the
composition of two dephasing channels:

$\frac{\rho_{\pm,\pm}}{\lvert
c_{\pm}\rvert^{2}}=(\mathcal{E}_{p_{1},E_{1}}\circ\mathcal{E}_{p_{2},E_{2}})(\outerproduct{\Xi_{\pm}}{\Xi_{\pm}}),$
(12)

where $\mathcal{E}_{p,E}(\rho)=(1-p)\rho+pE\rho E^{\dagger}$,
$E_{1}=Z_{2}Z_{3}$, $E_{2}=Z_{3}$, and where

$\displaystyle\begin{aligned}
&p_{1}=\frac{1}{2}(1-e^{-2\lvert\alpha_{1}\rvert^{2}})\simeq\eta_{1}\lvert\alpha\rvert^{2},\\\
&p_{2}=\frac{1}{2}(1-e^{-2\lvert\alpha_{2}\rvert^{2}})\simeq\eta_{2}\lvert\alpha\rvert^{2}.\end{aligned}$
(13)

The approximations in Eq. (13) are valid to leading order in $\eta_{j}\ll 1$.

The operators $E_{1,2}$ can be understood as follows: In the cat-state basis
$\ket{\mathrm{C}_{\pm}}\propto\ket{\alpha}\pm\ket{-\alpha}$, Eq. (1) describes
a CZ gate for any nonzero $\alpha$. Since $\ket{\mathrm{C}_{\pm}}$ are
eigenstates of the photon-number parity operator $e^{i\pi a^{\dagger}a}$, the
loss of a photon constitutes an $X$ error,
$\ket{\mathrm{C}_{\pm}}\to\ket{\mathrm{C}_{\mp}^{\prime}}$, up to a change in
coherent-state amplitude, $\ket{\alpha}\rightarrow\ket{\alpha^{\prime}}$ [53].
Hence, as a result of the CZ gates involved in the parity check, photon loss
leads to $Z$ errors on the code qubits. (Generally speaking, an $X$ error on
qubit $a$, propagated through a CZ acting on qubits $a$ and $b$, leads to a
$Z$ error on qubit $b$.) These $Z$ errors appear in the combination
$E_{1}=Z_{2}Z_{3}$ for photon loss occurring between qubits 1 and 2 (Fig. 3),
and as $E_{2}=Z_{3}$ for photon loss occurring between qubits 2 and 3.

In a surface code where stabilizer parity checks are performed using
propagating pulses of light, photon loss would therefore lead to correlated
(horizontal hook) errors on code qubits. However, due to the gauge degree of
freedom introduced by the gauge group, the horizontal hook error $Z_{2}Z_{3}$
introduced by the parity check [cf. Eq. (12)] is equivalent to the single-
qubit error $Z_{1}$ [9]. This is because $Z_{2}Z_{3}(Z_{1}Z_{2}Z_{3})=Z_{1}$,
where $Z_{1}Z_{2}Z_{3}\in\mathcal{G}$. Since $X$-basis parity checks differ
only by some additional Hadamards applied to $\rho$ [Eq. (10)], the effects of
photon loss during $X$-basis parity checks are instead equivalent, up to a
gauge operator, to single-qubit $X_{1}$ and $X_{3}$ errors occurring with the
same probabilities (Table 1). These errors are not detected by the parity
check in which they are introduced (since $Z$-basis parity checks only detect
$X$ errors and vice versa), but may be detected in the next parity check of
the other basis.

Operator | Error $E$ introduced | Prob($E$)
---|---|---
being measured | by photon loss |
$Z^{\otimes 3}$ | $Z_{1}$ | $p_{1}$
| $Z_{3}$ | $p_{2}$
$X^{\otimes 3}$ | $X_{1}$ | $p_{1}$
| $X_{3}$ | $p_{2}$

Table 1: Errors arising due to photon loss during a $Z^{\otimes 3}$ gauge-
operator parity check are equivalent (up to a gauge transformation) to $Z_{1}$
or $Z_{3}$ errors arising with probabilities $p_{1}$ and $p_{2}$,
respectively. During an $X^{\otimes 3}$ parity check, these errors are instead
equivalent to $X_{1}$ or $X_{3}$ errors.

###  III.2 Measurement errors

Determining the parity of two or more qubits using this approach requires a
phase-sensitive measurement of the electromagnetic field (to infer the sign of
$\alpha$). Such a measurement can be implemented via homodyne detection [54].
In homodyne detection, a signal field is first mixed on a 50:50 beam splitter
with a local oscillator in coherent state $\ket{\beta}$, and the output
intensity is then measured, giving information about the phase of the signal
relative to the local oscillator. In this case, the signal field is the
propagating light pulse entangled with the state of the qubits [Eq. (10)].
Without loss of generality, we assume that $\alpha$ is real and positive,
$\alpha\in\mathbb{R}^{+}$. With a strong local oscillator
($\beta\in\mathbb{R}\gg\alpha)$, homodyne detection implements a projective
measurement $\outerproduct{x}{x}$ onto the $x$ eigenbasis [54], where here,
$\ket{x}$ is an eigenstate of $\hat{x}=(a^{\dagger}+a)/\sqrt{2}$. The sign
$\pm=\mathrm{sign}(x)$ of the measured displacement can then be taken as an
inference of even ($+$) or odd ($-$) parity for the state of the qubits. In
the event that $\alpha$ is complex, the measured quadrature can be adjusted by
introducing a phase on the local oscillator: $\beta\rightarrow\beta
e^{i(\varphi-\pi)}$. In that case, the projection is instead onto an
eigenstate of $\hat{x}_{\varphi}=\hat{x}\cos{\varphi}+\hat{p}\sin{\varphi}$,
where $[\hat{x},\hat{p}]=i$ [54].

Given $\rho$ [Eq. (10)], the state $\rho_{x}$ of the code qubits conditioned
on the measurement outcome $x$ is then given by

$\rho_{x}=\frac{\mathrm{Tr}_{\mathrm{EM}}\\{\outerproduct{x}{x}\rho\outerproduct{x}{x}\\}}{p(x)},$
(14)

where $p(x)=\mathrm{Tr}\\{\outerproduct{x}{x}\rho\\}$, and where
$\mathrm{Tr}_{\mathrm{EM}}\\{\dotsm\\}$ denotes a partial trace over the state
of the electromagnetic field. This quantity can be evaluated straightforwardly
using the representation $\langle
x|\bar{\alpha}\rangle=\pi^{-1/4}e^{-\frac{1}{2}(x-\sqrt{2}\bar{\alpha})^{2}}$
of $\ket{\bar{\alpha}}$. For a thresholded decision, where $x>0$ ($x<0$) is
taken to indicate even parity (odd parity), the post-measurement state for an
inference of even parity is given by

$\varrho_{+}=\int_{0}^{\infty}dx\>p(x)\rho_{x},$ (15)

with a similar expression for $\varrho_{-}$ involving an integral from
$-\infty$ to 0. The joint probability $P(``\pm",\mp)$ of the state being
odd/even ($\mp$) for an inference of even/odd parity ($``\pm"$) is

$P(``\pm",\mp)=\mathrm{Tr}\\{\hat{\Pi}_{\mp}\varrho_{\pm}\\}=\frac{\lvert
c_{\mp}\rvert^{2}}{2}\mathrm{erfc}(\sqrt{2}\bar{\alpha}),$ (16)

where $\hat{\Pi}_{\pm}$ is a projector onto $\mathcal{H}_{\pm}$ and
$\mathrm{erfc}(x)$ is the complementary error function. The error
$p_{\mathrm{M}}$ associated with the measurement is then given by

$\displaystyle\begin{aligned}
p_{\mathrm{M}}=P(``+",-)+P(``-",+)=\frac{1}{2}\mathrm{erfc}(\sqrt{2}\bar{\alpha}).\end{aligned}$
(17)

The total probability of an error occurring as part of a parity check—either
an incorrect parity inference or a code-qubit error—is then (neglecting
corrections of order $p_{i}p_{j}$ for $p_{i}\ll 1$):

$p_{\mathrm{tot}}=p_{\mathrm{M}}+p_{1}+p_{2}\simeq\frac{1}{2}\frac{e^{-2\alpha^{2}}}{\alpha\sqrt{2\pi}}+\frac{1}{2}\sum_{j=1,2}\eta_{j}\lvert\alpha\rvert^{2}.$
(18)

In the second, approximate equality, we have neglected subleading corrections
in $\eta_{j}|\alpha|^{2}<1$ and approximated the error function by its
asymptotic form for $|\alpha|>1$. Since errors introduced by photon loss occur
at a rate that increases with $\alpha$ ($\sim\eta_{j}\lvert\alpha\rvert^{2}$),
while measurement errors are suppressed for large $\alpha$
($\sim\alpha^{-1}e^{-2\alpha^{2}}$), there is an inherent trade-off between
the two types of error as a function of coherent-state amplitude. Their total
impact can therefore be minimized by optimizing $\alpha$ (Fig. 4).

In light of the nonlinear (exponential) suppression of $p_{\mathrm{M}}$ with
$|\alpha|^{2}$, it is natural to consider whether repeating the measurement
can lead to a smaller error. In this case, a single-shot measurement with a
coherent state of amplitude $\alpha=\sqrt{N}\alpha_{0}$ leads to the same
measurement error $p_{\mathrm{M}}$ as a sequence of $N$ measurements performed
with smaller coherent state amplitudes $\alpha_{0}$. However, this equivalent
performance can only be reached if a soft decision is made, where each
measurement outcome $x_{k}$ $(k=1,2,\ldots,N)$ is assigned a confidence
$p(\pm|x_{k})=\mathrm{Tr}\\{\hat{\Pi}_{\pm}\rho_{x_{k}}\\}$, and where the
individual outcomes are correlated in determining the most likely parity. If
instead, each of the $N$ measurements is thresholded independently leading to
a majority-vote decision, the error is less favorable,
$p_{\mathrm{M,th}}\sim\sqrt{p_{\mathrm{M}}}$ (up to logarithmic corrections
for large $N$) [55]. The large discrepancy between the error for a thresholded
(hard) decision and the soft decision is a feature of the Gaussian
distribution of measurement outcomes $x_{k}$ [55, 56], a model that is
accurately realized for measurement of a coherent state quadrature. Exploiting
this soft-decision advantage in repeated measurements with weak pulses may
prove useful if the amplitude of each coherent state is limited for technical
reasons (nonlinearity, unwanted qubit excitation, amplifier noise, etc.).

Figure 4: The total error $p_{\mathrm{tot}}$ [Eq. (18)] is given by the sum of
competing terms: measurement errors, which decrease with $\alpha$, and errors
due to photon loss, which increase with $\alpha$. For simplicity, we assume
that $\eta_{i}=\eta$ for all $i$.

##  IV Entangled resource states

In addition to enabling quantum error correction, flying-cat parity checks
could also be used to distribute entanglement in a quantum network, with
applications in both quantum computing and quantum communication. In this
section, we discuss the concrete example of generating three-qubit
Greenberger-Horne-Zeilinger (GHZ) states. GHZ states have applications in
precision sensing and multi-party quantum communication. They could also be
used as the central resource for weight-3, ancilla-based inter-node parity
checks using the approach of Refs. 7, 8 (an alternative to the strategy for
direct parity checks described above). We further show how to create a
particular six-qubit entangled “tetrahedron” state using only $X$\- and
$Z$-basis weight-3 parity checks and single-qubit gates. This state can be
used as a resource for three-party controlled quantum teleportation of an
arbitrary two-qubit state. Its preparation also provides a convenient setting
in which to benchmark the performance of the parity checks themselves.

###  IV.1 A three-qubit GHZ state

An alternative, established approach to long-range parity checks involves the
generation and consumption of four-qubit GHZ states [7, 8]. In this setup,
each node houses a code qubit and (at least) one ancilla. To perform a parity
check, four ancilla qubits located at four different nodes are prepared in a
GHZ state. The parity of the code qubits located at these nodes can then be
inferred by applying controlled-NOT gates between the code qubit and the
ancilla located at each node, then measuring all ancillae. The distributed GHZ
state in this proposal could be generated, in principle, via flying-cat parity
checks. Although the original proposal of Refs. 7, 8 focused on weight-4
parity checks, weight-3 checks are sufficient to operate a subsystem surface
code, as described above, so in this section we consider the preparation of
three-qubit GHZ states. The ancilla-based approach of Refs. 7, 8 may be
preferable over a direct flying-cat parity check in the case where the photon
loss rate is high, leading to a high probability of errors being introduced on
the code qubits. The low-fidelity GHZ states produced in the presence of high
photon loss could then be purified “offline” prior to being consumed as part
of a parity check, as considered in Refs. [7, 8].

A simple measurement-based strategy for three-qubit GHZ-state preparation
begins with preparation of the three qubits in $\ket{+++}$. Measuring
$Z_{1}Z_{2}$ and $Z_{2}Z_{3}$ projects the qubits into an entangled state,
which can then be transformed into
$\ket{\mathrm{GHZ}}\propto\ket{000}+\ket{111}$ with an $X$-gate correction on
the appropriate qubit. For a measurement outcome $Z_{1}Z_{2}=-1$, this is
qubit 1; for $Z_{2}Z_{3}=-1$, it is qubit 3; if both outcomes are $-1$, it is
qubit 2. However, as described above in Sec. III, a $Z$-basis flying-cat
parity check may introduce $Z$ errors on the qubits. For a measurement of
$Z_{i}Z_{j}$, these are single-qubit $Z_{j}$ errors occurring with probability
$p_{ij}=(1-e^{-2\eta_{ij}\lvert\alpha\rvert^{2}})/2$, where here, $\eta_{ij}$
is the beam splitter reflectivity quantifying the strength of the losses
incurred while traveling between qubits $i$ and $j$. Measurement errors
occurring with probability
$p_{\mathrm{M}}^{ij}=\mathrm{erfc}[\sqrt{2}(1-\eta_{ij})\alpha]/2$ will also
result in the wrong correction operator being applied. In the presence of
photon loss and accounting for measurement errors, the two parity checks and
correction step described above will lead to preparation of the following
mixed state:

$\sigma=(1-p)\outerproduct{\mathrm{GHZ}}{\mathrm{GHZ}}+\sum_{i}p_{i}\outerproduct{i_{\perp}}{i_{\perp}},$
(19)

where
$p=\sum_{i}p_{i}=p_{12}+p_{23}-p_{12}p_{23}+p_{\mathrm{M}}^{12}+p_{\mathrm{M}}^{23}+p_{\mathrm{M}}^{12}p_{\mathrm{M}}^{23}$,
and where $\langle\mathrm{GHZ}|i_{\perp}\rangle=0$ for all $i$.

Since $\ket{\mathrm{GHZ}}$ can be described as the simultaneous $+1$
eigenstate of a set of CSS stabilizer generators—namely, $Z_{1}Z_{2}$,
$Z_{2}Z_{3}$, and $X_{1}X_{2}X_{3}$—several noisy copies of $\sigma$
distributed across three nodes (“Alice”, “Bob”, and “Charlie”) could be
purified using any CSS error-correcting code, e.g., $C_{4}$ [57]. In the case
of $C_{4}$, this protocol would require that Alice, Bob, and Charlie share
four copies of $\sigma$. They would each measure the stabilizers of $C_{4}$ on
their four qubits, and then use local operations and classical communication
to produce a less noisy, encoded version of $\sigma$ [57]. Since the six-qubit
“tetrahedron” state discussed in the following subsection also has a
description in terms of a set of CSS stabilizer generators, the same protocol
[57] could be used to purify this state as well.

###  IV.2 A six-qubit entangled state

Figure 5: (a) Six qubits can be associated with the six edges of a
tetrahedron. Each of the stabilizers in Eq. (LABEL:stabilizers) is a product
of three Pauli-$Z$ or Pauli-$X$ operators acting on qubits that neighbor a
common face ($Z$) or vertex ($X$). (b) A potential real-space setup for
applications discussed in the main text: Each node ($A,B,C$) contains a pair
of qubits whose associated tetrahedron edges have no face nor vertex in
common. Every parity check required for preparation of $\ket{\mathrm{T}}$ [Eq.
(LABEL:stabilizers)] involves only one qubit at each node. For nodes $(A,B,C)$
implemented via the cavity-QED realization shown in Fig. 1, the three qubits
not being measured during a given parity check can be made inactive by
detuning them from their respective cavities.

In this section, we show how weight-3 parity checks can be used to prepare a
“tetrahedron” state—a six-qubit entangled state where each qubit can be
associated with one edge of a tetrahedron [Fig. 5(a)]. This state provides a
source of shared randomness with potential applications for three-party
quantum key distribution. A tetrahedron state can also be used as a resource
for controlled quantum teleportation of an arbitrary two-qubit state.
Independent of these potential applications, the tetrahedron state provides a
natural testbed for benchmarking the performance of weight-3 flying-cat parity
checks (since its preparation requires both $X$\- and $Z$-basis parity
checks).

The tetrahedron state spans the codespace of a (trivial) CSS code admitting a
set of stabilizer generators that can be associated with the faces and
vertices of a tetrahedron [Fig. 5(a)]:

$\displaystyle\begin{aligned} &S_{1}=Z_{1}Z_{3}Z_{5},\quad\quad
S_{4}=X_{1}X_{3}X_{6},\\\ &S_{2}=Z_{1}Z_{4}Z_{6},\quad\quad
S_{5}=X_{1}X_{4}X_{5},\\\ &S_{3}=Z_{2}Z_{4}Z_{5},\quad\quad
S_{6}=X_{2}X_{3}X_{5}.\ \end{aligned}$ (20)

It may be verified that the stabilizers associated with the remaining face and
vertex are simply given by $S_{1}S_{2}S_{3}$ and $S_{4}S_{5}S_{6}$.

The stabilizer eigenvector relations $S_{i}\ket{\mathrm{T}}=\ket{\mathrm{T}}$,
$i=1,2,\dots,6$, admit one common eigenstate, the tetrahedron state:

$\displaystyle\begin{aligned}
\ket{\mathrm{T}}=\frac{1}{2}\sum_{\beta=\Phi^{\pm},\Psi^{\pm}}\ket{\beta}_{12}\ket{\beta}_{34}\ket{\beta}_{56},\end{aligned}$
(21)

where $\ket{\Phi^{\pm}}=(\ket{00}\pm\ket{11})/\sqrt{2}$ and
$\ket{\Psi^{\pm}}=(\ket{01}\pm\ket{10})/\sqrt{2}$. If node $A$ houses qubits 1
and 2, node $B$ qubits 3 and 4, and node $C$ qubits 5 and 6 [Fig. 5(b)], then
the outcomes of Bell measurements at nodes $A$-$C$ are uniformly distributed
across Bell states but perfectly correlated across nodes. A tetrahedron state
can therefore provide Alice, Bob, and Charlie with two bits of shared
randomness, which could be used to build a shared private key for
cryptography.

With the qubits partitioned in the manner described above, every stabilizer
$S_{i}$ in Eq. (LABEL:stabilizers) involves one qubit at each node.
Preparation of $\ket{\mathrm{T}}$ can be achieved in two steps, starting from
a random initial state. In the first step, all $S_{i}$ are measured using
flying-cat parity checks. At this point, each stabilizer measurement returns a
random outcome, $\sigma_{i}=\pm 1$. In the second step, the $-1$ outcomes are
treated like an error syndrome and the appropriate correction operator
(identified in the manner described below) is applied, transforming the post-
measurement state into the target state $\ket{\mathrm{T}}$.

Since $\ket{\mathrm{T}}$ [Eq. (21)] is the non-degenerate ground state of
$-\sum_{i}S_{i}$, the code defined by $S_{i}$ is trivial, reflecting the
trivial genus-0 topology of the tetrahedron. Despite the absence of logical
qubits (which, in a toric-code construction [58], requires that
$-\sum_{i}S_{i}$ have degenerate ground states), the code nonetheless serves
as a quantum memory for $\ket{\mathrm{T}}$: Given full syndrome information,
consisting of the six eigenvalues $\sigma_{i}$ of $S_{i}$, any number of $X$
and $Z$ errors can be corrected using a simple syndrome-decoding algorithm.
The procedure is identical for $X$ and $Z$ errors, so without loss of
generality, we give the decoding procedure for $X$ errors:

  1. 1.

Measure the $Z$-basis stabilizers $S_{1-3}$. If the syndrome is $(1,1,1)$,
then no correction is required. All other errors can be divided into two
classes:

  2. 2.

Class I errors: If syndrome $j$ in Table 2 is obtained, then apply an $X$ gate
to qubit $j$.

  3. 3.

Class II errors: If the syndrome consists entirely of $-1$ outcomes, then
apply $X$ gates to any pair $(1,2)$, $(3,4)$, or $(5,6)$ of qubits having no
face nor vertex in common.

$j\;\;$ | $\;\;\sigma_{1}\;\;$ | $\;\;\sigma_{2}\;\;$ | $\;\;\sigma_{3}\;\;$ | $\;\;\sigma_{4}\;\;$ | $\;\;\sigma_{5}\;\;$ | $\;\;\sigma_{6}\;\;$
---|---|---|---|---|---|---
$1\;\;$ | $-1$ | $-1$ | $+1$ | $-1$ | $-1$ | $+1$
$2\;\;$ | $+1$ | $+1$ | $-1$ | $+1$ | $+1$ | $-1$
$3\;\;$ | $-1$ | $+1$ | $+1$ | $-1$ | $+1$ | $-1$
$4\;\;$ | $+1$ | $-1$ | $-1$ | $+1$ | $-1$ | $+1$
$5\;\;$ | $-1$ | $+1$ | $-1$ | $+1$ | $-1$ | $-1$
$6\;\;$ | $+1$ | $-1$ | $+1$ | $-1$ | $+1$ | $+1$

Table 2: Syndrome data for decoding Class I errors.

Any number of $X$ errors can be corrected in this way: Single-qubit errors are
reversed, while multi-qubit $X$ errors corrected as single-qubit $X$ errors
are transformed either into $S_{4-6}$ or into products thereof, all of which
act trivially on $\ket{\mathrm{T}}$. As an example, consider the scenario
where qubits 1 and 3 have undergone bitflip errors. The syndrome
$(\sigma_{1},\sigma_{2},\sigma_{3})=(1,-1,1)$ dictates that an $X$ gate be
applied to qubit 6 (Table 2). This operation transforms the state
$X_{1}X_{3}\ket{\mathrm{T}}$ into $S_{4}\ket{\mathrm{T}}=\ket{\mathrm{T}}$.
For an example of a Class II error, consider the scenario where both qubits 1
and 2 have undergone $X$ errors. In this case,
$(\sigma_{1},\sigma_{2},\sigma_{3})=(-1,-1,-1)$, requiring that $X$ gates be
applied to any pair $(1,2)$, $(3,4)$, or $(5,6)$ of qubits. If $(1,2)$ is
selected, then the errors are reversed. If $(3,4)$ is selected instead, then
the correction operator $X_{3}X_{4}$ transforms $X_{1}X_{2}\ket{\mathrm{T}}$
into $S_{5}S_{6}\ket{\mathrm{T}}=\ket{\mathrm{T}}$.

Multipartite entanglement of $\ket{\mathrm{T}}$ can be detected by measuring
the entanglement witness [59, 60]

$\mathcal{W}=\alpha_{\ket{\mathrm{T}}}\mathbbm{1}-\outerproduct{\mathrm{T}}{\mathrm{T}},$
(22)

where here, $\alpha_{\ket{\mathrm{T}}}$ is the largest Schmidt coefficient
across bipartitions of $\ket{\mathrm{T}}$. This construction is designed to
have a non-negative expectation value $\langle\mathcal{W}\rangle\geq 0$ for
all biseparable states. A negative expectation value therefore signals
detection of genuine multipartite entanglement.

For states (like $\ket{\mathrm{T}}$) having a description in terms of a set of
CSS stabilizer generators, it is known that the maximal Schmidt coefficient
across bipartitions is $\alpha_{\ket{\mathrm{T}}}=1/2$, and that the witness
$\mathcal{W}$ can be expressed in terms of the CSS stabilizer generators [61]:

$\mathcal{W}=\frac{3}{2}\mathbbm{1}-\prod_{i=1}^{3}\frac{\mathbbm{1}+S_{i}}{2}-\prod_{i=4}^{6}\frac{\mathbbm{1}+S_{i}}{2}.$
(23)

The expectation value $\langle\mathcal{W}\rangle$ can be estimated using two
measurement settings only: one where all qubits are measured in the $Z$ basis,
and one where all qubits are measured in the $X$ basis. The fidelity
$F=\langle\mathrm{T}\lvert\rho\rvert\mathrm{T}\rangle=1/2-\langle\mathcal{W}\rangle$
of some noisy state $\rho$ relative to $\ket{\mathrm{T}}$ can therefore be
estimated using single-qubit measurements, independent of the parity checks
that were used to create the state. This provides an avenue for benchmarking
the quality of the weight-3 flying-cat parity checks used to prepare
$\ket{\mathrm{T}}$.

When a complete set of parity checks is performed starting from some initial
state, a set of six eigenvalues $\left\\{\sigma_{i}\right\\}$ is found. In the
absence of measurement errors or errors due to photon loss, the operation
required to transform the initial state into $\ket{\mathrm{T}}$ can be
correctly identified based on these eigenvalues. A measurement error, however,
will lead to the wrong operation being applied, resulting in the creation of a
state orthogonal to $\ket{\mathrm{T}}$. Errors on physical qubits can also be
introduced by photon loss during a parity check. This loss-induced backaction
takes the form of stochastic $Z$ errors ($X$ errors) during measurement of
$Z$-basis ($X$-basis) parity checks. Such errors could be corrected in
subsequent error-correction cycles using the decoding procedure explained
above. However, any errors introduced by the last set of parity checks (prior
to any measurements occurring at nodes $A$-$C$) remain uncorrected. If, for
instance, this last round of checks consists of $Z$-basis parity checks, then
$\ket{\mathrm{T}}$ may be affected by uncorrected $Z$ errors, which also
produce states orthogonal to $\ket{\mathrm{T}}$. To leading order in
$p_{1,2}$, and neglecting corrections $O(p_{1,2}p_{\mathrm{M}})$, we then have

$\langle\mathcal{W}\rangle\simeq-\frac{1}{2}+[1-(1-p_{\mathrm{M}})^{6}]+3(p_{1}+p_{2}).$
(24)

Here, the term $[1-(1-p_{\mathrm{M}})^{6}]$ gives the probability of making at
least one measurement error across the six stabilizer measurements, while the
term $3(p_{1}+p_{2})$ gives the probability of photon loss introducing an
error during the last set of parity checks (consisting of either three
$X$-basis or three $Z$-basis checks). Notably, in the limit $\alpha\rightarrow
0$, we have $\langle\mathcal{W}\rangle=1/2-1/2^{6}$, giving $F=1/2^{6}$. This
is the fidelity expected from randomly guessing the six stabilizer outcomes
and applying a correction based on these guesses.

Figure 6: The preparation fidelity
$F=\langle\mathrm{T}\lvert\rho\rvert\mathrm{T}\rangle$ accounting for
measurement errors and errors due to photon loss.

####  IV.2.1 Controlled quantum teleportation

The tetrahedron state can be used as a resource state for three-party
controlled quantum teleportation. Controlled quantum teleportation of a
single-qubit state using a three-qubit GHZ state was introduced in Ref. [62],
and since then, many protocols extending the scheme to multiqubit states using
various resource states have been proposed [63, 64, 65, 66, 67, 68]. The
premise underlying this protocol is that Alice and Bob should be able to
teleport quantum states with unit fidelity only with Charlie’s cooperation.
Here, the fidelity of a teleportation protocol is given by the fidelity of the
state reconstructed by Bob relative to Alice’s original state. In Ref. [69],
Li and Ghose introduced the notion of control power to quantify Charlie’s
influence over the fidelity of the teleportation protocol. The control power
is equal to $1-\bar{F}$ for an average teleportation fidelity $\bar{F}$. For
teleportation of a two-qubit state, Charlie’s control power must be
$1-\bar{F}\geq 3/5$ in order to prevent Alice and Bob from benefiting from
their shared entanglement [70]. Not all entangled states meet this criterion,
but we now show that the state $\ket{\mathrm{T}}$ does.

We imagine that Alice, Bob, and Charlie share one copy of $\ket{\mathrm{T}}$,
with each party holding two qubits according to the partitioning shown in Fig.
5(b). Alice has two additional qubits $A_{1},A_{2}$ in state
$\ket{\varphi}=a\ket{\Phi^{+}}+b\ket{\Psi^{+}}+c\ket{\Psi^{-}}+d\ket{\Phi^{-}}$,
which she wants to teleport to Bob. She performs Bell measurements on pairs of
qubits $(A_{1},1)$ and $(A_{2},2)$ and transmits the results to Bob, who
applies the correction operators given in Table 3. Once these correction
operators have been applied, Bob and Charlie share one of four possible
states:

$\displaystyle\begin{aligned}
&a\ket{\Phi^{+},\Phi^{+}}+b\ket{\Psi^{+},\Psi^{+}}+c\ket{\Psi^{-},\Psi^{-}}+d\ket{\Phi^{-},\Phi^{-}},\\\
&a\ket{\Phi^{+},\Psi^{+}}+b\ket{\Psi^{+},\Phi^{+}}+c\ket{\Psi^{-},\Phi^{-}}+d\ket{\Phi^{-},\Psi^{-}},\\\
&a\ket{\Phi^{+},\Psi^{-}}+b\ket{\Psi^{+},\Phi^{-}}+c\ket{\Psi^{-},\Phi^{+}}+d\ket{\Phi^{-},\Psi^{+}},\\\
&a\ket{\Phi^{+},\Phi^{-}}+b\ket{\Psi^{+},\Psi^{-}}+c\ket{\Psi^{-},\Psi^{+}}+d\ket{\Phi^{-},\Phi^{+}}.\end{aligned}$
(25)

$\mathrm{Alice}_{1}$ | Bob | $\mathrm{Alice}_{2}$ | Bob | Charlie | Bob
---|---|---|---|---|---
$\ket{\Phi^{+}}$ |  | $\ket{\Phi^{+}}$ |  | $\ket{0}\ket{+}$ |
$\ket{\Psi^{+}}$ | $X_{3}$ | $\ket{\Psi^{+}}$ | $X_{4}$ | $\ket{0}\ket{-}$ | $Z_{3}Z_{4}$
$\ket{\Phi^{-}}$ | $Z_{3}$ | $\ket{\Phi^{-}}$ | $Z_{4}$ | $\ket{1}\ket{+}$ | $X_{3}X_{4}$
$\ket{\Psi^{-}}$ | $Z_{3}X_{3}$ | $\ket{\Psi^{-}}$ | $Z_{4}X_{4}$ | $\ket{1}\ket{-}$ | $Z_{3}Z_{4}X_{3}X_{4}$

Table 3: Correction operators for a two-qubit controlled quantum teleportation
scheme using $\ket{\mathrm{T}}$ as the entangled resource state.

If Charlie wished to complete the teleportation, he would measure one of his
qubits in the $Z$ basis and the other in the $X$ basis. He would then send the
measurement outcomes to Bob, who would conditionally apply further corrections
to his qubits according to Table 3. When Bob is given both Alice’s and
Charlie’s measurement outcomes, the fidelity of the teleportation protocol is
unity. With only Alice’s information, however, Bob’s qubits are in a mixed
state $\rho_{\mathrm{B}}$ obtained by tracing over Charlie’s qubits. The
fidelity $\langle\varphi\lvert\rho_{\mathrm{B}}\rvert\varphi\rangle$ of the
teleportation protocol for state $\ket{\varphi}$ can then be averaged over the
Haar measure $d\varphi$ to obtain the average fidelity

$\bar{F}=\int
d\varphi\>\langle\varphi\lvert\rho_{\mathrm{B}}\rvert\varphi\rangle=\frac{2}{5},$
(26)

corresponding to a control power of $1-\bar{F}=3/5$. The average teleportation
fidelity that Alice and Bob achieve without Charlie’s help is therefore only
equal to the fidelity of the best classical strategy [71], in which Alice
performs an optimal measurement of her two-qubit state and sends the result to
Bob, who then attempts to reconstruct Alice’s state on the basis of this
classical information. The maximal fidelity for this classical strategy is
given by the optimal fidelity for estimation of a quantum state, equal to
$2/5$ for a two-qubit system [72, 73, 74]. Charlie therefore has the power to
prevent Alice and Bob from deriving any advantage (over classical strategies)
from their shared entanglement.

We note that the notion of control power given above only has meaning when the
three parties share a pure state. Noise can, however, be incorporated under
appropriate conditions [75]. Whether the noisy tetrahedron state resulting
from a sequence of flying-cat parity checks with finite measurement errors and
finite photon loss provides sufficient control power is a question for future
study, beyond the scope of the present work.

##  V Conclusion

High-quality flying-cat parity checks could provide the ability to perform
error correction in distributed architectures, where native qubit-qubit
interactions are not typically available across nodes. However, operations
involving propagating quasimodes of the electromagnetic field will introduce
photon loss as a source of error on code qubits.

In this work, we have analyzed the trade-off between measurement errors and
errors due to photon loss in the context of weight-3 flying-cat parity checks.
We have shown how the total impact of these error sources could be minimized
as a function of the coherent-state amplitude. In addition, we have verified
that the horizontal hook errors due to photon loss remain gauge-equivalent to
single-qubit errors for gauge operators of the form $X^{\otimes 3}$,
$Z^{\otimes 3}$, so that flying-cat parity checks preserve this key advantage
of the subsystem surface code. In the context of entanglement distribution, we
also described how flying-cat parity checks can be used to generate three-
qubit GHZ states, as well as a six-qubit entangled “tetrahedron” state that
can be used for controlled quantum teleportation of an arbitrary two-qubit
state. Another important result of this paper is the conditional post-
measurement state $\rho_{x}$ of the data qubits following a phase-sensitive
measurement of the electromagnetic field, incorporating both measurement
errors and photon-loss-induced errors. This object can be used to find the
conditional probabilities for error patterns on the code qubits, shot-by-shot
for each measured value of $x$. In a quantum error correcting code, this
probability could be passed to a decoding scheme that takes full advantage of
this soft information [76, 77, 78].

Flying-cat parity checks rely on sources and detectors for (classical)
coherent states of light, rather than the single photons used to link nodes in
several other quantum-network implementations. These parity checks therefore
do not require single-photon sources or detectors, although they do require a
high-quality stable source of coherent light and low-noise phase-sensitive
measurements. We have only considered the simplest encoding (a single-mode
input coherent state). In future work, it may be interesting to consider
whether a more elaborate encoding could be used to simultaneously detect
photon loss and parity using, e.g., a cat-code variant [79]. Additionally,
rapid qubit gates and dynamical decoupling sequences performed to preserve
code qubits coupled to cavities will generically lead to bursts of light that
can carry away information [80]. Whether dynamical decoupling can be made
compatible with flying-cat parity checks is an open question.

The flexible qubit connectivity enabled by the use of propagating light pulses
could also provide an avenue towards realizing the nonlocal interactions
required by certain quantum low-density parity-check codes [81, 82, 83, 84,
85]. Quantum LDPC codes have shown promise for reducing the number of physical
qubits required to encode a logical qubit compared to leading candidates like
the surface code.

###### Acknowledgements.

We thank K. Wang for useful discussions. We also acknowledge funding from the
Natural Sciences and Engineering Research Council of Canada (NSERC) and from
the Fonds de Recherche du Québec–Nature et technologies (FRQNT).

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