††thanks: * and $\dagger$ contributed equally to this work.

# All-optical Quantum State Engineering for Rotation-symmetric Bosonic States

Rajveer Nehra1<EMAIL_ADDRESS>Miller Eaton2<EMAIL_ADDRESS>Olivier
Pfister2<EMAIL_ADDRESS>Alireza Marandi1<EMAIL_ADDRESS>1
Department of Electrical Engineering, California Institute of Technology,
Pasadena, CA 91125, USA  2 Department of Physics, University of Virginia,
Charlottesville, VA 22904, USA

###### Abstract

Continuous-variable quantum information processing through quantum optics
offers a promising platform for building the next generation of scalable
fault-tolerant information processors. To achieve quantum computational
advantages and fault tolerance, non-Gaussian resources are essential. In this
work, we propose and analyze a method to generate a variety of non-Gaussian
states using coherent photon subtraction from a two-mode squeezed state
followed by photon-number-resolving measurements. The proposed method offers a
promising way to generate rotation-symmetric states conventionally used for
quantum error correction with binomial codes and truncated Schrödinger cat
codes. We consider the deleterious effects of experimental imperfections such
as detection inefficiencies and losses in the state engineering protocol. Our
method can be readily implemented with current quantum photonic technologies.

##  I Introduction

Quantum information processing (QIP) opens a new paradigm for next generation
information processors, offering significant advantages over classical
analogues for various fields including computation, communication, metrology,
and sensing. In the last couple of decades, QIP has been widely explored
mostly over discrete variables with qubits on many physical platforms such as
superconducting circuits, photonics, trapped ions, quantum dots, nuclear
spins, and neutral atoms Clarke and Wilhelm (2008); O’brien  _et al._ (2009);
Cirac and Zoller (1995); Imamog¯lu  _et al._ (1999); Saffman  _et al._ (2010);
Vandersypen  _et al._ (2001). Another universal paradigm for QIP makes use of
continuous variables (CV) such as the amplitude- and phase-quadratures of the
quantized electromagnetic field for information encoding and processing Lloyd
and Braunstein (1999). The key interest in CVQIP comes primarily from its
unprecedented potential to generate a massively entangled scalable quantum
states known as cluster states at room temperature Chen  _et al._ (2014);
Larsen  _et al._ (2019); Asavanant  _et al._ (2019).
While the large cluster states have been deterministically produced, they do
not offer any exponential speedups with Gaussian gates and Gaussian
measurements alone Walls and Milburn (1994). Therefore it is of paramount
importance to include some non-Gaussian elements to achieve quantum
computational advantages Bartlett  _et al._ (2002); Menicucci  _et al._
(2006). Non-Gaussianity can be achieved through measurements such as photon-
number-resolved detection (PNRD) Lita  _et al._ (2008); Nehra  _et al._
(2019), non-Gaussian gates such as the cubic phase gate Ghose and Sanders
(2007); Yanagimoto  _et al._ (2020), or by the inclusion of non-Gaussian
resources such as binomial states Michael  _et al._ (2016a), Schrödinger cat
states Ourjoumtsev  _et al._ (2006); Eaton  _et al._ (2019a), and Gottesman-
Kitaev-Preskill (GKP) state Gottesman  _et al._ (2001). In optical systems, a
deterministic implementation of such non-Gaussian gates and non-Gaussian
states is experimentally challenging due to weak optical nonlinearities.
However, these non-Gaussian elements can be probabilistically realized using
Gaussian resources (squeezed states and linear optics) with non-Gaussian
measurements such as PNRD. Various schemes based on photon- addition,
subtraction, parity projection, and catalysis for generating quantum states
with non-Gaussian Wigner functions have been proposed and demonstrated Dakna
_et al._ (1998); Ourjoumtsev  _et al._ (2006); Eaton  _et al._ (2019a);
Zavatta  _et al._ (2004); Thekkadath  _et al._ (2020); Takase  _et al._
(2021).
Not only are non-Gaussian states required for exponential speedup, they are
also crucial for quantum error correction to achieve fault-tolerance in
quantum computation as Gaussian operations cannot protect against Gaussian
errors Niset  _et al._ (2009). A particular instance of useful non-Gaussianity
is the GKP state, which encodes discrete quantum information in a CV Hilbert
space, and thereby can be used for qubit based computation Gottesman  _et al._
(2001); Tzitrin  _et al._ (2020). The ideal form of GKP states requires
infinite energy, but universality can still be achieve with approximated GKP
states and Gaussian operations Baragiola  _et al._ (2019). While quantum error
correction (QEC) with GKP states has recently been experimentally implemented
in superconducting and trapped ion systems Campagne-Ibarcq  _et al._ (2020);
Flühmann  _et al._ (2019), an all-optical means to generate GKP states has
remained a challenge despite the existence of several proposals Gol’tsman  _et
al._ (2001); Vasconcelos  _et al._ (2010); Weigand and Terhal (2018); Etesse
_et al._ (2014); Eaton  _et al._ (2019b); Shi  _et al._ (2019); Fabre  _et
al._ (2020); Hastrup  _et al._ (2021).

Another promising avenue to exploit non-Gaussianity for CVQIP include quantum
states that are rotationally symmetric in phase space in analog to translation
phase space symmetry in GKP states. Error correcting codes designed with such
states take advantage of the fact that states with $K$-fold rotational
symmetry have decompositions over the Fock-basis with $K$-periodic spacing,
making it easier to ascertain when photon loss and dephasing errors occur.
Such codes include binomial codes and cat codes, and Pegg-Barnett codes
Grimsmo  _et al._ (2020); Michael  _et al._ (2016a); Bergmann and van Loock
(2016). While binomial and cat codes have been recently demonstrated in
superconducting quantum circuits Reinhold  _et al._ (2020); Hu  _et al._
(2019), their implementations in optical domain have remained elusive.
In this work, we introduce an all-optical method to generate rotationally
symmetric states with 2-fold and 4-fold-symmetry, in particular, binomial code
states and truncated cat code states. Our method requires resources as minimal
as a two-mode squeezed vacuum (TMSV) state, easily accessible linear optics,
and PNRD at low photon numbers. Our proposal requires squeezing levels below
what has already been demonstrated Vahlbruch  _et al._ (2016a), and low PNR
detection which is now feasible due to advances in highly efficient
transition-edge sensors Lita  _et al._ (2008) and number-resolving
superconducting nanowire single-photon detectors detectors Cahall  _et al._
(2017); Nicolich  _et al._ (2019); Endo  _et al._ (2021), and time-multiplexed
detection schemes Achilles  _et al._ (2003).
We show that by tuning the initial available resource squeezing and post-
selecting on desired the PNR outcomes, our method can generate a variety of
states in addition to exact binomial states and approximated cat states with
the generation rates in the KHz-MHz depending on the detection scheme using
state-of-the-art PNR detectors with high count rates. Moreover, these
rotationally symmetric states can be exactly produced as opposed to GKP
states, which are only approximated due to available finite squeezing limit.
Our paper is structured as follows. In Section II, we provide an overview of
bosonic quantum error correction with binomial codes. Section III discusses
the general framework of our method to generate a various rotation-symmetric
non-Gaussian states. In Section V, we focus on generating binomial and cat-
like codes in the ideal case as well as in the presence of experimental
imperfections. In Section VI, we derive the conditions for desired detection
efficiency for faithful error correction. We present an architectural analysis
in Section VII and we conclude and offer an outlook in Section VIII.

##  II Rotation-symmetric Bosonic Codes: Recap

A detailed discussion on rotationally symmetric quantum error-correcting codes
can be found in Refs. Chuang  _et al._ (1997); Michael  _et al._ (2016b); Li
_et al._ (2017); Grimsmo  _et al._ (2020); here we briefly summerize the main
concepts. The rotation-symmetric bosonic codes include the bosonic encodings
that remain invariant under discrete rotations in phase space in the same
manner as GKP encodings are invariant under phase space translations. Some
representative examples of rotation-symmetric bosonic codes are binomial codes
and cat codes, which have been recently demonstrated in circuit quantum
electrodynamics (cQED) architectures Michael  _et al._ (2016c); Albert  _et
al._ (2018); Hu  _et al._ (2019). The logical code words are stabilized by the
photon-number super parity operator defined as

$\hat{\Pi}_{K}:=e^{\frac{i2\pi\hat{n}}{K}},$ (1)

where $\hat{n}=\hat{a}^{\dagger}\hat{a}$ is the photon-number operator.
Therefore, in order to satisfy $\hat{\Pi}_{K}|\psi\rangle=|\psi\rangle$, the
state $|\psi\rangle$ must have support on every K-th Fock basis.

For a given error set
$\mathcal{E}=\\{\hat{E}_{1},\hat{E}_{2},...,\hat{E}_{l}\\}$, a necessary and
sufficient criterion for a faithful error correction is known as the Knill-
Laflamme condition, mathematically defined as:

$\hat{P}_{\mathcal{C}}\hat{E}^{\dagger}_{l}\hat{E}_{m}\hat{P}_{\mathcal{C}}=\alpha_{l,m}\hat{P}_{\mathcal{C}};\hskip
14.22636pt\forall\hskip 5.69054pt\hat{E}_{l},\hat{E}_{m}\in\mathcal{E},$ (2)

where $\hat{P}_{\mathcal{C}}$ is the projector defined over the code space
$\mathcal{C}$ and $\alpha_{l,m}$ are entries of a Hermitian matrix Knill and
Laflamme (1997). For example, let’s consider the (4-fold symmetric) binomial
code words

${|0_{L}\rangle}=\frac{1}{2}\ket{0}+\frac{\sqrt{3}}{2}\ket{4},\hskip
11.38109pt{|1_{L}\rangle}=\frac{\sqrt{3}}{2}\ket{2}+\frac{1}{2}\ket{6},$ (3)

whose amplitudes are the square roots of binomial coefficients. One can easily
see that these code words are able to detect and faithfully correct the errors
given by error set
$\mathcal{E}=\\{\mathbb{I},\hat{a},\hat{a}^{\dagger}\hat{a}\\}$ as per Knill-
Laflamme condition.

Physically, the Knill-Laflamme conditions ensure that the probability for a
single-photon loss or a dephasing error to occur are the same for each code
word, making it impossible for the environment to distinguish between the
logical basis states $\ket{0_{L}}$ and $\ket{1_{L}}$. This preserves the
encoded quantum information (the state’s amplitudes in the logical basis), as
it is not deformed as a result of the error and can thus be recovered by means
of unitary operations. To understand this further, let us consider a single-
photon loss error for quantum state $\ket{\psi}$ encoded using the code words
defined in 3. The logical state
$|\psi\rangle=\alpha|0_{L}\rangle+\beta\ket{1_{L}}$ is an even photon-number
parity state which transforms to odd parity state
$\ket{\psi^{\prime}}=\sqrt{3}\alpha\ket{3}+\sqrt{3/2}\beta(\ket{1}+\ket{5})$
under a single-photon loss error. Thus we get:

$\hat{a}\ket{\psi}\rightarrow\ket{\psi^{\prime}}=\alpha\ket{\tilde{0}_{L}}+\beta\ket{\tilde{1}_{L}},$
(4)

where $\ket{\tilde{0}_{L}}=\ket{3}$ and
$\ket{\tilde{1}_{L}}=1/\sqrt{2}(\ket{1}+\ket{5})$ are the error words, which
are orthogonal to the original code words. We note that the quantum
information encoded in the complex amplitudes $\alpha$ and $\beta$ is
preserved and can be faithfully recovered by mapping the error words
$\\{\ket{\tilde{0}_{L}},\ket{\tilde{1}_{L}}\\}$ to code words
$\\{\ket{0_{L}},\ket{1_{L}}\\}$ by means of unitary operations Michael  _et
al._ (2016b). Since the single-photon loss changes the photon-number parity of
the original state from even to odd, the parity measurements can be used as an
error syndrome measurement. In optics, error syndrome measurements can be
performed using highly efficient photon-number-resolving transition-edge
sensors Lita  _et al._ (2008); Nehra  _et al._ (2019); Morais  _et al._
(2020).

In general, the photon loss process can be viewed more precisely by subjecting
a quantum state to a completely-positive and trace-preserving (CPTP) bosonic
channel with non-unity transmission. In this case, the effect of the channel
is given by the Kraus operator-sum representation formulated as

$\rho^{\prime}=\mathcal{L}(\rho_{0})=\sum_{k=0}^{\infty}\hat{E_{k}}\rho_{0}\hat{E_{k}}^{\dagger},$
(5)

where $\rho^{\prime}$ and $\rho_{0}$ are the output and input states,
respectively, and to ensure the CPTP channel, one needs
${\sum_{k=0}^{\infty}E_{k}^{\dagger}E_{k}=\mathbb{I}}$. In this work, we treat
loss as an amplitude-damping Gaussian channel, which can be modeled as an
optical mode traveling a distance $L$ through a medium with loss coefficient
$\alpha$ dB/cm. The Kraus operator associated with losing $k$ photons is

$\hat{E_{k}}=\sqrt{\frac{(1-e^{-{\gamma}})^{k}}{k!}}e^{-\tfrac{1}{2}\gamma\hat{a}^{\dagger}\hat{a}}\hat{a}^{k},$
(6)

where we define $\gamma=\alpha L$. A $k$-photon loss to $\rho$ occurs with the
probability of

$P_{k}=\langle\hat{E_{k}}^{\dagger}\hat{E_{k}}\rangle=\frac{\gamma^{k}}{k!}\text{Tr}[\hat{a}^{k}\rho\,\hat{a}^{\dagger}{}^{k}]+\mathcal{}{O[\gamma^{k+1}]}.$
(7)

With this in mind, QEC codes with the ability to correct up to $k$-photon
losses can be reframed as codes that correct operators $E_{k}$ up to $k$-th
order in $\gamma$. As a result, the code words we propose to create in Eq. 3
have 4-fold-symmetry in phase space with the ability to correct single photon
losses, making it effective up to first order in $\gamma$ and corrects for
$E_{0}$ and $E_{1}$ corresponding to ‘no-jump’ and ‘single-jump’ errors,
respectively.

##  III Proposed Method: Analytical Model

In this section, we detail the proposed method displayed in Fig. 1. Our method
starts by preparing by a two-mode squeezed vacuum (TMSV) state by interfering
two orthogonal single-mode squeezed vacuum (SMSV) states produced by optical
parametric amplifiers (OPAs) at the first balanced beamsplitter labeled as BS
1 in Fig. 1. This is followed by two highly unbalanced beamsplitters (BS 2)
used for photon subtractions from each mode of the TMSV state. Next, a
balanced beamsplitter (BS 1) is placed to interfere the subtracted photons in
order to erase the information about from which mode the subtracted photons
came from. As a result, this combination of photon subtractions and the
balanced interference allows one to coherently subtract photons from the TMSV
state. Finally, photon-number-resolving (PNR) measurements are performed on
three output modes prepares the desired state $|\psi\rangle$ in the fourth
mode for a certain combination of PNR measurement outcomes.
Interfering two SMSV states at a balanced beamsplitter is equivalent to
preparing a TMSV state by action of the two-mode squeezed operator
$S(z)_{ab}=e^{[z\hat{a}^{\dagger}\hat{b}^{\dagger}-z^{*}\hat{a}\hat{b}]}$ on
two-mode vacuum, where $z=\mathcal{}{R}e^{i\phi}$ with real squeezing
parameter $\mathcal{}{R}$ and phase $\phi$. In the photon-number basis, the
TMSV state prepared in modes ‘a’ and ‘b’ after the first BS 1 is given as
Walls and Milburn (1994)

$|0,\mathcal{}{R}\rangle_{ab}=\frac{1}{\cosh{{R}}}\sum_{n=0}^{\infty}e^{in\phi}\tanh^{n}R|nn\rangle_{ab},$
(8)

After encountering the highly unbalanced beamsplitters (BS 2), the state
interferes with vacuum modes ‘c’ and ‘d’, which leads to the four-mode state

$|\psi\rangle_{\text{abcd}}=\hat{U}_{ac}(\theta)\hat{U}_{bd}(\theta)|0,\mathcal{}{R}\rangle_{ab}|00\rangle_{cd},$
(9)

where $\hat{U}_{ac}(\theta),\hat{U}_{bd}(\theta)$ are beamsplitter unitary
operators where $(t,r)=(\text{cos}\theta,\text{sin}\theta)$ are the
transmission and reflections coefficients, respectively. We first consider the
simplest case where the probability of subtracting more than one photon is
negligible, i.e., at most one photon is subtracted from each mode. For single
photon subtraction, we set both beamsplitters such that $t\rightarrow 1$
($\theta<<1$) which allows one to approximate the unitary operators as

$\displaystyle\hat{U}_{ac}(\theta)$
$\displaystyle=\text{exp}[\theta(\hat{a}^{\dagger}\hat{c}-\hat{a}\hat{c}^{\dagger})]\approx\mathbb{I}+\theta[\hat{a}^{\dagger}\hat{c}-\hat{a}\hat{c}^{\dagger}]+\mathcal{}{O}(\theta^{2}),$
$\displaystyle\hat{U}_{bd}(\theta)$
$\displaystyle=\text{exp}[\theta(\hat{b}^{\dagger}\hat{d}-\hat{b}\hat{d}^{\dagger})]\approx\mathbb{I}+\theta[\hat{b}^{\dagger}\hat{d}-\hat{b}\hat{d}^{\dagger}]+\mathcal{}{O}(\theta^{2}),$
(10)

where we consider $\mathcal{O}(\theta^{2})$ terms negligible to ensure that
mostly single photons are subtracted from each mode. From Eq. 9 and Eq. 10, we
get the unnormalized four-mode state

$\hat{U}_{ac}(\theta)\hat{U}_{bd}(\theta)|0,\mathcal{}{R}\rangle_{ab}|00\rangle_{cd}\propto\sum_{n}^{\infty}e^{in\phi}\text{tanh}^{n}R[\mathbb{I}+\theta(\hat{a}^{\dagger}\hat{c}-\hat{a}\hat{c}^{\dagger}+\hat{b}^{\dagger}\hat{d}-\hat{b}\hat{d}^{\dagger})]|nn\rangle_{ab}|00\rangle_{cd},\\\
$ (11)

Finally, the action of last balanced beamsplitter (BS 2) leads to the
resulting state

$|\psi\rangle_{\text{abcd}}\propto\sum_{n}^{\infty}e^{in\phi}\text{tanh}^{n}R\bigg{[}|00\rangle_{cd}-\frac{\theta}{\sqrt{2}}\big{[}(\hat{a}-\hat{b})|10\rangle_{cd}+(\hat{a}+\hat{b})|01\rangle_{cd}\big{]}\bigg{]}|nn\rangle_{ab}.$
(12)

We now perform the PNR measurements on three output modes as depicted in Fig.
1. Looking at the state in Eq. 12, it can be immediately seen that for
$n_{1}=n_{2}=0$, the initial TMSV state is unchanged as one expects. We now
consider two cases of $n_{1}=1,n_{2}=0$ and $n_{1}=0,n_{2}=1$, i.e., only one
of the two detectors detects a single photon. The two orthogonal output states
in these two cases are

$|\psi\rangle^{\pm}_{ab}\propto\sum_{n}^{\infty}\sqrt{n}e^{in\phi}\text{tanh}^{n}R[\ket{n-1,n}\pm\ket{n,n-1}],$
(13)

where the positive (negative) sign corresponds to [$n_{1},n_{2}]=[1,0]$
([$n_{1},n_{2}]=[0,1]$).

It is worth mentioning that states like $\ket{\psi}^{\pm}$ have been shown to
achieve Heisenberg-limit of phase measurements in quantum interferometry
Hofmann (2006); Eaton  _et al._ (2021).

Figure 1: The proposed scheme for generating rotation-symmetric error-
correcting codes. A two-mode squeezed vacuum (TMSV) state is generated by
combining two orthogonal single-mode squeezed vacuum (SMSV) states at the
first balanced beamsplitter (BS 1). Photons are then subtracted from each
modes of the TMSV state using highly unbalanced beamsplitters (BS 2). The
subtracted photons are interfered at the second balanced beamsplitter followed
by PNR measurements on three output modes, which prepares the fourth mode in
the desired code for a certain measurement combination of $n_{1},n_{2}$, and
$n_{3}$. The dotted red box represents the coherent photon subtraction
process. A particular case shown here considers $[n_{1},n_{2},n_{3}]$ =
$[1,1,4]$, resulting in $\ket{\psi}\propto\ket{2}+\ket{6}$ with 4-fold
rotational symmetry as evident in the displayed Wigner function. OPA: Optical
parametric amplifier. BS 1: Balanced beamsplitter. BS 2: Unbalanced
beamsplitter for photon subtraction. EOM: Electro-optic modulator.

We now consider a third PNR detector placed in the path of mode ‘b’ to detect
$n_{3}$ photons. The third PNR measurement projects the $\rho^{\pm}$ to the
pure state given by

$\displaystyle|\psi\rangle^{\pm}_{a}=$
$\displaystyle\mathcal{N}[\sqrt{n_{3}c_{n_{3},n_{3}}}|n_{3}-1\rangle$
$\displaystyle\pm\sqrt{(n_{3}+1)c_{n_{3}+1,n_{3}+1}}|n_{3}+1\rangle],$ (14)

where

$\mathcal{N}=\sqrt{n_{3}c_{n_{3},n_{3}}+(n_{3}+1)c_{n_{3}+1,n_{3}+1}}$ (15)

$c_{n,n^{\prime}}={e^{i\phi(n+n^{\prime})}\tanh^{n+n^{\prime}}{\mathcal{}{R}}}/{\cosh^{2}{\mathcal{}{R}}}$
(16)

From Eq. 14, one can see that for $n_{3}$ = odd (even), the final state has
two consecutive even (odd) Fock components corresponding to $n_{3}-1$ and
$n_{3}+1$. We note that this particular task of coherent single-photon
subtraction with highly unbalanced beamsplitters can be performed with click
detectors by ensuring that the probability of reflecting more than one photon
is vanishing. However, the third detector needs to be PNR detector. In Section
V, we show that how one can generate rotation-symmetric states with 2-fold and
4-fold phase space symmetry in the Wigner function.
We now generalize the proposed method for arbitrary detection of $n_{1}$ and
$n_{2}$ photons. In this case, to the leading order approximation, only the
terms involving up to $\theta^{n_{1}+n_{2}}$ will contribute in the expansion
of unitary operators of the photon subtracting beamsplitters (BS 2 in Fig. 1)
in Eq. 11. As a result, the Taylor expansion can be approximated to

$\hat{U}_{ac}(\theta)\hat{U}_{bd}(\theta)\approx\sum_{k=0}^{N}\frac{\theta^{N}}{k!(N-k)!}(\hat{a}^{\dagger}\hat{c}-\hat{a}\hat{c}^{\dagger})^{N-k}(\hat{b}^{\dagger}\hat{d}-\hat{b}\hat{d}^{\dagger})^{k},$
(17)

where $N=n_{1}+n_{2}$ is the total number of photons detected after the photon
subtraction step. Since we have vacuum inputs to modes $c$ and $d$, and as a
consequence of detecting $N$ photons in total, Eq. 17 reduces to

$\hat{U}_{ac}(\theta)\hat{U}_{bd}(\theta)\approx\sum_{k=0}^{N}\frac{(-\theta)^{N}}{k!(N-k)!}(\hat{a}\hat{c}^{\dagger})^{N-k}(\hat{b}\hat{d}^{\dagger})^{k}.$
(18)

This approximation can now be transformed by the second balanced beamsplitter,
$\hat{U}_{cd}(\theta)$, into

$\hat{U}_{cd}\hat{U}_{ac}(\theta)\hat{U}_{bd}(\theta)\hat{U}_{cd}^{\dagger}\approx\bigg{(}\frac{-\theta}{\sqrt{2}}\bigg{)}^{N}\sum_{k=0}^{N}\frac{\hat{{a}}^{N-k}\hat{b}^{k}}{k!(N-k)!}(\hat{c}^{\dagger}+\hat{d}^{\dagger})^{N-k}(-\hat{c}^{\dagger}+\hat{d}^{\dagger})^{k}.$
(19)

The action of these beamsplitters to the TMS resource state and vacuum modes
$c$ and $d$ followed by detecting $n_{2}$ and $n_{1}$ photons in the
transformed modes $c$ and $d$, respectively, leads to the final state

$|\psi\rangle_{ab}=\hat{U}_{cd}\hat{U}_{ac}(\theta)\hat{U}_{bd}(\theta)\hat{U}_{cd}^{\dagger}|0,R\rangle_{a,b}\otimes|0,0\rangle_{c,d}\propto\sum_{k=0}^{N}A_{n_{1},n_{2},k}\hat{a}^{N-k}\hat{b}^{k}|0,\mathcal{}{R}\rangle_{ab},$
(20)

where the coefficient $A_{n_{1},n_{2},k}$ is

$A_{n_{1},n_{2},k}=\sum_{i_{min}}^{i_{max}}\frac{(-1)^{i+k-n_{>}}\sqrt{n_{1}!n_{2}!}}{i!(i+k-n_{>})!(n_{>}-i)!(N-i-k)!},$
(21)

and the summation goes from $i_{min}=\text{Max}(0,n_{>}-k)$ to
$i_{max}=\text{Min}(n_{>},N-k)$, where $n_{>}=\text{Max}(n_{1},n_{2})$. The
overall success probability of this combination of PNR measurements is

$P([n_{1},n_{2}])=\Big{(}\frac{\theta}{\sqrt{2}}\Big{)}^{2N}\text{Tr}\Big{[}\sum_{k=0}^{N}\sum_{j=0}^{N}A_{n_{1},n_{2},k}A_{n_{1},n_{2},j}\hat{a}^{N-k}\hat{b}^{k}|0,\mathcal{}{R}\rangle_{ab}\langle
0,\mathcal{}{R}|\hat{a}^{{\dagger}^{N-j}}\hat{b}^{{\dagger}^{j}}\Big{]}.$ (22)

One salient point to note about the coherent photon subtraction is that the
approximation of weakly reflective subtraction beamsplitters used in Eq. 18 is
not strictly necessary. In fact, as we show in appendix IX.2, the full
treatment of perfect coherent photon subtraction from a TMSV state with
arbitrary beamsplitter parameter $\theta$ reduces to the same two-mode state
in Eq. 20, but with a different effective squeezing parameter. If the initial
experimental squeezing parameter is $\mathcal{}{R}$, then the nonzero
beamsplitter reflectivity acts to replace $\mathcal{}{R}$ in Eq. 20 with an
effective squeezing parameter of
$\mathcal{}{R}^{\prime}=\text{tanh}^{-1}{(t^{2}\tanh{\mathcal{}{R}})}$, where
$t^{2}=\cos^{2}{\theta}$ is the transmission of each subtraction beamsplitter.
Later we show how one can utilize this feature of coherent photon subtraction
to increase the success rates of preparing the desired states. We now
investigate the third PNR measurement on the coherently photon-subtracted
state represented by Eq. 20. A measurement outcome of $n_{3}$ photons projects
the two-mode state $|\psi\rangle_{a,b}$ to the following parity state

$\displaystyle|\psi\rangle_{a}$
$\displaystyle=\mathcal{N}\sum_{k=k_{min}}^{N+n_{3}}A^{\prime}_{n_{1},n_{2},k}|2k-N-
n_{3}\rangle_{a},$ (23)

where we have

$A^{\prime}_{n_{1},n_{2},k}=A_{n_{1},n_{2},(k-n_{3})}\frac{(e^{i\phi}\tanh{\mathcal{}{R}})^{k}k!}{\sqrt{(2k-N-
n_{3})!}},$ (24)

and $k_{min}=\text{Max}(n_{3},\lceil\tfrac{1}{2}N+\tfrac{1}{2}n_{3}\rceil)$
with $\lceil.\rceil$ being the ceiling function. The normalization coefficient
is given by

$\mathcal{N}=\frac{\left(\tfrac{1}{2}\theta^{2}\right)^{N}}{n_{3}!(\cosh{\mathcal{}{R}})^{2}\sqrt{P_{\text{succ}}}},$
(25)

where $P_{\text{succ}}$ is the overall success probability of the measurement
outcome configuration $[n_{1},n_{2},n_{3}]$ given as

$P_{\text{succ}}=\frac{\left(\tfrac{1}{2}\theta^{2}\right)^{N}}{n_{3}!(\cosh{\mathcal{}{R}})^{2}}\sum_{k=k_{min}}^{N+n_{3}}(A^{\prime}_{n_{1},n_{2},k})^{2}.$
(26)

This completes the generalized derivation for coherent photon subtraction
scheme for generating rotation-symmetric error correcting codes. Examining Eq.
23, we see that in addition to the state having definite photon-number parity,
the number of Fock components in the superposition is determined by the number
of subtracted photons to reach a maximum of $N+1$ when $n_{1}\neq n_{2}$ and
$n_{3}>N$. When $n_{3}<N$, the maximum number of components drops to
$\lfloor\tfrac{1}{2}N+\tfrac{1}{2}n_{3}\rfloor+1$. A particularly interesting
case occurs when $n_{1}=n_{2}$. In this case the coefficient
$A_{n_{1},n_{2},k}$ vanishes for $k=\text{odd}$ as seen below:

$\displaystyle A_{n_{1},n_{2},k_{odd}}=\sum_{i=0}^{n_{2}-\frac{k-1}{2}}$
$\displaystyle(-1)^{n_{2}+k-i}\big{[}(2n_{2}-k-i)!(k+i-n_{2})!i!(n_{2}-i)!)\big{]}^{-1}$
$\displaystyle+(-1)^{-n_{2}-i}\big{[}i!(n_{2}-i)!(2n_{2}-k-i)!(k+i-n_{2})!)\big{]}^{-1}=0.$
(27)

Therefore, only the terms where $k=\text{even}$ contribute to the state in Eq.
23 meaning that the final state has half as many Fock-basis components as the
$n_{1}\neq n_{2}$ case, and each of these components is separated by integer
multiples of 4. Indeed, this is a consequence of extended Hong-Ou-Mandel (HOM)
type interference between the subtracted photons Hong  _et al._ (1987). Since
we are considering only the case where $n_{1}=n_{2}$, the subtracted photons
from each mode of the TMSV can only occur in even numbers. The simplest case
of such scenario is $n_{1}=n_{2}=1$, which means both the subtracted photons
came from either mode ‘a’ or ‘b’, as one expects in a typical HOM experiment.
As we show later, this extended HOM interference property can be used to
prepare rotationally-symmetric states which can be used in bosonic quantum
error correction schemes Michael  _et al._ (2016a); Grimsmo  _et al._ (2020).

##  IV Numerical Experiments

In general for arbitrary $n_{1},n_{2},$ and $n_{3}$, we show in Fig. 2 how the
probability to successfully generate any non-Fock state with either 2-or
4-fold phase space symmetry can be optimized by varying the subtraction
beamsplitter (BS 2 in Fig. 1) reflectivies based on the initial resource
squeezing level. Since changing the subtraction beamsplitter reflectivity has
no effect on the form of the output state other than to reduce the original
TMS state squeezing parameter to a smaller effective squeezing, this feature
can be used as an external experimental knob to tune the mean output state
energy and boost the generation rates. The ‘non-Fock’ qualifier means we
exclude generated states that are composed of only a single Fock-basis
component. Thus, the probabilities plotted simply sum the probabilites to
obtain any interesting superposition state of the desired symmetry for a given
subtraction beamsplitter reflectivity at each initial squeezing. Additionally,
4-fold rotation symmetric are also posses 2-fold symmetry, Fig. 2a shows the
probability to generate either type of states. In Fig. 2b, we exclusively show
the case of $n_{1}=n_{2}$ where 4-fold symmetric states are generated.

Figure 2: (a) The probability to successfully generate any non-Fock parity
state (2-fold or 4-fold-symmetry) as the subtraction beamsplitter reflectivity
is changed for many values of initial two-mode squeezing. (b) Probability to
successfully generate only 4-fold-symmetry states. (c),(d) The average of the
mean-photon number of the successfully generated states. The large dots
indicate the mean photon number of a single mode of input state at each
respective squeezing. Regions on the curves to the left of the dots show it is
likely to increase the average energy of the mode when the state is
successfully generated.

The plots demonstrate that there is a threshold value for the beamsplitter
reflectivity beneath which the generation probability drops precipitously, but
increasing the reflectivity beyond about $10\%$ ($r^{2}=0.1$) gives limited
returns. Furthermore, in Fig. 2c and 2d we show the expected mean photon
number, $\langle a^{\dagger}a\rangle_{\text{exp}}$ of the output state given
the specified initial conditions. These values are obtained by averaging over
the mean photon number of each output state weighted by the probability of
obtaining that state while excluding the non-superposition states. As one can
see, the value of $\langle a^{\dagger}a\rangle_{exp}$ for the generated states
only decreases as reflectivity increases, which intuitively follows
considering more of the original TMS state energy is diverted toward the
subtraction detectors. In all cases however, it is interesting to see that for
all values of squeezing considered, the energy of the generated state can be
increased beyond what was present in a single mode of the initial TMS state.
The large dot corresponding to each squeezing curve in the plots shows the
mean photon-number of each single mode of the initial resource state and thus
the regions to the left of this point indicate that on average, successfully
generating the desired symmetry state with the indicated squeezing and
beamsplitter reflectivities will increase the mean photon-number of the output
state. This effect is attributable to subtracting photons from two-mode
squeezed vacuum, and has been noted previously in the context of quantum state
interferometry Carranza and Gerry (2012).

##  V State Engineering for Bosonic Error Correcting Codes

In this section, we show how the proposed scheme can be tuned to generate
exact binomial- and truncated cat-like codes by choosing certain combinations
of PNR measurements. We provide a detailed analysis on required resource
squeezing, success probabilities, and how lossy PNR detection affects the
quality of the generated bosonic codes.

###  V.1 General two-component states

As mentioned earlier in Sec. III, simply by tuning the initial squeezing
parameter and post-selecting for either $n_{1}=1,n_{2}=0$ or $n_{1}=0,n_{2}=1$
produces any 2-fold symmetric two-component state of the form

$\ket{\psi_{\pi}}\propto\ket{m-1}+\alpha\ket{m+1},$ (28)

where the only requirement is that $|\alpha|<\sqrt{\tfrac{m+1}{m}}$.
Similarly, detecting one photon in each of modes $c$ and $d$ ($n_{1}=n_{2}=1$)
allows the production of arbitrary 4-fold symmetric two-component states of
the form

$\ket{\psi_{\pi/2}}\propto\ket{m-2}+\beta\ket{m+2},$ (29)

where $|\beta|<\sqrt{\tfrac{(m+1)(m+2)}{m(m-1)}}$. In the above formulas,
$\alpha$ and $\beta$ are complex numbers determined by the amplitude and phase
of the squeezing parameter for the initial resource TMSV state. If the initial
squeezing parameter is written in complex form as $z=\mathcal{}{R}e^{i\phi}$,
then beginning with a state that satisfies

$e^{i\phi}\tanh{\mathcal{}{R}}=-\alpha\bigg{(}\frac{{m}}{{m+1}}\bigg{)}^{\frac{1}{2}}$
(30)

and performing a PNR detection of $n_{3}=m$ after the coherent subtraction
leads to the creation of the 2-fold-symmetry state with parameter $\alpha$ in
Eq.28. In order to generate the 4-fold symmetric state in Eq. 29, the
condition on the initial squeezing is

$e^{i\phi}\tanh{\mathcal{}{R}}=\sqrt{-\beta}\left(\frac{m(m-1)}{(m+1)(m+2)}\right)^{\frac{1}{4}},$
(31)

where again a PNR detection of $n_{3}=m$ must follow the coherent photon
subtraction of the $n_{1}=n_{2}=1$ case.

###  V.2 Binomial Code State Generation

We now consider two particular cases of 4-fold symmetric two-component states,
in particular the binomial code words of Eq. 3. These binomial code words
${|0_{L}\rangle}$ and ${|1_{L}\rangle}$ can be exactly created by coherently
subtracting $n_{1}=n_{2}=1$ photons and performing respective PNR measurement
of $n_{3}=2$ and $n_{3}=4$, provided the effective squeezing is chosen to
satisfy the relation in Eq. 31 for $\beta_{\ket{0_{L}}}=\sqrt{3}$ and
$\beta_{\ket{1_{L}}}=1/\sqrt{3}$. In order to satisfy these requirements, the
experimental resources must achieve squeezing thresholds of $10.63$ dB to
perfectly create the ${|0_{L}\rangle}$ state and $6.08$ dB to create the
${|1_{L}\rangle}$ state. Note that for experimental squeezing above these
thresholds, the beamsplitter reflectivity must be tuned to decrease the
effective squeezing back to these quantities. However, squeezing below these
thresholds does not mean that QEC with rotationally-symmetric code words is no
longer possible. Lower squeezing means that the lower Fock-components of the
code states will contribute more to the superposition, but as long as the
coefficients for the two code words are chosen such that they have the equal
photon number moments, error correction is still possible as in accordance
with Knill-Laflamme condition. In fact, the coefficients deviating from a
binomial distribution may prove beneficial in specific circumstances if they
are optimized to reduce specific error rates Michael  _et al._ (2016b).
For the present discussion, we limit our focus to the binomial code states of
Eq. 3. We show in Fig. 3a how the probability to successfully generated the
binomial code states varies with initial resource squeezing. When the
squeezing is exactly at the thresholds (10.63 dB for $\ket{0_{L}}$ and 6.08 dB
for $\ket{1_{L}}$) to generate the desired state, the subtraction
beamsplitters must have vanishing reflectivities, resulting in small success
probabilities as shown on the left most parts of the plots in Fig. 3. For
larger available initial squeezing, however, increasing the subtraction
reflectivity until the effective squeezing is at the required thresholds
boosts the success probability by three orders of magnitude for both code
states. Note that these larger squeezing values for optimized success
probability are within the realm of realistically achievable squeezing
Vahlbruch  _et al._ (2016a).

Figure 3: (a) Probability to successfully generate the code states
$\ket{0_{L}}$ and $\ket{1_{L}}$ as initial squeezing is increased. The
subtraction beamsplitters are tuned to satisfy Eq. 31. (b) Probability to
successfully generate any 4-fold symmetric state and (c) and 2-foldsymmetric
state with $k$ Fock-basis components in the each superposition as subtraction
beamsplitter reflectivity is increased given an initial squeezing of 12 dB.

In Fig. 3b, we display the success probability to generate multi-component
4-fold symmetric states with $k$ Fock-basis components in the superposition
given an initial resource squeezing value of $12$ dB as the subtraction
beamsplitter reflectivity varies. Generating states with $k$ components in the
superposition is a higher order process in the beamsplitter reflectivity as it
requires $n_{1}=n_{2}\geq k-1$, so the two component state described by Eq. 29
is the most common result. It is worth mentioning that the growing number of
components in the superposition is a consequence of extended HOM interference.
The post-selection is done for the particular case of $n_{1}=n_{2}$ after the
HOM interference, but this can happen in many ways as long as we have
$n_{1}^{(-)}=N-2l,n_{2}^{(-)}=2l;l\in[0,N/2]$, where $n_{1}^{(-)}$ and
$n_{2}^{(-)}$ denotes the number of photons subtracted from modes ‘$a$’ and
‘$b$’, respectively before the HOM inferences at the second balanced
beamsplitter, $BS$ 1\. Likewise, Fig. 3c shows the success probability to
generate $k$-component 2-fold-symmetry states with an initial $12$ dB of
squeezing.
In Fig. 4, we plot the Wigner functions of the generated binomial code words
with the optimal effective squeezing of 10.63 dB and 6.08 dB for $\ket{0_{L}}$
and $\ket{1_{L}}$, respectively. On the left, the resultant Wigner functions
(a) and (c) are for the case of perfect detection ($\eta=1$) for all three PNR
detectors. As expected, the Wigner functions display 4-fold symmetry owing to
the photon-number support on every 4th Fock state. In an experimental
realization of such state engineering protocol, the photon losses due to
propagation though lossy optical components and inefficient PNR detections are
inevitable and lead to decoherence of the desired state.
Next, we investigate the effects of imperfect detection ($\eta=0.90$) for
$n_{3}$ measurement for generating both the binomial code words. As evident
from the Wigner functions in Fig. 4 (b) and (d), the negativity of the Wigner
functions is reduced while the phase space symmetry is still preserved. Note
that we have considered the ideal detection for $n_{1}$ and $n_{2}$ PNR
measurements since the deleterious effects of their imperfect detection can be
reduced by lowering the reflectivity of the subtracting beamsplitters to
ensure that the probability of subtracting more than one photon is negligible.
While this prevents the decoherence of the generated code words caused by
imperfect detection of $n_{1}$ and $n_{2}$, it lowers the overall success
rate. Therefore, the PNR detectors with high detection efficiency are desired
in such state engineering protocols.

Figure 4: Wigner function visualization of binomial codewords. (a) and (c) are
the ideal codewords for $\ket{0_{L}}$ and $\ket{1_{L}}$ when no losses are
considered in our protocol. (b) and (d) are in the presence of 10% losses in
the PNR measurement of $n_{3}$ photons. We see that the features such as phase
space interference and the negativity of the Wigner functions are preserved
with 10% losses.

We then evaluate the state Fidelity, Wigner Negativity, and Wigner Log
Negativity (WLN) against the imperfect detection efficiency, $\eta$ for
$n_{3}$ detection. The losses are modeled by setting up a fictitious
beamsplittier of transmissivity $\eta$ in the path of $n_{3}$ photons.
Numerical simulations are displayed in Fig. 5. The generated state fidelity,
defined as
$F(\rho,\sigma)=\big{(}\text{Tr}[\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}]\big{)}^{2}$,
with the ideal binomial code words in Eq. 3 is plotted against the overall
detection efficiency in Fig. 5a. We see that for a given detection efficiency,
$|1_{L}\rangle$ performs better than $|0_{L}\rangle$, as evident from the
larger fidelity difference at lower overall detection efficiency.

Figure 5: Effects of lossy PNR detection for $n_{3}$ photons. (a) State
Fidelity with the ideal codewords. The minimum Wigner function amplitude and
Wigner Log Negativity (WLN), (b) and (c), respectively.

In CV resource theories, the Wigner Negativity (WN), i.e., the minimum value
of the negativity of Wigner function, and Wigner Log Negativity (WLN) are the
key measures to quantify the non-Gaussian nature of quantum resources which
are essential for quantum computational advantages and fault-tolerance Takagi
and Zhuang (2018). The WLN is defined as

$\text{WLN}:=\text{ln}\bigg{[}\int|W(q,p)|dqdp\bigg{]},$ (32)

and it immediately follows that a non-negative Wigner function has zero WLN.
The WN and WLN are plotted in Fig. 5b and Fig. 5b, respectively. We found that
the non-Gaussian nature of the code words is preserved even for the detection
efficiency as low as $\eta=0.5$, as seen from the negative minimum amplitude
of the Wigner function, $W(Q,P)_{\text{min}}$ and non-zero value of WLN. It is
worth pointing out that the PNR detection efficiency of 98% is a
experimentally feasible with state-of-the-art transition-edge sensors Fukuda
_et al._ (2011), and methods to obtain $>99\%$ are possible Gerrits  _et al._
(2016). This concludes the realistic implementations of our state engineering
scheme for binomial codes.

###  V.3 Truncated Cat Code State Generation

In this section, we investigate the feasibility of generating code words
having support on higher photon-number. We show that such codes satisfy the
Knill-Laflamme condition for faithful error correction against single-photon
loss. Consider two events where we perform the detection combinations of
$[n_{1},n_{2},n_{3}]=[0,1,m]$ and $[n_{1},n_{2},n_{3}]=[0,2,m]$ with $m\geq
2$, which we denote as $\ket{\psi_{0,1,m}}$ and $\ket{\psi_{0,2,m}}$,
respectively. Using the results from Eqs. 23 and 24, the former case yields a
two component 2-fold symmetric state of the form

$\ket{\psi_{0,1,m}}\propto
e^{-i\frac{\phi}{2}}(\tanh{R})^{-\frac{1}{2}}\sqrt{m}\ket{m-1}-e^{i\frac{\phi}{2}}(\tanh{R})^{\frac{1}{2}}\sqrt{m+1}\ket{m+1},$
(33)

while the latter case gives us

$\ket{\psi_{0,2,m}}\propto
e^{-i\phi}(\tanh{R})^{-1}\frac{\sqrt{m}}{\sqrt{m+1}}\ket{m-2}+2\frac{\sqrt{m+1}}{\sqrt{m-1}}\ket{m}+e^{i\phi}\tanh{R}\frac{\sqrt{m+2}}{\sqrt{m-1}}\ket{m+2}.$
(34)

By inspection, it is easy to see that these states are mutually orthogonal
where $\ket{\psi_{0,2,m}}$ has parity of $m$ and $\ket{\psi_{0,1,m}}$ has the
opposite parity. As seen from the Wigner function plots of the example cases
of $m=5$ and $m=7$, these types of states resemble truncated cat-like states
with the associated fast interference fringes near the origin in phase-space.
If we can arrange for the states to satisfy the Knill-Laflamme condition
defined in Eq. 2

$\bra{\psi_{0,1,m}}\hat{a}^{\dagger}\hat{a}\ket{\psi_{0,1,m}}=\bra{\psi_{0,2,m}}\hat{a}^{\dagger}\hat{a}\ket{\psi_{0,2,m}},$
(35)

then the two states can be used as code words to protect against single-photon
losses. Using the level of squeezing as a tuning experimental knob, this
condition can be exactly satisfied. In Fig. 7, we show how each pair of states
can be made with the same experimental scheme without changing the subtraction
beamsplitter reflectivities, and that for the correct squeezing (indicated by
the solid dot where the solid and dotted lines cross in Fig. 7a ), the two
orthogonal states each have the same mean photon number. This means that the
scheme can be set with some initial squeezing and beamsplitter parameters and
is capable of generating either of the pair of orthogonal states. For higher
detected $m$, the two orthogonal states have nearly the same mean photon
number for values of squeezing past the exact point as well, leading to
slightly larger success probabilities for both $\ket{\psi_{0,1,m}}$ and
$\ket{\psi_{0,2,m}}$ as shown in the respective plots in Fig. 7b and 7c. The
general form of Eq. 23 can be used to find other pairs of orthogonal truncated
cat-like states with more terms in the superposition beyond the simple case of
measuring only zero and one or zero and two photons in the subtraction step.
However, states created by subtracting larger numbers of photon generally
arise less frequently due to decreasing success probabilities with unbalanced
beamsplitters, and for arbitrary subtractions of $n_{1}$ and $n_{2}$, cat-like
state production may not happen for every end PNR detection of $n_{3}$. Next,
we consider how to enlarge the success probability by using multiplexing
techniques.

Figure 6: Wigner functions and photon-number distributions of codewords with
2-fold phase space symmetry. (a) and (b) are two orthogonal codewords with
mean photon-number $\bar{n}\approx 4.6$ for the measurement outcomes of
$[n_{1},n_{2},n_{3}]=[0,2,5]$ and $[n_{1},n_{2},n_{3}]=[0,1,5]$, respectively.
Likewise, (c) and (d) are for $[n_{1},n_{2},n_{3}]=[0,2,7]$ and
$[n_{1},n_{2},n_{3}]=[0,1,7]$ with mean photon number $\bar{n}\approx 6.7$.
Insets show the photon-number distributions having support on odd and even
photon numbers. The effective squeezing for (a) and (b) is 6.42 dB and 8.30 dB
for (c) and (d). Figure 7: (a) Mean photon number of the orthogonal states
generated by detecting $[0,1,m]$ (dashed lines) and $[0,2,m]$ (solid lines)
photons as initial squeezing increases for a fixed subtraction beamsplitter
reflectivity of $10\%$. For each $m$, there is a squeezing value that yields
orthogonal states with the same mean photon number (indicated by solid dots)
which is necessary to satisfy the Knill-Laflamme criteria. (b) The probability
to generate the $[0,1,m]$ or $[1,0,m]$ states, which differ only by an optical
phase, and (c) the probability to generate the $[0,2,m]$ or $[2,0,m]$ states.

###  V.4 Success probability enhancement by resource multiplexing

Thus far, we have considered the generation of bosonic codes using only one
setup presented in Fig. 1. To boost the success probability of generating a
particular code one can employ multiplexing schemes, which have been used to
enlarge the heralding probability of single-photons Bonneau  _et al._ (2015);
Joshi  _et al._ (2018); Kaneda and Kwiat (2019). For a given success
probability $P_{\text{succ}}$ per state generation setup, the probability of
generating the desired code with $N_{\text{MUX}}$ multiplexed resources is

$P_{\text{MUX}}=1-(1-P_{\text{succ}})^{N_{\text{MUX}}}$ (36)

For a given success probability tolerance $\delta=1-P_{\text{MUX}}$, one can
estimate the number of multiplexed sources as

${N_{\text{MUX}}}=\bigg{\lceil}{\frac{\text{log}_{10}(\delta)}{\text{log}_{10}(1-P_{\text{succ}})}}\bigg{\rceil},$
(37)

where $\lceil.\rceil$ is the ceiling function. In Fig.8(a), we plot the
success probability against the number of multiplexed sources for states with
both 2-fold (solid line) and 4-fold (dotted line) symmetry at assumed
experimental squeezing values of $15$ dB (red) and $6$ dB (blue). The
probabilities are calculated using beamsplitter parameters that maximize the
single-scheme success probabilities for each case according to Fig. 2. Fig.
8(a) also shows the the multiplexed probability to generate the $\ket{0_{L}}$
(dashed green) and $\ket{1_{L}}$ (dashed black) states where again the other
parameters are chosen to maximize individual scheme success probabilities. The
figure inset zooms in to show where the multiplexed probabilities exceed 0.90.
For respective probability tolerances of $0.1$ and $0.01$, only $N_{MUX}=4$
and 8 schemes are needed to successfully generation a 2-fold symmetric
superposition when initial squeezing is $15$ dB. The second panel of the
figure, Fig. 8(b) performs the same multiplexing analysis on the truncated
cat-like states whose Wigner function are shown in Fig. 6. The solid lines
show the $P_{MUX}$ for the cases where the initial squeezing is chosen such
that the mean photon number of orthogonal pairs
($\ket{\psi_{0,1,m}}\text{and}\ket{\psi_{0,2,m}}$) is the same. However, the
success probability can be significantly increased when larger initial
squeezing is available, and orthogonal states still have approximately the
same mean number of photons as seen in Fig. 7. The dotted lines in Fig. 8(b)
show $P_{\text{MUX}}$ when the input squeezing is 12 dB.

Figure 8: Including $N_{MUX}$ multiplexed protocols increases the overall
success probability, $P_{MUX}$. (a) Curves are shown for the probability to
generate any superposition 2-fold symmetric state using 15 dB and 6 dB initial
squeezing (solid red and blue, respectively), any 4-fold symmetric at 15 and 6
dB (dotted red and blue), and the binomial code words $\ket{1_{L}}$ and
$\ket{0_{L}}$ (dashed black and green) when initial squeezing maximizes the
single-shot success probability. (b) The $P_{MUX}$ values are plotted for the
truncated cat-like states of the Wigner functions shown in Fig. 6. Solid lines
correspond to states generated from initial squeezing values that ensure the
mean photon number is the same for each orthogonal state in the pair of
$\ket{\psi_{0,1,m}}$ and $\ket{\psi_{0,2,m}}$. Dotted lines assumed a flat
$12$ dB of initial squeezed, and the mean photon numbers of the pairs of
states are now only approximately equal.

##  VI Imperfect binomial code words

In order to engineer the states ${|0_{L}\rangle}$ and ${|1_{L}\rangle}$, the
final PNR detector must register two and four photons, respectively. However,
when the detector is imperfect and has efficiency $\eta<1$, this leads to
mixtures instead of ideal code words. We wish to evaluate how well error
correction can proceed and try to find a bound on the detector inefficiency
beneath which QEC is still possible. In our analysis, we ignore the dark
counts of the detector which is a reasonable assumption for superconducting
single-photon detectors. Let’s begin with the state after coherent photon
subtraction where a single photon was measured in each of the subtraction
detectors, i.e., $n_{1}=n_{2}=1$. The state after photon subtraction is

$\ket{\psi}\propto(\hat{a}^{2}-\hat{b}^{2})\ket{0,\mathcal{}{R}}_{ab},$ (38)

Note that this process can be assumed to be ideal, as slight inefficiencies in
the subtraction detectors are less important than the end PNR efficiency since
the probability to detect values of $n_{1}$ and $n_{2}$ larger than one is an
order of magnitude smaller than $n_{1}=n_{2}=1$ due to highly unbalanced
beamsplitters. It is, however, desirable to have near-unity efficiency in
order to subtract photons at higher rates.

For the ${|0_{L}\rangle}$ code word, performing lossy PNR detection of two
photons on mode $`a^{\prime}$ of Eq. 38 yields the mixtures

$\displaystyle\rho_{0_{L}}$
$\displaystyle=\frac{\text{Tr}_{a}\left[\Pi_{2}\ket{\psi}\bra{\psi}\right]}{\text{Tr}\left[\Pi_{2}\ket{\psi}\bra{\psi}\right]}$
$\displaystyle\propto\text{Tr}_{a}\Bigg{\\{}\sum^{\infty}_{n,n^{\prime}=0}e^{i\phi(n-n^{\prime})}(\tanh{\mathcal{}{R_{0_{L}}}})^{n+n^{\prime}}\Big{(}\bra{2}_{a}(a^{2}-b^{2})\ket{n,n}\bra{n^{\prime},n^{\prime}}({a^{\dagger}}^{2}-{b^{\dagger}}^{2})\ket{2}_{a}$
(39) $\displaystyle\hskip
34.1433pt+3(1-\eta)\bra{3}_{a}(a^{2}-b^{2})\ket{n,n}\bra{n^{\prime},n^{\prime}}({a^{\dagger}}^{2}-{b^{\dagger}}^{2})\ket{3}_{a}+\mathcal{}{O}[(1-\eta)^{2}]\Big{)}\Bigg{\\}},$
(40)

where $\Pi_{n}=\sum_{m=n}p(n|m)|m\rangle\langle m|$ is the POVM for ‘m’
photons detection from a PNR detector with efficiency $\eta$ and
$p(n|m)={m\choose n}\eta^{n}(1-\eta)^{m-n}$ is the conditional probability of
detecting ‘n’ photons out of ‘m’ photons incidented to the detector. For high
efficiency PNR detectors with near unit efficiency, we can neglect orders of
$(1-\eta)^{2}$ and higher. If we assume that the effective squeezing was
chosen properly to for the desired state, then we can write the final output
as the mixture

$\rho_{0_{L}}=(1-\delta_{0_{L}}){|0_{L}\rangle}\bra{0_{L}}+\delta_{0_{L}}\rho_{E_{0_{L}}},$
(42)

where the erroneous component is

$\rho_{E_{0_{L}}}=\frac{1}{8}\left(\sqrt{3}\ket{1}+\sqrt{5}\ket{5}\right)\left(\sqrt{3}\bra{1}+\sqrt{5}\bra{5}\right)$
(43)

and the ratio

$\frac{\delta_{0_{L}}}{1-\delta_{0_{L}}}=3\sqrt{2}(1-\eta).$ (44)

Similarly, one can go through the same procedure for the other code word with
a lossly PNR detection of four photons. In this case we have

$\displaystyle\rho_{1_{L}}$
$\displaystyle=(1-\delta_{1_{L}}){|1_{L}\rangle}\bra{1_{L}}+\delta_{1_{L}}\rho_{E_{1_{L}}},$
$\displaystyle\rho_{E_{1_{L}}}$
$\displaystyle=\frac{1}{32}\left(5\ket{3}+\sqrt{7}\ket{7}\right)\left(5\bra{3}+\sqrt{7}\bra{7}\right),$
$\displaystyle\frac{\delta_{1_{L}}}{1-\delta_{1_{L}}}$
$\displaystyle=\frac{8\sqrt{2}}{\sqrt{15}}(1-\eta).$ (45)

The Wigner functions plotted in Fig. 4(b) and 4(d) show the effects the
detector inefficiency when $\eta=0.90$. Additionally, Fig. 5 plots the
fidelity and Wigner log negativity of the imperfect code states as efficiency
deviates from unity.

Now, we wish to examine how the error and recovery process is affected by the
imperfections introduced by the non-ideal detector. Because the chosen code
only protects against single photon loss, we need only consider effects up to
first order in $\gamma$ from the Kraus operators describing the loss. We will
use the approximate code word of $\rho_{0_{L}}$ as an example and treat the
effects of no-jump errors ($E_{0}$) and single-jump errors ($E_{1}$)
separately.

####  VI.0.1 No-jump errors

Expanding in factors of $\gamma$, the Kraus operator for a no-jump event is

$E_{0}=1-\frac{1}{2}\gamma\hat{a}^{\dagger}\hat{a}+\mathcal{}{O}[(\gamma)^{2}].$
(46)

To leading order in $\gamma$, this error acts on $\rho_{0_{L}}$ to give

$\displaystyle E_{0}\rho_{0_{L}}E_{0}^{\dagger}=$
$\displaystyle(1-\delta_{0_{L}})E_{0}{|0_{L}\rangle}\bra{0_{L}}E_{0}^{\dagger}+\delta_{0_{L}}\rho_{E_{0_{L}}}$
$\displaystyle-\tfrac{1}{2}\delta_{0_{L}}\gamma\left(\hat{a}^{\dagger}\hat{a}\rho_{E_{0_{L}}}+\rho_{E_{0_{L}}}\hat{a}^{\dagger}\hat{a}\right).$
(47)

The first term in the above expression is the ideal code word transformed by
$E_{0}$ and is proven to be correctable with unitary operations Michael  _et
al._ (2016b). The middle term contains the error from state generation and is
orthogonal to the ideal code word. Due to the trace preserving nature of
unitary matrices and the orthogonality of the first two components in the
mixture, applying the unitary recovery operations will preserve the
probability ratio of the error component with the desired corrected component.
This means that after the recovery, the error from this term will not grow in
magnitude but have the same contribution as the original state preparation
error. Thus, if we can neglect the last term in Eq. 47, then error in the
final code word after error and recovery will be the same as that of state
generation and not grow in magnitude. In order to neglect this final term, we
need $\delta_{0_{L}}$ to be of the same order as $\gamma$. From Eq. 44, we can
determine a condition on the PNR detector efficiency in order for the error
correction to be feasible up to $\mathcal{}{O}(\gamma)$. This condition is

$1-\eta\sim\gamma.$ (48)

####  VI.0.2 Single-jump errors

The expanded single-jump Kraus operator is

$E_{1}=\left(\gamma-\tfrac{1}{2}(\gamma)^{2}\right)^{\frac{1}{2}}a+\mathcal{}{O}[(\gamma)^{\frac{3}{2}}],$
(49)

and acts on $\rho_{0_{L}}$ to yield

$\displaystyle
E_{1}\rho_{0_{L}}E_{1}^{\dagger}=(\gamma-\tfrac{1}{2}(\gamma)^{2})\Big{[}$
$\displaystyle(1-\delta_{0_{L}})\hat{a}{|0_{L}\rangle}\bra{0_{L}}\hat{a}^{\dagger}$
$\displaystyle+\delta_{0_{L}}\hat{a}\rho_{E_{0_{L}}}\hat{a}^{\dagger}\Big{]}.$
(50)

The photon loss error is detected by measuring the photon number mod two for
our chosen code space, which in this case, is a parity measurement. The error
portion of the mixture, $\rho_{E_{0_{L}}}$, is orthogonal to the ideal state
${|0_{L}\rangle}\bra{0_{L}}$, so a measurement after a photon loss might
incorrectly yield an even parity result since
$\hat{a}\rho_{E_{0_{L}}}\hat{a}^{\dagger}$ has the same parity as
${|0_{L}\rangle}\bra{0_{L}}$. The probability to incorrectly diagnose the loss
is thus given by the ratio

$P_{inc}=\frac{\text{Tr}[\delta_{0_{L}}\hat{a}\rho_{E_{0_{L}}}\hat{a}^{\dagger}]}{\text{Tr}[(1-\delta_{0_{L}})\hat{a}{|0_{L}\rangle}\bra{0_{L}}\hat{a}^{\dagger}]}\approx\frac{9\delta_{0_{L}}}{10(1-\delta_{0_{L}})}.$
(51)

Again, since the accuracy of our error correcting code is only up to first
order in $\gamma$, if the probability to incorrectly diagnose the error is
negligible compared to $\gamma$ then our code still has efficacy. Using Eq.
44, this leads to the slightly stronger condition on the detector efficiency
of

$\eta\gg(1-\gamma).$ (52)

This condition states that in order for the QEC to be effective, the losses
from detector inefficiency must be considerably smaller than the expected
losses the code words will protect against.

##  VII Architectural analysis

In this section, we discuss the experimental feasibility of our proposal with
currently available resources. Optical Parametric Oscillators (OPOs) below
threshold have been demonstrated to achieve as high as 15 dB of desired
squeezing for our scheme Vahlbruch  _et al._ (2016b). Along with low loss
free-space linear optics and fiber coupled highly efficient PNR detectors such
as transition-edge sensors (TESs) and superconducting nanowire single photon
detector (SNSPDs) Lita  _et al._ (2008); Nehra  _et al._ (2019); Cahall  _et
al._ (2017); Fukuda  _et al._ (2011), our proposal can be readily realized.
However, this approach requires tight control and active stabilization of the
optical cavity lengths and precise free-space alignment, making it challenging
to scale for multiplexed experimental setups for boosting the success
probability for state generation. Integrated quantum photonics offers a
robust, compact, ultra-low loss scalable platform for building larger quantum
circuits including both linear and nonlinear components Peruzzo  _et al._
(2010); Metcalf  _et al._ (2013); Najafi  _et al._ (2015); Spring  _et al._
(2013). Thanks to the miniaturization offered by photonic integrated circuit
(PIC) technology, quantum states can be generated and processed within the
same PIC.

The tight transverse field confinement in integrated waveguide structures
leads to large effective nonlinearity and alleviates deleterious effects such
as gain-induced diffraction that degrades the obtainable squeezing in pulsed
pump experiments Kim  _et al._ (1994); Kim and Kumar (1994); Eto  _et al._
(2008). Owing to the large nonlinearity, high levels of squeezing can be
achieved in the waveguide structures over a bandwidth that is several orders
of magnitude larger than OPOs.
Over the last couple of decades, the integrated structures based on materials
such as Lithium Niobate (LN), Silicon Nitride (SiN) and Silica have been the
key enabler for many quantum photonics experiments Sanaka  _et al._ (2001);
Kanter  _et al._ (2002); Guerreiro  _et al._ (2014); Saglamyurek  _et al._
(2011); Arrazola  _et al._ (2021); Dutt  _et al._ (2015). While SiN and Silica
based integrated architectures have the scalability advantage in nanophotonic
integration over LN based diffused waveguides with large cross-sections, they
suffers from the weak third-order nonlinearity and unwanted noise, which
limits the measured squeezing to only of few dBs Dutt  _et al._ (2015); Yang
_et al._ (2021). Recently, the thin-film Lithium Niobate-On-Insulator (LNOI)
has lead to several breakthroughs in nanophotonics, owing to sub-wavelength
confinement, dispersion engineering and efficient quasi-phasematching using
high quality periodic poling, more than an order of magnitude improvements in
the nonlinear efficiency compared to traditional LN waveguides have been
achieved Wang  _et al._ (2018); Jankowski  _et al._ (2020); Zhao  _et al._
(2020). Furthermore, ultralow-threshold OPOs Lu  _et al._ (2021); McKenna  _et
al._ (2021) and high gain optical parametric amplification and generation have
been recently demonstrated Ledezma  _et al._ (2021); Jankowski  _et al._
(2021a). In the quantum domain, thin-film LN waveguide structures with pulsed
pump operation offer simple non-resonant and efficient sources for ultra-
broadband single-pass traveling-wave phase-sensitive degenerate optical
parametric amplifiers (OPAs) for generating squeezed light Caves (1982). In
addition to a strong second-order nonlinear interaction, LN has large electro-
optic coefficients and ultra-low loss propagation of $\alpha=3$ dB/m, which
allows for efficient and scalable linear optics implementations within the
same chip Zhang  _et al._ (2017); Zhu  _et al._ (2021). Considering the
measured nonlinear efficiency and propagation loss in thin-film LN platform,
on-chip squeezing levels of $>15$ dB are within the reach of current
experimental capabilities Rao  _et al._ (2019); Jankowski  _et al._ (2021b).
On the measurement side, significant progress has been made both in developing
low loss chip-to-fiber interconnects Hu  _et al._ (2020); He  _et al._ (2019)
and in monolithic integration of single-photon detectors on the LN thin-film
Sayem  _et al._ (2020); Colangelo  _et al._ (2020); Lomonte  _et al._ (2021).
Our proposal can thus be realized with the current quantum photonics
technology.

##  VIII Conclusions and outlook

In this paper, we explored the coherent photon subtraction from a two-mode
squeezed vacuum to generate non-Gaussian quantum states with 2-fold and 4-fold
phase space symmetry in the Wigner function, including exact binomial code
states and truncated cat-like states. Such states have the desired Wigner
function negativity required for quantum advantages in CV quantum information
processing. While the proposed method is not a push-button, desired codes can
be generated with rates of KHz-MHz by pairing up with state-of-the-art PNR
detectors with high count rates. Furthermore, the success probabilities can be
enlarged by starting with higher initial resource squeezing and multiplexing
schemes, which can be achieved by monolithic integration on the same
nanophotonic chip. From architectural analysis we believe that our proposal
can be readily realized with already demonstrated squeezing and high quantum
efficiency PNR detection.
Moreover, our protocol can be extended to increase the rotation symmetry by
utilizing PNR based breeding methods. Higher rotation symmetry leads to
enlarged spacing in the photon number basis, offering to correct for larger
photon loss, gain, and dephasing errors. The truncated finite dimensional
Hilbert space makes these codes hardware efficient and amenable for high
fidelity unitary operations, thereby opening promising avenues for their
applications in some other areas, in addition to fault-tolerant quantum
computation, such as quantum metrology and quantum communication in optical
quantum information processing. An important future research is to investigate
such state engineering protocol for generating higher order codes allowing to
correct for higher photon loss, photon gain, and dephasing errors.

## Acknowledgments

RN thanks Prof. Joshua Combes for fruitful discussions. ME thanks Chun-Hung
Chang and Prof. Nicolas Menicucci for helpful feedback and conversation. RN
and AM gratefully acknowledge support from ARO grant no. W911NF-18-1-0285, and
NSF grant no. 1846273 and 1918549. The authors wish to thank NTT Research for
their financial and technical support. ME and OP gratefully acknowledge
support from NSF grant PHY-1708023, Jefferson Laboratory LDRD grant. We are
grateful to developers of QuTip Python Package Johansson  _et al._ (2013),
which was used to perform numerical simulations in this work.

##  IX Appendix

###  IX.1 Numerical modeling

Numerical simulations were performed using the QuTip Python Package Johansson
_et al._ (2013). We define beamsplitter and two-mode squeezing operations as

$\displaystyle U_{ab}$
$\displaystyle=e^{\theta(\hat{a}\hat{b}^{\dagger}-\hat{a}^{\dagger}\hat{b})}$
(53) $\displaystyle S(z)_{ab}$
$\displaystyle=e^{z\hat{a}^{\dagger}\hat{b}^{\dagger}-z^{*}\hat{a}\hat{b}}$
(54)

where $r=\cos{\theta}$ is the beamsplitter reflection coefficient and
$z=Re^{i\pi}$ is the complex squeezing parameter. After applying the squeezing
operation to vacuum and sending the two modes $a$ and $b$ through
beamsplitters, PNR detection is performed using the POVM for a lossy PNR
detector detector, which is given as

$\Pi_{n}=\sum_{m=n}^{N_{trunc}}p(n|m)|m\rangle\langle m|,$ (56)

where $N_{trunc}=40$ is the size of the Hilbert space used, and
$p(n|m)={m\choose n}\eta^{n}(1-\eta)^{m-n}$ is the conditional probability of
detecting ‘n’ photons if ‘m’ photons are incident to a PNR detector with the
detection efficiency of $\eta$.

The overall success probability for different states was determined by summing
Eq. 26 over all possible values of $n_{1},n_{2},$ and $n_{3}$ such that the
symmetry condition being tested was satisfied, and the sum was truncated when
all remaining probabilities were below $\mathcal{}{O}[(10^{-8})]$.

###  IX.2 Perfect coherent photon subtraction

To this point we have made use of the approximation that the subtraction
beamsplitters are highly unbalanced so we can assume perfect photon
subtraction for subsequent derivations. However, even in a laboratory setting
where beamsplitters with non-vanishing reflectivity are used, we can recover
the ideal subtraction case with a modification to the value of initial
squeezing. In the case where subtraction beamsplitters are given by the
operators $U_{ac}=U_{bd}=e^{-\theta(a^{\dagger}b-ab^{\dagger})}$ and the third
beamsplitter to make the subtraction coherent is
balanced,$U_{cd}=e^{-\frac{\pi}{4}(a^{\dagger}b-ab^{\dagger})}$, applying
these operators to a TMS vacuum yields

$\displaystyle
U_{cd}U_{ac}U_{bd}|0,\mathcal{}{R}\rangle_{ab}=\sum_{n=0}^{\infty}\frac{(e^{i\phi}\tanh{\mathcal{}{R}})^{n}}{n!}\left(ta^{\dagger}+\frac{r}{\sqrt{2}}(c^{\dagger}+d^{\dagger})\right)^{n}\left(tb^{\dagger}+\frac{r}{\sqrt{2}}(-c^{\dagger}+d^{\dagger})\right)^{n}|0\rangle_{abcd},$
(57)

where $r=\sin{\theta}$ and $t=\cos{\theta}$ are the real reflectivity and
transmissivity coefficients of the subtraction beamsplitters. Detecting
$n_{1}$ photons in mode $c$ and $n_{2}$ photons in mode $d$ leads to the
unnormalized output state of

$\displaystyle|\phi\rangle$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=0}^{n_{1}+n_{2}}B\frac{(t^{2}\tanh{\mathcal{}{R}})^{n}}{k!(n_{1}+n_{2}-k)!}a^{n_{1}+n_{2}-k}b^{k}|nn\rangle_{ab}$
$\displaystyle=\sum_{k=0}^{N}Ba^{N-k}b^{k}|0,\mathcal{}{R^{\prime}}\rangle_{ab},$
(58)

where $N\equiv n_{1}+n_{2}$ and $B=A_{n_{1},n_{2},k}$ from Eq. 21. The above
equation is then exactly the same as the perfect coherent subtraction case of
Eq. 20 with an effective decreased squeezing parameter of

$\mathcal{}{R}^{\prime}=\text{tanh}^{-1}{(t^{2}\tanh{\mathcal{}{R}})}.$ (59)

This shows that the idealized states discuss ed are in fact equivalent to
realistic states with modified squeezing. Furthermore, this fact can be used
to increase the overall success probability of a protocol if one has the
ability to generate an initial TMS vacuum state with squeezing in excess of
what is required to produce a specific target. By beginning with a large
squeezing parameter, $\mathcal{}{R}$, one can increase the reflectivity of the
subtraction beamsplitters to increase the likelihood of successfully
subtracting $n_{1}$ and $n_{2}$ photons until the effective squeezing has been
reduced to the $\mathcal{}{R}^{\prime}$ value required to produce the target.

###  IX.3 Numerical data

In Table 1, squeezing values at the intersection points in Fig. 7 are given.

$m$ |  $\langle a^{\dagger}a\rangle$ |  Squeezing (dB) |  Prob. [0,1,m] |  Prob. [0,2,m]
---|---|---|---|---
2 |  1.21 |  2.78 |  $1.39\times 10^{-3}$ |  $8.46\times 10^{-5}$
3 |  2.36 |  4.22 |  $1.41\times 10^{-3}$ |  $1.82\times 10^{-4}$
4 |  3.47 |  5.40 |  $1.51\times 10^{-3}$ |  $3.08\times 10^{-4}$
5 |  4.56 |  6.44 |  $1.53\times 10^{-3}$ |  $4.32\times 10^{-4}$
6 |  5.62 |  7.40 |  $1.72\times 10^{-3}$ |  $6.28\times 10^{-4}$
7 |  6.68 |  8.30 |  $1.86\times 10^{-3}$ |  $8.34\times 10^{-4}$
8 |  7.72 |  9.12 |  $1.94\times 10^{-3}$ |  $1.03\times 10^{-3}$
Table 1: Numerical data for the experimental parameters when the mean photon-
number is equal for both the code words in Fig. 7

## References

  * Clarke and Wilhelm (2008) J. Clarke and F. K. Wilhelm, Nature 453, 1031 (2008).
  * O’brien  _et al._ (2009) J. L. O’brien, A. Furusawa, and J. Vučković, Nature Photonics 3, 687 (2009).
  * Cirac and Zoller (1995) J. I. Cirac and P. Zoller, Physical review letters 74, 4091 (1995).
  * Imamog¯lu  _et al._ (1999) A. Imamog¯lu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, Phys. Rev. Lett. 83, 4204 (1999).
  * Saffman  _et al._ (2010) M. Saffman, T. G. Walker, and K. Mølmer, Reviews of modern physics 82, 2313 (2010).
  * Vandersypen  _et al._ (2001) L. M. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, Nature 414, 883 (2001).
  * Lloyd and Braunstein (1999) S. Lloyd and S. L. Braunstein, Phys. Rev. Lett. 82, 1784 (1999).
  * Chen  _et al._ (2014) M. Chen, N. C. Menicucci, and O. Pfister, Phys. Rev. Lett. 112, 120505 (2014).
  * Larsen  _et al._ (2019) M. V. Larsen, X. Guo, C. R. Breum, J. S. Neergaard-Nielsen, and U. L. Andersen, Science 366, 369 (2019), https://science.sciencemag.org/content/366/6463/369.full.pdf .
  * Asavanant  _et al._ (2019) W. Asavanant, Y. Shiozawa, S. Yokoyama, B. Charoensombutamon, H. Emura, R. N. Alexander, S. Takeda, J.-i. Yoshikawa, N. C. Menicucci, H. Yonezawa, and A. Furusawa, Science 366, 373 (2019), https://science.sciencemag.org/content/366/6463/373.full.pdf .
  * Walls and Milburn (1994) D. F. Walls and G. J. Milburn,  _Quantum Optics_ (Springer, Berlin, 1994).
  * Bartlett  _et al._ (2002) S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, Phys. Rev. Lett. 88, 097904 (2002).
  * Menicucci  _et al._ (2006) N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, Phys. Rev. Lett. 97, 110501 (2006).
  * Lita  _et al._ (2008) A. E. Lita, A. J. Miller, and S. W. Nam, Opt. Expr. 16, 3032 (2008).
  * Nehra  _et al._ (2019) R. Nehra, A. Win, M. Eaton, R. Shahrokhshahi, N. Sridhar, T. Gerrits, A. Lita, S. W. Nam, and O. Pfister, Optica 6, 1356 (2019).
  * Ghose and Sanders (2007) S. Ghose and B. C. Sanders, Journal of Modern Optics 54, 855 (2007), https://doi.org/10.1080/09500340601101575 .
  * Yanagimoto  _et al._ (2020) R. Yanagimoto, T. Onodera, E. Ng, L. G. Wright, P. L. McMahon, and H. Mabuchi, Phys. Rev. Lett. 124, 240503 (2020).
  * Michael  _et al._ (2016a) M. H. Michael, M. Silveri, R. T. Brierley, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, Phys. Rev. X 6, 031006 (2016a).
  * Ourjoumtsev  _et al._ (2006) A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, Science 312, 83 (2006).
  * Eaton  _et al._ (2019a) M. Eaton, R. Nehra, and O. Pfister, New Journal of Physics 21, 113034 (2019a).
  * Gottesman  _et al._ (2001) D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64, 012310 (2001).
  * Dakna  _et al._ (1998) M. Dakna, L. Knöll, and D.-G. Welsch, Eur. Phys. J. D 3, 295 (1998).
  * Zavatta  _et al._ (2004) A. Zavatta, S. Viciani, and M. Bellini, Science 306, 660 (2004).
  * Thekkadath  _et al._ (2020) G. Thekkadath, B. Bell, I. Walmsley, and A. Lvovsky, Quantum 4, 239 (2020).
  * Takase  _et al._ (2021) K. Takase, J.-i. Yoshikawa, W. Asavanant, M. Endo, and A. Furusawa, Physical Review A 103, 013710 (2021).
  * Niset  _et al._ (2009) J. Niset, J. Fiurášek, and N. J. Cerf, Phys. Rev. Lett. 102, 120501 (2009).
  * Tzitrin  _et al._ (2020) I. Tzitrin, J. E. Bourassa, N. C. Menicucci, and K. K. Sabapathy, Physical Review A 101, 032315 (2020).
  * Baragiola  _et al._ (2019) B. Q. Baragiola, G. Pantaleoni, R. N. Alexander, A. Karanjai, and N. C. Menicucci, Phys. Rev. Lett. 123, 200502 (2019).
  * Campagne-Ibarcq  _et al._ (2020) P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys-Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, _et al._ , Nature 584, 368 (2020).
  * Flühmann  _et al._ (2019) C. Flühmann, T. L. Nguyen, M. Marinelli, V. Negnevitsky, K. Mehta, and J. P. Home, Nature 566, 513 (2019).
  * Gol’tsman  _et al._ (2001) G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, Appl. Phys. Lett. 79, 705 (2001), https://doi.org/10.1063/1.1388868 .
  * Vasconcelos  _et al._ (2010) H. M. Vasconcelos, L. Sanz, and S. Glancy, Opt. Lett. 35, 3261 (2010).
  * Weigand and Terhal (2018) D. J. Weigand and B. M. Terhal, Phys. Rev. A 97, 022341 (2018).
  * Etesse  _et al._ (2014) J. Etesse, R. Blandino, B. Kanseri, and R. Tualle-Brouri, New Journal of Physics 16, 053001 (2014).
  * Eaton  _et al._ (2019b) M. Eaton, R. Nehra, and O. Pfister, New Journal of Physics, in press; arxiv:1903.01925 [quant-ph]. (2019b), 10.1088/1367-2630/ab5330.
  * Shi  _et al._ (2019) Y. Shi, C. Chamberland, and A. Cross, New Journal of Physics 21, 093007 (2019).
  * Fabre  _et al._ (2020) N. Fabre, G. Maltese, F. Appas, S. Felicetti, A. Ketterer, A. Keller, T. Coudreau, F. Baboux, M. Amanti, S. Ducci, _et al._ , Physical Review A 102, 012607 (2020).
  * Hastrup  _et al._ (2021) J. Hastrup, K. Park, J. B. Brask, R. Filip, and U. L. Andersen, npj Quantum Information 7, 1 (2021).
  * Grimsmo  _et al._ (2020) A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Physical Review X 10, 011058 (2020).
  * Bergmann and van Loock (2016) M. Bergmann and P. van Loock, Phys. Rev. A 94, 042332 (2016).
  * Reinhold  _et al._ (2020) P. Reinhold, S. Rosenblum, W.-L. Ma, L. Frunzio, L. Jiang, and R. J. Schoelkopf, Nature Physics 16, 822 (2020).
  * Hu  _et al._ (2019) L. Hu, Y. Ma, W. Cai, X. Mu, Y. Xu, W. Wang, Y. Wu, H. Wang, Y. Song, C.-L. Zou, _et al._ , Nature Physics 15, 503 (2019).
  * Vahlbruch  _et al._ (2016a) H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 117, 110801 (2016a).
  * Cahall  _et al._ (2017) C. Cahall, K. L. Nicolich, N. T. Islam, G. P. Lafyatis, A. J. Miller, D. J. Gauthier, and J. Kim, Optica 4, 1534 (2017).
  * Nicolich  _et al._ (2019) K. L. Nicolich, C. Cahall, N. T. Islam, G. P. Lafyatis, J. Kim, A. J. Miller, and D. J. Gauthier, Phys. Rev. Applied 12, 034020 (2019).
  * Endo  _et al._ (2021) M. Endo, T. Sonoyama, M. Matsuyama, F. Okamoto, S. Miki, M. Yabuno, F. China, H. Terai, and A. Furusawa, arXiv preprint arXiv:2102.09712 (2021).
  * Achilles  _et al._ (2003) D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, and I. A. Walmsley, Opt. Lett. 28, 2387 (2003).
  * Chuang  _et al._ (1997) I. L. Chuang, D. W. Leung, and Y. Yamamoto, Physical Review A 56, 1114 (1997).
  * Michael  _et al._ (2016b) M. H. Michael, M. Silveri, R. T. Brierley, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, Phys. Rev. X 6, 031006 (2016b).
  * Li  _et al._ (2017) L. Li, C.-L. Zou, V. V. Albert, S. Muralidharan, S. Girvin, and L. Jiang, Physical review letters 119, 030502 (2017).
  * Michael  _et al._ (2016c) M. H. Michael, M. Silveri, R. Brierley, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, Physical Review X 6, 031006 (2016c).
  * Albert  _et al._ (2018) V. V. Albert, K. Noh, K. Duivenvoorden, D. J. Young, R. Brierley, P. Reinhold, C. Vuillot, L. Li, C. Shen, S. Girvin, _et al._ , Physical Review A 97, 032346 (2018).
  * Knill and Laflamme (1997) E. Knill and R. Laflamme, Physical Review A 55, 900 (1997).
  * Morais  _et al._ (2020) L. A. Morais, T. Weinhold, M. P. de Almeida, A. Lita, T. Gerrits, S. W. Nam, A. G. White, and G. Gillett, arXiv preprint arXiv:2012.10158 (2020).
  * Hofmann (2006) H. F. Hofmann, Phys. Rev. A 74, 013808 (2006).
  * Eaton  _et al._ (2021) M. Eaton, R. Nehra, A. Win, and O. Pfister, Phys. Rev. A 103, 013726 (2021).
  * Hong  _et al._ (1987) C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).
  * Carranza and Gerry (2012) R. Carranza and C. C. Gerry, J. Opt. Soc. Am. B 29, 2581 (2012).
  * Takagi and Zhuang (2018) R. Takagi and Q. Zhuang, Phys. Rev. A 97, 062337 (2018).
  * Fukuda  _et al._ (2011) D. Fukuda, G. Fujii, T. Numata, K. Amemiya, A. Yoshizawa, H. Tsuchida, H. Fujino, H. Ishii, T. Itatani, S. Inoue, _et al._ , Optics express 19, 870 (2011).
  * Gerrits  _et al._ (2016) T. Gerrits, A. Lita, B. Calkins, and S. W. Nam, in  _Superconducting devices in quantum optics_ (Springer, 2016) pp. 31–60.
  * Bonneau  _et al._ (2015) D. Bonneau, G. J. Mendoza, J. L. O’Brien, and M. G. Thompson, New Journal of Physics 17, 043057 (2015).
  * Joshi  _et al._ (2018) C. Joshi, A. Farsi, S. Clemmen, S. Ramelow, and A. L. Gaeta, Nature communications 9, 1 (2018).
  * Kaneda and Kwiat (2019) F. Kaneda and P. G. Kwiat, Science Advances 5 (2019), 10.1126/sciadv.aaw8586, https://advances.sciencemag.org/content/5/10/eaaw8586.full.pdf .
  * Vahlbruch  _et al._ (2016b) H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 117, 110801 (2016b).
  * Peruzzo  _et al._ (2010) A. Peruzzo, M. Lobino, J. C. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, _et al._ , Science 329, 1500 (2010).
  * Metcalf  _et al._ (2013) B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, _et al._ , Nature communications 4, 1 (2013).
  * Najafi  _et al._ (2015) F. Najafi, J. Mower, N. C. Harris, F. Bellei, A. Dane, C. Lee, X. Hu, P. Kharel, F. Marsili, S. Assefa, _et al._ , Nature communications 6, 1 (2015).
  * Spring  _et al._ (2013) J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, _et al._ , Science 339, 798 (2013).
  * Kim  _et al._ (1994) C. Kim, R.-D. Li, and P. Kumar, Optics letters 19, 132 (1994).
  * Kim and Kumar (1994) C. Kim and P. Kumar, Physical review letters 73, 1605 (1994).
  * Eto  _et al._ (2008) Y. Eto, T. Tajima, Y. Zhang, and T. Hirano, Optics Express 16, 10650 (2008).
  * Sanaka  _et al._ (2001) K. Sanaka, K. Kawahara, and T. Kuga, Physical Review Letters 86, 5620 (2001).
  * Kanter  _et al._ (2002) G. S. Kanter, P. Kumar, R. V. Roussev, J. Kurz, K. R. Parameswaran, and M. M. Fejer, Optics express 10, 177 (2002).
  * Guerreiro  _et al._ (2014) T. Guerreiro, A. Martin, B. Sanguinetti, J. S. Pelc, C. Langrock, M. M. Fejer, N. Gisin, H. Zbinden, N. Sangouard, and R. T. Thew, Phys. Rev. Lett. 113, 173601 (2014).
  * Saglamyurek  _et al._ (2011) E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussieres, M. George, R. Ricken, W. Sohler, and W. Tittel, Nature 469, 512 (2011).
  * Arrazola  _et al._ (2021) J. Arrazola, V. Bergholm, K. Brádler, T. Bromley, M. Collins, I. Dhand, A. Fumagalli, T. Gerrits, A. Goussev, L. Helt, _et al._ , Nature 591, 54 (2021).
  * Dutt  _et al._ (2015) A. Dutt, K. Luke, S. Manipatruni, A. L. Gaeta, P. Nussenzveig, and M. Lipson, Physical Review Applied 3, 044005 (2015).
  * Yang  _et al._ (2021) Z. Yang, M. Jahanbozorgi, D. Jeong, S. Sun, O. Pfister, H. Lee, and X. Yi, arXiv preprint arXiv:2103.03380 (2021).
  * Wang  _et al._ (2018) C. Wang, C. Langrock, A. Marandi, M. Jankowski, M. Zhang, B. Desiatov, M. M. Fejer, and M. Lončar, Optica 5, 1438 (2018).
  * Jankowski  _et al._ (2020) M. Jankowski, C. Langrock, B. Desiatov, A. Marandi, C. Wang, M. Zhang, C. R. Phillips, M. Lončar, and M. M. Fejer, Optica 7, 40 (2020).
  * Zhao  _et al._ (2020) J. Zhao, M. Rüsing, U. A. Javid, J. Ling, M. Li, Q. Lin, and S. Mookherjea, Optics Express 28, 19669 (2020).
  * Lu  _et al._ (2021) J. Lu, A. A. Sayem, Z. Gong, J. B. Surya, C.-L. Zou, and H. X. Tang, arXiv preprint arXiv:2101.04735 (2021).
  * McKenna  _et al._ (2021) T. P. McKenna, H. S. Stokowski, V. Ansari, J. Mishra, M. Jankowski, C. J. Sarabalis, J. F. Herrmann, C. Langrock, M. M. Fejer, and A. H. Safavi-Naeini, arXiv preprint arXiv:2102.05617 (2021).
  * Ledezma  _et al._ (2021) L. Ledezma, R. Sekine, Q. Guo, R. Nehra, S. Jahani, and A. Marandi, arXiv preprint arXiv:2104.08262 (2021).
  * Jankowski  _et al._ (2021a) M. Jankowski, N. Jornod, C. Langrock, B. Desiatov, A. Marandi, M. Lončar, and M. M. Fejer, arXiv preprint arXiv:2104.07928 (2021a).
  * Caves (1982) C. M. Caves, Physical Review D 26, 1817 (1982).
  * Zhang  _et al._ (2017) M. Zhang, C. Wang, R. Cheng, A. Shams-Ansari, and M. Lončar, Optica 4, 1536 (2017).
  * Zhu  _et al._ (2021) D. Zhu, L. Shao, M. Yu, R. Cheng, B. Desiatov, C. Xin, Y. Hu, J. Holzgrafe, S. Ghosh, A. Shams-Ansari, _et al._ , arXiv preprint arXiv:2102.11956 (2021).
  * Rao  _et al._ (2019) A. Rao, K. Abdelsalam, T. Sjaardema, A. Honardoost, G. F. Camacho-Gonzalez, and S. Fathpour, Optics express 27, 25920 (2019).
  * Jankowski  _et al._ (2021b) M. Jankowski, J. Mishra, and M. Fejer, arXiv preprint arXiv:2103.02296 (2021b).
  * Hu  _et al._ (2020) C. Hu, A. Pan, T. Li, X. Wang, Y. Liu, S. Tao, C. Zeng, and J. Xia, arXiv preprint arXiv:2009.02855 (2020).
  * He  _et al._ (2019) L. He, M. Zhang, A. Shams-Ansari, R. Zhu, C. Wang, and L. Marko, Optics letters 44, 2314 (2019).
  * Sayem  _et al._ (2020) A. A. Sayem, R. Cheng, S. Wang, and H. X. Tang, Applied Physics Letters 116, 151102 (2020).
  * Colangelo  _et al._ (2020) M. Colangelo, B. Desiatov, D. Zhu, J. Holzgrafe, O. Medeiros, M. Loncar, and K. Berggren, in  _CLEO: Science and Innovations_ (Optical Society of America, 2020) pp. SM4O–4.
  * Lomonte  _et al._ (2021) E. Lomonte, M. A. Wolff, F. Beutel, S. Ferrari, C. Schuck, W. H. Pernice, and F. Lenzini, arXiv preprint arXiv:2103.10973 (2021).
  * Johansson  _et al._ (2013) J. R. Johansson, P. D. Nation, and F. Nori, Computer Physics Communications 184, 1234 (2013).