# The Roles of Kerr Nonlinearity in a Bosonic Quantum Neural Network

Huawen Xu<EMAIL_ADDRESS>Division of Physics and Applied Physics,
School of Physical and Mathematical Sciences, Nanyang Technological
University, 21 Nanyang Link, Singapore 637371, Singapore  Tanjung Krisnanda
Division of Physics and Applied Physics, School of Physical and Mathematical
Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371,
Singapore  Ruiqi Bao  Division of Physics and Applied Physics, School of
Physical and Mathematical Sciences, Nanyang Technological University, 21
Nanyang Link, Singapore 637371, Singapore  Timothy C. H. Liew
<EMAIL_ADDRESS>Division of Physics and Applied Physics, School of
Physical and Mathematical Sciences, Nanyang Technological University, 21
Nanyang Link, Singapore 637371, Singapore  MajuLab, International Joint
Research Unit UMI 3654, CNRS, Université Côte d’Azur, Sorbonne Université,
National University of Singapore, Nanyang Technological University, Singapore

###### Abstract

The emerging technology of quantum neural networks (QNNs) attracts great
attention from both the fields of machine learning and quantum physics with
the capability to gain quantum advantage from an artificial neural network
(ANN) system. Comparing to the classical counterparts, QNNs have been proven
to be able to speed up the information processing, enhance the prediction or
classification efficiency as well as offer versatile and experimentally
friendly platforms. It is well established that Kerr nonlinearity is an
indispensable element in a classical ANN, while, in a QNN, the roles of Kerr
nonlinearity are not yet fully understood. In this work, we consider a bosonic
QNN and investigate both classical (simulating an XOR gate) and quantum
(generating Schrödinger cat states) tasks to demonstrate that the Kerr
nonlinearity not only enables non-trivial tasks but also makes the system more
robust to errors.

_Introduction._ –Biologically inspired artificial neural networks (ANN) have
shown great accomplishments in processing information with the ability to
break through the von Neumann bottleneck (referring to the delay between
processor and memory) Jelena  _et al._ (2015); Chouard and Venema (2015);
LeCun  _et al._ (2015); Butler  _et al._ (2018). Various types of ANN
architectures have been proposed in the field of machine learning, for
example, feedforward neural networks Bebis and Georgiopoulos (1994) and
recurrent neural networks LeCun  _et al._ (2015); Herbert and Harald (2004);
Maass  _et al._ (2002); Enel  _et al._ (2016); Lukosevicius and Jaeger (2009)
that are shown to be powerful in speech/pattern/fingerprint classification
Deng  _et al._ (2013); Chan  _et al._ (2016); Ciresan  _et al._ (2011); Leung
_et al._ (1991), financial forecasting Kaastra and Boyd (1996) and nonlinear
series prediction Zhang  _et al._ (2001). We normally aim for more complexity
from an ANN to obtain richer dynamics. This in turn allows for a better
performance for the tasks mentioned above. For example, this can be achieved
by considering more layers/nodes Xu (2017), more completed architectures Liu
_et al._ (2016); Zhang  _et al._ (2019) and stronger nonlinearity Andrew L.
Maas (2013), all contribute to performance.

Based on these requirements of an ANN, many physical systems have been
proposed to be hardware implementation platforms for ANNs, for example:
memristors Krzysteczko  _et al._ (2012), spintronics Locatelli  _et al._
(2014); Quang Diep  _et al._ (2014), microcavity exciton-polaritons Xu  _et
al._ (2020); Opala  _et al._ (2019); Ballarini  _et al._ (2020); Matuszewski
_et al._ (2021), etc. Among these platforms, a nonlinear activation function
is an indispensable part for even basic machine learning tasks. The role of
the activation function is to determine the output of a specific node given a
set of corresponding inputs. This means that if we allow only a linear
activation function in an ANN, the outputs will simply be a linear
transformation of the inputs (no matter how many network layers one
implements). Such an ANN can be represented just by a matrix multiplication
corresponding to a simple input-to-output process. Nonlinearity is required
from a physical system for hardware implementation for more general
transformations.

Recently, quantum neural networks (QNNs) Biamonte  _et al._ (2017); Dunjko and
Briegel (2018); Altaisky  _et al._ (2016) emerged as promising platforms
combining the characteristics of ANN and quantum physics, which aim to gain
advantages of quantum mechanics, for example, having more degrees of freedom
from the large Hilbert space McClean  _et al._ (2018) and quantum correlations
between quantum modes Shen  _et al._ (2020) in performing either classical or
quantum tasks. In this direction, QNNs have been shown to offer speedup in
solving classical tasks Szegedy (2004); F.  _et al._ (2014); Dunjko  _et al._
(2016); Paparo  _et al._ (2014) and improvement in learning efficiency Xu  _et
al._ (2021); Nakajima  _et al._ (2019); Fujii and Nakajima (2017); Neigovzen
_et al._ (2009), compared to their classical counterparts. On the other hand,
quantum tasks such as entanglement recognition Ghosh  _et al._ (2019a), phase
estimation Krisnanda  _et al._ (2021a), quantum state preparation Krisnanda
_et al._ (2021b); Ghosh  _et al._ (2019b) and quantum state tomography Ghosh
_et al._ (2021) have also been proposed with different kinds of QNNs.

Intuitively, one expects that nonlinearity is also important in a QNN similar
to the classical ANN, which requires further demonstration. In this work, we
take a bosonic QNN as an example with different architectures and simple
nearest neighbour hopping between the quantum network nodes. The architectures
shall be kept simple as the aim is to demonstrate how the Kerr nonlinearity
plays a role in the QNN for both classical and quantum tasks. We first choose
the classical task as simulating an XOR gate, where having nonlinearity in the
input-to-output process is necessary. One should note that here both the input
signal and output signal are classical. We show that Kerr nonlinearity indeed
allows for the XOR gate to function. It is worth noting that this is done
without nonlinear elements from anywhere else. The latter can emerge simply
from considering different quantities for the inputs and outputs, for example
amplitude and intensity, which in turn clouds the roles of Kerr nonlinearity.
Moreover, by considering a more realistic situation where the measurement
error is unavoidable in practice, we found that the Kerr nonlinearity has an
error correction effect for classical tasks. In particular, we consider errors
on the measured outputs, and found that stronger Kerr nonlinearity leads to
reduction of errors for the XOR gate. Next, we extend this investigation to
the quantum regime where we consider a quantum operation generating a
corresponding Schrödinger cat state given an arbitrary coherent state. Here
the outputs are quantum states. Since a Schrödinger cat is usually
characterized by its Wigner function, we choose the cost function to minimize
the difference between the Wigner function of the obtained states and target
Schrödinger cat states. We show that Kerr nonlinearity plays an essential role
that allows this quantum operation and makes the QNN noise/error resistant.

_The model._ The considered quantum neural network (QNN) consists of $n$
bosonic modes with random nearest neighbour coupling. The Hamiltonian can be
expressed as:

$\displaystyle\mathcal{\hat{H}}_{0}=\sum_{i=1}^{n}\left(E\hat{a}_{i}^{\dagger}\hat{a}_{i}+\alpha\hat{a}_{i}^{\dagger}\hat{a}_{i}^{\dagger}\hat{a}_{i}\hat{a}_{i}\right)+\sum_{ij}J_{ij}(\hat{a}_{i}\hat{a}_{j}^{\dagger}+\hat{a}_{i}^{\dagger}\hat{a}_{j}),$
(1)

where we consider an onsite energy $E$ and strength of Kerr nonlinearity
$\alpha$ the same for each bosonic mode. Also, $J_{ij}$ represents the nearest
neighbour coupling strength between modes $i$ and $j$. In our scheme, the QNN
is evolved following the quantum master equation:

$\displaystyle
i\hbar{\dot{\rho}}=[\mathcal{\hat{H}}_{0},\rho]+\sum_{i=1}^{n}\frac{i\gamma}{2}\mathscr{L}(\rho,\hat{a}_{i}),$
(2)

where $\mathcal{\hat{H}}_{0}$ is the Hamiltonian in Eq. 1 and the Lindblad
operator $\mathscr{L}$ is defined as $\mathscr{L}(\rho,\hat{x})\equiv
2\hat{x}\rho\hat{x}^{\dagger}-\hat{x}^{\dagger}\hat{x}\rho-\rho\hat{x}^{\dagger}\hat{x}$.
The second term on the right-hand side of Eq. 2 determines the decaying
process of the modes in the QNN with a rate $\gamma/\hbar$. We obtain the
density matrix $\rho(\tau)$ and occupation number $\left\langle
n_{i}\right\rangle=\text{Tr}\\{\hat{a}_{i}^{\dagger}\hat{a}_{i}\rho(\tau)\\}$
of each mode at a certain time $\tau$ from the evolution of Eq. 2.

_Classical Task._ Here we consider the classical task of simulating a
classical XOR gate with a four bosonic mode QNN; the input and target output
relation is shown in Fig. 1(b). The first two modes ($\hat{a}_{1,2}$) are the
input modes and the last two modes ($\hat{a}_{3,4}$) are the output modes (see
Fig. 1(a)). The input signals can be introduced as amplitudes of coherent pump
or encoded as the occupation numbers for the input modes. After the evolution
for a time $\tau$, we measure the occupation numbers on the output modes. We
first investigate the case of injecting the input signals via coherent pump,
in which the Hamiltoinian in Eq. 1 now reads:

$\displaystyle\mathcal{\hat{H}}_{0}\rightarrow\mathcal{\hat{H}}_{0}+\sum_{i=1}^{2}(P_{i}\hat{a}_{i}^{\dagger}+P_{i}^{*}\hat{a}_{i}),$
(3)

Figure 1: (a) Schematic of a quantum neural network with nearest neighbour
coupling (brown part), where $\hat{a}_{1}$ and $\hat{a}_{2}$ are the input
modes and $\hat{a}_{3,4}$ are the output modes. The occupation number on
$\hat{a}_{3,4}$ are weighted to obtain the final outputs ($\hat{b}_{1,2}$).
(b) Input and corresponding target outputs for a XOR gate. The results from
the QNN are shown in the "Output" columns. (c) Output error as a function of
the strength of Kerr nonlinearity (in the unit of $\gamma$) for two input
signal encoding methods. $1$ represents the occupation number input encoding
scheme whereas $2$ means the coherent pumping amplitude encoding method.

where the amplitudes of the coherent pump correspond to the binary input in
Fig. 1(b). This form of pumping is consistent with photonic neural network
prototypes based on exciton-polaritons Ballarini  _et al._ (2020). It turns
out that without Kerr nonlinearity
($\alpha\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}$), the QNN can
already simulate an XOR gate, i.e., the trained outputs can match the target
with negligible error, see the last two columns in the table of Fig. 1(b).
This somewhat unexpected result can be explained by noting that the inputs and
outputs in the QNN are nonlinearly related. That is, the inputs are encoded as
pumping amplitudes while the output as occupation numbers. As occupation
number is a nonlinear function of amplitude, the system is capable of
nonlinear input-to-output mapping corresponding to the XOR gate. Similar
results are also demonstrated in Ref. Mujal  _et al._ (2021). Given that the
Kerr nonlinearity is not required, we also verified that an evolution of
classical states treated within a mean-field model of the system can reproduce
the considered XOR gate (see the results in Fig. 2), which also rules out the
importance of quantum correlations in this task. Details of simulating the
classical neural networks can be found in the Supplementary Materials (SM).
Let us also note that if we were to instead consider particle intensities
rather than pumping amplitudes as the inputs, the system would no longer be
capable of simulating a nonlinear map and the XOR gate would not be possible.
While the simple investigation so far suggests that Kerr nonlinearity is not
necessary for the QNNs and ANNs, as one might obtain nonlinear elements in the
system by other means, we will show below that it does offer an essential
element in situations accounting for measurement errors, noise, or more
complicated tasks.

Figure 2: (a) Input output correspondence when the intensity $|\psi_{n}|^{2}$
is measured as the outputs for cases when Kerr nonlinearity $\alpha$ is set to
be non-zero (Output1) or zero (Output2). (b) Input output correspondence when
the wavefunction amplitude $\psi_{n}$ is measured as the outputs for cases
when Kerr nonlinearity $\alpha$ is set to be non-zero (Output1) or zero
(Output2).

Let us consider noise introduced on the measured occupation number of each
mode. Instead of taking $\left\langle n_{i}\right\rangle$ as the output,
$\left\langle n_{i}\right\rangle+\delta_{i}$ is considered, where $\delta_{i}$
is the measurement error on the $i$-th mode. We consider a uniformly
distributed random error $\delta_{i}=[0,0.8]\times\left\langle
n_{i}\right\rangle$. This model is such that the errors are fractions of the
corresponding occupation numbers. The error of the trained outputs is defined
as the difference between the targets and the actual trained outputs. Note
that since in a logic gate, both the inputs and outputs are typically binary
digits, the error must be below $0.5$ to operate successfully. Now, we encode
the input information with occupation numbers in modes $\hat{a}_{1}$ and
$\hat{a}_{2}$, which means instead of starting from vacuum state for all
inputs, we start with coherent states with average occupations $0/1$ in modes
$1$ and $2$ according to the different inputs in Fig. 1(b). In this case, no
pumping scheme is considered in the system (set $P=0$ in Eq. 2). In Fig. 1(c)
(blue curve), we can see the error drops rapidly to insignificant as the
nonlinearity increases. However, the error is non-negligible when zero Kerr
nonlinear strength is considered ($>0.5$).

Now, we revisit the scheme where the inputs are encoded in the amplitudes of
the coherent pump $P_{1,2}$. In this case, we start the system from vacuum
states for all the modes. In Fig. 1(c) (red curve), we can see the error is
already insignificant even when $\alpha$ is set to zero. As mentioned above,
this is because there is nonlinearity already present, i.e., that between the
input and output quantities. One can still see that the error drops with
increasing of the strength of Kerr nonlinearity. Intensity encoded input is
also consistent with neural networks based on non-resonantly pumped polariton
condensates Mirek  _et al._ (2021), which are considered explicitly in the SM.
This concludes that Kerr nonlinearity offers an essential element for QNNs to
simulate nonlinear processes such as the XOR gate, in the presence of
measurement errors.

Figure 3: (a) Schematic of the linear mixing process of the QNN. The photons
emitted from the QNN modes $\hat{a_{i}}$ (red parts) can be recombined to form
new modes $\hat{A}$ (blue parts). Since we are targeting at one output mode,
the other modes are assumed to be vacuum (gray parts). (b) The Wigner function
of the target Schrödinger cat state. (c) The Wigner function of the prepared
Schrödinger cat state with an error $\delta=0.07$. Here we consider a five
bosonic mode QNN with random nearest neighbour coupling. Parameters:
$P=1000\gamma,J=1000\gamma,\tau=0.01/\gamma$, and $\alpha=450\gamma$.

_Schrödinger cat state generating operation._ Here, we consider a quantum task
utilising the QNN, that is, generating a Schrödinger cat state from a given
coherent state $\left|\beta\right\rangle$. The target Schrödinger cat state is
defined as:

$\displaystyle\left|Cat_{\beta,k}\right\rangle=\frac{1}{N_{\beta,k}}[\left|\beta\right\rangle+\left|-\beta\right\rangle],$
(4)

where the coherent state is expressed as
$\left|\beta\right\rangle=e^{\beta\hat{a}^{\dagger}-\beta^{*}\hat{a}}\left|0\right\rangle$,
the coefficient $N_{\beta,k}=\sqrt{2[1+(-1)^{k}e^{-2\beta^{2}}]}$ and
$\left|0\right\rangle$ denotes the vacuum state.

In this task, we consider five bosonic modes in the QNN and start the system
from a coherent state ($\left|\beta\right\rangle$) for mode $1$ and vacuum
states ($\left|0\right\rangle$) for other modes, see the schematic in Fig.
2(a). Each mode is under a coherent pump and the system evolves following Eq.
2 until time $\tau$. The reservoir modes can emit, e.g., photons, which can be
recombined through linear mixing to form a set of new bosonic modes
($\hat{b}_{i}$) satisfying $[\hat{b}_{i},\hat{b}_{j}^{\dagger}]=\delta_{ij}$.
The linear mixing is energy conserving and can be done with linear optics
elements, i.e., beam splitters and phase shifters Carolan  _et al._ (2015).
The modes after the linear mixing process can be represented by their
annihilation (creation) operator $\hat{c}_{i}$ ($\hat{c}_{i}^{\dagger}$),
where $\hat{c}_{i}=\sum_{j}\textbf{W}_{ij}\hat{b}_{j}$, where $W_{ij}$ is the
weight matrix that is unitary, following the commutation conditions from the
output modes Krisnanda  _et al._ (2021b). For a QNN composed of $N$ modes, one
can construct at most $N$ output modes with the linear mixing process. In what
follows, we will focus on one output mode, while the other $N-1$ are assumed
to be vacuum, which can be realized with conditional measurements. These are
routinely used to generate Fock states in a strongly coupled oscillator-spin
system Cirac  _et al._ (1993), motional Fock states of an atom de Matos Filho
and Vogel (1996), superposition of pure states Parkins  _et al._ (1993); Song
_et al._ (1990); Ogawa  _et al._ (1991), photon added states Dakna  _et al._
(1999), etc. Conditional measurement is also demonstrated to have improvement
on teleportation of continuous variables Opatrný  _et al._ (2000); Cochrane
_et al._ (2002); Olivares  _et al._ (2003). Related experiments based on beam
splitters and two photon-number-resolving detectors Allevi  _et al._ (2010)
and silicon photomultipliers Chesi  _et al._ (2021) have also been realized.
We note that apart from linear optics, the linear mixing transformation can be
obtained with simple tunable hopping interactions between bosonic modes. An
exemplary experimental setup in this case has been demonstrated with two
interacting microwave cavities Gao  _et al._ (2018).

Since the task is generating a corresponding Schrödinger cat state, we define
the Wigner function of the obtained quantum states from the QNN as
$W^{O}(x,p)$ and the target Schrödinger cat state as $W^{T}(x,p)$, the error
can be defined as:

$\displaystyle\delta=\frac{\int(W^{O}(x,p)-W^{T}(x,p))^{2}dxdp}{\int(W^{O}(x,p)+W^{T}(x,p))^{2}dxdp}.$
(5)

In this case, the incoherent loss in the QNN plays a role similar to noise in
the classical task, i.e., a process that reduces the quality of the output. In
our simulation, we first start with different specific coherent states, for
example, with amplitude $\beta$ equals $1.4,1.3,1.2,1.1$ and $1$. The aim here
is to generate corresponding Schrödinger cat states from different initial
states $|\beta\rangle$ by optimizing the connection weights from the QNN to
the new constructed modes. Fig. 2(c) presents a prepared Schrödinger cat state
with an error $\delta=0.07$, while the conditional measurement probability is
$0.028$. One can compare it with Fig. 2(b) to see that it is well matched with
the target state. In Fig. 3, we can also see that with increasing strength of
Kerr nonlinearity ($\alpha$), the error decreases down to $\sim 0.03$ for
$\beta=1$, which proves that the Kerr nonlinearity is helping the QNN
functioning in a noisy environment. We also consider when the strength of Kerr
nonlinearity goes to infinity, in which case the bosonic modes are effective
fermions. Fig. 3 shows that with infinite Kerr nonlinearity, the QNN still
offers low enough error $\delta$ to generate the Schrödinger cat states. The
conditional measurement probability for infinite Kerr nonlinearity (fermions)
to achieve comparable error $\delta$ to the results in Fig. 2(c) is $0.085$.
Additionally, we note that with decreasing values of $\beta$, the error
decreases as well, which indicates that for smaller amplitude $\beta$, the
corresponding Schrödinger cat states can be easier to obtain.

In this process, we keep the nearest neighbour coupling $\\{J_{ij}\\}$ random
and fixed. Moreover, for a general assessment, we generate $10$ random
coherent states with $\beta\in[1,1.4]$ and use the same optimisation scheme as
described before. The average error under different strengths of nonlinearity
is presented in Fig. 3(b), which shows similar trend to that in Fig. 3(a). Our
results show that in general, i.e., for different coherent states as input,
the Kerr nonlinearity allows error resistant production of Schrödinger cat
states. We have also considered other quantum tasks, such as single photon
state generation, and found the same conclusion (see SM).

Figure 4: (a) The error $E$ as a function of different strength of Kerr
nonlinearity $\alpha$. Different color indicates different amplitudes of the
input coherent states $\left|\beta\right\rangle$. (b) Averaged error $E$ as a
function of the strength of Kerr nonlinearity $\alpha$ when the coherent
states are generated randomly ($\beta$ is randomly distributed between $1.0$
and $1.4$). Each data point is averaged by $10$ times.

_Conclusion._ We demonstrated how the Kerr nonlinearity functions in quantum
machine learning with a random nearest neighbour coupling bosonic QNN by
considering both classical and quantum tasks. Starting with a classical task,
we simulate the XOR gate. We show that the nonlinear input-output mapping is
able to perform nonlinear classical tasks, for example, XOR gate, even without
the Kerr nonlinearity. In a practical environment, the measurement error is
unavoidable. We then introduce error on the measured occupation number. The
Kerr nonlinearity is shown to be able to correct the error. When considering
the quantum task, we construct a quantum gate operating on coherent states and
generating corresponding Schrödinger cat states. The incoherent loss is
interpreted as error in this system, the results show that Kerr nonlinearity
offers the capability to resist the error/noise. These results give a clear
direction of how to construct and optimize a QNN for performing different
tasks (both in classical and quantum regime).

_Acknowledgement._ This work was supported by the Singapore Ministry of
Education under its AcRF Tier 2 grant MOE2019-T2-1-004.

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