# Non-Gaussian and Gottesman-Kitaev-Preskill state preparation by photon
catalysis

Miller Eaton<EMAIL_ADDRESS>Department of Physics, University of
Virginia, 382 McCormick Rd, Charlottesville, VA 22903, USA  Rajveer Nehra
Department of Physics, University of Virginia, 382 McCormick Rd,
Charlottesville, VA 22903, USA  Olivier Pfister  Department of Physics,
University of Virginia, 382 McCormick Rd, Charlottesville, VA 22903, USA

###### Abstract

Continuous-variable quantum-computing (CVQC) is the most scalable
implementation of QC to date but requires non-Gaussian resources to allow
exponential speedup and quantum correction, using error encoding such as
Gottesman-Kitaev-Preskill (GKP) states. However, GKP state generation is still
an experimental challenge. We show theoretically that photon catalysis, the
interference of coherent states with single-photon states followed by photon-
number-resolved detection, is a powerful enabler for non-Gaussian quantum
state engineering such as exactly displaced single-photon states and
$M$-symmetric superpositions of squeezed vacuum (SSV), including squeezed cat
states ($M=2$). By including photon-counting based state breeding, we
demonstrate the potential to enlarge SSV states and produce GKP states.

##  I Introduction

Quantum Computing (QC) offers the possibility to solve certain computational
problems which are intractable in the realm of classical computation
Feynman1982 ; Shor1994 . In the last couple of decades, QC has been widely
explored over discrete variables, mostly qubits, and several architectures
have been proposed and experimentally realized Ladd2010 . Another equally
universal flavor of QC makes use of continuous variables (CV) Lloyd1999 ;
Braunstein2005a ; Weedbrook2012 ; pfister2019continuous , such as the position
and momentum of a quantum harmonic oscillator or, analogously, the amplitude-
and phase-quadrature amplitudes of the quantized electromagnetic field. The
interest of CVQC comes primarily from the large-scale, and highly scalable,
implementations that have been experimentally demonstrated of measurement-
based QC substrates such as 1D cluster entangled states with at least 60
simultaneously entangled qumodes Chen2014  and one million sequentially
entangled qumodes, accessible two at a time Yoshikawa2016 . Recently, 2D
cluster states have been demonstrated Larsen2019 ; Asavanant2019 . A fault
tolerance threshold has been proven to exist for CVQC Menicucci2014ft  for the
Gottesman-Kitaev-Preskill (GKP) quantum error correction protocol
Gottesman2001 ; Ghose2007 . This protocol uses error code states which have a
non-Gaussian Wigner function, a required resource in CV quantum information
for CVQC exponential speedup Bartlett2002 , for entanglement distillation
Eisert2002  and Bell inequality violation Bell1987 , and for quantum error
correction Niset2009 .

Figure 1: Various feasible techniques for generating quantum states with non-
Gaussian Wigner functions. Diagram (a) depicts $n$ photon subtraction, (b) and
(b’) denote variants of $n$ and single-photon additions, respectively, and (c)
generalizes the previous cases for an $m$ fock state with arbitrary
beampslitter parameters and $n$ photon detection.

The experimental generation of non-Gaussian states is therefore a key effort
in quantum information science. It has been shown that GKP states could be
probabilistically generated from squeezed Schrodinger cat states
Vasconcelos2010 , and this process has been made deterministic
weigand_generating_2018 . Single-photon states, which have been generated and
characterized using heralding detection of downconverted photon pairs
Lvovsky2001 ; Laiho2010 ; Morin2012 ; Nehra2019 , are excellent non-Gaussian
ingredients. Sophisticated techniques, such as photon subtraction Dakna1998 ;
Ourjoumtsev2006  and addition Zavatta2004 , have led to very promising
advances. In photon subtraction, Fig.1(a), a nonclassical state of light
impinges onto a highly unbalanced, say transmissive, beamsplitter and a
photon-number-resolving (PNR) detector, ideally, or at least a single-photon
sensitive one, detects the reflected light. Conditioned on the detection of
$n$ photons, one can show that the transmitted light is, to a good
approximation, the state $a^{n}\left|\,\psi\,\right\rangle$, where $\psi$
denotes the initial state of light and $a$ the photon annihilation operator
Wenger2004 ; Ourjoumtsev2006 . This was recently generalized to multimode
light Averchenko2016 . Photon subtraction cannot work on coherent states, of
course, but photon addition, Figures(b,b’), does Zavatta2004 . In Fig.1(c), a
more general process is presented, called photon catalysis Lvovsky2002 .

The technique of photon catalysis is derived from the “quantum scissors”
scheme Pegg1998  and consists in interfering a quantum input (not necessarily
pure) with a Fock state and performing PNR detection of one beamsplitter
output. The beamsplitter is no longer necessarily balanced, and its
coefficients are parameters of the process as well. Bartley et al. explored
this process and showed its potential for the creation of non-Gaussian
coherent state superpositions and squeezed states bartley_multiphoton_2012 .
By varying the parameters of the beamsplitter, Bartley et. al experimentally
performed single-photon catalysis on a coherent state input to demonstrate the
creation of states exhibiting nonclassical photon statistics. Photon catalysis
has also been explored in utilizing multi photon detection hu_multiphoton_2016
, with general single-mode Gaussian inputs Birrittella2018 , and for use as an
entanglement enhancer bartley_directly_2015 . Note that near-unity efficiency
PNR is now an experimentally available resource, thanks to superconducting
transition-edge sensors Lita2008 .

In this paper, we explore the use of single-step and sequential photon
catalysis to generate non-Gaussian states of interest to CVQC, in particular
exact displaced single-photon states and phase-symmetric superpositions of
squeezed vacuum (SSV) states, of which squeezed Schrödinger cat states are a
subset. By including a PNR-based breeding protocol, we demonstrate the
potential to enlarge SSV states and generate GKP states. In this method, no
squeezed states are required, the needed resources being coherent states and
linear optics, single-photon states, and PNR detection.

##  II Photon catalysis

In the rest of this paper, we take the restriction of photon catalysis in
Fig.1(c) to $m$=1, i.e., to single-photon input resource states, but we keep
the option of variable $n$ for PNR detection. For an arbitrary input mode
$\left|\,\psi\,\right\rangle_{a}\left|\,1\,\right\rangle_{b}=\sum_{m=0}^{\infty}\psi_{m}|m\rangle_{a}|1\rangle_{b}$,
the output state is Caves1980 ; BarnettRadmore1997

$\left|\,\text{out}\,\right\rangle_{ab}=\sum_{m=0}^{\infty}\psi_{m}\sum_{k=0}^{m}\binom{m}{k}^{\frac{1}{2}}r^{m-k}t^{k}\Bigl{[}t\sqrt{m-k+1}|m-k+1\big{\rangle}_{a}|k\big{\rangle}_{b}-r\sqrt{k+1}|m-k\big{\rangle}_{a}|k+1\big{\rangle}_{b}\Bigr{]},$
(1)

where the beamsplitter operator is defined by
$U_{ab}=\exp[\theta(ab^{\dagger}-a^{\dagger}b)]$ with the reflection and
transmission coefficients being $r=\cos{\theta}$ and $t=\sin{\theta}$. If,
say, output $a$ is sent to a PNR detector which measures $n$ photons, then
output $b$ is projected into the state

$\displaystyle\left|\,\phi\,\right\rangle_{b}$
$\displaystyle=\,_{a}\\!\left\langle\,n\,\right|out\,\rangle_{ab}$ (2)
$\displaystyle=\sum_{\ell=0}^{\infty}\frac{\psi_{\ell+n-1}}{\sqrt{n}}\binom{\ell+n-1}{\ell}^{\frac{1}{2}}\,r^{n-1}t^{\ell-1}\left(nt^{2}-\ell
r^{2}\right)\left|\,\ell\,\right\rangle_{b}.$

If the beamsplitter is designed so that destructive quantum interference
$nt^{2}=\ell r^{2}$ occurs, then the corresponding Fock state
$\left|\,nt^{2}/r^{2}\,\right\rangle$ is absent from the output. An
application of this situation is given in the next section on exact state
displacement by such Fock-state filtering. Note that the state amplitudes are
shifted by $n-1$ in the process. When postselecting on $n=1$, this shift
disappears and, by setting $r^{2}=1/(q+1)$ for the beamsplitter, one can
remove the $q$-photon amplitude:

$\displaystyle\left|\,\phi_{\bar{q}}\,\right\rangle_{b}=\sum_{\ell=0}^{\infty}\psi_{\ell}\,\left(\frac{q}{q+1}\right)^{\frac{\ell+1}{2}}\left(1-\frac{\ell}{q}\right)\left|\,\ell\,\right\rangle_{b}.$
(3)

If $\left|\,\psi\,\right\rangle$ has a maximum number $\ell_{\text{max}}$ of
amplitudes and if $q\gg\ell_{\text{max}}$, then the Fock-state filtering is
almost perfect: only $\psi_{q}$ is removed and the other amplitudes are
practically unchanged.

However, the number of free parameters allows us to Fock-filter a state in
several different ways. Rather than postselecting $n=1$ and tuning the
beamsplitter, it would be much more advantageous, from an experimental point
of view, to postselect as little as possible and use PNR detection to the
fullest. Let us also fix $r=t=\frac{1}{\sqrt{2}}$, then

$\displaystyle\left|\,\phi\,\right\rangle_{b}\propto\sum_{\ell=0}^{\infty}\psi_{\ell+n-1}\binom{\ell+n-1}{\ell}^{\frac{1}{2}}\,2^{-\frac{\ell}{2}}\left(n-\ell\right)\left|\,\ell\,\right\rangle_{b},$
(4)

thereby removing the $n$-photon Fock state from the $b$ output if $n$ photons
are detected in port $a$. Again, one should keep in mind that the state
amplitudes $\psi_{k}$ are shifted by $n-1$ in the process, hence this
operation of photon catalysis is more complex than just Fock-state filtering.
However, as we now show, photon catalysis can nontrivially generate exact
displaced single-photon states, as well as arbitrarily good approximations of
non-Gaussian states that are of interest in quantum information processing,
such as Schrödinger-cat states and GKP quantum error code resource states. In
all the rest of the paper, we will consider either a coherent-state input,
$\left|\,\psi\,\right\rangle=\left|\,\alpha\,\right\rangle$, or inputs derived
from previous photon catalysis steps,
$\left|\,\psi^{\prime}\,\right\rangle=\left|\,\phi\,\right\rangle$.

To practically apply photon catalysis to arbitrary quantum states, we
developed a numerical procedure to take an input density matrix, possibly
impure, and transform it by photon catalysis. These calculation details are
described in Appendix A.

##  III Exact displaced single-photon states

The use of phase space displacements is very important for implementing a
variety of operations in continuous-variable quantum information, since the
Weyl-Heisenberg group of field quadrature shifts is the CV analog of the Pauli
group for qubits Bartlett2002 . These displacements can be experimentally
realized by combining the state to be displaced with a coherent state at a
highly unbalanced beamsplitter Paris1996 . However, this method does require a
partial trace over the other output port, yielding a statistical mixture that
only approaches the exact displaced state as the reflectivity approaches
unity, which in turn limits the amount of displacement. Moreover, as discussed
in kunal2018 , the displacement operation becomes less accurate as the energy
constraint for the input states increases. By contrast, photon catalysis is
not subject to this limitation and can generate an exact displaced single-
photon state.

###  III.1 Lossless case

We recall the derivation of Ref. 39 for an
$\left|\,\alpha\,\right\rangle\left|\,1\,\right\rangle$ input. The photon
catalysis output is, before PNR detection,

$\displaystyle\left|\,\text{out}\,\right\rangle_{ab}=D_{a}(r\alpha)D_{b}(t\alpha)(ta^{\dagger}-rb^{\dagger})\left|\,0\,\right\rangle_{a}\left|\,0\,\right\rangle_{b}.$
(5)

Performing a partial trace over mode $a$ results in a mixture of displaced
single-photon and weakened coherent given by

$\rho_{out}=t^{2}\left|\,t\alpha\,\right\rangle\left\langle\,t\alpha\,\right|+r^{2}D(t\alpha)\left|\,1\,\right\rangle\left\langle\,1\,\right|D^{\dagger}(t\alpha).$
(6)

In the limiting case of $\alpha\rightarrow\infty$ while $t\rightarrow 0$, we
recover a pure single-photon state displaced by $t\alpha$. In all experimental
cases where $\alpha$ is finite and $t>0$, we see that there is unavoidable
mixing with a coherent state of the same amplitude as the desired displacement
Paris1996 . Defining the fidelity between $\rho_{out}$ and a target state,
$\rho_{T}$, by

$\displaystyle F$
$\displaystyle=\Bigg{|}Tr\sqrt{\sqrt{\rho_{out}}\rho_{T}\sqrt{\rho_{out}}}\Bigg{|}^{2},$
(7)

we see that the fidelity of the approximate displaced photon given with the
ideal pure state,
$\rho_{T}=D(t\alpha)\left|\,1\,\right\rangle\left\langle\,1\,\right|D^{\dagger}(t\alpha)$,
is simply $F=r^{2}$.

However, if we don‘t perform the partial trace and instead send mode $a$ of
Eq. 5 to a PNR detector and condition on the detection of $n$ photons, we get

$\displaystyle\left|\,\phi\,\right\rangle_{b}$
$\displaystyle\propto{}_{a}\\!\left\langle\,n\,\right|\left.\text{out}\,\right\rangle_{ab}$
(8) $\displaystyle\propto(nt-\alpha
r^{2}b^{\dagger})\left|\,t\alpha\,\right\rangle_{b}.$ (9)

We can then evaluate the overlap of $\left|\,\phi\,\right\rangle_{b}$ with a
single-photon state displaced by $\beta$:

$\displaystyle\left\langle\,1\,\right|D^{\dagger}(\beta)\left|\,\phi\,\right\rangle$
$\displaystyle\propto nt(t\alpha-\beta)-\alpha
r^{2}[1+\beta^{*}(t\alpha-\beta)].$ (10)

Normalizing Eq. (9) and taking $\alpha,\beta\in\mathbb{R}$ yields the fidelity

$\displaystyle F$
$\displaystyle=e^{-|\beta-t\alpha|^{2}}\frac{\big{|}nt(\beta-t\alpha)+r^{2}\alpha(1+t\alpha\beta-|\beta|^{2})\big{|}^{2}}{r^{4}|\alpha|^{2}+t^{2}(n-r^{2}|\alpha|^{2})^{2}}.$
(11)

Examining this result, we see that if $r^{2}|\alpha|^{2}$ is specifically
chosen to be an integer so that $n=r^{2}|\alpha|^{2}$ photons are detected,
then $F=1$ for $\beta=\sqrt{|\alpha|^{2}-n}=\alpha t$. While this result holds
for any value of displacement, $\beta$, the specific case of taking an integer
value for $|\beta|^{2}$ allows us to view the method of photon catalysis to
enact displacements as a process that removes a single Fock component as per
Eq. 2. To see this, consider the displaced single-photon state,

$\displaystyle D(\beta)\left|\,1\,\right\rangle$
$\displaystyle=D(\beta)a^{\dagger}D^{\dagger}(\beta)\left|\,\beta\,\right\rangle$
(12)
$\displaystyle=a^{\dagger}\left|\,\beta\,\right\rangle-\beta^{*}\left|\,\beta\,\right\rangle$
(13)
$\displaystyle\propto\sum_{m=1}^{\infty}\frac{\beta^{m}}{\sqrt{m!}}\bigl{(}m-|\beta|^{2}\bigr{)}\left|\,m\,\right\rangle-|\beta|^{2}\left|\,0\,\right\rangle.$
(14)

Clearly, the $m=|\beta|^{2}$ Fock-component is absent, which is precisely the
component that was removed from the initial coherent state, $\alpha$, by the
photon catalysis process, i.e., the component
$nt^{2}/r^{2}=|\alpha|^{2}t^{2}=|\beta|^{2}$.

Our method gives the same result as the limit case given by Ref. Paris1996 ;
however, here, the beamsplitter retains $t>0$ and the limit case is not needed
to reach exact displacement. Therefore, we see that by tuning the reflectivity
of a beamsplitter and post-selecting on the desired $n$ detection, it is
possible to use photon catalysis with a coherent state to prepare an exact
displaced single-photon state of displacement amplitude $\beta=t\alpha$.

###  III.2 Lossy case

If we now consider an imperfect detector of quantum efficiency $\eta<1$, the
final state given by Eq. (42) is no longer pure and the fidelity with the
target displaced Fock-state is no longer unity. However, the fidelity can be
improved slightly by modifying the displacement of the target state,
$\rho_{T}=D(\beta)\left|\,1\,\right\rangle\left\langle\,1\,\right|D(\beta)^{\dagger}$.
In this case, we postulate, and numerically verify, that the maximum fidelity
is achieved when

$\displaystyle\alpha$ $\displaystyle=\sqrt{\frac{n}{\eta r^{2}}}$ (15)
$\displaystyle\beta$ $\displaystyle=\sqrt{\alpha^{2}-\frac{n}{\eta}}$ (16)

and this value approaches unity as $t\rightarrow 0$ ($r\rightarrow 1$) as
illustrated in Fig.2(c),

Figure 2: (a): Wigner functions for lossless photon catalysis per Eq. (9). As
the amplitude of the initial coherent state approaches the optimal ratio, the
output Wigner function becomes that of a displaced single-photon Fock State.
When $\alpha^{2}=n/r^{2}$, the fidelity between the two states is unity. (b):
Wigner functions for photon catalysis with detector efficiency $\eta=0.9$.
(Amplitudes have the same ratio of optimal values as in (a)) (c): Maximum
fidelity achievable with a displaced Fock-state of displacement
$\beta=\sqrt{\alpha^{2}-n/\eta}$ for detector efficiency $\eta$ when the
beamsplitter parameter $r$ is varied. The black dot-dashed line indicates the
fidelity of an ideal displacement with that obtained by the usual technique
Paris1996  for $r^{2}=0.97$, where the PNR detector is replaced by a partial
trace.

where the Wigner functions for various parameters are evaluated numerically
using the open-source Python modules QuTip Johansson2012  and Strawberry
Fields Killoran2019 . In order to verify that the codes perform as expected,
we computed the above photon catalysis step using QuTip and confirmed that the
output state is identically a displaced Fock state for correctly chosen
parameters.

Fig.2 shows the resulting Wigner functions after photon catalysis as the
coherent state amplitude is tuned. If the PNR detector is perfect (Fig.2(a)),
then the negativity of the Wigner function increases and becomes maximal as
$|\alpha|^{2}$ approaches the optimal value to achieve an ideal displacement.
Fig.2(b) demonstrates that when $|\alpha|^{2}$ is the same fraction of the
optimal value as in Fig.2(a), but the detector is no longer perfect
($\eta=0.9$), the Wigner function has the same qualitative shape in each case
but exhibits an overall decreased negativity due to the effective loss at the
detector. When comparing to a typical experimental displacement such as using
the beamsplitter parameter $r^{2}=0.97$ Nehra2019 (black dot-dashed line in
Fig.2(c)), the addition of even imperfect PNR detectors improves the
attainable fidelity with the ideal displaced single-photon state.

##  IV Schrödinger cat states

We now turn to Schrödinger-cat coherent superpositions (SCSs), which are of
the type

$\displaystyle\left|\,SCS_{\pm}(\zeta)\,\right\rangle=N(\left|\,\zeta\,\right\rangle\pm\left|\,-\zeta\,\right\rangle),$
(17)

where $N$ is the normalization constant. These non-Gaussian states have been
proposed for quantum computing Ralph2003  and small-amplitude optical SCSs
have been created by several methods Ourjoumtsev2006 ; Ourjoumtsev2007 ;
Neergaard-Nielsen2006 ; Takahashi2008 ; Gerrits2010 ; etesse_experimental_2015
, but there has yet to be a reliable approach to generate larger photon-number
SCSs. Methods have been proposed to “breed” SCSs using two smaller SCSs, a
beamsplitter, and conditional homodyne detection to create a larger SCS of
amplitude $\sqrt{2}\zeta$ Lund2004 ; Oh2018 , including an approach of
particular interest involving an iterative process similar to photon catalysis
where the PNR detectors are replaced with homodyne measurements
etesse_proposal_2014 . These approaches will also require quantum memories. To
our best knowledge, the largest created optical SCS to-date made use of two
squeezed vacuum resources and a breeding step to create a squeezed SCS with
$\zeta=2.15$ and a fidelity of $0.74$ (according to the fidelity defined by
Eq. 7) Sychev2017 .

As exemplified by the work of Sychev et al. Sychev2017 , most SCS preparation
methods result in a squeezed cat state,
$S(r)\left|\,SCS_{\pm}(\zeta)\,\right\rangle$, where
$S(r)=\exp[r(a^{{\dagger}\,2}-a^{2})/2]$ is the single-mode squeezing
operator. Without loss of generality, we take the squeezing parameter
$r\in\mathbb{R}$ for the numerical simulations in this work.

Using a procedure of iterated photon catalysis steps with only a coherent
state and single photons as inputs, we show that one can create states that
approach exact squeezed SCS, where the final amplitude and squeezing of the
state increases with the number of photon catalysis steps. We numerically
demonstrate that for low photon numbers, the final SCS amplitude after photon
catalysis increases more rapidly with each additional step than with the
number of steps required by the homodyne detection based breeding protocol
Lund2004 .

The general idea to generate squeezed SCS states from multiple photon
catalysis steps is somewhat counterintuitive: consider, for example, the “odd”
SCS,

$\displaystyle|SCS_{-}(\zeta)\rangle=\frac{1}{\sqrt{\sinh{|\zeta|^{2}}}}\sum_{n=0}^{\infty}\frac{\zeta^{2n+1}}{\sqrt{(2n+1)!}}\left|\,2n+1\,\right\rangle,$
(18)

which only contains odd photon numbers. Naïvely, it would be tempting to
consider using photon catalysis to filter the even Fock components from a
coherent state in an attempt to approximate an odd SCS. However, this approach
would require cascaded stages which would “undo” one another in general
because of the shift of the probability amplitudes by $n-1$, at each stage,
when $n>1$. Thus, the previously filtered Fock amplitudes in a cascaded scheme
do reappear, in general, after the next stage. Nonetheless, and remarkably,
cascaded photon catalysis can be used to generate excellent approximations to
squeezed SCSs — which are useful to generate the GKP resource states crucial
to CV quantum error correction, as we will see in the next section.

Figure 3: Protocol for generating an approximate squeezed SCS using photon
catalysis. The initial coherent state of amplitude $\alpha$ is successively
interfered with single-photon Fock states on a tree of variable beamsplitters,
each of which is followed by a PNR detector. These steps are sequential in
that each prior PNR detection occurs before the intermediate state is
interfered at the next beamsplitter in the tree. By tuning the beamsplitter
parameters and initial coherent state amplitude, the output conditioned by $N$
PNR detection steps is very near a squeezed SCS, up to a final displacement.

###  IV.1 Cascaded photon catalysis

We turn to the squeezed SCS generation protocol, which consists of photon
catalysis iterated over $N$ steps as shown in Fig.3. If we consider allowing
the beamsplitter parameters and PNR measurements to vary at each step, then
each iteration gives us two additional degrees of freedom: the
deterministically controllable beamsplitter reflectivity, $r_{i}$, and the
probabilistic detection, $n_{i}$. We limit the scope of this paper to
measurements that are subsequently performed, which means that after being
post-selected for an outcome $n_{i}$ at the $i^{th}$ step, the output state
will be the input to the $(i+1)^{th}$ step along with the single-photon as
displayed in Fig.3. The possibility of using measurement results as feed-
forward parameters, so as to optimize protocol efficiency, will be
investigated in future work.

Using the general output given by Eq. 44 for a coherent state,
$\left|\,\alpha\,\right\rangle$, we arrive at the $N$-catalyzed state given by

$\displaystyle\left|\,\phi{{}_{N}}\,\right\rangle\propto\sum_{m=0}^{\infty}\frac{\alpha^{m}}{\sqrt{m!}}\prod_{k=0}^{N-1}t^{m}_{k+1}\Biggl{\\{}n_{N-k}t_{N-k}^{2}-r_{N-k}^{2}\Bigl{[}m+\sum_{j=0}^{k-1}(n_{N-j}-1)\Bigr{]}\Biggr{\\}}\left|\,m\,\right\rangle.$
(19)

Following the intuition that filtering Fock components may lead to an SCS, we
wish to compare the fidelity of the state given by Eq.
LABEL:eq:cascaded_coherent with that of a squeezed SCS. However, in order to
simplify the resulting expression and decouple the squeezing parameter from
the SCS amplitude, we instead choose to compare the photon-catalyzed state
with a superposition of squeezed vacuum (SSV) given by

$\left|\,SSV_{\pm}(\beta)\,\right\rangle\propto\left[D(\beta)\pm
D(-\beta)\right]S(r)\left|\,0\,\right\rangle.$ (20)

Using the property Walls1994

$\displaystyle D(\beta)S(r)$ $\displaystyle=S(r)D(\zeta)$ (21)
$\displaystyle\zeta$ $\displaystyle=\beta\cosh{r}+\beta^{*}\sinh{r},$ (22)

it is easy to see that the squeezed SCS and SSV states are equivalent when

$\zeta=e^{-r}\beta$ (23)

where $\beta$ is chosen to be real without loss of generality. By allowing for
a potential displacement following the multiple photon catalysis procedure, we
can derive an expression for the fidelity with the $N$ step catalyzed state
and the target state,

$\rho_{T}=D(\delta)\left|\,SSV\,\right\rangle\left\langle\,SSV\,\right|{D^{\dagger}(\delta)}.$
(24)

Here, $\delta$ is the amplitude by which the final catalyzed state must be
displaced back to the origin of phase-space in order to recover the desired
$|SSV\rangle$ state defined in Eq. 20. The analytical expression for the
fidelity with the target state is derived in Appendix C and given by

$\displaystyle F$
$\displaystyle=\frac{C_{SSV\pm}^{2}C_{\left|\,\phi_{N}\,\right\rangle}^{2}}{\cosh{r}}\Biggl{|}\sum_{\ell,m=0}^{\infty}\Biggl{\\{}(\tanh{r})^{\ell}\frac{(2\ell-1)!!\alpha^{m}}{\sqrt{(2\ell)!m!}}[A(2\ell,m,-\beta-\delta)-A(2\ell,m,\beta-\delta))]$
(25)
$\displaystyle\times\prod_{k=0}^{N-1}\Bigl{[}t_{k+1}^{m}\Bigl{(}n_{N-k}t_{N-k}^{2}-r_{N-k}^{2}(m+\sum_{j=0}^{k-1}(n_{N-j}-1)\Bigr{)}\Bigr{]}\Biggr{\\}}\Biggr{|}^{2},$

where $C_{SSV\pm}$ normalizes Eq. 20, $C_{\left|\,\phi_{N}\,\right\rangle}$
normalizes Eq. LABEL:eq:cascaded_coherent, and we define
$A(n,m,\gamma)\equiv\left\langle\,n\,\right|D(\gamma)\left|\,m\,\right\rangle$.
This expression can be optimized with respect to all available parameters. In
order to limit the computation time and adhere to realistic PNR detection
capabilities, we constrain each $n_{i}<10$. For each combination of possible
detections subject to the constraints, we implemented the Nelder-Mead
algorithm to perform a multivariate numerical optimization in Mathematica.
Additionally, although all optimizations were performed with respect to the
SSV amplitude, $\beta$, we will give the results in terms of an ideal squeezed
SCS amplitude for easier comparison with other work.

Figure 4: Calculated Wigner functions and density matrix elements for the
approximate squeezed cat states resulting from varying the number of cascaded
photon catalysis iterations with an initial coherent state input. Row (a)
shows the results after two iterations, (b) after three iterations, and (c)
after four. In each separate case (a-c), all beamsplitter parameters, PNR
detections, and the input coherent state amplitude are tuned independently to
optimize the fidelity with the nearest squeezed cat state. Column (i) shows
the plotted Wigner functions for the photon catalyzed states, and (ii) gives
the density matrix elements. Column (iii) gives the density matrix elements of
the ideal associated squeezed cat, and (iv) shows the magnitude of the
difference between (ii) and (iii).

We performed the fidelity optimization for a given coherent state passing
through two, three, and four-step cascaded photon catalysis process. We
noticed that in each case, there are multiple sets of parameters that generate
a high-fidelity SSV state. In hindsight, this may not be a surprise, as the
number of free parameters likely under-constrains the system for low amplitude
target states. We selected the results that have the highest probability of
success, which are depicted both in terms of the plotted Wigner function and
density matrix elements in Fig.4. We find that we can produce a nearly ideal
SCS with $\zeta=1.50$ squeezed by 1.91 dB after a two step process
($F>0.999$). Optimizing a three-step process yields an SCS with $\zeta=2.18$
squeezed by 4.17 dB ($F=0.984$), and using $N=4$ results in $\zeta=2.67$ with
squeezing of 4.52 dB ($F=0.977$). The full list of optimized parameters for
each case is provided in Appendix D. We note that scaling from $N=2$ to $N=3$
in this procedure already increases the cat amplitude by a factor of
$\sqrt{2}$. This approach only includes a single additional interference and
detection stage whereas the other breeding procedure discussed earlier
requires three additional steps provided that the two smaller cats to be bred
came from an $N=2$ photon catalysis procedure. This indicates that an
iterative photon-catalysis process might scale better for larger SCS creation
than homodyne-based breeding.

Figure 5: Photon number probability amplitudes for various states. Top:
distributions for coherent state input and the quantum state after the first
detection step. Bottom: distributions for the quantum state after the second
PNR detection and for the final catalyzed state after a displacement(blue)
compared to the target SSV state(red).

It is important to keep in mind that each $N$-step photon catalysis process is
optimized to produce an approximate SSV state as a final result; simply adding
an additional catalysis step to any of these final approximate cats will not
increase the cat amplitude or squeezing under the current approach. As an
example, consider Fig.5, which shows the photon number distributions before
and after each catalysis step for $N=2$. The filtering of Fock components at
each step is not perfect due to the numerical procedure, but it is easy to see
that the first projective PNR measurement, $|1\rangle\langle 1|$, acts to
nearly eliminate the $\left|\,2\,\right\rangle$ component from the initial
coherent state, while the second step effectively shifts and re-scales the
distribution. A final displacement of the Wigner function back to the origin
of phase-space results in a very near approximation to the target state as
shown by the comparison of probability amplitudes in the rightmost panel of
Fig.5. Although the first catalysis step acts to de-Gaussify the coherent
state, applying the displacement at this point would result in poor overlap
with any class of SSV.

###  IV.2 Loss tolerance

Experimental imperfections and associated optical losses will contribute to
non-ideal single-photon resources, in which case we must consider a mixture of
single-photon and vacuum as the realistic input at each interference
beamsplitter Lvovsky2001 ; Nehra2019 . This can be modeled by sending each
pure single photon through a lossy channel with transmission $\gamma$, so that
each input is now

$\rho_{mix}=\gamma\left|\,1\,\right\rangle\left\langle\,1\,\right|+(1-\gamma)\left|\,0\,\right\rangle\left\langle\,0\,\right|.$
(26)

By using the optimized parameters for the ideal procedure and replacing the
pure single-photons with $\rho_{mix}$ at each step, we see in Fig.6 how
including a vacuum contribution adversely affects the fidelity with the target
states in each case. The decreased purity more greatly harms the procedures
with larger $N$, unsurprisingly.

However, we note that just as in Ref. etesse_proposal_2014 , our protocol in
of itself adds little imperfection to the result, which can be seen directly
in Fig.6 by comparing the drop in fidelity between neighboring values of $N$.
For purities above $\gamma=0.8$, the decrease in the fidelity of our output
state with the target between $N=2$ and $N=3$ step processes is nearly
equivalent to the fidelity difference between $N=3$ and $N=4$ step procedures,
which indicates the change in fidelity is nearly linear with the number of
impure single-photon inputs.

Fig.6 indicates the importance of pure single photons for our protocol;
fortunately, there has been much work in the area of on-demand, highly
efficient single-photon emitters using solid state devices aharonovich_solid-
state_2016 ; loredo2016scalable ; somaschi2016near . On-demand quantum dot
emitters have already demonstrated single-photon purities above $99\%$ ding_-
demand_2016 , and a recent proposal making use of quantum feedback control
suggests that on-chip sources with greater than $99\%$ on-demand extraction
efficiency are possible with current technology heuck_temporally_2018 .

Figure 6: The effects of vacuum mixing due to (a) impure single-photon input
and (b) imperfect detectors are shown separately for $N$-step photon catalysis
with $N=2,3$ and $4$. The fidelity is calculated with respect to the same
target SSV states as for the optimized procedure shown in Fig.4.

As in Sec. III, we consider the effects of imperfect detectors in each step,
where we assume identical detector efficiencies of $\eta\leq 1$. Similarly to
having impure single-photons, each additional imperfect PNR measurement acts
to further decrease the fidelity with the target state in a way that appears
linear with increasing photon catalysis steps. We see from Fig.6 that
generating high-fidelity cat states would require the use of highly efficient
PNR detectors, as $\eta\leq 0.85$ prohibits even $N=2$ step procedures from
yielding states above a fidelity of 0.9. Thanks to recent advances, high
efficiency PNR is achievable with current transition-edge sensors Lita2008
and superconducting nanowire single-photon detector arrays dauler_review_2014
, which have been recently reported to have PNR capabilities Nicolich2019 .

###  IV.3 Success probability

For methods of non-Gaussian state engineering that use homodyne detection as
in Ref. etesse_proposal_2014 , one can select a confidence window of $\Delta
x$ on acceptable post-selection measurement results that allow for an increase
in procedure success probability at the cost of decreasing fidelity. In our
case, however, the projective operator of PNR detection lies in a discrete
subspace, so we cannot allow for continuous deviations from the ideal
detection scheme. Nonetheless, we can still examine the fidelity and success
probability in cases where the initial coherent state amplitude and $N$
deterministic beamsplitter parameters are held fixed at values optimized for
SSV generation, but some $n_{i}$ deviate, necessarily in integer steps, from
the ideal scenario.

For each combination of detection events that deviates, we can re-optimize the
fidelity from Eq. 25, where now only the target state parameters are allowed
to vary (SSV amplitude, squeezing, and the displacement amplitude). In this
way, it is possible to generate a finite list of PNR detection combinations
that differ from the combination that best approximates an ideal SSV state,
yet still yields a final state above a given threshold fidelity, $F_{thr}$,
with a target state. This suggests that approaches involving feed-forward
based on measurement results might also be fruitful. If one only discards the
photon-catalyzed states where the detected combination
$\\{n_{1},n_{2},\ldots,n_{N}\\}$ is outside of the acceptable list, then the
photon-catalyzed state is guaranteed to have a fidelity $F\geq F_{thr}$ with
an appropriate SSV state. This target state may vary slightly in amplitude and
squeezing from the SSV target for the optimal detection scheme (removing a
different total photon number changes the total energy of the state), but the
target is still required to be an SSV state, and we show later how two similar
SSV states can be enlarged by utilizing PNR detection. By decreasing
$F_{thr}$, more combinations of possible PNR detections are included in the
“accept” list, thereby increasing the probability of successfully generating
an SSV state. The effects of lowering $F_{thr}$ for the two, three, and four-
step photon catalysis procedures considered in the previous section are
depicted in Fig.7. For the $N=2$ and $N=3$ cases, we see that setting
$F_{thr}=0.9$ yields nearly an order of magnitude increase in success
probability from the perfect detection scenario, while the $N=4$ case shows an
increase of nearly two orders of magnitude for the same fidelity threshold.
Paired with state-of-the-art technology in deterministic single-photon sources
ding_-demand_2016 ; hanschke2018quantum  with emission rates on the order of
$10^{6}$ counts/sec aharonovich_solid-state_2016 , an $N=3$ step photon
catalysis process would yield nearly perfect SSV states at rates in the
kilohertz regime.

Figure 7: The success probability to generate an approximate SSV state with at
least a given threshold fidelity is shown for a cascaded photon catalysis
procedure of two(green), three(red), and four(blue) steps. Allowing the PNR
detection results to deviate from the optimized values yields an increased
probability of success at a cost to fidelity.

###  IV.4 Higher-symmetry SSV states

The versatility of cascaded photon catalysis allows us to examine classes of
SSV states that exhibit beyond two-fold phase space rotational symmetry. These
states can be more generally expressed as superpositions in the form of

$\\\ \left|\,SSV^{(M)}\,\right\rangle\propto\sum_{n=0}^{M-1}D(\beta
e^{\frac{2n}{M}i\pi})S(re^{\frac{4n}{M}i\pi})\left|\,0\,\right\rangle,$ (27)

where $M$ is the number of components in the superposition and degree of
rotational symmetry. Each component consists of squeezed vacuum that has been
displaced with an integer multiple of phase increment of
$\nicefrac{{2\pi}}{{M}}$, where the squeezer anti-squeezes the vacuum
perpendicularly to the axis of the subsequent displacement. Although there is
no restriction on $r$ for this class of states, we consider $r<0$, as this
corresponds to the states we generated with cascaded photon catalysis. While
we only explicitly consider protocols to generate states with $M=2$ and $M=3$
in this work, we expect that it is possible to further generalize the
generation of $SSV^{(M>3)}$ by optimizing photon-catalysis.

For the next highest symmetry state (three-fold symmetry), we can again use
the protocol depicted in Fig.3 with three photon catalysis steps, but with
different input coherent state amplitude and beamsplitter parameters, to
create the state

$\Bigl{(}D(\beta)S(r)+D(\beta e^{\frac{2\pi i}{3}})S(re^{\frac{4\pi
i}{3}})+D(\beta e^{\frac{4\pi i}{3}})S(re^{\frac{2\pi
i}{3}})\Big{)}\left|\,0\,\right\rangle,$ (28)

where $r=-0.24$, $\beta=1.255$, and the fidelity is greater that $0.99$. This
is achieved with detection events of
$\\{n_{1},n_{2},n_{3}\\}=\\{6,\,4,\,2\\}$, which were found serendipitously.
The comparison of the ideal and procedurally generated states is depicted in
Fig.8(a), and Fig.8(b) demonstrates the $120^{\circ}$ rotational symmetry of
the Wigner function. As this state is created with a cascaded photon catalysis
procedure, the success probability and dependence on purity of single-photon
inputs and detector efficiency is very similar to the other $N=3$ case
considered earlier. The utility of the $M=3$ SSV state is developed further in
the next section.

Figure 8: Photon-number distribution, (a), and Wigner function, (b), of the
approximate $SSV^{(3)}$ state generated by a three-step photon catalysis
protocol, where the squeezing parameter is $r=-0.24$ and amplitude is
$\beta=1.255$. The Wigner function nicely displays $120^{\circ}$ phase-space
rotational symmetry.

##  V Breeding Protocols

Methods to combine SCS with breeding protocols have been widely explored to
generate larger amplitude SCS states Sychev2017 ; etesse_experimental_2015 ;
Lund2004 ; Oh2018 ; laghaout_amplification_2013  and other classes of comb
states etesse_proposal_2014 ; Vasconcelos2010 . A recent work shows how feed-
forward displacements can allow for deterministic creation of optical GKP
states from a supply of squeezed SCS weigand_generating_2018 . Here, we
motivate the extension of these breeding protocols to PNR detection as opposed
to traditional homodyne measurements. Additionally, we make use of the phase-
insensitivity of direct photon-counting measurements to demonstrate how PNR
detectors can be used to breed higher symmetry states.

###  V.1 SSV states

First, we begin by examining a procedure with the aim to take two lower
amplitude Mth-order SSV states and produce a larger SSV state of the same
order of symmetry. Taking two identical states given by Eq. 27 and interfering
them on a balanced beamsplitter, we have the state

$\left|\,\phi\,\right\rangle\propto
U_{ab}\sum_{n=0}^{M-1}D_{a}(\beta_{n})S_{a}(\xi_{n})\left|\,0\,\right\rangle_{a}\otimes\sum_{n=0}^{M-1}D_{b}(\beta_{n})S_{b}(\xi_{n})\left|\,0\,\right\rangle_{b},$
(29)

where $\beta_{n}=|\beta|e^{\frac{2n}{M}i\pi}$ and
$\xi_{n}=re^{\frac{4n}{M}i\pi}$, with the beamsplitter operation given by
$U_{ab}=e^{\frac{\pi}{4}(ab^{\dagger}-a^{\dagger}b)}$ . Developed further in
Appendix E and numerically verified for the first few SSV states, it can be
seen that for finite squeezing and relatively larger values of $|\beta|$, a
PNR measurement of zero photon, $|0\rangle\langle 0|$, on output mode $b$
leads to

${}_{b}\langle
0\left|\,\phi\,\right\rangle_{ab}\approx\sum_{n=0}^{M-1}D_{a}(\sqrt{2}\beta_{n})S_{a}(\xi_{n})\left|\,0\,\right\rangle_{a}.$
(30)

Figure 9: Fidelity and success probability of the PNR detection-based breeding
protocol for various initial amplitudes, $\beta$, and squeezing, $\xi$. The
colors bars indicate fidelity (top panel) and success probability (bottom
panel) for the respective plots, where dark green is unity. $M=2$ (a), $M=3$
(b), and $M=4$ (c) symmetric SSV states are bred into larger states in the
same class with amplitude $\sqrt{2}\beta$ and the same squeezing parameter.

This means that using PNR detectors, SSV states can be bred into states with
the same squeezing, but with increased amplitude
$|\beta|\rightarrow\sqrt{2}|\beta|$. The factor of $\sqrt{2}$ increase is the
same result as achieved for SSV states with $M=2$ using homodyne detection-
based breeding protocols, but the nature of performing a homodyne measurment
along a particular axis in phase-space means that higher symmetry features are
not preserved, thus indicating that PNR-based protocols such as this will be
necessary to breed general SSV-type states. The fidelity of the bred states
with the enlarged target states as well as success probabilities are shown in
Fig.9. In all cases examined, the fidelity quickly approaches unity for values
of $|\beta|^{2}\gtrapprox M$. Additionally, the success probabilities are
greater than $10^{-1}$ in all regions where fidelities are numerically
indistinguishable from one, and the probability for success in the $M=2$ case
holds steady at $0.5$ for squeezing $|\xi|\lessapprox 0.3$ and
$|\beta|^{2}>4$.

###  V.2 Square-lattice GKP states

Here, we adapt the experimental proposal of Vasconcelos et al. Vasconcelos2010
to our cascaded photon catalysis method of producing SCSs followed by PNR
detector-based breeding. The essence of their original idea is as follows: by
entangling two squeezed SCS on a balanced beamsplitter and performing a
homodyne measurement of $p=0$ on one output, the output of the other port is
projected into a state exhibiting a series of evenly spaced Gaussian peaks
along the $x$-quadrature axis where the peak amplitudes follow a binomial
distribution about a central peak at $x=0$, and the peak width is determined
by the SCS squeezing. Repeating an identical protocol with two of these
outputs serves to increase the number of peaks by elevating the order of the
binomial distribution. Recently, Weigand et. al have shown that including
feed-forward displacements based on the homodyne measurement results makes
this process deterministic weigand_generating_2018 . After many iterations and
in the limit of large initial SCS squeezing, this method accurately
approximates an ideal square-lattice GKP state, which can be explicitly
written as Gottesman2001

$\displaystyle|\mu_{\text{sqr}}\rangle\propto\sum_{n_{1},n_{2}=-\infty}^{\infty}e^{-i\hat{p}\sqrt{\frac{2\pi}{d}}(dn_{1}+\mu)}e^{i\hat{q}\sqrt{\frac{2\pi}{d}}n_{2}}|0\rangle,$
(31)

where $d$ is the dimensionality of the code space and $\mu=0,\dots,d-1$ labels
the specific GKP state in the code space. To make the state physically
realizable in the case of finite energy, the state can be modulated by a
Gaussian envelope Noh2018 . The finite energy GKP state is thus given by

$\displaystyle\left|\,\mu^{\Delta}_{sqr}\,\right\rangle\propto
e^{-\Delta^{2}\hat{N}}\left|\,\mu_{sqr}\,\right\rangle$ (32)

where $\hat{N}=a^{\dagger}a$ is the photon-number operator. These finite-
energy GKP states compose a periodic square-lattice grid in phase-space, where
each peak has width $\Delta$. The result of following this homodyne-detection
based procedure with two states generated by our four-step photon catalysis
procedure is shown in the top left plot of Fig.10.

If we perform PNR detection on one mode following the entangling beamsplitter
instead of a homodyne measurement, we can obtain different final states
according to the selected measurement result that each exhibit the periodic
comb-like structure of a GKP state as demonstrated in Fig.10. Using a balanced
beamsplitter, $U_{ab}$, and phase-shift operator $R(\Theta)=e^{i\Theta
a^{\dagger}a}$, the overall result can be concisely written as

$\displaystyle\left|\,\phi\,\right\rangle\propto{}_{b}\\!\left\langle\,n\,\right|U_{ab}R_{a}(\Theta)\left|\,SSV{{}^{(2)}}\,\right\rangle_{a}\left|\,SSV^{(2)}\,\right\rangle_{b}.$
(33)

This is exactly the process that we used earlier to breed SSV states, but with
two important changes. Because PNR detection is phase-insensitive, precisely
controlling the relative phases between the states impinging on the entangling
beamsplitter becomes vital, and here we set the phase difference to
$\Theta=\frac{\pi}{2}$ to achieve the square-grid comb-like periodicity.
Additionally we allow for PNR detection other than vacuum.

Figure 10: Wigner functions of bred two-fold symmetry SSV states (squeezed SCS
states) achieved by the photon catalysis protocol after various detector
measurements. The homodyne plot demonstrates the grid state of the result
after a $p=0$ quadrature measurement, while the other plots show the resultant
state after a PNR detection of $n$ photons. The $n=4$ case has a fidelity
$F=0.996$ with a square-grid GKP state with finite-energy peak-width
$\Delta=0.545$.

As a concrete example, if we entangle two SSV states generated from photon
catalysis (Fig.4(c)) and detect $n=4$ photons in one mode, we achieve a state
that has a fidelity of $F=0.996$ with a finite-energy square GKP state of
width $\Delta=0.545$. This single event occurs with a success probability of
$0.09$, but the probability to achieve at least one of the comb states shown
in Fig.10 exceeds $0.60$.

Since finite-energy GKP code states are no longer orthogonal within the
$d$-dimensional code space, error correction protocols become probabilistic
due to the non-zero overlap between the logically encoded states. In the
original work, the probability to successfully correct an error hinges solely
on the ability to accurately distinguish code states, leading to a $99\%$
success rate for $\Delta\sim 0.5$ Gottesman2001 , such as is the case for our
states. However, a more realistic error-correcting protocol considers
potential shift errors on the ancilla qubits as well as the encoded qubit to
be corrected, leading to a more stringent requirement of $\Delta<0.15$ to
achieve the same success rate glancy2006error . Utilizing more photon
catalysis iterations to increase the initial SCS squeezing and performing
multiple breeding operations as in weigand_generating_2018  will result in
suitable states for realistic quantum error correction.

###  V.3 Hexagonal GKP states

One principle advantage exhibited by the GKP encoding scheme is an inherent
resistance to shift errors in the conjugate position and momentum quadratures
$x$ and $p$. The hex GKP state, formulated in the original work Gottesman2001
and recently expanded upon in greater detail in Refs. Noh2018  and
albert_performance_2018 , improves upon the initial square lattice state by
being able to correct larger displacement errors. As a hexagonal lattice in
phase space, the hex GKP state can now account for shift errors with a new
upper-bound phase space radius of $r\leq\sqrt{\frac{\pi}{\sqrt{3}d}}$, as
opposed to $r\leq\sqrt{\frac{\pi}{2d}}$ intrinsic to the original, where $d$
is the dimensionality of the code space. This hex GKP state can be explicitly
written as

$\displaystyle|\mu_{\text{hex}}\rangle\propto\sum_{n_{1},n_{2}=-\infty}^{\infty}e^{-\tfrac{i}{2}(\hat{q}+\sqrt{3}\hat{p})\sqrt{\frac{4\pi}{d\sqrt{3}}}(dn_{1}+\mu)}e^{i\hat{q}\sqrt{\frac{4\pi}{d\sqrt{3}}}n_{2}}|0\rangle,$
(34)

where as in the square-lattice case, applying the non-unitary operator
$e^{-\Delta^{2}\hat{N}}$ makes the state physically realizable Noh2018 .
Fig.11 shows the Wigner function for a linear combination of hex GKP states of
dimension $d=2$ plotted with different peak-widths.

Figure 11: Comparison of finite energy hex GKP states to that of two
$120^{\circ}$ rotationally symmetric SSV states after breeding. (a) and (b)
show the Wigner function of a mixture of hex GKP states of code dimension 2
with peak widths $\Delta=0.2$ and $\Delta=0.46$, respectively. (c) shows the
Wigner function of bred SSV states each developed by an $N=3$ photon catalysis
protocol.

Noting that the SSV(2) states (squeezed SCS) can be bred into square-lattice
GKP states, we are motivated to examine what happens when breeding SSV(3)
states, such as those generated in Sec. IVIV.4. Because the states have
$\nicefrac{{2\pi}}{{3}}$ rotational symmetry, we fix the phase difference of
the two initial states to be $\Theta=\nicefrac{{\pi}}{{3}}$ at the entangling
beamsplitter, and send one output mode to a PNR detector. Just as with the
SSV(2) breeding protocol, we find that several possible measurement results
lead to a final state exhibiting the desired hexagonal symmetry. Fig.11(c)
shows the result of breeding two of the states depicted in Fig.8 after post-
selecting on a PNR measurement of $n=0$ with a probability of 0.31. While only
having a fidelity of $0.8$ with the mixed hex GKP state in Fig.11(b), the
visual similarity and relatively high success probability indicate that there
may be a suitable SSV(3) for more realistic hex GKP synthesis.

##  VI Conclusion

In this work, we have further explored the photon catalysis process
Lvovsky2002  where interfering a single-photon with an input state on a
beamsplitter can result in specifically engineered non-Gaussian quantum states
contingent upon beamsplitter parameters and conditioned by PNR detection. By
using a coherent state as an input, we demonstrated that one photon catalysis
step can implement arbitrary displacements on the single photon Fock state and
preserve state purity. Further, by cascading the photon catalysis process, we
are able to filter out multiple photon number components and accurately
approximate relatively large superpositions of squeezed vacuum, including what
is to our knowledge the largest proposed optical SCS without the need for
large-number Fock states or breeding. Furthermore, we adapted breeding
protocols based on homodyne detection etesse_proposal_2014 ; Sychev2017 ;
Lund2004 ; Oh2018 ; laghaout_amplification_2013  to a PNR-based scheme that
allows for breeding $M$-fold symmetry SSV states. By changing the phase
between input modes of the PNR-based breeding, we showed that it is possible
to reconstruct square and hexagonal lattice comb-states that exhibit the
structure of finite-energy GKP-class code states. Future avenues of
exploration may include utilizing PNR measurement results to implement feed-
forward modifications to subsequent photon catalysis steps in the hopes to
increase protocol success rates.

It has been recently shown that combining a supply of GKP states with Gaussain
QC alone is sufficient for fault-tolerant universal QC; therefore, developing
a technique to sythesize GKP states is vital baragiola_all-gaussian_2019 .
Machine learning optimization algorithms have demonstrated the potential to
create high-fidelity GKP states Arrazola2019 , but these methods require in
excess of 100 gate operations. The method of photon catalysis and PNR-based
breeding we describe here provides an experimentally feasible approach to
generate various nonclassical states, including GKP, important for quantum
information and quantum computing.

## Acknowledgments

RN would like to thank Lu Zhang for discussions on photon catalysis at QCMC
2018. ME would like to thank Chun-Hung Chang and Jacob Higgins for pointing
out an inconsistency in figure scaling. OP would like to thank Rafael
Alexander and Mirko Lobino for discussions. The work is supported by NSF grant
No. PHY-1708023.

##  Appendix A Numerical modeling

We begin by defining a generic beamsplitter operation, projector, and input
density matrices as

$\displaystyle U_{ab}$ $\displaystyle=e^{\theta(ab^{\dagger}-a^{\dagger}b)}$
(35) $\displaystyle P_{n}$ $\displaystyle=(|n\rangle\langle
n|)_{a}\otimes\mathbb{1}_{b}$ (36) $\displaystyle\rho^{in}_{a}$
$\displaystyle=(|\psi\rangle\langle\psi|)_{a}$ (37)
$\displaystyle\rho^{in}_{b}$ $\displaystyle=(|1\rangle\langle 1|)_{b}$ (38)

where $r=\cos{\theta}$ is the reflection coefficient. After the beamsplitter,
but before the detection event, the new density matrix is

$\displaystyle\rho_{ab}$
$\displaystyle=U_{ab}\rho^{in}_{a}\otimes\rho^{in}_{b}U_{ab}^{\dagger}.$ (39)

We then apply the projective measurement of mode $a$ by an ideal detector and
normalize to obtain the resulting density matrix of mode $b$, given by

$\displaystyle\rho_{out}$
$\displaystyle=\frac{Tr_{a}[\rho_{ab}P_{n}]}{Tr[\rho_{ab}P_{n}]}.$ (40)

Imperfect detection can be modeled by including a loss beamsplitter followed
by a perfect detector as depicted in Fig.12, where we trace out over the lost
mode. In this case we now have a vacuum input mode to consider, so a new input
state of $\rho^{in}_{c}=(|0\rangle\langle 0|)_{c}$, a loss beamsplitter
$U_{ac}=e^{\xi(ac^{\dagger}-a^{\dagger}c)}$ where $\eta=\cos{\xi}$ is the
detector efficiency, and a modified projector, $P^{\eta}_{n}=(|n\rangle\langle
n|)_{a}\otimes\mathbb{1}_{b}\otimes\mathbb{1}_{c}$. Eq. (39) becomes

$\displaystyle\rho_{abc}$
$\displaystyle=U_{ac}\rho_{ab}\otimes\rho^{in}_{c}U_{ac}^{\dagger}.$ (41)

As before, we apply the projective measurement of the detector, but must now
also trace out over the lost beamsplitter port in addition to the detected
mode. Our final state is now

$\displaystyle\rho_{out,\eta}$
$\displaystyle=\frac{Tr_{ac}[\rho_{abc}P^{\eta}_{n}]}{Tr[\rho_{abc}P^{\eta}_{n}]}.$
(42)

Figure 12: Loss model. Photon catalysis with an imperfect detector can be
modeled by placing a beamsplitter of reflectivity $\eta$ with vacuum input
before an ideal detector, where the other output is lost via a partial trace.

In several sections of this work, we explore the possibility to perform
cascaded photon catalysis by using the resulting density matrix from the $i$th
step as the input state to the $i+1$ interference beamsplitter,

$\displaystyle{\rho^{in}_{a}}^{(i+1)}={\rho^{(i)}_{out}}$ (43)

This procedure allows us to independently vary the beamsplitter parameter
$r_{j}$ and PNR detection $n_{j}$ at each step and obtain the intermediate
resulting density matrix. After repeated the process as many times as desired,
we can compute the fidelity between a target state, $\rho_{T}$, and the result
of our process, $\rho_{out}$, which is defined by Eq. 7.

When performing cascaded photon catalysis calculations, we we limited the
Hilbert space dimension to 50 as the tensor product between Hilbert spaces
causes an exponential increase in computational expense with each additional
mode added.

##  Appendix B Cascaded photon catalysis

Consider iterating the photon catalysis process by using the output from one
step as the input to the next. Here, we derive the general quantum state at
the output of an N-step photon catalysis procedure where the initial input is
taken to be an arbitrary pure state, $\psi$, and each step uses a single-
photon Fock state as the second input. The $i$th step is described by a
beamsplitter of real reflectivity and transmissivity, $r_{i}$ and $t_{i}$,
followed by a perfect PNR detector that detects $n_{i}$ photons. We start by
giving the result and then proceed with mathematical induction as proof. After
N steps, we posit that the final quantum state, up to a normalization, will be

$\displaystyle\left|\,\phi{{}_{N}}\,\right\rangle$
$\displaystyle\propto\sum_{m=0}^{\infty}\Biggl{\\{}\psi_{m-N+\sum_{j=1}^{N}n_{j}}\left[\frac{[m-N+\sum_{j=1}^{N}n_{j}]!}{m!}\right]^{\frac{1}{2}}\prod_{k=0}^{N-1}\Biggl{[}\frac{r_{k+1}^{n_{k+1}-1}}{\sqrt{n_{k+1}!}}t_{N-k}^{m-1+\sum_{j=0}^{k-1}(n_{N-j}-1)}$
(44)
$\displaystyle\times\Bigl{(}n_{N-k}t_{N-k}^{2}-r_{N-k}^{2}(m+\sum_{j=0}^{k-1}(n_{N-j}-1))\Bigr{)}\Biggr{]}\Biggr{\\}}\left|\,m\,\right\rangle.$

By taking $N=1$, it is easy to see that this reduces to Eq. 2, thus 44 is
verified for a single photon catalysis step. For the proof, we now assume 44
to be true and feed $\phi_{N}$ into an $(N+1)$th step. The resulting quantum
state after the PNR detection of $n_{N+1}$ photons can be obtained from
equation 2 by using the $m+n_{N+1}-1$th component of $\phi_{N}$ to arrive at

b
$\displaystyle\left\langle\,n_{N+1}\,\right|U_{ab}\left|\,\phi_{N+1}\,\right\rangle_{a}\otimes\left|\,1\,\right\rangle_{b}\propto\sum_{m=0}^{\infty}\Biggl{\\{}\frac{\psi_{m+n_{N+1}-1-N+\sum_{j=1}^{N}n_{j}}}{\sqrt{n_{N+1}}}\binom{m+n_{N+1}-1}{m}^{\frac{1}{2}}$
(45)
$\displaystyle\times\left[\frac{[m+n_{N+1}-1-N+\sum_{j=1}^{N}n_{j}]!}{(m+n_{N+1}-1)!}\right]^{\frac{1}{2}}r_{N+1}^{n_{N+1}-1}t_{N+1}^{m-1}[n_{N+1}t_{N+1}^{2}-mr_{N+1}^{2}]$
$\displaystyle\times\prod_{k=0}^{N-1}\Biggl{[}\frac{r_{k+1}^{n_{k+1}-1}}{\sqrt{n_{k+1}!}}t_{N-k}^{m+n_{N+1}-2+\sum_{j=0}^{k-1}(n_{N-j}-1)}$
$\displaystyle\times\Bigl{(}n_{N-k}t_{N-k}^{2}-r_{N-k}^{2}(m+n_{N+1}-1+\sum_{j=0}^{k-1}(n_{N-j}-1))\Bigr{)}\Biggr{]}\Biggr{\\}}\left|\,m\,\right\rangle.$

Reordering the sum and product indices allows us to arrive at an expression
identical to Eq. 44, but with $N\rightarrow N+1$, and thereby complete the
proof. From Eq. 44, we can also extract the overall success probability for a
particular set of detection events:

$\displaystyle
P(n_{1},\dots,n_{N})=\sum_{m=0}^{\infty}\Biggl{\\{}\psi_{m-N+\sum_{j=1}^{N}n_{j}}^{2}\left[\frac{[m-N+\sum_{j=1}^{N}n_{j}]!}{m!}\right]$
(46)
$\displaystyle\times\prod_{k=0}^{N-1}\Biggl{[}\frac{r_{k+1}^{2n_{k+1}-2}}{n_{k+1}!}t_{N-k}^{2m-2+2\sum_{j=0}^{k-1}(n_{N-j}-1)}\Bigl{(}n_{N-k}t_{N-k}^{2}-r_{N-k}^{2}(m+\sum_{j=0}^{k-1}(n_{N-j}-1))\Bigr{)}^{2}\Biggr{]}\Biggr{\\}}.$

##  Appendix C Fidelity with SSV

We now derive an expression for the fidelity between a pure SSV state and the
result from an $N$-step filtered coherent state. The SSV states are given by

$\left|\,SSV_{\pm}\,\right\rangle=C_{SSV\pm}\left[D(\beta)\pm
D(-\beta)\right]S(r)\left|\,0\,\right\rangle,$ (47)

where $r$ is the single-mode squeezing parameter, $\beta$ is the displacement
magnitude to create the superposition, and the normalization coefficient is

$C_{SSV\pm}=\left[\frac{1}{2\pm
2e^{-2|\beta|^{2}e^{-2r}}}\right]^{\frac{1}{2}}.$ (48)

As a first step, we derive the overlap coefficients between an n-photon Fock
state and a displaced m-photon Fock state, which will be useful later. This is
given by

$\displaystyle A(n,m,\gamma)$
$\displaystyle\equiv\left\langle\,n\,\right|D(\gamma)\left|\,m\,\right\rangle$
(49)
$\displaystyle=\left\langle\,n\,\right|D(\gamma)\frac{{a^{\dagger}}^{m}}{\sqrt{m!}}D^{\dagger}(\gamma)\left|\,m\,\right\rangle$
(50)
$\displaystyle=\left\langle\,n\,\right|\frac{(a^{\dagger}-\gamma^{*})^{m}}{\sqrt{m!}}\left|\,\gamma\,\right\rangle$
(51)
$\displaystyle=e^{-|\gamma|^{2}/2}\sum_{k=0}^{m}\binom{m}{k}\frac{(-1)^{k}\sqrt{n!}\,|\gamma|^{2k+n-m}}{m!(n-m+k)!}.$
(52)

Because the photon-catalyzed coherent state must be displaced back to the
origin of phase space before comparing to an SSV state (see main text), we are
interested in a fidelity of the form

$F=|\left\langle\,SSV_{\pm}\,\right|D^{\dagger}(\delta)\left|\,\phi_{N}\,\right\rangle|^{2},$
(53)

where $\delta$ here is the amplitude of displacement needed after the photon
catalysis procedure. By substituting the coefficients of a coherent state of
magnitude $\alpha$ for $\psi$ in Eq. 44 and using the overlap coefficients
given by 52, we can arrive at the overall expression

$\displaystyle F$
$\displaystyle=\frac{C_{SSV\pm}^{2}C_{\left|\,\phi_{N}\,\right\rangle}^{2}}{\cosh{r}}\Biggl{|}\sum_{\ell,m=0}^{\infty}\Biggl{\\{}(\tanh{r})^{\ell}\frac{(2\ell-1)!!\alpha^{m}}{\sqrt{(2\ell)!m!}}[A(2\ell,m,-\beta-\delta)-A(2\ell,m,\beta-\delta))]$
(54)
$\displaystyle\times\prod_{k=0}^{N-1}\Bigl{[}t_{k+1}^{m}\Bigl{(}n_{N-k}t_{N-k}^{2}-r_{N-k}^{2}(m+\sum_{j=0}^{k-1}(n_{N-j}-1)\Bigr{)}\Bigr{]}\Biggr{\\}}\Biggr{|}^{2},$

where $C_{\left|\,\phi_{N}\,\right\rangle}$ is determined by normalizing Eq.
44 with a coherent state input. This equation for fidelity can now be
numerically maximized for the experimentally accessible parameters of $r_{i}$,
$\alpha$, and $\delta$, as well as the desired post-selected PNR measurements
of $n_{i}$ for $N$ photon catalysis steps.

##  Appendix D Optimized parameters

The cascaded photon catalysis protocol for $N=2,3,$ and $4$ steps was
numerically optimized to yield the high-fidelity SSV(2) states discussed in
the main text. The full list of parameters for the protocols and resultant
states are displayed in Table 1.

$N$ |  2 |  3 |  4
---|---|---|---
Fidelity |  $>0.999$ |  0.984 |  0.977
Success probability |  $1.5\times 10^{-2}$ |  $1.8\times 10^{-3}$ |  $4.2\times 10^{-5}$
$\beta$ |  0.90 |  1.35 |  1.59
Squeezing parameter |  -0.22 |  -0.48 |  -0.52
$\alpha$ |  1.20 |  3.54 |  4.66
{$r_{1}$, $r_{2}$, $r_{3}$, $r_{4}$} |  {0.60, 0.85, - -, - -} |  {0.64, 0.49, 0.52, - -} |  {0.58, 0.55, 0.70, 0.42}
{$n_{1}$, $n_{2}$, $n_{3}$, $n_{4}$} |  {1, 2, - -,- -} |  {5, 2, 1, - -} |  {6, 4, 2, 1}
Table 1: Parameters of $N$-step photon-catalysis that optimize the fidelity
with the nearest SSV(2) state. The fidelity was numerically optimized for all
combinations of integer $n_{i}$ below 10, with all other parameters allowed to
vary independently. The combinations with the highest fidelities and success
probabilities are displayed. $\alpha$: input coherent state amplitude.
$\beta$: target SSV state amplitude. $r_{i}$: real-valued reflectiviy of the
$i$th beamsplitter in cascaded photon catalysis where $r_{i}^{2}+t_{i}^{2}=1$.
$n_{i}$: Measurement result of the $i$th PNR detector.

##  Appendix E SSV enlargement by PNR

As described in the main text, the breeding protocol for enlarging a general
$M$th order symmetry SSV state consists of sending two identical $SSV^{(M)}$
states in respective modes $a$ and $b$ to a balanced beamsplitter, and
performing a PNR measurement on the output mode $b$. Before the detection
step, the state is given by Eq. 29 which can be rewritten as

$\displaystyle\left|\,\phi\,\right\rangle$
$\displaystyle\propto\sum_{n=0}^{M-1}U_{ab}D_{a}(\beta_{n})D_{b}(\beta_{n})S_{a}(\xi_{n})S_{b}(\xi_{n})\left|\,0\,\right\rangle_{a}\left|\,0\,\right\rangle_{b}$
(55) $\displaystyle+\sum_{n_{1}\neq
n_{2}}^{M-1}U_{ab}D_{a}(\beta_{n_{1}})D_{b}(\beta_{n_{2}})S_{a}(\xi_{n_{1}})S_{b}(\xi_{n_{2}})\left|\,0\,\right\rangle_{a}\left|\,0\,\right\rangle_{b},$

where we note that $|\beta_{n_{1}}|=|\beta_{n_{2}}|$ and
$|\xi_{n_{1}}|=|\xi_{n_{2}}|$ for all values of $n_{1}$ and $n_{2}$. Using a
balanced beamsplitter defined by
$U_{ab}=e^{\frac{\pi}{4}(ab^{\dagger}-a^{\dagger}b)}$, it can be seen that
identical displacers and squeezers on separate modes are transformed in the
Heisenberg picture according to

$\displaystyle U_{ab}D_{a}(\beta_{n})D_{b}(\beta_{n})U^{\dagger}_{ab}$
$\displaystyle=D_{a}(\sqrt{2}\beta_{n})$ (56) $\displaystyle
U_{ab}S_{a}(\xi_{n})S_{b}(\xi_{n})U^{\dagger}_{ab}$
$\displaystyle=S_{a}(\xi_{n})S_{b}(\xi_{n}),$ (57)

so after detecting zero photons in mode $b$, the first sum in Eq. 55 becomes

$\frac{1}{\sqrt{\cosh{|\xi|}}}\sum_{n=0}^{M-1}D_{a}(\sqrt{2}\beta_{n})S_{a}(\xi_{n})\left|\,0\,\right\rangle_{a}.$
(58)

If we can show that the terms from the second sum in Eq. 55 are negligible,
then normalizing Eq. 58 will give us the desired enlarged $SSV^{(M)}$ state.
The beamsplitter acting on the two unequal displacers gives

$\displaystyle U_{ab}D_{a}(\beta_{n_{1}})D_{b}(\beta_{n_{2}})U^{\dagger}_{ab}$
$\displaystyle=D_{a}(\frac{\beta_{+}}{\sqrt{2}})D_{b}(\frac{\beta_{-}}{\sqrt{2}})$
(59)

where $\beta_{+}=\beta_{n_{1}}+\beta_{n_{2}}$ and
$\beta_{-}=\beta_{n_{1}}-\beta_{n_{2}}$. Applying Eq. (13) and (14) from Ref.
Kim2002  and writing the squeezing arguments as $\xi_{n1}=-re^{2i\varphi}$ and
$\xi_{n2}=-re^{2i\theta}$, we see that the two unequal squeezers on vacuum are
transformed by

$\displaystyle
U_{ab}S_{a}(-re^{2i\varphi})S_{b}(-re^{2i\theta})\left|\,0\,\right\rangle_{a}\left|\,0\,\right\rangle_{b}=R_{a}(\varphi)R_{b}(\theta)S_{a}\Big{(}-\frac{r}{2}(1+e^{2i(\varphi-\theta)})\Big{)}$
(60) $\displaystyle\times
S_{b}\Big{(}-\frac{r}{2}(1+e^{-2i(\varphi-\theta)})\Big{)}S_{ab}\Big{(}-2r\sinh{(\varphi-\theta)}\Big{)}\left|\,0\,\right\rangle_{a}\left|\,0\,\right\rangle_{b},$

where $R(\Theta)=e^{i\Theta a^{\dagger}a}$ is the phase-shift operator.
Detecting zero photon in mode $b$ leads to each term in the second sum of Eq.
55 taking the form:

$\displaystyle
D_{a}(\beta_{+})S_{a}(\varsigma)\,{}_{b}\\!\left\langle\,0\,\right|D_{b}(\beta_{-})S_{b}(\varsigma^{*})S_{ab}(\Delta)\left|\,0\,\right\rangle_{b}\left|\,0\,\right\rangle_{a},$
(61)

where $\varsigma=-\frac{r}{2}(1+e^{2i(\varphi-\theta)})$ and
$\triangle=-2r\sinh{(\varphi-\theta)}.$ Now, realizing that
$\beta_{-}=\frac{|\beta|}{\sqrt{2}}(e^{\frac{2\pi in_{1}}{M}}-e^{\frac{2\pi
in_{2}}{M}})$ is always nonzero for $n_{1}\neq n_{2}$ when $n_{1},n_{2}<M$, we
see that each of the terms having the form of Eq. 61 consists of the overlap
of vacuum with two and single-mode squeezed vacuum that has been displaced.
For finite squeezing, it is clear that this overlap term can always be made
arbitrarily small for a large enough displacement amplitude $|\beta_{-}|$. If
this is the case, then the entire second sum in Eq. 55 vanishes, and detecting
zero photons in mode $b$ reduces 55 to 58.

The only question that remains is how large of an initial displacement
amplitude, $|\beta|$, is considered large enough, as it can be seen that
increasing $M$ leads to more terms in the sum that we would wish to neglect.
In practice, this can be determined numerically for a desired fidelity
threshold with the target enlarged $SSV^{(M)}$, and we see from Fig.9 in the
main text that $|\beta|\geq\sqrt{2}$ for $M=2$ and $3$, and
$|\beta|\geq\sqrt{5}$ for $M=4$ is sufficient for high-fidelity breeding.

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