11institutetext: Department of Computer Science, Aberystwyth University,
Wales, UK
11email<EMAIL_ADDRESS>22institutetext: Department of Computer
Science, Edge Hill University, Lancashire, UK
22email<EMAIL_ADDRESS>
# Orderly Disorder in Point Cloud Domain

Morteza Ghahremani 11 Bernard Tiddeman 11 Yonghuai Liu 22 and Ardhendu Behera
22

###### Abstract

In the real world, out-of-distribution samples, noise and distortions exist in
test data. Existing deep networks developed for point cloud data analysis are
prone to overfitting and a partial change in test data leads to unpredictable
behaviour of the networks. In this paper, we propose a smart yet simple deep
network for analysis of 3D models using ‘orderly disorder’ theory. Orderly
disorder is a way of describing the complex structure of disorders within
complex systems. Our method extracts the deep patterns inside a 3D object via
creating a dynamic link to seek the most stable patterns and at once, throws
away the unstable ones. Patterns are more robust to changes in data
distribution, especially those that appear in the top layers. Features are
extracted via an innovative cloning decomposition technique and then linked to
each other to form stable complex patterns. Our model alleviates the
vanishing-gradient problem, strengthens dynamic link propagation and
substantially reduces the number of parameters. Extensive experiments on
challenging benchmark datasets verify the superiority of our light network on
the segmentation and classification tasks, especially in the presence of noise
wherein our network’s performance drops less than 10% while the state-of-the-
art networks fail to work.

###### Keywords:

Point cloud, deep neural network, orderly disorder, segmentation,
classification

##  1 Introduction

Object classification and semantic segmentation of 3D models are foundations
of numerous computer vision applications like autonomous driving and robot
manipulation. Thus far, a considerable number of convolutional neural networks
(CNNs) have been developed for such tasks [1, 2, 3, 4, 5, 6] and in most cases
they yield promising results, especially when the distributions of test and
train datasets are similar. However, 3D models in the real world contain out-
of-distribution samples, different samplings, noise and distortions that
significantly influence their performance. Fig. 1 shows a few examples of
wrong classification in the presence of noise.

Figure 1: Classification results of most existing networks highly depend on
the distribution of training and test data. A partial change in the
distribution of test data by adding Gaussian noise $\mathcal{N}(0,0.02)$ leads
to misclassification.

Using a large number of parameters for modeling and training results in
overfitting and exponentially growing computational cost. It also renders the
networks to being more data-driven which makes them unable to work under a
small change in the point set. Although several works [7, 8, 9] show less
tendency towards overfitting, their performance is highly dependent on the
samples. Most models are susceptible to irregular samples and work only under
certain conditions. Meanwhile, developing highly accurate, robust and fast
models for the processing of 3D data is demanded by many practical
applications like autonomous driving.

Our studies have shown that patterns of an object represent the structural
information of that object and remain almost unchanged under different
samplings of the data and so are a more stable basis on which to build
classification and segmentation algorithms. In this paper, we propose a novel
cloning technique aiming at extraction of the stable patterns from an object.
Such features can efficiently improve the network’s performance. Our robust
network is the first successful attempt to tackle and to investigate
classification and segmentation under irregular samplings of point cloud data.
Additionally, it needs a relatively low number of parameters so it is fast and
less prone to overfitting. The key contributions of this paper are as follows:

  * •

We design a robust deep neural network whose performance is not significantly
affected by data grid, thus it is invulnerable to noise, out-of-distribution
samples and distortions.

  * •

The proposed model mitigates the problem of distance saturation in KNN-based
models.

  * •

The architecture of the proposed network allows the user to go deeper and
deeper without the problem of vanishing-gradient. This scheme is capable of
analysis of highly complex objects.

  * •

We provide thorough empirical and theoretical analysis on the stability and
efficiency of the proposed method using the 3D benchmark datasets: ModelNet
and ShapeNet [2, 10].

In order to be robust to undesirable factors, our pattern-based network does
not require any annotated samples, and it just trains once on noise-free point
cloud data and then runs over any distorted ones111Supplementary materials are
available at https://github.com/mogvision/pattern-net.

##  2 Related work

Deep learning methods for 3D shape analysis and understanding can be broadly
divided into view-based, volumetric and point cloud-based categories. View-
based techniques [3, 4, 11] map 3D models into 2D view scenes and then employ
image-based CNNs for further analysis. Self-occlusions and information loss
often occur during mapping. Volumetric methods [1, 2, 12, 13, 14] quantize the
input 3D models into a regular grid before applying 3D CNNs. Loss of
resolution, high memory and computational demands are the main limitations of
voxelization.

Instead of converting or mapping 3D models into other domains, they can be
analyzed in the point cloud domain directly. Due to the impressive results of
PointNet [5], most studies have been devoted to learning directly in the point
cloud domain. Since PointNet does not consider the local pattern of a given 3D
point cloud, PointNet++ [15] was proposed, which uses a hierarchical
application of PointNet to multiple subsets of a 3D point cloud. Inspired by
DenseNet [16], DensePoint [17] was introduced that learns a dense contextual
representation for point cloud processing via a deep hierarchy architecture.
Exploitation of other aspects of local structure with PointNet are also
reported in [6, 18]. Superpoint [19] partitions the point cloud into
geometrical homogeneous elements and then a graph convolution network is
applied to such local elements. The main drawback of these methods is their
lack of shape awareness. More precisely, such methods do not explicitly model
the local spatial layout of points. To this end, several works have been
developed [9, 20, 21, 22] that capture the spatial layout of a point cloud by
learning a high-level relation expression among 3D points.

Although the approach of explicitly modeling the relation improves the
segmentation results, isolating high-level relation features from low-level
ones may not strengthen relation propagation, subsequently there is a
vanishing-gradient problem [16]. The appropriate distance or neighbour count
parameter to use in the KNN-based networks [9, 20, 17] is often obtained via
trial-and-error, which is saturated at a certain number of neighbours and even
drastically drops by increasing the number of neighbours or neighbourhood
radius. This aspect of such networks is not favorable, especially in dense 3D
models.

Another main weakness of the existing networks that makes them impractical for
real world data is their vulnerability to noise, distortions and out-of-
distribution sampling schemes. Robustness is a key property that allows
applying the same model to different irregular point clouds. Enriching the
training step with annotated data, like adding noise, distortions etc, is not
a straightforward solution to the above problem. This is because increasing
the volume of input data with annotated samples makes the training convergence
difficult to achieve. Additionally, a large-scale input data renders a high
number of parameters, resulting in overfitting. Hence there is interest in
developing CNNs that work efficiently under disparity between the training and
test data.

##  3 Proposed Pattern-Wise Network

The goal is to establish and learn a deep neural network that converts an
input point cloud $P=\\{p_{m}\in\mathbb{R}^{d},m=1,...,M\\}$ into a set of
segmentation labels $\Gamma_{s}=\\{\gamma_{s}\in\mathbb{R},s=1,...,S\\}$ or a
set of classification labels
$\Gamma_{c}=\\{\gamma_{c}\in\mathbb{R},c=1,...,C\\}$. Here, $M$ is the total
number of 3D points and $d$ is the dimension of the point set that can be
represented as a set of 3D coordinates plus other measured features like
color, normal vectors etc
$p_{m}=\\{coordinate:(x_{m},y_{m},z_{m}),color:(r_{m},g_{m},b_{m}),normal:(N^{x}_{m},N^{y}_{m},N^{z}_{m})\\}$.
In this paper, we just consider the 3D coordinates and extension of the
network over color and normal is straightforward. The output of the network
for segmentation tasks is a vector of labels $\gamma_{s}\in\\{1,...,S\\}$,
where $S$ is the number of segmentation labels. Likewise, for the
classification task, 3D points are labelled as $\gamma_{c}\in\\{1,...,C\\}$
with $C$ classes.

###  3.1 Network Properties

A segmentation/classification network for a point set must meet the following
four requirements [5, 15, 23]:

Property i (Permutation Invariant): It states that segmentation/classification
scores must be invariant to changes in the order of 3D points. Unlike pixels
in images or voxels in volumetric grids, 3D point cloud has no order and due
to its irregular format, the network must be invariant to the order of points.

Property ii (Transformation Invariant): The labels/classifications of points
must not be varied by their changes in rotation, scale and translation.

Property iii (Points Relations): In the point cloud, the relation between
points is determined by their distance from each other. The distance metrics
could be Euclidean distance, Manhattan distance, cosine distance, etc. Points
in the point cloud are not isolated and their neighbours make a meaningful
subset that can be measured by an appropriate metric.

Property iv (Robustness): The segmentation parts or classification labels of
points must not be varied under different samplings. In practice, the
distribution of test data is not close to that of the training data.
Additionally, presence of out-of-distribution samples, noise and distortions
in test data is inevitable so point cloud networks must be robust to the
irregular samples.

The above properties are the backbone of our network.

###  3.2 Network Architecture

(a) Cloning analysis

(b) Entropy of patterns

(c) Categories of patterns

Figure 2: (a) $P$ is the input point set which is constituted of three subsets
shown in blue ($P\\{1\\}$), black ($P\\{2\\}$) and red ($P\\{3\\}$). The
abstract/structural information (solid green lines) remains unchanged across
different samplings. (b) Entropy of patterns with different characteristics.
(c) Categorization of potential patterns inside a rigid/non-rigid object in
point clouds.

The orderly disorder theory was introduced in physics [24, 25] and it refers
to a way of describing the complex structure of disorders within complex
systems. Unpredictable disorders could occur just under external disturbances
not because of internal reasons. Ordered/predictable disorders may not be seen
by human vision and this increases the ambiguity between the predictable and
unpredictable disorders. However, the entropy metric could give us the degree
of chaos inside a complex structure. Chaos theory has been well studied in
mathematics, behavioral science, management, sociology etc. With the success
of CNNs in solving high-order problems, our aim is to deeply analyze the links
between points in the given point cloud.

For a rigid/non-rigid 3D object in the given point cloud, the location of
points may change under different sampling operations, external disturbances
etc. but its abstract information does not vary. Such abstract information can
be named as stationary information and they are predictable while non-
stationary information refers to those features that do not obey a regular
pattern and under different conditions show different behaviours. The best way
is to encode a 3D object by its stationary patterns which always show
predictable behaviour. Contrary to stationary patterns, the behaviour of non-
stationary patterns is completely unpredictable. For example, the minor
details of a 3D model can vary under various samplings but its skeleton
remains unchanged [Fig. 2(a)]. Thus, we can claim that the classification
score of a 3D object must not be varied under changes in the density and
distribution of points if the number of points is sufficiently large, i.e.

$\Gamma_{[p_{1},...,p_{N}]}=\Gamma_{[p_{1},...,p_{M}]}\qquad if\qquad N<M\ \&\
N\gg 1.$ (1)

In other words, if our observation of an object is sufficiently detailed, then
few sample variations must not change the label of the enquiry object. Let’s
assume $M=LN$ for a sufficiently large $N$, where $L$ is a positive integer.
According to Eq. 1, we can say that

$\Gamma_{[p_{1},...,p_{N}]}=\Gamma_{[p_{(l-1)N+1},...,p_{lN}]},\quad\forall
l\in\\{2,...,L\\}.$ (2)

One possible solution to the above equation is to decompose the input point
cloud $P$ into $L$ levels via a random down-sampling operator in such a way
that all $L$ point subsets $P{\\{l\\}},l\in\\{1,...,L\\}$, are completely
different while their overall schemes/abstracts are similar to each other.
Under these conditions, Eq. 2 is asserted. If we apply a random down-sampling
operator to point cloud $P$ that provides

$\displaystyle P{\\{l\\}}\cap
P\\{j\\}=\emptyset\quad\forall\,l,j\in\\{1,...,L\\}\ \&\ l\neq j,$ (3)
$\displaystyle\bigcup^{L}_{l=1}P{\\{l\\}}=P,$ (4) $\displaystyle
H(P{\\{l\\}})\simeq H(P{\\{j\\}})\quad\quad\forall\,l,j\in\\{1,...,L\\}\ \&\
l\neq j,$ (5)

then we can assert that all the $L$ point subsets have similar stationary
structures/patterns. In Eq. 5, ‘$H$’ denotes the entropy of each subset and
this equation assures that all the subsets have approximately similar
information content. If $R$ denotes the entropy of redundant/predictable
patterns in a point cloud set and similarly, the entropy of random/less-
redundant/unpredictable ones is denoted by $\overline{R}$, then the entropy
$H$ of each subset is equal to

$\displaystyle
H\big{(}P{\\{l\\}}\big{)}=R\big{(}P{\\{l\\}}\big{)}+\overline{R}\big{(}P{\\{l\\}}\big{)},\quad\quad\forall\,l\in\\{1,...,L\\}.$
(6)

(a) $P$

(b) $P{\\{1\\}}\cap P\\{2\\}=\emptyset$ and $P{\\{1\\}}\cup P\\{2\\}=P$

Figure 3: An example of how the proposed cloning decomposition can
successfully remedy chaos in the given point cloud. For better visualisation,
axis ‘z’ was set to 0. (a) Conventional KNN-based networks yield asymmetric,
complex, and dissimilar patterns for two adjacent points; (b) The cloning
decomposition technique increases the probability of extracting symmetric,
simple, and similar patterns.

The entropy of redundant patterns is close to zero and the large part of the
overall entropy ‘$H$’ is assigned to unpredictable patterns $\overline{R}$.
Possible values of entropy for patterns with different characteristics is
depicted in Fig. 2(b). It is worth noting that the randomness of the down-
sampling operator ensures that each point subset includes all parts/organs of
an object. We name this strategy as ‘cloning decomposition’ and an example is
illustrated in Fig. 2(a), where all the point subsets of the given 3D object
share approximately similar structures across multiple decomposed levels while
none of them shares identical 3D samples. Overall, the feature space of an
object could be categorized into a number of undesirable non-stationary
patterns in which the patterns are chaotic and a series of stationary
patterns, wherein complex and simple patterns are learnt from orderly stable
relations [Fig. 2(c)].

The simplified point subsets can markedly help the network to extract
stationary patterns, enhancing robustness as stated by Property iv above. An
illustrative example is depicted in Fig. 3. This example aims at drawing the
KNN responses (here, $K=7$) of two adjacent points. According to Fig. 3(a),
conventional KNN-based networks rely heavily on the density of 3D points and
also they may not find reliable neighbours as the radius of the neighbourhood
for an enquiry point depends on the density and distribution of the point
cloud data. The proposed technique decomposes the given point cloud in Fig.
3(a) into two subsets via Eqs. 3-5. As shown in Fig. 3(b), it could provide
similar patterns for a complex point cloud. The radius of the neighbourhood
for both the blue and the red points is almost equal. As will be discussed
later, this strategy efficiently helps the network not to be saturated with
its K nearest neighbours while keeping the radius of the neighbourhood
reasonable.

The proposed network is depicted in Fig. 4. The framework has four main layers
including cloning decomposition (CD), searching relations (SR), learning
relations (LR) and linkage patterns (LP) layers. In the following, we detail
the functionality of each layer.

  * •

Cloning Decomposition (CD) Layer: Image acquisition is often made under
various conditions that directly affect quality and quantity of 3D models and
subsequently, their point cloud samples. This layer decomposes the input
object points into multiple subsets via Eqs. 3-5. Since high-level patterns
are directly deduced from the low-level ones, this layer plays a key role in
the extraction of reliable patterns. Cloning analysis reduces the number of
parameters, alleviates the vanishing-gradient problem and improves the
convergence pace.

Figure 4: Proposed network architecture for classification of point cloud
data.

  * •

Searching Relations (SR) Layer: The task of this layer is to search all
possible links between the feature vectors (or equivalently potential low-
level patterns) according to Property iii above. This is done via the KNN
algorithm in the Hilbert space. Instead of using the Euclidean space
dominantly employed by most existing deep networks, we use a Hilbert kernel.
Given two low-/high-level feature vectors $X$ and $Y$, a measure/relation of
similarity between them in the Euclidean space can be expressed as

$\displaystyle\mathcal{E}(X,Y)=||X||^{2}_{2}+||Y||^{2}_{2}-2<X,Y>,$ (7)

while the Hilbert kernel yields

$\displaystyle\mathcal{H}(X,Y)=2\big{(}1-<\frac{X}{||X||_{2}},\frac{Y}{||Y||_{2}}>\big{)}.$
(8)

In the above equations, the angle brackets denote an inner product operator.
The Hilbert kernel emphasizes the cross-similarity while the self-similarity
remains the same, i.e. unit. Unlike the Euclidean space that is biased by
self-similarity, the Hilbert space just considers the cross-similarity between
enquiry feature vectors and thus, it is expected that it can find reliable
relations between feature vectors.

  * •

Learning Relations (LR) Layer: This layer seeks and learns potential relations
between all input feature vectors via a convolution kernel followed by a batch
normalization operator. A max pooling operator is then applied to the outputs
to obtain the global feature of the input features. The max pooling is a
symmetric function that guarantees that the extracted features are
permutation-invariant, as stated by Property i above. The combination of the
convolution kernel, batch normalization and max pooling operators is often
called multi-layer perceptron (MLP) [5].

  * •

Linkage Patterns (LP) Layer: This layer comprises several MLP operators and
its aim is to aggregate the relations across all the subsets and extract the
most stable patterns from them. All the subsets have different samples and the
LP layer is applied to the patterns for extracting the common ones. Such
patterns carry stationary information of the object and they are robust to
irregular samples and density.

Now, we detail the proposed pattern-based CNN (hereafter it is abbreviated as
Pattern-Net) for the classification and segmentation of given point clouds.

###  3.3 Classification and Segmentation Networks

In the classifier model depicted in Fig. 4, the input point cloud data is
first decomposed into ‘$L$’ subsets via the cloning technique described in the
previous section. Inside each subset, relations between each query point and
its neighbours is sought by the KNN algorithm. This is done by applying four
MLPs $\\{64,64,64,64\\}$ to each cloning subset. Similar to KNN-based networks
[26, 20, 9], we compute K nearest neighbour responses of edges emanating from
the enquiry feature point and stack them with the enquiry feature point. The
KNN algorithm is considerably affected by the density of points. If the 3D
model is sparse, then the best K responses will lie within a large volume
neighbourhood while such responses in a high density model may fall into a
small radius [Fig. 3(a)]. Moreover, adjacent points in high density 3D samples
would share similar KNN responses and this makes the network data-dependent.
In other words, the network may fail to work for test point cloud data with
different samplings from the training data. This problem can be seen in almost
all KNN-based networks which here is solved by the cloning layer. In the
proposed model, the LP layer finds the most repeated relations across
different subsets and then labels them as stable patterns. Even in the
presence of changes in point coordinates, the patterns remain approximately
unchanged as we consider overall behaviour of a group of points rather than
the exact behaviour of each point.

Each cloning subset yields a description vector of length 256 and they are
called cloning description vectors $\psi_{l},l\in\\{1,...,L\\}$. All the $L$
cloning description vectors are arranged in a matrix $\Psi$. An MLP is applied
to all cube features over all the subsets to yield a global description vector
$\phi$, shown in light green in the figure. Here, the goal is to make each
cloning description vector $\psi_{l}$ similar to the global one $\phi$ as much
as possible. If we consider a linear relationship between the cloning and
global description vectors, i.e. $\phi=\Psi\omega$, then the estimated
coefficients $\omega$ can be computed by the Moore–Penrose inverse, i.e.
$\omega=\Psi^{\dagger}\phi=(\Psi^{T}\Psi)^{-1}\Psi^{T}\phi$. Parameter
$\omega$ determines the contribution of each cloning vector in the resultant
global vector. The deviation of $\omega$ elements should approach zero if all
the cloning description vectors are completely similar to the global one. We
add this term into the loss function:

$L(\theta)=\underbrace{-\frac{1}{n}\sum_{i=1}^{n}\sum_{c=1}^{C}y_{ic}\log{p_{ic}}}_{classification\>loss}+\lambda\underbrace{\sigma(\Psi^{\dagger}\phi)}_{linear\>mapping\>loss}$
(9)

Table 1: Classification accuracy in percentage (%) on ModelNet40 (‘-’:unknown) Method | Input | Avg. classes | Overall
---|---|---|---
PointNet [5] | 1k-xyz | 86.0 | 89.2
PointNet++ [15] | 5k-xyz | - | 91.9
PointCNN[23] | 1k-xyz | 88.1 | 92.2
ECC[27] | 1k-xyz | 83.2 | 87.4
DGCNN[20] | 2k-xyz | 90.7 | 93.5
SO-Net[26] | 2k-xyz | 88.7 | 90.9
DensePoint[17] | 1k-xyz | - | 93.2
RS-CNN[9] | 1k-xyz | - | 93.6
| 1k-xyz | 90.3 | 92.9
Pattern-Net | 2k-xyz | 90.7 | 93.6
| 4k-xyz | 90.8 | 93.9

In the above equation, the first term is the categorical cross-entropy
function for computing the loss of the predicted labels and the second term
enforces the network to yield zero standard deviation for weights obtained by
the linear mapping between the cloning and global description vectors.
$\lambda$ is a predetermined constant, whose value is determined by the
smoothing label’s value in the one-hot encoded $y_{ic}$. Finally, $p_{ic}$ is
a scaled (softmax) logits. The Moore–Penrose pseudo-inverse can be simply
implemented by singular value decomposition (SVD) [28]. The ultimate cloning
vector is obtained by applying a max-pooling operator to the cloning vectors.
The resulting vector is aggregated with the global vector to yield the
description vector for the given point set which is of length 512. Finally,
the description vector is fed into three MLPs $\\{256,256,C\\}$, configured
for classification. The resultant description vector is also used in
segmentation. For this task, four MLPs {64,64,64,64} are applied to the input
data to extract low- and high-level features. They are then concatenated with
the description vector to encode each point. Similar to the classification
task, three MLPs $\\{256,256,S\\}$ are employed for segmentation. The drop-
rate of all decoding MLPs except the last one is fixed at $0.5$.

##  4 Experimental Results

Figure 5: (a) Influence of parameter K and number of points on the
classification accuracy. (b) Linear mapping loss for different cloning levels
(#cloning levels$\times$#points).

We have evaluated Pattern-Net on the ModelNet40 dataset [2] for the
classification task. It contains 12311 meshed CAD models from 40 categories.
Similar to the other work, 9843 models were used for training and the rest for
testing and the models were normalized to a unit sphere. Each model is
uniformly sampled from the mesh faces in 1k, 2k and 4k samples. During the
training step, the points are augmented by randomly rotating, scaling and
translating for being transformation invariant (Property ii above). The
quantitative comparisons with the state-of-the-art point-based methods are
presented in Table 1. Our method for 1k and 2k points is on par with the other
methods and gives the best result for 4k points.

Table 2: Classification accuracy in percentage (%) on ModelNet40 in the presence of noise [$2k-xyz+\mathcal{N}(0,\sigma)$] Method |  | $\mathcal{N}(0,0.02)$ | $\mathcal{N}(0,0.05)$ | $\mathcal{N}(0,0.08)$ | $\mathcal{N}(0,0.1)$ | $\mathcal{N}(0,0.15)$
---|---|---|---|---|---|---
PointCNN | 78.7 | 40.8 | 18.6 | 10.5 | 4.7
DGCNN |  | 92.9 | 69.1 | 29.9 | 11.4 | 4.2
SO-Net |  | 70.6 | 35.4 | 11.9 | 9.8 | 5.8
Pattern-Net(4x512) |  | 93.5 | 92.4 | 89.1 | 84.2 | 32.6

Unlike the existing methods, where the performance is quickly saturated by a
specific number of points, our network is saturated slowly as can be seen in
Fig. 5. For a fixed number of decomposed levels, an increase in the number of
points improves the classification accuracy. There is also a direct link
between the accuracy and parameter K in KNN. Tuning parameter K in the KNN-
based networks is not straightforward and it is often obtained by trial-and-
error. According to the figure, further increasing K yields better results but
the performance increase slows down after $K=30$. In order to make a balance
between computational time and performance, parameter K was set to 30
throughout this study.

Figure 6: Classification results on ModelNet40 with added Gaussian noise
$\mathcal{N}(0,0.05)$. First 10 shapes shown are for each query, with the
first line for our Pattern-Net and the second line for DGCNN. The
misclassified objects are highlighted in red.

A suitable value for the number of cloning levels relies on the number of
input points, parameter K and the convergence of Eq. 9. As shown in Fig. 5,
the rate of convergence is reduced by increasing cloning numbers and the
network requires more epochs to find similar patterns between different
cloning levels. However, lower mapping loss does not guarantee high accuracy
as the loss equation is constituted of two terms including the mapping loss
and the classification loss. Our experiment shows that {3,4,5} cloning levels
often yield good results. In Table 1, this parameter was set to 4.

As mentioned previously, the ability to tolerate noise is a necessity for
robust and practical deep learning methods. In the following experiment, we
added zero-mean white Gaussian noise with different standard deviation values
$\mathcal{N}(0,\sigma)$ to the test samples. The classification accuracy
results are reported in Table LABEL:table::noise and a part of results is
depicted in Fig. 6. The table shows that our networks can tolerate noise up to
$\mathcal{N}(0,0.1)$ while the state-of-the-art methods failed to work. The
drop is less than 10% for $\mathcal{N}(0,0.1)$, which is impressive.

Table 3: Segmentation results (%) on ShapeNet Category (#) |  | PointNet | PointNet++ | PointCNN | DGCNN | SO-Net | RS-CNN | Pattern-Net
---|---|---|---|---|---|---|---|---
Areo (2690) |  | 83.4 | 82.4 | 82.4 | 84.0 | 82.8 | 83.5 | 84.3
Bag (76) |  | 78.7 | 79.0 | 80.1 | 83.4 | 77.8 | 84.8 | 81.0
Cap (55) |  | 82.5 | 87.7 | 85.5 | 86.7 | 88.0 | 88.8 | 87.4
Car (898) |  | 74.9 | 77.3 | 79.5 | 77.8 | 77.3 | 79.6 | 80.1
Chair (3758) |  | 89.6 | 90.8 | 90.8 | 90.6 | 90.6 | 91.2 | 91.4
Ear (69) |  | 73.0 | 71.8 | 73.2 | 74.7 | 73.5 | 81.1 | 79.7
Guitar (787) |  | 91.5 | 91.0 | 91.3 | 91.2 | 90.7 | 91.6 | 91.4
Knife (392) |  | 85.9 | 85.9 | 86.0 | 87.5 | 83.9 | 88.4 | 88.1
Lamp (1547) |  | 80.8 | 83.7 | 85.0 | 82.8 | 82.8 | 86.0 | 86.3
Laptop (451) |  | 95.3 | 95.3 | 95.7 | 95.7 | 94.8 | 96.0 | 95.8
Motor (202) |  | 65.2 | 71.6 | 73.2 | 66.3 | 69.1 | 73.7 | 72.1
Mug (184) |  | 93.0 | 94.1 | 94.8 | 94.9 | 94.2 | 94.1 | 94.1
Pistol (283) |  | 81.2 | 81.3 | 83.3 | 81.1 | 80.9 | 83.4 | 82.2
Rocket (66) |  | 57.9 | 58.7 | 51.0 | 63.5 | 53.1 | 60.5 | 62.4
Skate (152) |  | 72.8 | 76.4 | 75.0 | 74.5 | 72.9 | 77.7 | 72.4
Table (5271) |  | 80.6 | 82.6 | 81.8 | 82.6 | 83.0 | 83.6 | 83.9
Avg. |  | 83.7 | 85.1 | 85.1 | 85.2 | 84.9 | 86.2 | 86.4
Figure 7: Segmentation results on ShapeNet. First shape of each category is
selected, where the left shape stands for the ground truth and the right one
for our Pattern-Net.

Segmentation of given point clouds: Segmentation of point cloud data is one of
the popular 3D tasks. We carried out this experiment on the ShapeNet benchmark
[10] and followed the data split in [5]. ShapeNet contains 16881 models of 16
categories and they are labelled in 50 parts in total. Like [10], the
Intersection-over-Union (IoU) of a shape is computed by averaging the IoUs of
different parts in that shape, and the IoU of each category is obtained by
averaging the IoUs of all the shapes belonging to that category. The results
are summarized in Table 3. When the training set is small, the performance of
our network is on par with the other methods. Our technique extracts common
deep features between shapes so it needs a sufficiently large number of
training samples. From the table, it can be seen that Pattern-Net outperforms
the existing methods on relatively large categories like airplane, car, chair
and lamp. A sample result of each category is illustrated in Fig. 7.

Table 4: Complexity of different methods for point cloud classification task Method | PointNet++ | PointCNN | DGCNN | SO-Net | RS-CNN | DensePoint | Pattern-Net
---|---|---|---|---|---|---|---
#params. | 1.48M | 8.2M | 11.8M | 11.5M | 1.41M | 670k | 399k

Complexity analysis: Table LABEL:table::complexity reports the complexity of
Pattern-Net as well as those of the existing techniques for the classification
task. The number of input points was set to 1024. Thanks to the cloning
decomposition, our technique needs less than 0.4M parameters which is much
lower than 670k of DensePoint. This characteristic of our method is more
appealing for real-time applications like mobile robotics and autonomous
driving.

##  5 Conclusions

In this study, we have proposed a novel technique for efficiently learning
stationary patterns in the given point clouds, which is less susceptible to
noise, distortions and overfitting. It is also invariant to changes in
translation, rotation and scale. The key idea is to decompose the point cloud
into multiple subsets with similar structural information. Then we enforce the
network to learn stable patterns. Compared with noise that is order-less with
unpredictable behaviour, natural objects have complex structures accompanied
with irregularities in some parts due to the external disturbances.
Informative patterns could be successfully extracted if the level of
randomness and uncertainty is diminished. Unpredictable disorders cause
inaccurate representation of given objects and this concept is known as
‘orderly disorder’ theory. To this end, we have proposed the cloning
decomposing technique. Since our network learns just stable patterns, it is
less prone to overfitting, which means it needs to train only once and can
then run over a variety of data (e.g. different noise and sampling patterns).
This method could provide a promising direction for robust representation of
point cloud data.

## Acknowledgements

The authors gratefully acknowledge the HPC resources provided by
Supercomputing Wales (SCW) and Aberystwyth University. M. Ghahremani
acknowledges his AberDoc and President scholarships awarded by Aberystwyth
University. Y. Liu and A. Behera are partially supported by BBSRC grant
BB/R02118X/1 and UKIERI-DST grant CHARM (DST UKIERI-2018-19-10) respectively.

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# Protocol for nonlinear state discrimination in rotating condensate

Michael R. Geller Department of Physics and Astronomy, University of Georgia,
Athens, Georgia 30602, USA

(April 24, 2024)

###### Abstract

Abstract

Nonlinear mean field dynamics enables quantum information processing
operations that are impossible in linear one-particle quantum mechanics. In
this approach, a register of bosonic qubits (such as neutral atoms or
polaritons) is initialized into a symmetric product state
$|\psi\rangle^{\\!\otimes n}$ through condensation, then subsequently
controlled by varying the qubit-qubit interaction. We propose an experimental
implementation of quantum state discrimination, an important subroutine in
quantum computation, with a toroidal Bose-Einstein condensate. The condensed
bosons here are atoms, each in the same superposition of angular momenta 0 and
$\hbar$, encoding a qubit. A nice feature of the protocol is that only readout
of individual quantized circulation states (not superpositions) is required.

##  I Introduction

A variety of atomtronic architectures have been proposed for quantum computing
and quantum technology applications Amico  _et al._ (2021, 2022). Two main
Bose-Einstein condensate (BEC) types have been considered for realizing
qubits: multi-component condensates and multi-mode condensates. Multi-
component and spinor condensate approaches Cirac  _et al._ (1998); Tian and
Zoller (2003); Byrnes  _et al._ (2012, 2015); Luiz  _et al._ (2015); Großardt
encode a single qubit in two (or more) metastable atomic states, such as spin
or hyperfine levels, with all atoms in the same translational mode (for
example the motional ground state). Multi-mode approaches Solenov and Mozyrsky
(2011); Amico  _et al._ (2014); Aghamalyan  _et al._ (2016) encode a single
qubit using two (or more) translational modes in a scalar condensate, such as
a BEC in a double-well trapping potential. In the limit where there are a
large number of condensed bosons in each well, the system becomes equivalent
to two (or more) BECs, each with a well defined phase, connected by tunneling
barriers that act as Josephson junctions Solenov and Mozyrsky (2011); Amico
_et al._ (2014); Aghamalyan  _et al._ (2016). Arrays of such BECs can be
produced in optical lattices and are described by the Bose-Hubbard model Amico
_et al._ (2014); Aghamalyan  _et al._ (2016). Another multi-mode approach,
which we adopt here, uses circulating states in a ring geometry Ryu  _et al._
(2020); Kapale and Dowling (2005); Ryu  _et al._ (2007); Ramanathan  _et al._
(2011); Eckel  _et al._ (2014); Kim  _et al._ (2018); Pezzè  _et al._ for the
translational modes.

Given the demonstrated high performance and scalability of trapped ion qubits
Moses  _et al._ (2023), superconducting qubits Krinner  _et al._ (2022);
Google Quantum AI (2023); McKay  _et al._ , and of neutral atom arrays
Bluvstein  _et al._ (2022); Graham  _et al._ (2022), what does a BEC qubit
offer? We argue that it offers a platform for an alternative approach to
quantum information processing that leverages the special properties of
condensates. In this approach, a BEC is used to prepare a register of qubits
in a product state $|\psi\rangle^{\\!\otimes n}$ and control its subsequent
evolution. From a quantum computing perspective, having multiple identical
copies of an unknown input is already a useful resource (whereas classical
information is freely cloned). In standard circuit-model quantum computation,
illustrated in Fig. 1a, initialized qubits are subsequently entangled using
two-qubit gates. Here we do the opposite and try to suppress entanglement,
Fig. 1b. This is achieved by making $n$ large, interactions weak, and by
preserving permutation symmetry. In this limit entanglement monogamy Coffman
_et al._ (2000); Zong  _et al._ bounds the pairwise concurrence to zero, and
the BEC is exactly described by a nonlinear Schrödinger equation (the
Gross–Pitaevskii equation Gross (1961); Pitaevskii (1961)), enabling novel
dynamics Abrams and Lloyd (1998); Wu and Niu (2000); Riedel  _et al._ (2010);
Meyer and Wong (2013); Amico  _et al._ (2014); Childs and Young (2016);
Aghamalyan  _et al._ (2016); Ryu  _et al._ (2020); Xu  _et al._ (2022);
Deffner (2022); Geller (2023a). The theory is developed in a large $n$ limit
with a rigorous bound on the error resulting from the mean field
approximation. The nonlinear approach trades exponential time complexity for
space complexity, requiring $n$ to be large.

Figure 1: Nonlinear quantum information processing with a BEC. (a) In circuit-
model quantum computation, a register of qubits is initialized to a product
state (such as $|0\rangle^{\\!\otimes n}$), after which gates are applied,
entangling the qubits. (b) In the nonlinear approach, the qubits ideally
remain in a product state $|\psi(t)\rangle^{\\!\otimes n}$ throughout the
computation. The BEC simulates a single nonlinear qubit.

Does this mean that $n$ has to be exponentially large? Actually the
requirements on $n$ are not that bad. This is because the BEC is assumed to be
initialized in a product state, and it takes time $t_{\rm ent}$ for the atomic
collisions to produce entanglement. Ideally, the whole experiment is performed
in a short-time regime. We measure the accuracy of mean field theory by
$\epsilon:=\|\rho_{\rm eff}(t)-\rho_{1}(t)\|_{1}$, and call this the model
error. Here $\rho_{\rm eff}$ is the mean field state, $\rho_{1}$ is the exact
state traced over all atoms but one, $t$ is the gate duration, and
$\|\cdot\|_{1}$ is the trace norm. In a large family of condensate models
Erdős and Schlein (2009); Geller (2023b)

$\displaystyle\epsilon\leq c\frac{e^{t/t_{\rm ent}}-1}{n},$ (1)

where $c$ and $t_{\rm ent}$ are positive constants (model-dependent quantities
independent of $t$ and $n$). Although the error might grow exponentially in
time, there is always a short-time window $t<t_{\rm ent}$ where the required
number of condensed atoms $n\approx ct/t_{\rm ent}\epsilon$ is sub-exponential
in $t$.

We propose a demonstration of quantum information processing using this
nonlinearity. In the remainder of this section we discuss the qubit encoding
and state discrimination subroutine. The protocol is explained in Sec. II.
Conclusions are given in Sec. III, and additional information about the BEC
model and large $n$ limit are provided in the Appendix.

###  I.1 Qubit encoding

BEC-based qubits necessarily encode a small number of parameters
($\psi_{0,1}\in{\mathbb{C}}$) into a large number of degrees of freedom and
the map is not unique. However two encodings can often be considered: Let
$a_{l}^{\dagger}$ create an atom in BEC component $l\in\\{0,1\\}$ (in a two-
component condensate) or in translational mode $l\in\\{0,1\\}$ (in a two-mode
condensate), and let $\psi_{0,1}$ be complex coordinates satisfying
$|\psi_{0}|^{2}+|\psi_{1}|^{2}=1$. One encoding that is interesting from a
quantum foundations perspective is

$\displaystyle|{\rm
CAT}_{n}\rangle:=\frac{\psi_{0}(a_{0}^{\dagger})^{n}+\psi_{1}(a_{1}^{\dagger})^{n}}{\sqrt{n!}}|{\rm
vac}\rangle,\ \ \langle{\rm CAT}_{n}|{\rm CAT}_{n}\rangle=1,\ \ n\geq 1,$ (2)

but this is a superposition of two macroscopically distinct BECs (a
Schrödinger cat state) which would be highly susceptible to decoherence Cirac
_et al._ (1998). Instead we use the encoding

$\displaystyle|F_{n}\rangle:=\frac{(\psi_{0}\,a_{0}^{\dagger}+\psi_{1}\,a_{1}^{\dagger})^{n}}{\sqrt{n!}}|{\rm
vac}\rangle,\ \ \langle F_{n}|F_{n}\rangle=1,\ \ n\geq 1,$ (3)

which is a condensate of $n$ bosons
$\psi_{0}\,a_{0}^{\dagger}+\psi_{1}\,a_{1}^{\dagger}$, each a single atom in a
superposition of components or modes. (While $|{\rm CAT}_{n}\rangle$ and
$|{F}_{n}\rangle$ depend on both $n$ and $\psi_{0,1}$, the latter dependence
is suppressed.) The encoding (3) was originally proposed by Cirac et al. Cirac
_et al._ (1998) and by Byrnes et al. Byrnes  _et al._ (2012, 2015) for two-
component condensates; in that case $|F_{n}\rangle$ is a pseudospin coherent
state Byrnes  _et al._ (2015). But our $a^{\dagger}_{0}$ and $a^{\dagger}_{1}$
create atoms in circulating states of orbital angular momentum $0$ and
$\hbar$, respectively, and it is better to regard $|F_{n}\rangle$ as a
coherent state of atoms in angular momenta superpositions. The states (3) are
mean field states since the atoms are not entangled. They satisfy

$\displaystyle a_{l}|F_{n}\rangle=\psi_{l}\sqrt{n}|F_{n-1}\rangle\ \ \ {\rm
and}\ \ \
a_{l}a_{l^{\prime}}|F_{n}\rangle=\psi_{l}\psi_{l^{\prime}}\sqrt{n(n-1)}|F_{n-2}\rangle.$
(4)

Because each atom in (3) carries a copy of the qubit state
$|\psi\rangle=\psi_{0}|0\rangle+\psi_{1}|1\rangle$, the state $|F_{n}\rangle$
exhibits a bosonic orthogonality catastrophe Guenther  _et al._ (2021) in the
large $n$ limit, meaning that close qubit states $|\psi\rangle$ and
$|\psi^{\prime}\rangle$ encode to orthogonal $|F_{n}\rangle$ and
$|F^{\prime}_{n}\rangle$ as $n\rightarrow\infty$ (the semiclassical limit in
the spin coherent state picture Byrnes  _et al._ (2015)). Furthermore, due to
the polynomial encoding in $|F_{n}\rangle$, the single-particle superposition
principle with respect to $\psi_{0,1}$ is violated (see below).

In the atomtronic implementation we assume a toroidal BEC operated in a regime
supporting metastable quantized circulation states with $l$ trapped vortices

$\displaystyle|\Phi_{l}^{n}\rangle=\frac{(a_{l}^{\dagger})^{n}}{\sqrt{n}}|\rm
vac\rangle,$ (5)

where $a_{l}^{\dagger}$ creates an atom in the ring with angular momentum
$l\in{\mathbb{Z}}$. These states are stabilized by the repulsive atomic
interactions Mueller (2002); Baharian and Baym (2013). An atom with mass $m$
and $l=1$ has velocity $\hbar/mR$ and circles the ring with angular velocity
$\Omega_{0}=\hbar/mR^{2}$. We construct a low-energy effective description for
the BEC within the manifold of states (3). This is possible because they are
selected out by the path integral in the large $n$ limit, due to their
diverging contribution to the action. The action in the subspace spanned by
these states is

$\displaystyle S_{\rm eff}[{\bar{\psi}}_{l},\psi_{l}]=\int\\!dt\,\langle
F_{n}|i\partial_{t}-H_{\rm rot}|F_{n}\rangle,$ (6)

where $H_{\rm rot}$ is the BEC Hamiltonian in the rotating frame. The BEC is
rotated with frequency $\Omega\approx\Omega_{0}/2$ to bring the 0-vortex (no
circulation) state $|\Phi_{0}^{n}\rangle$ and the 1-vortex state
$|\Phi_{1}^{n}\rangle$ close in energy. Higher energy $l$ are then neglected,
leading to a two-mode model. In the large $n$ limit (see Appendix) the saddle
point equations are

$\displaystyle\frac{d}{dt}\\!\begin{pmatrix}\psi_{0}\\\
\psi_{1}\end{pmatrix}\\!=\\!-iH_{\rm eff}\\!\begin{pmatrix}\psi_{0}\\\
\psi_{1}\end{pmatrix}\\!,\ \ H_{\rm
eff}=V_{01}\sigma^{x}+B_{z}\sigma^{z}+g(|\psi_{0}|^{2}\\!-\\!|\psi_{1}|^{2})\,\sigma^{z}\\!.$
(7)

The first two terms in $H_{\rm eff}$ generate rigid $x$ and $z$ rotations of
the Bloch sphere. Rotations about $x$ couple $l=0$ and $l=1$ angular momenta
and are produced by breaking rotational symmetry. Here $V_{01}$ is a matrix
element for an applied potential energy barrier. The parameter $B_{z}$ is
controlled by the frequency $\Omega$ of the BEC rotation discussed above. The
nonlinear term describes a $z$ rotation with a rate that increases with
increasing Bloch sphere coordinate ${\rm
tr}(\rho\sigma^{z})=|\psi_{0}|^{2}\\!-\\!|\psi_{1}|^{2}$, vanishes on the
equator, and reverses direction for ${\rm tr}(\rho\sigma^{z})<0$. This
$z$-axis torsion Mielnik (1980) (1-axis twisting Kitagawa and Ueda (1993)) of
the Bloch sphere is the key to fast state discrimination, but is prohibited in
ordinary single-particle quantum mechanics. Although the qubit here is
informational and not associated with any physical 2-state system, we can
define a logical basis $\\{|0\rangle,|1\rangle\\}$ and treat it like any other
qubit:

$\displaystyle|\psi\rangle=\psi_{0}|0\rangle+\psi_{1}|1\rangle=\begin{pmatrix}\psi_{0}\\\
\psi_{1}\end{pmatrix}\\!,\ \ |0\rangle:=\begin{pmatrix}1\\\
0\end{pmatrix}=|\Phi_{0}^{n}\rangle,\ \ |1\rangle:=\begin{pmatrix}0\\\
1\end{pmatrix}=|\Phi_{1}^{n}\rangle.$ (8)

It should be emphasized that (3) is the physical state of the quantum gas, not
(8). However the basis states $|0\rangle,|1\rangle$ are the quantized
circulation states $|\Phi_{0,1}^{n}\rangle$, which is important for the
readout step.

###  I.2 Single-input state discrimination

As an application, we consider the problem of quantum state discrimination
Barnet and Croke ; Bae and Hwang (2013); Bae and Kwek (2015); Rouhbakhsh and
Ghoreishi , a basic task in quantum information science. In the two-state
variant considered here, a quantum state
$|\psi\rangle\in\\{|a\rangle,|b\rangle\\}$ is input to a processor, which
knows the values of $|a\rangle$ and $|b\rangle$ ahead of time and tries
determine which was provided (with a bounded failure probability). This is
easy if $|a\rangle$ and $|b\rangle$ are orthogonal: For a qubit, a single
unitary $U_{\rm read}=|0\rangle\langle a|+|1\rangle\langle b|$ rotates
$\alpha|a\rangle+\beta|b\rangle$ to $\alpha|0\rangle+\beta|1\rangle$, which is
then measured in the standard basis. The challenging case is when $|a\rangle$
and $|b\rangle$ are similar, $|\langle a|b\rangle|^{2}=1-2^{-k}\ {\rm with}\
k\gg 1$, where $n>2^{k}$ identical copies of the input are required Helstrom
(1976). In minimum-error discrimination, the subroutine selects $|a\rangle$ or
$|b\rangle$, each with some probability of error, and the objective is to
minimize the average error. In unambiguous state discrimination, the
subroutine identifies $|a\rangle$ or $|b\rangle$ perfectly, but has the
possibility of abstaining, returning an inconclusive result. State
discrimination can be used to solve NP-complete (and harder) problems Abrams
and Lloyd (1998); Aaronson ; Childs and Young (2016), at the expense of
$2^{k}$ input copies and exponential runtime. This cost reflects the limited
information gained from measurement.

Abrams and Lloyd Abrams and Lloyd (1998) showed that certain nonlinearity in
the Schrödinger equation would bypass this exponential cost, allowing NP-
complete problems to be solved efficiency (in an idealized setting with no
errors or decoherence). But the presence of such nonlinearity would constitute
a fundamental modification of quantum mechanics that is not supported by
experiment Bollinger  _et al._ (1989); Chupp and Hoare (1990); Walsworth  _et
al._ (1990); Majumder  _et al._ (1990). In a condensate, the nonlinearity is
not fundamental, but effective. Although we can realize nonlinear gates, this
doesn’t constitute a complexity violation, due to the large $n$ requirement of
mean field theory.

##  II Protocol

Figure 2: State discrimination channel. Here $\langle a|b\rangle\neq 0$ but
$\langle 0|1\rangle=0$, so the channel must be nonlinear. Note that the output
is always a basis state, $|0\rangle$ or $|1\rangle$, simplifying readout.

The process is illustrated in Fig. 2. A single state
$|\psi\rangle\in\\{|a\rangle,|b\rangle\\}$ is input to the discriminator,
which ideally returns output $|0\rangle$ if $|\psi\rangle\\!=\\!|a\rangle$, or
returns $|1\rangle$ if $|\psi\rangle\\!=\\!|b\rangle$. The single-input
discriminator regarded as a channel must be nonunitary, because the overlap
$\langle a|b\rangle$ is not preserved. Equivalently, the distance
$\|\rho_{a}-\rho_{b}\|_{1}$ between their density matrices in trace norm is
not preserved in time (here $\|X\|_{1}:={\rm tr}\sqrt{X^{\dagger}X}$). For
pure states, $\|\rho_{a}-\rho_{b}\|_{1}=2|\sin(\theta_{ab}/2)|$, where
$\theta_{ab}$ is the angle between their Block vectors. Linear completely-
positive trace preserving (CPTP) channels satisfy
$\frac{d}{dt}\|\rho_{a}-\rho_{b}\|_{1}\leq 0$; they are either distance
preserving or strictly contractive on the inputs Ruskai (1994). Because the
discriminator orthogonalizes the potential inputs, it is expansive on those
inputs: $\frac{d}{dt}\|\rho_{a}-\rho_{b}\|_{1}>0$. Thus, the discriminator is
described by a nonlinear PTP channel Abrams and Lloyd (1998); Mielnik (1980);
Geller (2023a).

The implementation proposed here does not discriminate an unknown input
(produced by a previous computation), but instead uses a black box state
preparation step to randomly prepare $|a\rangle$ or $|b\rangle$, with a small
Bloch vector angle $\theta_{ab}\geq 0$ between them. Then $|\langle
a|b\rangle|^{2}=\cos^{2}(\theta_{ab}/2)\approx 1-(\theta_{ab}/2)^{2}$. This
can be accomplished by initializing in the $|\Phi^{n}_{0}\rangle$ state and
using $V_{01}$ and $B_{z}$ in (7) to apply $x$ and $z$ rotations. (Ideally,
this step is hidden from the remainder of the experiment.) The discrimination
gate itself follows Refs. Abrams and Lloyd (1998); Childs and Young (2016) and
uses the $z$-axis torsion to increase the angle between $|a\rangle$ and
$|b\rangle$. It’s clear that $|a\rangle$ and $|b\rangle$ should begin with
equal and opposite $z$ components $z_{a}=-z_{b}$ [here $r^{\mu}_{a,b}={\rm
tr}(\rho_{a,b}\sigma^{\mu}),\ \mu\in\\{1,2,3\\}$]. Consider a simple option
with $y_{a,b}=0$, namely

$\displaystyle|a\rangle$ $\displaystyle=$
$\displaystyle\cos\bigg{(}\frac{\pi-\theta_{ab}}{4}\bigg{)}|0\rangle+\sin\bigg{(}\frac{\pi-\theta_{ab}}{4}\bigg{)}|1\rangle,$
(9) $\displaystyle|b\rangle$ $\displaystyle=$
$\displaystyle\cos\bigg{(}\frac{\pi+\theta_{ab}}{4}\bigg{)}|0\rangle+\sin\bigg{(}\frac{\pi+\theta_{ab}}{4}\bigg{)}|1\rangle,$
(10)

which has

$\displaystyle
x_{a}=x_{b}=\bigg{|}\\!\cos\bigg{(}\frac{\theta_{ab}}{2}\bigg{)}\bigg{|},\ \
y_{a}=y_{b}=0,\ \ z_{a}=\sin\bigg{(}\frac{\theta_{ab}}{2}\bigg{)},\ \
z_{b}=-\sin\bigg{(}\frac{\theta_{ab}}{2}\bigg{)}.$ (11)

After switching on $g$, the two input options evolve as $R_{z}(\pm
gt\theta_{ab})$ and orthogonalize after a time $t\approx\pi/g\theta_{ab}$.
However this implementation does not have a favorable scaling with
$\theta_{ab}$. The optimal protocol for nonlinear discrimination was derived
by Childs and Young (CY) in Childs and Young (2016). Instead of (11), the CY
gate begins with

$\displaystyle
x_{a}=x_{b}=\bigg{|}\\!\cos\bigg{(}\frac{\theta_{ab}}{2}\bigg{)}\bigg{|},\ \
y_{a}=z_{a}=\frac{\sin(\frac{\theta_{ab}}{2})}{\sqrt{2}},\ \
y_{b}=z_{b}=-\frac{\sin(\frac{\theta_{ab}}{2})}{\sqrt{2}},$ (12)

and applies $x$ rotations to hold $y_{a,b}=z_{a,b}$ during the subsequent
evolution in order to reach antipodal points on the Bloch sphere. The options
orthogonalize in a time $t=O(\log\frac{1}{\theta_{ab}})$, after which a
readout gate $U_{\rm read}$ (defined with respect to time-evolved
$|a\rangle,|b\rangle$) transforms them to circulation states
$|\Phi_{0}^{n}\rangle$ or $|\Phi_{1}^{n}\rangle$, which are then measured via
time-of-flight Amico  _et al._ (2005); Moulder  _et al._ (2012).

In an idealized context where (7) is regarded as exact, and where there are no
control errors, readout errors, decoherence errors, or noise, the nonlinear
discriminator works perfectly every time. We refer to this idealization as a
single-input discriminator to distinguish it from the more familiar minimum
error and unambiguous discriminators based on linear CPTP channels Barnet and
Croke ; Bae and Hwang (2013); Bae and Kwek (2015); Rouhbakhsh and Ghoreishi .
Of course any actual atomtronic realization is likely to suffer from all such
errors, and may fail to give the correct answer or return an answer at all.
Although the theoretically achievable performance depends sensitively on the
system and device details, and is beyond the scope of this work, we note that
combining torsion with non-CP dissipation is predicted to implement an
autonomous discriminator Geller (2023a), whose control sequence and operation
is (mostly) independent of $|a\rangle$ and $|b\rangle$. In this
implementation, the nonlinearity and dissipation create two basins of
attraction with a shared boundary in the Bloch ball, one with an attracting
fixed point near $|0\rangle$ and the other with an attracting fixed point near
$|1\rangle$, giving the discriminator a degree of intrinsic fault-tolerance.

Figure 3: Linear versus nonlinear qubit evolution. (a) Unitary evolution.
Vectors show the isometric flow of states on the Bloch sphere generated by
linear Hamiltonian $H=\sigma^{z}$. (b) Torsion dynamics generated by nonlinear
Hamiltonian $H=\langle\psi|\sigma^{z}|\psi\rangle\,\sigma^{z}$.

The presence of channel nonlinearity indicates a breakdown of the
superposition principle. Figure 3 illustrates a nice example of this effect:
In Fig. 3a, the evolution of a superposition
$\psi_{0}|0\rangle+\psi_{1}|1\rangle$ is given by a superposition of evolved
basis states $e^{-it}|0\rangle$ and $e^{it}|1\rangle$, shown as a velocity
field. However in Fig. 3b, the evolved states $e^{-it}|0\rangle$ and
$e^{-it}|1\rangle$ are now static (phase factors are a global phase), whereas
the actual dynamics is not, except on the equatorial plane.

##  III Conclusions

We have discussed an approach to quantum information processing that leverages
the special properties of condensates, including their nonlinearity, and
proposed an atomtronic implementation of a “nonlinear” qubit. An experimental
demonstration of nonlinear state discrimination, while striking, would not by
itself constitute a computation, because the qubit isn’t coupled to anything.
To implement a useful computation, the BEC qubit must be entangled with other
qubits (for example trapped ions) in a scalable circuit-model quantum
computer, which is not addressed here.

The standard models of quantum computation assume gates and errors based on
linear CPTP channels. Physical hardware, however, might admit initial
correlation and be better described by more general maps Dominy  _et al._
(2016); Dominy and Lidar (2016). It is therefore interesting to investigate
any additional computational power enabled by quantum channels beyond the
linear CPTP paradigm, as we did here. Another example was investigated by Chen
et al. Chen  _et al._ , who experimentally demonstrated unambiguous state
discrimination in a linear but non-Hermitian optical system. After completing
this work, Großardt posted a preprint Großardt  proposing the use of a two-
component BEC coupled to a neutral atom computer to simulate a large family of
nonlinear Schrödinger equations. Given their potential for fast quantum state
discrimination and simulation of the nonlinear Schrödinger equation, the non-
Hermitian and nonlinear approaches to quantum information processing deserve
further exploration.

###### Acknowledgements.

This work was partly supported by the NSF under grant no. DGE-2152159.

##  Appendix A BEC model

Here we derive the qubit equation of motion (7). We consider a toroidal BEC
(thin circular ring with radius $R$) with rotating tunneling barriers that act
as Josephson junctions Amico  _et al._ (2014); Eckel  _et al._ (2014);
Aghamalyan  _et al._ (2016); Ryu  _et al._ (2020). Thin means the dynamics is
quasi-1d in the azimuthal direction, $\theta$. This requires the energy,
temperature, and effective interaction strength to be below an energy scale
$\Delta\epsilon$ determined by the confining potential. The shape of the
potential (without barriers) is mostly arbitrary as long as it is invariant
under rotations about the axis threading the ring, which we call the $z$ axis.
The angular momentum eigenfunctions on the ring are
$\varphi_{l}(\theta)=e^{il\theta}/\sqrt{2\pi R},$ with $l\in{\mathbb{Z}}$ the
angular momentum. In the absence of the tunnel barriers and interaction, these
are stationary states. The condensate consists of $n$ weakly interacting
bosonic atoms of mass $m$, each in their electronic ground state
$|\Psi_{0}\rangle$. At sufficiently low energy and densitiy, the atomic
collisions are elastic, and the Hamiltonian is

$\displaystyle H(t)=\int_{{\rm
Vol}}\\!\\!d^{3}r\,\bigg{\\{}\frac{\hbar^{2}\nabla\phi^{\dagger}\\!\cdot\\!\nabla\phi}{2m}+\frac{U}{2}\,\phi^{\dagger}\phi^{\dagger}\phi\,\phi+V\phi^{\dagger}\phi\bigg{\\}},\
\ [\phi({\bf r}),\phi^{\dagger}({\bf r}^{\prime})]=\delta({\bf r}-{\bf
r}^{\prime}).$ (13)

Here ${\rm Vol}$ is the volume of the ring, $U=4\pi\hbar^{2}a_{\rm s}/m$ is a
short-range interaction strength (proportional to the s-wave scattering length
$a_{\rm s}$), and $V({\bf r},t)$ is a confining potential, including the
rotating barriers. Acting on the vacuum, $\phi^{\dagger}({\bf r})$ creates a
bosonic atom in state $|\Psi_{0}\rangle$ at point ${\bf r}$. We assume a
tunable repulsive interaction with $a_{\rm s}\geq 0$. We also assume zero
temperature, no dissipation, and no disorder.

Two rotating tunnel barriers are used to implement an atomtronic quantum
interference device Aghamalyan  _et al._ (2016); Ryu  _et al._ (2020). When
the barriers are turned on, the Hamiltonian (13) is time dependent. Assuming
the barriers are rigidly rotated about the $z$ axis with frequency $\Omega$,
we have $V({\bf r},t)=e^{-i\Omega tL_{z}/\hbar}\,V({\bf r},0)\,e^{i\Omega
tL_{z}/\hbar}$, where $L_{z}$ is the angular momentum.

However, we can transform to a noninertial reference frame in which the
Hamiltonian, $H_{\rm rot}$, is time independent. Decomposing the time-
evolution operator in the lab frame as

$\displaystyle U_{\rm
lab}=Te^{-\frac{i}{\hbar}\int_{0}^{t}Hdt^{\prime}}=e^{-i\Omega
tL_{z}/\hbar}U_{\rm rot},$ (14)

we obtain

$\displaystyle\frac{dU_{\rm rot}}{dt}=-\frac{i}{\hbar}H_{\rm rot}U_{\rm rot},\
\ H_{\rm rot}=e^{i\Omega tL_{z}/\hbar}(H-\Omega L_{z})e^{-i\Omega
tL_{z}/\hbar}=H(0)-\Omega L_{z}.$ (15)

Next we discuss the two-mode limit: In the low energy, thin ring limit, we can
expand the field operators and angular momentum as

$\displaystyle\phi({\bf r})=\sum_{l}\frac{e^{il\theta}}{\sqrt{{\rm
Vol}}}\,a_{l},\ \ L_{z}=\sum_{l}\hbar la^{\dagger}_{l}a_{l},\ \
[a_{l},a_{l^{\prime}}^{\dagger}]=\delta_{ll^{\prime}},$ (16)

which leads to

$\displaystyle H_{\rm rot}$ $\displaystyle=$
$\displaystyle\frac{\hbar\Omega_{0}}{2}\sum_{l}l^{2}a_{l}^{\dagger}a_{l}+\frac{U}{2\,{\rm
Vol}}\sum_{l_{1},l_{2},l_{3}}a_{l_{1}+l_{3}}^{\dagger}a_{l_{2}-l_{3}}^{\dagger}a_{l_{2}}a_{l_{1}}+\sum_{l_{1},l_{2}}\,V_{l_{1}l_{2}}\,a_{l_{1}}^{\dagger}a_{l_{2}}-\hbar\Omega\sum_{l}la^{\dagger}_{l}a_{l},\
\ \ $ (17)

where

$\displaystyle V_{l_{1}l_{2}}$ $\displaystyle=$
$\displaystyle\oint\\!\frac{d\theta}{2\pi}\,V(\theta,t\\!=\\!0)\,e^{-i(l_{1}-l_{2})\theta}.$
(18)

Nonzero $V_{l_{1}l_{2}}$ induce transitions between angular momentum states.
Then we have

$\displaystyle H_{\rm
rot}=\sum_{l}\hbar\omega_{l}a_{l}^{\dagger}a_{l}+\frac{U}{2\,{\rm
Vol}}\sum_{l_{1},l_{2},l_{3}}a_{l_{1}+l_{3}}^{\dagger}a_{l_{2}-l_{3}}^{\dagger}a_{l_{2}}a_{l_{1}}+\sum_{l_{1},l_{2}}\,V_{l_{1}l_{2}}\,a_{l_{1}}^{\dagger}a_{l_{2}}-\frac{n\hbar\Omega^{2}}{2\Omega_{0}},$
(19)

where

$\displaystyle\omega_{l}=\frac{(\Omega-l\Omega_{0})^{2}}{2\Omega_{0}},\ \
\omega_{0}=\frac{\Omega^{2}}{2\Omega_{0}},\ \
\omega_{1}=\frac{(\Omega-\Omega_{0})^{2}}{2\Omega_{0}}.$ (20)

As explained above, the BEC is rotated with frequency
$\Omega\approx\Omega_{0}/2$ to bring the $l=0$ and $l=1$ states close in
energy. We restrict (19) to angular momenta $l=0,1$ neglecting the others on
the basis of their higher energy. Then

$\displaystyle\sum_{l_{1},l_{2},l_{3}}a_{l_{1}+l_{3}}^{\dagger}a_{l_{2}-l_{3}}^{\dagger}a_{l_{2}}a_{l_{1}}$
$\displaystyle=$
$\displaystyle\sum_{l\in{\mathbb{Z}}}\bigg{\\{}a_{l}^{\dagger}a_{-l}^{\dagger}a_{0}a_{0}+a_{l+1}^{\dagger}a_{-l}^{\dagger}a_{0}a_{1}+a_{l}^{\dagger}a_{1-l}^{\dagger}a_{1}a_{0}+a_{1+l}^{\dagger}a_{1-l}^{\dagger}a_{1}a_{1}\bigg{\\}}\
\ \ \ $ (21) $\displaystyle=$ $\displaystyle
a_{0}^{\dagger}a_{0}^{\dagger}a_{0}a_{0}+a_{1}^{\dagger}a_{1}^{\dagger}a_{1}a_{1}+4a_{0}^{\dagger}a_{1}^{\dagger}a_{1}a_{0}$
(22) $\displaystyle=$
$\displaystyle(a_{0}^{\dagger}a_{0})^{2}-a_{0}^{\dagger}a_{0}+(a_{1}^{\dagger}a_{1})^{2}-a_{1}^{\dagger}a_{1}+4a_{0}^{\dagger}a_{0}a_{1}^{\dagger}a_{1}.$
(23)

This leads to a two-mode model

$\displaystyle H_{\rm
rot}\\!=\\!\sum_{l=0,1}\bigg{(}\\!\hbar\omega_{l}+V_{ll}+\gamma
a_{l}^{\dagger}a_{l}-\gamma\\!\bigg{)}a_{l}^{\dagger}a_{l}+\gamma^{\prime}a_{0}^{\dagger}a_{0}a_{1}^{\dagger}a_{1}+\bigg{(}\\!V_{01}a_{0}^{\dagger}a_{1}\\!+\\!{\bar{V}_{01}}a_{1}^{\dagger}a_{0}\\!\bigg{)}-\frac{n\hbar\Omega^{2}}{2\Omega_{0}},\
\ \ \ \ \ \ $ (24)

where

$\displaystyle\gamma=\frac{U}{2\,{\rm Vol}},\ \
\gamma^{\prime}=4\gamma=\frac{2U}{{\rm Vol}}.$ (25)

In what follows we will treat $\gamma,\gamma^{\prime}\geq 0$ as independent
parameters, allowing (24) to apply to other systems as well. The last term in
(24) subtracts the classical kinetic energy of the spinning ring:
$n\hbar\Omega^{2}\\!/2\Omega_{0}=\frac{1}{2}I_{\rm ring}\Omega^{2},\ I_{\rm
ring}=nmR^{2}\\!.$

Finally we discuss the large $n$ limit: Condensates feature an enhanced two-
particle interaction $\langle a^{\dagger}a^{\dagger}aa\rangle\approx n(n-1)$
caused by the effectively infinite-ranged interaction between condensed atoms.
This makes a naive large $n$ limit unphysical, because the energy per particle
diverges Lieb  _et al._ (2000), and our low-energy assumptions would be
violated. The framework discussed here is instead based on a modified large
$n$ limit where the interaction simultaneously weakens as $1/n$ (a standard
assumption in rigorous studies of mean field theory Lieb  _et al._ (2000);
Benedikter  _et al._ (2016)). This allows for a rigorous study of the large
$n$ limit including bounds on the accuracy of mean field theory. For real
$V_{01}\\!=\\!\oint\\!\frac{d\theta}{2\pi}\,V(\theta)\,e^{i\theta}$, and after
dropping the classical kinetic energy term (which does not affect the qubit
dynamics) we obtain

$\displaystyle H_{\rm
rot}=\sum_{l=0,1}\bigg{(}\\!(\\!\hbar\omega_{l}+V_{ll})a_{l}^{\dagger}a_{l}+\gamma
a_{l}^{\dagger}a_{l}^{\dagger}a_{l}a_{l}-\gamma
a_{l}^{\dagger}a_{l}\\!\bigg{)}+\gamma^{\prime}a_{0}^{\dagger}a_{0}a_{1}^{\dagger}a_{1}+V_{01}\bigg{(}\\!a_{0}^{\dagger}a_{1}\\!+\\!a_{1}^{\dagger}a_{0}\\!\bigg{)}.\
\ \ $ (26)

Evaluating (6) and assuming $n\gg 1$ leads to (setting $\hbar=1$)

$\displaystyle S_{\rm
eff}\\!=\\!n\\!\\!\int\\!\\!dt\bigg{\\{}\\!\sum_{l=0,1}\\!\bigg{(}\\!{\bar{\psi}}_{l}i\partial_{t}\psi_{l}\\!-\\!(\\!\omega_{l}+V_{ll})|\psi_{l}|^{2}\\!-\\!n\gamma\,|\psi_{l}|^{4}\\!\bigg{)}\\!-\\!n\gamma^{\prime}|\psi_{0}\psi_{1}|^{2}\\!-\\!V_{01}({\bar{\psi}}_{0}\psi_{1}\\!+\\!{\bar{\psi}}_{1}\psi_{0})\\!\bigg{\\}}.\
\ \ \ \ \ \ \ $ (27)

Here ${\bar{z}}$ denotes complex conjugation. Due to the $O(n^{2})$
interaction energies in (26), we cannot take the $n\rightarrow\infty$ limit in
(27) without violating our low-energy assumptions. Instead we consider a
modified limit defined by the simultaneous limits
$\gamma,\gamma^{\prime}\rightarrow 0$, $n\rightarrow\infty$, and low energy.
To evaluate (27) in this limit we assume that the interaction strengths
decrease with $n$ as $\gamma=K/n$ and $\gamma^{\prime}=K^{\prime}/n$, where
$K$ and $K^{\prime}$ are now fixed coupling constants. Then

$\displaystyle S_{\rm
eff}\\!=\\!n\\!\\!\int\\!\\!dt\bigg{\\{}\\!\sum_{l=0,1}\\!\bigg{(}\\!{\bar{\psi}}_{l}i\partial_{t}\psi_{l}\\!-\\!(\\!\hbar\omega_{l}+V_{ll})|\psi_{l}|^{2}\\!-\\!K|\psi_{l}|^{4}\bigg{)}\\!-\\!K^{\prime}|\psi_{0}\psi_{1}|^{2}\\!-\\!V_{01}({\bar{\psi}}_{0}\psi_{1}\\!+\\!{\bar{\psi}}_{1}\psi_{0})\\!\bigg{\\}}.\
\ \ \ \ \ \ \ $ (28)

To obtain (28) we have used the results summarized below in Table 1, which
also gives the corresponding results for encoding (2). In the large $n$ limit
the stationary phase approximation leads to (7) with

$\displaystyle H_{\rm eff}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\hbar\omega_{0}+V_{00}+2K|\psi_{0}|^{2}+K^{\prime}\,|\psi_{1}|^{2}&V_{01}\\\
V_{01}&\hbar\omega_{1}+V_{11}+2K|\psi_{1}|^{2}+K^{\prime}\,|\psi_{0}|^{2}\end{pmatrix}$
(29) $\displaystyle=$ $\displaystyle V_{01}\sigma^{x}+B_{z}\sigma^{z}+g\,{\rm
tr}(\rho\sigma^{z})\,\sigma^{z}+{\rm const.},$ (30)

where

$\displaystyle B_{z}:=\frac{\hbar\omega_{0}-\hbar\omega_{1}+V_{00}-V_{11}}{2}\
\ {\rm and}\ \ g:=\frac{2K-K^{\prime}}{2}=-\frac{Un}{2\,{\rm Vol}}.$ (31)

This concludes the derivation of (7).

Table 1: One- and two-particle correlators versus encoding. | $\langle a_{l}^{\dagger}a_{l^{\prime}}\rangle$ | $\langle a_{l}^{\dagger}a_{l}a_{l^{\prime}}^{\dagger}a_{l^{\prime}}\rangle$
---|---|---
$|{\rm CAT}_{n}\rangle$ | $n\,|\psi_{l}|^{2}\delta_{ll^{\prime}}$ | $n(n-1)\,|\psi_{l}|^{2}\delta_{ll^{\prime}}$
$|{F}_{n}\rangle$ | $n\,\psi_{l}^{*}\psi_{l^{\prime}}$ | $n(n-1)\,|\psi_{l}|^{2}|\psi_{l^{\prime}}|^{2}$

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