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11institutetext: Karlsruhe Institute of Technology (KIT), FZI11institutetext:
Karlsruhe Institute of Technology (KIT) 22institutetext: FZI Research Center
for Information Technology

# JointMotion: Joint Self-supervision for
Joint Motion Prediction

Royden Wagner 11 Ömer Şahin Taş 1122 Marvin Klemp 11 Carlos Fernandez 11

###### Abstract

We present JointMotion, a self-supervised learning method for joint motion
prediction in autonomous driving. Our method includes a scene-level objective
connecting motion and environments, and an instance-level objective to refine
learned representations. Our evaluations show that these objectives are
complementary and outperform recent contrastive and autoencoding methods as
pre-training for joint motion prediction. Furthermore, JointMotion adapts to
all common types of environment representations used for motion prediction
(i.e., agent-centric, scene-centric, and pairwise relative), and enables
effective transfer learning between the Waymo Open Motion and the Argoverse 2
Forecasting datasets. Notably, our method improves the joint final
displacement error of Wayformer, Scene Transformer, and HPTR by 3%, 7%, and
11%, respectively.

###### Keywords:

Self-supervised learning representation learning multi-modal pre-training
motion prediction

Figure 1: JointMotion. (a) Connecting motion and environments: Our scene-level
objective learns joint scene representations by matching embeddings of motion
sequences ($Z^{M}$) and environment context ($Z^{E}$). (b) Masked instance
modeling: Our instance-level objective includes a complete set of perception
predictions: agent position ($x$, $y$), dimensions ($l$, $w$, $h$),
acceleration (a), velocity (v), yaw angle ($\theta_{\text{yaw}}$), temporal
order ($n_{\text{t}}$), and class ($c_{\text{a}}$); lane node position (x, y)
and class ($c_{\text{l}}$); traffic light position ($x,y$), temporal order
($n_{\text{t}}$), and class ($c_{\text{tl}}$).

##  1 Introduction

Self-supervised learning (SSL), where supervision signals are generated from
unlabeled data, by design excels in applications with large amounts of
unlabeled data and limited labeled data (e.g., [4, 8, 15]). Recently, SSL
methods combined with supervised fine-tuning outperform plain supervised
learning with the same amount of data [5, 16] and shorter training times [12].
This makes SSL a versatile choice to improve existing methods in a wide range
of applications.

In this work, we focus on improving joint motion prediction for autonomous
driving. Motion prediction aims to predict the future motion of traffic agents
given their past motion and environment context, such as lane networks and
traffic light states. Therefore, motion prediction is essential for estimating
future interaction amongst traffic agents [6] and subsequently assessing the
risk of planned trajectories [23]. The majority of recent motion prediction
methods perform marginal prediction, where future motion is predicted for each
agent individually (e.g., [26, 31, 21]). Joint motion prediction predicts
scene-wide motion modes, i.e. a mode captures a joint set of future motion
sequences for multiple agents within a traffic scene, enabling interaction
modeling.

Since joint motion prediction requires scene-wide representations, we propose
a scene-level SSL objective connecting motion and environment. We complement
this by an instance-level SSL objective that refines these representations,
enhancing overall prediction accuracy. As scene-level objective, we match all
past motion sequences in a scene to the corresponding environment context,
such as map data and traffic light states. Thereby, a model learns which sets
of motion sequences are likely in a given environment, including interaction
amongst agents. As instance-level objective, we reconstruct masked motion
sequences, masked lane polylines, and masked sequences of traffic light
states. Unlike related work [7, 9], our instance-level objective includes a
complete set of object features (agent dimensions, position, orientation,
speed and acceleration profiles, and temporal order) to match to the
environment context, rather than just positions. Hence, our main contributions
are the two jointly optimized SSL objectives that accelerate the training and
improve the precision of motion prediction models:

  1. 1.

Connecting motion and environments via similarity learning, see Fig. 1 (a),

  2. 2.

Masked instance modeling with a complete set of object features, see Fig. 1
(b).

##  2 Related work

SSL for motion prediction in autonomous driving. Inspired by the success of
SSL in computer vision (e.g., [8, 15, 16]) and natural language processing
(e.g., [4, 24]), related methods apply SSL to pre-train motion prediction
models for autonomous driving. PreTraM [33] learns a joint embedding space for
trajectory and map data via contrastive learning. During pre-training, the
similarity of trajectory and map embeddings from the same traffic scene is
maximized, while the similarity to embeddings from other scenes is minimized
(i.e., negative examples). This objective is misleading when similar scenes
are sampled in a mini-batch, e.g., the map embeddings of two four-way stops
should be similar, but are erroneously used as negative examples against each
other. The concurrent works Traj-MAE [7], Forecast-MAE [9], and RMP [34] are
masked autoencoding methods for pre-training motion prediction models. They
mask trajectory and/or lane tokens and learn to reconstruct them. Traj-MAE and
Forecast-MAE represent motion sequences as trajectory polylines, where each
polyline point represents only the agents’ position omitting available
features such as velocity and acceleration profiles, agent dimensions, yaw
angles, and temporal order (see Fig. 1 (b)). Furthermore, they do not include
traffic light states as environment context. RMP limits the scope of pre-
training to motion sequences and omits environment context, such as lane data
and traffic light states, entirely. SSL-Lanes [3] extends lane masking with
the objectives of distance to intersection prediction, maneuver and
success/failure classification. Although these objectives are well adapted to
motion prediction, they require non-trivial heuristics (e.g., for clustering
maneuvers).

Joint motion prediction for autonomous driving. Joint motion predictions aims
to predict the joint distribution of future motion sequences over multiple
agents in a traffic scene. Thus, a predicted motion mode represents a scene-
wide set of motion sequences with one sequence per agent. Scene Transformer
[22] is an encoder-decoder transformer model for joint motion prediction. The
encoder learns global scene-centric representations (i.e., agent, lane, and
traffic light features), the decoder transforms these embeddings into joint
motion modes. Global position embeddings are used for the learned scene-
centric representations, which are more difficult to learn than pairwise
relative position embeddings (cf. [36, 10]). MotionLM [25] reformulates motion
prediction as language prediction task and learns a vocabulary of discrete
motion vectors. Joint prediction modes are generated by jointly decoding
sequences of motion vectors for multiple agents. MotionDiffuser [18] performs
joint motion prediction as conditional denoising diffusion process. A scene-
wide set of noisy trajectories (i.e., positions disturbed by Gaussian noise)
is transformed by a denoiser into a scene-wide set of trajectories that
approximates the ground truth joint prediction mode. The learned denoiser is
conditioned on environment context such as lane data and traffic light states.
At inference, trajectories are sampled from pure noise. Both MotionLM and
MotionDiffuser use an agent-centric encoder [21], which leads to repeated
computation since a traffic scene is encoded for each agent individually. To
achieve a scene-wide context for a shared decoder, these methods concatenate
multiple agent-centric embeddings with positional and rotation information
into a common reference frame. Furthermore, their best performing variants
exhibit high inference latency and are not real-time capable (MotionLM: 250
ms111This latency is linearly interpolated, as inference latency is only
reported up to the second best/largest model (see Table 8 in the Appendix of
[25]), MotionDiffuser: 409 ms).

Marginal motion prediction with auxiliary interactive prediction objectives. A
marginal motion mode represents a single motion sequence for one agent. Many
recent methods extend marginal prediction models with modules for interactive
prediction objectives (i.e., dense [14, 27] or conditional prediction [29,
20]). In such methods, the prediction modules are still trained in a marginal
manner, where training targets are marginal motion modes per agent. The
additional modules are trained to combine marginal predictions by minimizing
overlap between them. Thus, these methods do not learn to model scene-wide
joint motion modes from ground up, but how to combine marginal modes with less
overlap. This limits interaction modeling to the learned re-combination of
marginal motion predictions.

##  3 Method

We present our method for self-supervised pre-training of motion prediction
models in two steps. First, we describe our scene-level SSL objective
(connecting motion and environments), then, we describe our agent-level SSL
objective (masked instance modeling).

###  3.1 Connecting motion and environments

Joint motion prediction requires scene-wide representations to decode joint
modes, which include predictions of multiple agents within a scene. Therefore,
we propose pre-training motion prediction models by connecting motion and
environments. This scene-level objective aims to learn a joint embedding space
for motion sequences and environment context (see Fig. 1 (a)). Thus, a model
implicitly learns which motion is likely in a given environment, including
traffic rules and interaction amongst agents.

In detail, we use the past motion of all agents within a scene to generate a
scene-level motion embedding ($Z^{M}$ in Fig. 1) and combine lane data and
traffic light states to a corresponding environment embedding ($Z^{E}$ in Fig.
1). These embeddings are generated by modality-specific encoders (i.e., for
motion, lanes, and traffic lights) followed by global average pooling and an
MLP-based projector (Proj in Fig. 1). We perform average pooling on the
intermediate embeddings ($H^{M}$ and $H^{E}$ in Fig. 1) to be invariant to
variations in the number of agents and the complexity of environments.
Following [13], we use two separate MLPs with LayerNorm and ReLU activations
as projectors, each having a hidden dimension of 2048 and an output dimension
of 256. We use the modality-specific encoders of recent motion prediction
models (e.g., [22, 36]) without modifications and remove the additional
projector after pre-training. The joint embedding space is learned by
similarity learning via redundancy reduction. Following [35, 30], we reduce
the redundancy of vector elements per embedding (i.e., for $Z^{M}$ and $Z^{E}$
individually) and maximize their similarity by approximating the cross-
correlation matrix of $Z^{M}$ and $Z^{E}$ to the corresponding identity
matrix. The redundancy reduction term ensures that individual embedding
elements capture different features. Therefore, it prevents model collapse
with trivial solutions for all embeddings across all scenes (e.g., zero
vectors). Non-trivial yet identical solutions are not explicitly prevented,
but are unlikely, as empirically shown in [35]. In contrast to [33], our
scene-level objective does not require negative examples, which are difficult
to define in this context. Unlike [30], this objective maximizes the
similarity of embeddings from different modalities (i.e., motion and
environment) rather than augmented views of the same modality, removing the
requirement to develop suitable augmentations.

###  3.2 Masked instance modeling

Scene-level representations are well suited to provide an overview of traffic
scenes, but lack details at instance level. For example, the exact position of
traffic agents in the past or lane curvatures. Therefore, we combine our
scene-level objective with the instance-level objective of masked instance
modeling to refine learned representations.

Inspired by masked sequence modeling [4, 16], we mask elements of past motion
sequences, lane polylines, and past traffic light states and learn to
reconstruct them from non-masked elements and environment context. As shown in
Fig. 1, we represent traffic agents with 10 features rather than just past
positions (cf. [7, 9]). In detail, we reconstruct agent positions ($x$, $y$),
dimensions ($l$, $w$, $h$), acceleration (a), velocity (v), yaw angle
($\theta_{\text{yaw}}$), temporal order ($n_{\text{t}}$), and classes
($c_{\text{A}}$, i.e., vehicles, cyclists, and pedestrians). For lanes, we
reconstruct positions ($x,y$) and lane classes ($c_{\text{L}}$). For traffic
light state sequences, we reconstruct positions ($x$, $y$), temporal order
($n_{\text{t}}$), and state classes ($c_{\text{TL}}$, i.e., green, yellow, and
red). Following [36], positions, dimensions, accelerations, velocities, and
yaw angles are represented as float values and temporal order and agent
classes as boolean one-hot encodings.

For models that use late fusion mechanisms (e.g., [22, 36]), we use the
modality-specific encoders to generate embeddings per modality (see Fig. 2
(a)). Afterwards, we concatenate these embeddings and use a shared local
decoder to reconstruct masked sequence elements from non-masked elements and
context from other modalities. As local decoder, we use transformer blocks
with PreNorm, local attention [2] with an attention window of 32 tokens, 8
attention heads, rotary positional embeddings [28], and feed forward layers
with an input and hidden dimension of 256 and 1024. For models that employ
early fusion mechanisms (e.g., [21]), we use learned queries and a shared
decoder to reconstruct the input sequences (see Fig. 2 (b)). Such models learn
a compressed latent representation for multi-modal input. Therefore, we use
learned queries in the same number as input tokens to decompress these
representations and reconstruct the input sequences. We use a regular cross-
attention mechanism between the learned queries and compressed latent
representations and a local self-attention mechanism within the set of learned
queries. The resulting transformer blocks have the same hyperparameters as in
the late fusion setup. For both variants, we use random attention masks for
masking and a masking ratio 60%. As training target, we minimize the Huber
loss between the reconstructed sequences ($S^{R}$) and the input sequences
($S^{I}$):
$\mathcal{L}_{\text{recon}}=\lambda_{\text{A}}\mathcal{L}_{\text{A}}(S^{R}_{\text{A}},S^{I}_{\text{A}})+\lambda_{\text{L}}\mathcal{L}_{\text{L}}(S^{R}_{\text{L}},S^{I}_{\text{L}})+\lambda_{\text{TL}}\mathcal{L}_{\text{TL}}(S^{R}_{\text{TL}},S^{I}_{\text{TL}})$.
If not specified otherwise, we set
$\lambda_{\text{A}}=\lambda_{\text{L}}=\lambda_{\text{TL}}=1$.

Figure 2: Masked instance modeling with late and early fusion encoders. (a)
Late fusion with modality-specific encoders for agents
(Encoder${}^{\text{A}}$), lanes (Encoder${}^{\text{L}}$), and traffic lights
(Encoder${}^{\text{TL}}$). (b) Early fusion with a shared encoder for all
modalities.

##  4 Experiments

###  4.1 Datasets and evaluation metrics

The Waymo Open Motion Dataset (WOMD) [11] is comprised of over 1.1 million
data points extracted from 103,000 urban or suburban driving scenarios,
spanning 20 seconds each. The state of object-agents includes attributes like
position, dimensions, velocity, acceleration, orientation, angular velocity,
and the status of turn signals. Each data point captures 1 second of past
followed by 8 seconds of future data. We resample this time interval with
10Hz.

The Argoverse 2 Motion Forecasting Dataset (AV2) [32] is comprised of 250,000
urban or suburban driving scenarios each spanning 11 seconds, with 5 seconds
of past and a 6 seconds of future data. The datasets entails interactions in
over 2,000 km of roadways across six geographically diverse cities.

Evaluation metrics. Following [11], we use the mean average precision (mAP),
the average displacement error (minADE), and the final displacement error
(minFDE) to evaluate motion predictions. All metrics are computed using the
minimum mode for $K=6$ modes. Accordingly, the metrics for the mode closest to
the ground truth are measured. We use the official challenge website to
compute metrics on WOMD. For the AV2 dataset, we use the evaluation software
provided by [36]. Therefore, joint modes (best scene-wide mode) are evaluated
on the interactive splits and marginal modes (best mode for each agent
individually) on the regular splits.

###  4.2 Comparison of self-supervised pre-training methods for joint motion
prediction

In this experiment, we compare our method with recent self-supervised pre-
training methods for autonomous driving using the WOMD dataset. Specifically,
we compare with contrastive learning via PreTraM [33] and two masked
autoencoding approaches, Forecast-MAE [9] and Traj-MAE [7].

Experimental setup. We pre-train and fine-tune well-established Scene
Transformer [22] models on the WOMD training split. We use the publicly
available implementation by [36] with 3 modality-specific encoders (i.e., for
agents, lanes, and traffic lights) and a shared decoder for $K=6$ motion
modes. For pre-training, we add an shared local decoder for late fusion models
(see Fig. 2) with 3 transformer blocks and the hyperparamters described in
Sec. 3.2. For our method, we additionally add two projectors for scene-level
representations as described in Sec. 3.1. For PreTraM, we follow its
trajectory-map contrastive learning configuration and add two linear
projection layers as projectors for trajectory and map embeddings. For
Forecast-MAE, we use a masking ratio of 60% and reconstruct positions of lane
polylines and past trajectories by minimizing the MSE loss. We exclude future
trajectories, since self-supervised learning by design does not use the same
labels as the intended downstream task (cf., [1]). For Traj-MAE, we use a
masking ration of 60% and reconstruct positions of lane polylines and past
trajectories by minimizing the corresponding Huber loss. For our method, we
minimize the joint loss of our proposed objectives:
$\mathcal{L}_{\text{JointMotion}}=\lambda_{\text{con}}\mathcal{L}_{\text{con}}+\mathcal{L}_{\text{recon}}$.
We set $\lambda_{\text{con}}=0.01$ and following [35] the weight of the
redundancy reduction term $\lambda_{\text{red}}=0.005$. We fine-tune the model
using its joint configuration and hard loss assignment. Accordingly, the loss
is computed for the best scene-wide joint prediction mode. As post-processing,
we follow [19] and adjust the predicted confidences of redundant predictions.
For all methods, we perform pre-training for 10 hours and fine-tuning for 23.5
hours using a training server with 4 A100 GPUs. For pre-training and fine-
tuning, we use AdamW with an initial learning rate of 1e-4 and a step learning
rate scheduler with a reduction rate of 0.5 and a step size of 25 epochs.

Results. Tab. 1 shows the results of this experiments. Explicit scene-level
objectives (i.e., PreTraM and JointMotion) lead to better and more balanced
performance across all agent types, while implicit scene-wide masked
autoencoding with Forecast-MAE or Traj-MAE tends to focus more on the
pedestrian class than on the others (see mAP scores). Therefore, it is likely
that explict scene-wide objectives enforce learning interactions between
varying agent types more. Overall, our method achieves the best scores across
all agent types. Our method without the objective of connecting motion and
environments (JointMotion w/o CME), achieves worse performance. This shows
that the combination of both objective works best and that the objectives are
complementary. Fig. 3 shows the validation reconstruction losses of both
configurations during pre-training and further highlights the complementary
nature of our two objectives. Specifically, the reconstruction of past traffic
light sequences and past motion are learned similar with both configurations.
However, the lane reconstructions are more accurate without our scene-level
objective. We hypothesize that models pre-trained with our scene-level
objective tend to focus more on the overall lane structure than on specific
details of individual lane polylines.

Table 1: Comparison of self-supervised pre-training methods for joint motion
prediction. All methods are used to pre-train Scene Transformer models [22] on
the Waymo Open Motion dataset and evaluated on the validation split. Agent
types: cyclist (cyc), pedestrian (ped), and vehicle (veh). “w/o CME” denotes
our method without the objective of connecting motion and environments. Best
scores are bold, second best are underlined.

Pre-training | mAP $\uparrow$ | minADE $\downarrow$ | minFDE $\downarrow$
---|---|---|---
| avg | cyc | ped | veh | cyc | ped | veh | cyc | ped | veh
None | 0.1596 | 0.1465 | 0.1821 | 0.1504 | 1.522 | 0.6982 | 1.486 | 3.653 | 1.654 | 3.476
Forecast-MAE [9] | 0.1592 | 0.1420 | 0.1873 | 0.1482 | 1.529 | 0.6940 | 1.423 | 3.664 | 1.652 | 3.343
Traj-MAE [7] | 0.1677 | 0.1492 | 0.1917 | 0.1623 | 1.421 | 0.7084 | 1.338 | 3.320 | 1.647 | 3.090
PreTraM [33] | 0.1689 | 0.1724 | 0.1775 | 0.1569 | 1.488 | 0.7108 | 1.461 | 3.489 | 1.657 | 3.374
JointMotion w/o CME | 0.1784 | 0.1652 | 0.1903 | 0.1796 | 1.457 | 0.6996 | 1.317 | 3.363 | 1.630 | 3.033
JointMotion | 0.1940 | 0.1970 | 0.1964 | 0.1886 | 1.343 | 0.6767 | 1.288 | 3.095 | 1.583 | 2.941

$0$$20$$40$$60$$80$$100$$120$$140$$160$$180$$200$$220$$240$$260$$280$$300$$320$$340$$360$$2$$4$$6$$8$$\cdot
10^{-2}$Step$\mathcal{L}_{\text{L}}$

$0$$20$$40$$60$$80$$100$$120$$140$$160$$180$$200$$220$$240$$260$$280$$300$$320$$340$$360$$3$$4$$5$$\cdot
10^{-2}$Step$\mathcal{L}_{\text{TL}}$

$0$$20$$40$$60$$80$$100$$120$$140$$160$$180$$200$$220$$240$$260$$280$$300$$320$$340$$360$$0$$2$$4$$6$$8$$\cdot
10^{-2}$Step$\mathcal{L}_{\text{A}}$

Figure 3: Loss plots of complementary pre-training objectives. The green curve
represents JointMotion w/o CME, while the blue curve represents JointMotion
with the objective of connecting motion and environments. Consistent with the
remainder of the document, L stands for lanes, TL stands for traffic lights,
and A stands for agents.

(pgfplots) Package pgfplots Warning: Table
’data/joint_motion_scene_transformer_shift.csv’ has an empty column name;
replacing it by ’0” Pre-trainingFine-tuningTraining from
scratch$0$$0.2$$0.4$$0.6$$0.8$$1$$1.2$$1.4$$1.6$$1.8$$2$$2.2$$2.4$$1\cdot
10^{-1}$$0.15$$0.2$Step $\times 10^{5}$mAP

Figure 4: Accelerating and improving training via SSL. Scene Transformer
models pre-trained with JointMotion achieve higher mAP scores on WOMD than
models trained from scratch, even in the same total wall training time (pre-
training + fine-tuning).

Fig. 4 shows that Scene Transformer models pre-trained with our method achieve
higher mAP scores on WOMD than models trained from scratch, even in the same
total wall training time (pre-training + fine-tuning).

##  5 Comparing scene-level self-supervision methods

In this experiment, we further compare the scene-level objectives PreTraM [33]
and JointMotion. We train Scene Transformer, HPTR, and a joint configuration
of Wayformer to cover all common types of environment representations in
motion prediction (i.e., agent-centric, scene-centric, and pairwise relative).

Experimental setup. For Scene Transformer, we use the same configuration as in
the previous experiment. For HPTR, we add 3 modality-specific encoders and use
a shared decoder for $K=6$ motion modes. For the joint configuration of
Wayformer, we follow [18] and use a shared encoder for early fusion, which
compresses the multi-modal input to 128 tokens. We concatenate multiple such
agent-centric embeddings with positional and rotation information into a
common reference frame and use a shared decoder to predict scene-wide joint
modes. For pre-training the Wayformer model, we add our decoder for early
fusion configurations (see Fig. 2). For all models, we employ the same
optimizer and learning rate scheduling as in the previous experiment. All
models are pre-trained for 10 hours and fine-tuned for 23.5 hours using a
training server with 4 A100 GPUs. Evaluation is performed on the interactive
validation split of WOMD and AV2. Our comparison is structured according to
the performance metrics mAP (mean Average Precision), minSADE (minimum Social
Average Displacement Error), minSFDE (minimum Social Final Displacement
Error), MR (Miss Rate), and OR (Overlap Rate). minSADE measures the
displacement error between the ground-truth future trajectories of interacting
agents and the closest joint prediction out of a predefined set, averaged over
the future time horizon [6]. This metric emphasizes the model’s ability to
account for social interactions in predicting the average trajectory
displacement accurately. On the other hand, minSFDE focuses on the
displacement error at the time horizon endpoint.

Results. Table 2 shows the results of this experiment. Our method consistently
outperforms PreTraM using different models with varying environment
representations. Our method improves all models, while the improvement of the
HPTR model (e.g., 11% lower minSFDE) is most significant and the improvement
of the agent-centric Wayformer is least significant (e.g., 3% lower minSFDE).
The Wayformer model is not pre-trainable with PreTraM since an instance-level
objective is required to decode the modality-specific tokens from fused
representations (cf., Sec. 3.2). The lowest block shows that our method leads
to transferable representations. Specifically, pre-training on WOMD improves
fine-tuning on AV2.

Table 2: Comparing scene-level self-supervision methods. All metrics are
computed using the Waymo Open Motion interactive (WOMD) and Argoverse 2
Forecasting (AV2) validation splits. Best scores are bold, second best are
underlined.

Dataset | Pre-training | Model | mAP $\uparrow$ | minSADE $\downarrow$ | minSFDE $\downarrow$ | MR $\downarrow$ | OR $\downarrow$
---|---|---|---|---|---|---|---
WOMD | None | Scene Transformer | 0.0822 | 1.5255 | 3.6715 | 0.7372 | 0.2868
| PreTraM [33] | Scene Transformer | 0.0722 | 1.5415 | 3.6508 | 0.7385 | 0.2915
| JointMotion | Scene Transformer | 0.0876 | 1.3830 | 3.2400 | 0.7090 | 0.2847
WOMD | None | HPTR | 0.1033 | 1.1682 | 2.6003 | 0.6030 | 0.2331
| PreTraM [33] | HPTR | 0.1286 | 1.0981 | 2.5049 | 0.5863 | 0.2345
| JointMotion | HPTR | 0.1456 | 1.0564 | 2.4006 | 0.5591 | 0.2297
WOMD | None | Wayformer (joint) | 0.1533 | 1.0209 | 2.3529 | 0.5461 | 0.2273
| JointMotion | Wayformer (joint) | 0.1553 | 0.9939 | 2.2823 | 0.5270 | 0.2143
AV2 | None | HPTR | 0.2554 | 2.255 | 1.138 | - | 0.0988
| JointMotion WOMD | HPTR | 0.2697 | 2.153 | 1.137 | - | 0.1025

##  6 Comparison with state-of-the-art methods for joint motion prediction

In this experiment, we compare our method with state-of-the-art methods for
joint motion prediction in autonomous driving.

Experimental setup. We pre-train HPTR (configured as in Sec. 5) for 10 hours
using JointMotion and fine-tune for 100 hours. We evaluate the methods in test
and validation splits of WOMD. We use the official challenge website to
compute performance metrics.

Results. Table 3 presents a comparative analysis of various state-of-the-art
methods for joint motion prediction on interactive splits of WOMD. In the test
split, we compare four different approaches. The Scene Transformer has a mAP
of 0.1192 and is outperformed by the other methods in all metrics. GameFormer
achieves a higher mAP of 0.1376 and exhibits competitive performance in terms
of minSADE and minSFDE. MotionDiffuser and JointMotion (HPTR) are close
contenders, with MotionDiffuser showing slightly better in all of the metrics
compared to JointMotion (HPTR).

For the Validation split, the results are similar. GameFormer (joint) scores a
mAP of 0.1339, with MotionLM (single replica) slightly outperforming it in
terms of mAP, achieving 0.1687. JointMotion (HPTR) presents an improved mAP of
0.1761 compared to its counterparts. Following HPTR, we compare against single
replica model of MotionLM and show its ensembling version for reference.
MotionLM (ensemble) demonstrates superior overall performance, a result
anticipated given power of ensembling.

Table 3: Comparison with state-of-the-art methods for joint motion prediction.
All methods are evaluated on interactive splits of the Waymo Open Motion
dataset. Following [36] we compare against single replica versions of joint
prediction methods, ensemble versions are shown for reference. Best scores are
bold, second best are underlined.

Split | Method (config) | Venue | mAP $\uparrow$ | minSADE $\downarrow$ | minSFDE $\downarrow$ | MR $\downarrow$ | OR $\downarrow$
---|---|---|---|---|---|---|---
Test | Scene Transformer (joint) [22] | ICLR’22 | 0.1192 | 0.9774 | 2.1892 | 0.4942 | 0.2067
| GameFormer (joint) [17] | ICCV’23 | 0.1376 | 0.9161 | 1.9373 | 0.4531 | 0.2112
| MotionDiffuser [18] | CVPR’23 | 0.1952 | 0.8642 | 1.9482 | 0.4300 | 0.2004
| JointMotion (HPTR) |  | 0.1869 | 0.9129 | 2.0507 | 0.4763 | 0.2037
Val | GameFormer (joint) [17] | ICCV’23 | 0.1339 | 0.9133 | 1.9251 | 0.4564 | -
| MotionLM (single replica) [25] | ICCV’23 | 0.1687 | 1.0345 | 2.3886 | 0.4943 | -
| JointMotion (HPTR) |  | 0.1761 | 0.9689 | 2.2031 | 0.4915 | 0.1990
| MotionLM (ensemble) | ICCV’23 | 0.2150 | 0.8831 | 1.9825 | 0.4092 | -

##  7 Conclusion

In this work, we introduced a SSL method for joint motion prediction of all
agents within a traffic scene. Our SSL framework integrates scene-level and
agent-level objectives that operate complementarily, enhancing the training
speed and accuracy of motion prediction models. Notably, our approach
outperforms recent contrastive and autoencoding methods previously utilized
for pre-training in motion prediction. Moreover, our objectives enable
effective transfer learning between the Waymo Open Motion and the Argoverse 2
Forecasting datasets. Our evaluations demonstrate significant performance
improvements over non-ensembling state-of-the-art joint prediction methods or
joint-prediction variants of marginal-prediction architectures, underscoring
the robustness and effectiveness of our proposed method.

## References

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# Information-driven Nonlinear Quantum Neuron

Ufuk Korkmaz and Deniz Türkpençe The authors acknowledge support from the
Scientific and Technological Research Council of Turkey (TÜBİTAK-Grant No.
120F353). The authors also wish to extend special thanks to the Cognitive
Systems Lab in the Department of Electrical Engineering providing the
atmosphere for motivational and stimulating discussions.(Corresponding
author:Ufuk Korkmaz)Manuscript received January 30, 2023; revised ???? ??,
2023. The authors are with the Department of Electrical Engineering, İstanbul
Technical University, İstanbul, , 34469 TURKEY e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The promising performance increase offered by quantum computing has led to the
idea of applying it to neural networks. Studies in this regard can be divided
into two main categories: simulating quantum neural networks with the standard
quantum circuit model, and implementing them based on hardware. However, the
ability to capture the non-linear behavior in neural networks using a
computation process that usually involves linear quantum mechanics principles
remains a major challenge in both categories. In this study, a hardware-
efficient quantum neural network operating as an open quantum system is
proposed, which presents non-linear behaviour. The model’s compatibility with
learning processes is tested through the obtained analytical results. In other
words, we show that this dissipative model based on repeated interactions,
which allows for easy parametrization of input quantum information, exhibits
differentiable, non-linear activation functions.

###### Index Terms:

Quantum neural networks, quantum neuron, quantum learning, open quantum
system, cost function, quantum activation.

##  I Introduction

Artificial neural networks (ANNs) play a crucial role in machine learning.
Complex tasks have been achieved with artificial neural network models in the
past two decades, especially with the increase in the capacities of computer
processors [1, 2, 3, 4]. However, the increasing volume of data to be
processed and the reaching of the end of Moore’s law pose significant
challenges to the rate of performance increase of artificial neural networks
[5].

Quantum computing is a computational paradigm harnessing resources that do not
have classical counterparts [6]. With only a few algorithms presented by this
paradigm, it has been demonstrated that solving some problems believed to be
impossible with classical computation methods is possible [7, 8, 9]. Due to
the reasons mentioned above, proposals for machine learning and artificial
neural network models based on the existing advantages of quantum computing
have begun to emerge in parallel with the growing interest in quantum
computing [10, 11, 12, 13]. Despite various proposals regarding quantum
artificial neural networks [14, 15], the lack of agreement on a widely
accepted model has made the topic an open field of research. Particularly at
the very fundamental level, the problem of simulating the non-linearity of a
quantum neuron with the widely linear properties of quantum computing is a
major issue that needs to be resolved.

The implementation proposals for artificial neural networks [16, 17] and non-
linear quantum neurons [18, 19] on standard quantum circuit models often
require high resource costs. In addition, if multi-controlled gates for the
neuron gate are required in the algorithm specifically prepared for the
problem to be solved, the need for time-dependent optimization to construct
these gates has the potential to limit the efficiency of the algorithm.
Moreover, the current quantum computers being constructed are largely
influenced by quantum noise, which severely limits the efficiency of quantum
algorithms requiring high resources. This era, referred to as the Noisy
Intermediate Scale Quantum computing [20] (NISQ), is characterized by this
phenomenon and efforts are being made to address it [21]. The dissipation-
driven quantum computing model was introduced as an equivalent to the standard
circuit quantum computing model [22]. Thus, the dissipation-driven model
offers robustness against quantum noise and is also a standalone analog
quantum computing model.

In this study, we introduce a dissipation-assisted quantum neuron model with
non-linear response that operates as a binary classifier quantum perceptron,
where the binary decision is read out by a probe quantum system (PQS) with
spin angular momentum $J\geq 1/2$. In the model, the PQS is in contact with
multiple, distinct quantum environments in pure quantum states with arbitrary
coupling rates as input quantum data. In the scenario, PQS undergoes a
dissipative equilibration process and reaches a steady state where the binary
decision is encoded. Steady state magnetization is the merit quantifier in our
model. The dissipation process is characterized by a collision model based on
repeated interactions [23, 24, 25, 26], which effectively characterizes open
quantum systems and enables easy parametrization of input quantum information
contained by the reservoirs. The environments bearing quantum information is
referred to as information reservoirs [27, 28].

We show that the quantum neuron responds as a hyperbolic tangent-like
activation which can be controlled by the value of $J$. We derive a master
equation and analytically obtain the steady-state solutions. We also derive a
cost function based on the obtained results and examine the performance on
gradient descent-based learning tasks. We observe that the proposed neuron is
suitable for learning schemes as the dissipation based process results a
continuous dynamics with proper gradient minimization. Finally, the steady
response of the neuron, driven by non-equilibrium environments, contains non-
vanishing quantum coherence. This opens up the possibility for future work to
harness non-classical quantum resources in a steady state.

##  II Preliminaries

As discussed above, proposals related to quantum models of artificial neural
networks are still at the level of basic discussions, so we aim to start with
definitions from the most basic classical model and thus establish a
connection with the quantum model we propose.

###  II-A The classic model

The learning theory of artificial neural networks is based on mathematical
models that mimic the functioning of the human brain, as first proposed by
McCulloch, Pitts, and Rosenblatt [29, 30]. The perceptron is a foundational
component of artificial neural networks carrying out binary classification
tasks. In essence, it outputs a binary label of either 0 or 1 based on the
input, which is represented as $\varphi_{in}=\bf{x}^{T}\bf{w}$, where
$\bf{x}=[x_{1},\ldots x_{N}]^{T}$ is the set of input features and
$\bf{w}=[w_{1},\ldots w_{N}]^{T}$ the corresponding weight vectors. This
binary output is generated through a non-linear function $f(.)$, where
$z=f(\varphi_{in})$, and the decision rule is set as $z=0$ if
$z=f(\varphi_{in})\geq 0$ and $z=1$ otherwise. The specific binary labels can
vary as needed. It’s important to note that a perceptron with an identity
activation function can still produce linear classification, though using non-
linear activation functions are beneficial for multi-layer ANN applications.

Supervised learning involves creating a mapping from input data to binary
labels

$\mathcal{X,Y}\rightarrow\\{0,1\\}.$ (1)

The input data, represented as $\mathcal{X}$, and the desired output,
represented as $\mathcal{Y}$, are part of a training set
$\mathcal{S}=(\mathcal{X},\mathcal{Y})$. The cost function, $C$, measures the
difference between the actual output vector, $\bf{A}$, and the desired output
vector, $\bf{Y}$. The cost function expression follows the least squares
method, as

$C=\frac{1}{2}(\bf{Y}-\bf{A})^{2}.$ (2)

The weight instances are updated iteratively through back-propagation

$\bf{w}_{k+1}:=\bf{w}_{k}+\delta\bf{w}_{k}$ (3)

to minimize the cost. A gradient-descent based method is used for the training
process, and the change in the parameters is determined by the partial
derivative, as

$\delta{w}_{k}=-\eta\frac{\partial C}{\partial{w}_{k}}.$ (4)

The learning rate, represented by $\eta$, influences the speed of the learning
task and is non-negative. The partial derivative expresses the change in the
parameter that leads to the largest descent.

###  II-B The quantum dynamics

This section addresses the dynamics of open systems and gives preliminary
definitions. Our model operates based on a dissipative protocol as previously
noted. The input data, presented in a classical manner, is represented as the
weighted sum of input features

$\varphi_{in}=\bf{x}^{T}\bf{w}=\sum_{i}w_{i}x_{i}.$ (5)

We describe the dissipative quantum equivalent of the above expression as

$\displaystyle\Lambda_{t}[\varrho_{0}]=\sum_{i}P_{i}\Phi_{t}^{(i)}{[\varrho_{0}]}$
(6)

where $\Phi_{t}^{(i)}$ is a completely positive trace preserving (CPTP)
quantum dynamical map that acts on the probe system characterized by a density
matrix $\varrho_{0}$ and $P_{i}$ represents the probability of the map
interacting with the $i$th information reservoir. The subscript $t$ in Eq. (6)
represents the time-dependence of the maps, which are generated by a physical
process

$\displaystyle\Phi^{(i)}_{t}[\varrho_{0}]=\text{Tr}_{\mathcal{R}_{i}}\\{U_{t}(\varrho_{0}\otimes\varrho_{\mathcal{R}_{i}})U_{t}^{\dagger}\\}$
(7)

that has a unitary propagator $U_{t}$ that acts on both the PQS and the
reservoir. The $i$th reservoir quantum state is represented by
$\rho_{\mathcal{R}i}$, and the partial trace over the $i$th reservoir is
represented by $\text{Tr}_{\mathcal{R}_{i}}$.

The quantum reservoirs provide initial quantum data, each composed of non-
correlated, non-interacting two-level quantum systems (subunits) defined by
the tensor product

$\displaystyle\rho_{\mathcal{R}_{i}}=\bigotimes_{k=1}^{n}\rho_{k}(\theta_{i},\phi_{i}).$
(8)

These subunits are initially prepared in pure quantum states with identical
Bloch parameters $\rho_{k}(\theta_{i},\phi_{i})$, which enables the
dissipative equivalence of the model with parametrized quantum circuits.

##  III Collision model and the quantum neuron

The model described relies on a standard quantum collisional model for its
dynamics. The probe system experiences a dissipative process due to multiple
independent reservoirs with arbitrary couplings. The steady state of the probe
is read through spin observable $S_{z}$, providing a binary classification
output. The dynamics are given by

$\displaystyle\Phi^{(i)}_{n\tau}=$
$\displaystyle\text{Tr}_{n}\big{[}\mathcal{U}_{i_{n}}\ldots\text{Tr}_{1}[\mathcal{U}_{i_{1}}\left(\varrho_{0}\otimes\rho_{\mathcal{R}_{i_{1}}}\right)\mathcal{U}_{i_{1}}^{\dagger}]\otimes\ldots$
$\displaystyle\ldots\otimes\rho_{\mathcal{R}_{i_{n}}}\mathcal{U}_{i_{n}}^{\dagger}\big{]}.$
(9)

where $n\tau$ is the elapsed time of the map after $n$ collisions and
$\mathcal{U}_{i_{k}}=\text{exp}[-\text{i}\mathcal{H}^{k}_{i}\tau]$ is the
unitary propagator applying the $k$th collision to the $i$th reservoir. The
system and the $i$th reservoir are initially prepared in a tensor product
state $\varrho(0)=\varrho_{0}(0)\otimes\rho_{\mathcal{R}_{i}}$. Only the probe
system experiences time dependence, and the reservoir states are reset after
each collision.

The dynamics of the system and reservoir are governed by the Hamiltonian
$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{int}$, where

$\displaystyle\mathcal{H}_{0}=\hbar\omega_{0}S_{0}^{z}+\frac{\hbar\omega_{i}}{2}\sum_{k=1}^{n}\sigma_{k_{i}}^{z}$
(10)

is the free part and

$\mathcal{H}_{int}=\hbar\sum_{k=1}^{n}g_{i}(S_{0}^{+}\sigma_{k_{i}}^{-}+\text{H.c.}),$
(11)

is the interaction part. Respectively, the PQS raising operator, Pauli-$z$
operator, Pauli raising and lowering operators are represented as $S_{0}^{+}$,
$\sigma^{z}$, $\sigma^{+}$, and $\sigma^{-}$, with $\hbar$ Planck’s constant
divided by $2\pi$ set to 1. The coupling coefficients $g_{i}$ are in the weak
coupling regime ($g_{i}\ll\omega_{0}$), ensuring minimal cross-talk between
reservoirs. The coefficients are proportional to probabilities ($g_{i}\propto
P_{i}$) in Eq. (6), acting as the quantum equivalent of weights in classical
models.

In this section, we present numerical and analytical results. As mentioned
above, the output results are read from a quantum system, PQS, that behaves in
a spin state $J\geq 1/2$. In quantum mechanics, systems that behave as spin
with $J\geq 1/2$ have $2J+1$ orientations relative to the quantization axis.
This result suggests that high spin states can be adapted to a multi-class
classification problem with observables in our methods. However, this idea
requires further study and analysis, so it falls out of the scope of this
study where we focus on binary classification.

Regardless of the total $J$ value of the PQS system, it always gives a binary
classification result associated with the relevant $S_{z}$ observables, which
are always non-linear with input quantum data weighted by the coefficients
$g_{i}$. We use the methods from our previous study [31], where the derivation
of the master equation and the analytical results were limited to $J=1/2$ and
only gave linear activation results. In this study, we present how PQS behaves
like a non-linear activation function for $J\geq 1/2$ cases and its effects on
minimizing the cost function. In addition, we discuss how PQS can be
physically applied for $J\geq 1/2$ in a separate section at the end of the
study.

For any spin value $J$, the corresponding generalized density matrix
expression can be defined through the basis of polarization operators. The
density operator expression for PQS, therefore, reads

$\displaystyle\varrho_{0}$
$\displaystyle=\frac{1}{d}\bf{1}+\overrightarrow{r}\cdot\overrightarrow{T}$
$\displaystyle=\frac{1}{d}\bf{1}+\sum_{l=1}^{2J}\sum_{m=-l}^{l}r_{lm}T_{lm}$
(12)

where $\bf{1}$ is the d-dimensional identity operator, $\overrightarrow{r}$ is
the generalized Bloch vector whose elements are
$r_{lm}=\text{Tr}T_{lm}^{\dagger}\varrho_{0}$ and $\overrightarrow{T}$ is the
relevant spin polarization vector. Alternative generalized density matrix
representations are also possible through the generalized Gell–Mann matrix
basis or the Weyl operator basis [32]. However, we will present our analytical
results below through polarization operator basis representation.

###  III-A Analytical results

The unitary propagator is expressed as
$\mathcal{U}(\tau)=\text{exp}[-\text{i}\mathcal{H}_{\text{int}}\tau]$ in the
interaction picture (as detailed in Appendix A) with respect to
$\mathcal{H}_{0}$. This is used to derive the master equation,

$\displaystyle\dot{\varrho}_{0}=$
$\displaystyle-\text{i}[\mathcal{H}_{\text{eff}},\varrho]+\sum_{i=1}^{N}g_{i}^{2}\left(\xi^{+}\mathcal{L}[S_{0}^{+}]+\xi^{-}\mathcal{L}[S_{0}^{-}]\right)$
$\displaystyle+\sum_{i<j}^{N^{\prime}}g_{i}g_{j}\left(\xi^{+}_{s}\mathcal{L}_{s}[S_{0}^{-}]+\xi^{-}_{s}\mathcal{L}_{s}[S_{0}^{+}]\right)$
(13)

where
$\mathcal{H}_{\text{eff}}=r\tau\sum_{i}^{N}g_{i}\left(\langle\sigma_{i}^{-}\rangle
S_{0}^{+}+\langle\sigma_{i}^{+}\rangle S_{0}^{-}\right)$, representing a
coherent drive on the probe qubit as the effective Hamiltonian. The averages
are calculated for identical reservoir units, i.e.
$\langle\mathcal{O}i\rangle=\text{Tr}[\mathcal{O}\varrho_{\mathcal{R}_{i}}]$.
The Lindblad superoperator is defined as $\mathcal{L}[o]\equiv 2o\varrho
o^{\dagger}-o^{\dagger}o\varrho-\varrho o^{\dagger}o$, and
$\mathcal{L}_{s}[o]\equiv 2o\varrho o-o^{2}\varrho-\varrho o^{2}$ refers to
the effect of squeezing by the reservoir. The coefficients of the Lindbladians
contain information related to different entries of the density matrices of
the reservoir units. For example, the standard Lindbladian coefficients
$\xi^{\pm}=r\tau^{2}\langle\sigma_{i}^{\pm}\sigma_{i}^{\mp}\rangle/2$ include
the diagonal entries of the $i^{th}$ reservoir unit, while
$\xi_{s}^{\pm}=2r\tau^{2}\langle\sigma_{i}^{\pm}\rangle\langle\sigma_{j}^{\pm}\rangle$
includes the off-diagonal entries of the pairwise distinct reservoir units,
with a total of $N^{\prime}=N(N-1)/2$ terms in the summation. Here, $r$ is a
positive number used in forming the master equation and represents the success
of any collision (see Appendix B).

Based on the model we presented, we obtained the steady state solution of the
master equation, since the quantum neuron binary classification decision is
given in the steady state. The steady state solution of Eq. (III-A) can be
obtained straightforwardly for any $J$ by using the related spin operators
$S_{0}^{\mp}$. However, as the analytical results become too cumbersome to
generalize the result, we deal with the case for $J=1$ analytically and
evaluate the higher spin cases numerically. Therefore the steady state of PQS
reads

$\varrho_{0}^{\text{ss}}=\begin{bmatrix}P_{11}&\gamma^{-}_{1}(P_{11}-P_{22})&2\gamma^{-}_{3}P_{22}\\\
\gamma^{+}_{1}(P_{22}-P_{11})&P_{22}&\gamma^{-}_{1}(P_{22}-P_{33})\\\
2\gamma^{+}_{3}P_{22}&\gamma^{+}_{1}(P_{33}-P_{22})&P_{33}\end{bmatrix}$ (14)

where $P_{nn}$ are the steady state populations (see Appendix B for detailed
expressions) and
$\gamma^{\pm}_{1}=\text{i}r\tau\sum_{i=1}^{N}g_{i}\langle\sigma_{i}^{\pm}\rangle$,$\gamma^{\pm}_{2}=r\tau^{2}\sum_{i=1}^{N}g^{2}_{i}\langle\sigma_{i}^{\pm}\sigma_{i}^{\mp}\rangle/2$,
$\gamma^{\pm}_{3}=2r\tau^{2}\sum_{i<j}^{N^{\prime}}g_{i}g_{j}\langle\sigma_{i}^{\pm}\rangle\langle\sigma_{j}^{\pm}\rangle$.
Please note that the $\xi^{\pm}$ coefficients in Eq. (III-A) are directly
related to the elements of the density matrix of the reservoir units, and the
$\gamma^{\pm}$ coefficients in Eq. (14) are the weighted summation of these
elements. These results analytically prove that PQS in its steady state gives
the weighted sum of the input quantum data. We will evaluate the following
numerical results for the activation behaviour.

(a)

(b)

Figure 1: Equilibration dynamics of the probe qubit magnetization against the
collision number depending on the reservoir qubit amplitude parameter
$\theta$. The probe qubit initialized in the spin-coherent state
$\ket{\bf{n}}$. The reservoir qubit initialized as
$\rho_{\mathcal{R}}=\ket{\Psi(\theta,\phi)}\bra{\Psi(\theta,\phi)}$ for
$\theta=0^{o},60^{o},80^{o},90^{o},100^{o},120^{o},180^{o}$ (from top to
bottom) and $\phi=0$. The target qubit-reservoir interaction time $\tau=3$ and
the coupling coefficient $g=0.02$ are dimensionless and scaled by
$\omega_{r}=10^{9}$Hz. (a) The spin angular momentum value is $J=1/2$. (b) The
spin angular momentum value is $J=5/2$.

(a)

(b)

(c)

Figure 2: The activation function is in the form of equilibrium dynamics of
probe qubit magnetization versus reservoir qubit amplitude parameter $\theta$.
The probe qubit initialized in the spin-coherent state $\ket{\bf{n}}$. The
reservoir qubit initialized as
$\rho_{\mathcal{R}}=\ket{\Psi(\theta,\phi)}\bra{\Psi(\theta,\phi)}$ for
$\theta=0^{o},60^{o},80^{o},90^{o},100^{o},120^{o},180^{o}$ and $\phi=0$. The
target qubit-reservoir interaction time $\tau=3$ and the coupling coefficient
$g=0.02$ are dimensionless and scaled by $\omega_{r}$. (a) The spin angular
momentum value is $J=1/2$. (b) The spin angular momentum value is $J=5/2$. In
this case, the magnetization of the probe qubit is hyperbolic tangent
($\tanh(x)$). (c) The spin angular momentum value is $J=9/2$.

###  III-B Numerical results

The numerical analysis begins by examining the simplest scenario where the PQS
is connected to a single (only $g_{1}\neq 0$) information reservoir in the
quantum state
$\varrho_{1}=\ket{\Psi(\theta_{1},\phi_{1})}\bra{\Psi(\theta_{1},\phi_{1})}$.
The progression of events is monitored by the number of collisions with
quantum data parameters represented by $\theta$. Throughout the analysis,
$\phi$ is set to 0, without loss of the generalization. The initial state of
the PQS is a spin coherent state

$\ket{\bf{n}}=\sum_{m-j}^{j}\sqrt{\binom{2j}{j-m}}\left(\cos(\frac{\theta}{2})\right)^{j+m}\\\
\times\left(\sin(\frac{\theta}{2})e^{\text{i}\phi}\right)^{j+m}\ket{j,m}$ (15)

with null magnetization where $\ket{j,m}$ are the standard angular momentum
basis states, $\bf{n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)^{T}$
with $\theta\in[0,\pi]$ and $\phi\in[0,2\pi[$. Here, $m$ is the spin
projection number with $m=-J,\cdots J$. The simulations are carried out by
using QUTIP package in Python [33] and the parameters are based on the
superconducting circuits [34, 35, 36], which serve as a reliable platform for
quantum information processing.

In this architecture, transmon qubits are connected through a resonator bus
[37], allowing interaction through virtual photon exchange. The coupling
strength between qubits can be adjusted by their dispersive connection to the
transmission line resonator. A typical superconducting circuit with weakly
connected transmon qubits has a resonator frequency of $\omega_{r}$ ranging
from 1-10 GHz with a qubit-resonator coupling of $g_{r}$ from 1-500 MHz and an
effective qubit-qubit coupling of $g$ from 1-100 MHz, with a qubit energy
relaxation time of $T_{1}$ ranging from 40-150 $\mu s$ [34, 37].

Fig. 1a illustrates the temporal evolution of PQS (for $J=1/2$) as a function
of the number of collisions ($NC$) in the presence of varying $\theta$
parameters of a single information reservoir. The monitored quantity is the
normalized magnetization, defined as
$\langle\tilde{S}_{z}\rangle=\text{Tr}[\varrho(n\tau)S_{z}]/J$. It is noted
that PQS reaches a steady state after $n\sim 2\times 10^{3}$ collisions. With
an interaction time of $\tau=3$ ns, the time required to reach the steady
state is $n\tau\approx 6\mu$s, which is much smaller than the $T_{1}$ value.
It has been reported that the time required to reach the steady state
decreased when PQS interacted with environments containing more than a single
qubit [38]. By keeping all parameters constant except for the $J$ parameter
and setting $J=5/2$, as seen in Fig. 1b, all the $\langle\tilde{S}_{z}\rangle$
values corresponding to the $\theta$ parameters of the information reservoir
exhibit a non-linear behaviour and approach the steady state values of
$\langle\tilde{S}_{z}\rangle=1$ and -1. In principle, single-input quantum
data can still be classified. While we continue to analyse the model with a
single information reservoir, a study of the model’s behaviour in the presence
of multiple information environments will be performed.

To gain a comprehensive understanding of the behaviour of the proposed model,
alongside a temporal evaluation, we focus on examining the steady state values
with respect to $\theta$, as shown in Fig. 2. The steady state magnetization
values of PQS marked as dots in the figure correspond to different values of
the $J$ parameter, resembling the $tanh(x)$ activation function. For a better
comparison, we also added the $tanh(x)$ function represented by a continuous
line in the figure. As known, $\theta$ is a geometric parameter belonging to
the Bloch sphere that characterizes the input quantum data and can take values
between 0 and $\pi$. The $tanh(x)$ function takes any real $x$ value and
outputs values of 1 and -1. Therefore, these two functions cannot be directly
compared, but a relative comparison can be made with properly established
scales. To this end, Fig. 2 is established for two different scales such as
$x\in[-2\pi,2\pi]$ and $\theta\in[\pi,0]$. The starting value of $\theta=\pi$
is a conscious choice for convenient comparison of
$\langle\tilde{S}_{z}\rangle$ with $tanh(x)$.

In Fig. 2a, where PQS is expressed as $J=1/2$, we observe the linear behaviour
of the magnetization over a wide range of values. Although this behaviour is
sufficient for the model to perform linear classification tasks on its own, it
may be insufficient to model complex, non-linear relationships in multi-
layered networks established with this parameter. On the other hand, as seen
in Fig. 2b, the steady state response of the model for the PQS value of
$J=5/2$ is seen to coincide with the hyperbolic tangent function, $tanh(x)$,
at the given scales. As seen in Fig. 2c, for higher $J$ values, the system
response is seen to converge more strongly towards the values of 1 and -1.

Hyperbolic tangent is often used as a differentiable activation function in
artificial neural networks, where the gradient can easily be calculated [39,
40]. However, $tanh(x)$ saturates at large positive and negative values,
making it challenging to train deep neural networks. This issue is known as
the vanishing gradient problem, where the gradients become very small, making
it challenging for the optimization algorithm to update the weights. The
behaviour of the proposed model being similar to the aforementioned function
suggests that it may have similar advantages and disadvantages. Nevertheless,
note that the proposed model has the capacity to control its steady behaviour
through adjusting the $J$ parameter. Moreover, the analytical results indicate
that regardless of the number of quantum input data, the additivity of quantum
dynamical maps can be ensured through considering the convex structure of the
density matrix formalism. As a result, the model will consistently act as if
it is receiving input information from a single reservoir with an effective
$\theta$ value always between 0 and 2$\pi$.

Figure 3: The steady state magnetization of the target qubit depending on the
variation of the couplings to the reservoirs. The spin angular quantum number
is $J=1/2$ for the blue line, and $J=5/2$ for the orange line. Variations of
the coupling rates are $g_{1}=g/2+\Delta g$, $g_{2}=g/2-\Delta g$. Here,
$\Delta g$ represents a fraction of $g$. The probe qubit prepared initially in
the spin-coherent state $\ket{\bf{n}}$ and interacted collisionally with the
reservoir units $\ket{\Psi(\theta_{i},\phi_{i})}$ with $\theta_{1}=0$,
$\phi_{1}=0$ and $\theta_{2}=\pi$, $\phi_{2}=0$. The target qubit-reservoir
interaction time $\tau=3$ and the coupling coefficient $g=0.02$ are
dimensionless and scaled by $\omega_{r}$.

After observing the non-linear behaviour of a single input quantum reservoir,
we examine the simplest case of multiple reservoirs. Fig. 3 shows the
normalized steady magnetization of PQS with varying $\delta g$. The PQS is
connected to two distinct quantum reservoirs,
$\ket{\theta_{1}=0,\phi_{1}=0}\equiv\ket{\uparrow}$ and
$\ket{\theta_{2}=\pi,\phi_{2}=0}\equiv\ket{\downarrow}$. The coupling
strengths of PQS to each reservoir are $g_{1}=g/2+\delta g$ and
$g_{2}=g/2-\delta g$ with $g=0.01$ and $-0.5\leq\delta g\leq 0.5$. In the
limit case where $\delta g=-0.5g$, $g_{1}=0$ and $g_{2}=g$, meaning PQS is
only connected to the $\ket{\downarrow}$ reservoir. In this case, as seen in
Fig. 3, the magnetization is $\langle\tilde{S}_{z}\rangle=-1$ as expected. For
intermediate values of $\delta g$ within the given range, the monitored
magnetization varies as $-1<\langle\tilde{S}_{z}\rangle<1$, and it reaches its
other limit value $\langle\tilde{S}_{z}\rangle=1$ for $\delta g=0.5$. From now
on, we will neglect the tilde symbol we use to indicate magnetization for the
sake of simplicity.

Thus, we have also shown how the PQS output behaves with multiple quantum
information inputs weighted by $g_{i}$. We also note in Fig. 3 that the case
we studied exhibits different non-linear behaviours for $J=1/2$ and $J=5/2$
situations as a confirmation of Fig. 2. If we compare Fig. 2 and Fig. 3
further, in Fig. 2, where PQS is only connected to a single reservoir, the
altered parameter is necessarily $\theta$, whereas in Fig. 3, where PQS is
connected to two reservoirs and $\theta$ values are fixed, the $g_{i}$ values
corresponding to the weights against the classical cases change. The behaviour
of our output quantifier, magnetization, gives similar results for both cases
as the parameters change, indicating that our model is suitable for the
training and learning tasks we will discuss below.

(a)

(b)

Figure 4: Cost function minimization depending on learning rates against the
variation of $g$.(a) The initial magnetization values of the reservoirs are
$\langle\sigma_{z}^{1}\rangle=0.94$, $\langle\sigma_{z}^{2}\rangle=-0.10$,
$g_{1}=0.002$, $g_{2}=0.05$ and the desired PQS magnetization $\langle
S_{z}^{0}\rangle_{des}^{ss}=0.42$, respectively. The spin angular momentum
value of the PQS is $J=1/2$. (b) The spin angular quantum numbers are $J=1/2$
for the blue solid line, $J=1$ for the orange dotted line. Both of the the
initial magnetization values of the reservoirs are
$\langle\sigma_{z}^{1}\rangle=0.95$, $\langle\sigma_{z}^{2}\rangle=-0.11$,
$g_{1}=0.001$, $g_{2}=0.04$ and $\langle S_{z}^{0}\rangle_{des}^{ss}=0.42$,
respectively. The learning rate is $\eta=1.2\times 10^{-2}g_{1}$

##  IV Learning of the model

In what follows, we examine the gradient-descent based learning scheme of the
introduced model. First, we develop the cost function as [41]

$C=\frac{1}{2}(\langle S_{z}^{0}\rangle_{des}^{ss}-\langle
S_{z}^{0}\rangle_{act}^{ss})^{2}.$ (16)

Here, $\langle S_{z}^{0}\rangle_{des}^{ss}$ is the desired and $\langle
S_{z}^{0}\rangle_{act}^{ss}$ is the actual steady magnetization values of the
PQS for the spin observable $S_{z}$. As expressed in Appendix C, the actual
steady state magnetization of the PQS is a function of the coupling rates
$g_{i}$. In this scheme, in analogy with Eq. 5, the coupling rates are
iterated as

$g_{k+1}=g_{k}+\delta_{g_{k}}$ (17)

where

$\delta g_{k}=-\eta\frac{\partial C}{\partial g_{k}}.$ (18)

Here, $\eta$ is the learning rate determining the variation speed of the
relevant parameter in the direction of the greatest descent. (See Appendix C
for the derivation of the cost function and its derivative for $g_{i}$
depending on the value of the $J$ parameter.)

Figs. 4a and 4b illustrate the cost minimization against the iteration steps
where the PQS is connected to two distinct information reservoirs with
arbitrary couplings. In both Figures, we observe smooth convergence for cost
minimization due to the continuous dynamics of the dissipation-assisted model.
As obvious in Fig. 4a, a learning rate of two orders of magnitude smaller than
the coupling rate can reasonably minimize the cost function. Moreover, we
observe that by keeping the learning rate and all the remaining parameters
fixed, the speed of minimization increases with higher $J$, as shown in Fig.
4b.

Figure 5:  Cost function minimization with surface depiction against the
variation of $g_{1}$ and $g_{2}$. The spin angular momentum value is $J=1$.
The learning rate $\eta=2.4\times 10^{-5}$, $\langle S_{z}^{1}\rangle=0.96$,
$\langle S_{z}^{2}\rangle=-0.64$, the target magnetization is $\langle
S_{z}^{0}\rangle_{des}^{ss}=0.12$ and the initial coupling rates before
optimization is $g_{1}=0.001$, $g_{2}=0.04$.

The cost function minimization for a two-dimensional parameter space is
depicted in Fig. 5. The cost function has a smooth landscape without any local
minima or plateaus. The optimization is achieved by updates in the coupling
rate, as indicated by the dashed line. The results demonstrate that the model
can successfully optimize the cost function without overshooting.

##  V Physical model

The non-linear quantum neuron, which is represented by a d-dim. Hilbert space
with total angular momentum $J>1/2$, has been introduced above as a
theoretical model. Now, we aim to focus on its potential physical
applications. As previously mentioned, the central idea of the model is based
on a dynamic process that results from the repeated interaction of 2-level
quantum systems (qubits) with a d-level PQS (qudit) system. In summary, we can
describe this model as a quantum information processing that involves
interactions between qubits and a qudit. The use of qudits in quantum
information processing offers a larger Hilbert space due to its multi-level
structure, allowing for richer information storage and processing with
enhanced parallel processing capabilities [42, 43]. Despite receiving less
attention compared to qubits, there are numerous natural and artificial
physical applications related to qudits.

The physical systems with intrinsic angular momentum higher than $J=1/2$ are
ubiquitous in nuclear spin systems or in subatomic particles [44]. However,
they can also be obtained through effective Hamiltonians in set-ups such as
trapped ions [45], magnetic molecules [46], or solid state systems [47]. The
emulation of a quantum spin system with a maximum angular momentum of $J=3/2$
has been achieved using superconducting quantum circuits [48].

Previous research has shown promising results in implementing qubit-qudit
interactions efficiently through hybrid quantum systems, such as solid-state
[49] or spin ensembles [50, 51] coupled to superconducting circuits. Most
recently, the achievement of a [Yb(trensal)] molecular qudit with a nuclear
spin of $I=5/2$ coupled to an effective electronic spin of $S=1/2$ through
circuit quantum electrodynamics (c-QED) has been reported [52]. The
experimental architectures are constructed using lumped-element resonators,
whose properties can be adjusted precisely to match the diverse nuclear and
electronic spin transitions in the molecular system [53]. The studies
mentioned above demonstrate that the model we presented is experimentally
applicable.

##  VI Conclusions

In this study, we present a quantum model that has the potential to be the
basic processing unit of a feedforward artificial neural network. The model
operates within the framework of an open quantum system and is shown to be a
differentiable, non-linear activation quantum model that can perform binary
classification based on both numerical and analytical results. Our quantum
neural model exhibits a response behavior similar to the well-known hyperbolic
tangent function in the literature of classical artificial neural networks.
Based on the results obtained from our model, we show that the non-linear
behaviour is strengthened by the total angular momentum value $J$ of the
system, which we refer to as PQS.

We obtained analytical results for the steady state using a master equation
based on the collision model used for the description of the open quantum
system, which allows for the determination of the input quantum information
using the Bloch sphere parameter. Using the obtained results, we developed a
cost function to test the applicability of the proposed model to learning
processes. The cost function is shown to be continuously smooth and dependent
on the value of $J$ in order to minimize it.

Furthermore, the current state-of-the-art already allows for the experimental
implementation of small-scale models of the proposed quantum neural network.
We believe that the model we propose is suitable for hybrid quantum computing
processes and can find application in neuromorphic hardware-based quantum
computers that operate in the NISQ regime.

##  Appendix A The Unitary Propagator

The creation of the unitary propagator is the focus of our collision model-
based analytical calculations. First, we denote $U$ as the relevant
Hamiltonian in the interaction picture. The unitary time-evolution operator is
written as
$\mathcal{U}(\tau)=\text{exp}[-\text{i}\mathcal{H}_{\text{int}}\tau]=\text{exp}[-\text{i}\tau
U]$. Here,

$\displaystyle U=\sum_{i=1}^{N}g_{i}(S_{0}^{+}\sigma_{i}^{-}+\text{H.c.})$
(19)

with $\hbar=1$. Then the propagator reads

$\mathcal{U}(\tau)\simeq{\bf{1}}-\text{i}\tau U-\frac{\tau^{2}U^{2}}{2}$ (20)

by evaluating the approximation up to second order in $\tau$ where
$\text{i}=\sqrt{-1}$. $S_{0}^{\mp}=S_{x}\mp\text{i}S_{y}$ are the spin ladder
operators acting on the PQS. The expression for the propagator can be written
in matrix form for any value of the spin angular momentum $J$ of the PQS using
Eq. 20. Here, we only present the expression for $J=1$ for simplicity, with
the analytical calculations for higher values of $J$ ($J\geq 1$) given in the
next appendix sections. The spin operators for $J=1$ read as

$\displaystyle S_{x}=\frac{1}{\sqrt{2}}\begin{bmatrix}0&1&0\\\ 1&0&1\\\
0&1&0\\\ \end{bmatrix}$ (21a) $\displaystyle
S_{y}=\frac{1}{\sqrt{2}}\begin{bmatrix}0&-\text{i}\ &0\\\ \text{i}\
&0&-\text{i}\\\ \ 0&\text{i}\ &0\\\ \end{bmatrix}$ (21b) $\displaystyle
S_{z}=\begin{bmatrix}1&0&0\\\ 0&0&0\\\ 0&0&-1\end{bmatrix}$ (21c)

and the matrices for $U$ and $U^{2}$ are as follows:

$\displaystyle U=\begin{bmatrix}&\mathcal{S}^{-}_{g_{i}}&\\\
\mathcal{S}^{+}_{g_{i}}&&\mathcal{S}^{-}_{g_{i}}\\\
&\mathcal{S}^{+}_{g_{i}}&\\\ \end{bmatrix}$ (22a) $\displaystyle
U^{2}=\begin{bmatrix}\mathcal{S}^{-}_{g_{i}}\mathcal{S}^{+}_{g_{i}}&&\\\
&\mathcal{S}^{-}_{g_{i}}\mathcal{S}^{+}_{g_{i}}+\mathcal{S}^{+}_{g_{i}}\mathcal{S}^{-}_{g_{i}}&\\\
&&\mathcal{S}^{+}_{g_{i}}\mathcal{S}^{-}_{g_{i}}\end{bmatrix}$ (22b)

where $\mathcal{S}^{\pm}_{g_{i}}=\sum_{i=1}^{N}g_{i}\sigma_{i}^{\pm}$
represent the collective operators acting on the reservoir information units
weighted by $g_{i}$. Therefore one obtains the corresponding propagator as

$\mathcal{U}(\tau)=\begin{bmatrix}\bf{1}-\frac{\tau^{2}}{2}\mathcal{S}^{-}_{g_{i}}\mathcal{S}^{+}_{g_{i}}&-\text{i}\tau\mathcal{S}^{-}_{g_{i}}&\\\
-\text{i}\tau\mathcal{S}^{+}_{g_{i}}&\bf{1}-\frac{\tau^{2}}{2}\mathcal{S}^{+}_{g_{i}}\mathcal{S}^{-}_{g_{i}}&-\text{i}\tau\mathcal{S}^{-}_{g_{i}}\\\
&-\text{i}\tau\mathcal{S}^{+}_{g_{i}}&\bf{1}-\frac{\tau^{2}}{2}\mathcal{S}^{+}_{g_{i}}\mathcal{S}^{-}_{g_{i}}\end{bmatrix}.$
(23)

##  Appendix B The Master Equation

As stated in the text, our analytical results are obtained through the
derivation of a master equation based on repeated interactions. According to
the method we follow, repeated interactions can be represented as a random
process described by the Poisson distribution. Therefore, the density matrix
representing the entire system within a time interval $\delta t$ can be
expressed as

$\displaystyle\varrho(t+\delta t)=r\delta
t\mathcal{U}(\tau)\varrho(t)\mathcal{U}^{\dagger}(\tau)+(1-r\delta
t)\varrho(t)$ (24)

Here, $r\delta t$ represents the likelihood of an interaction occurring at a
rate of $r$, while $1-r\delta t$ represents the likelihood of a non-
interaction state. As $\delta t\rightarrow 0$, the following master equation

$\displaystyle\dot{\varrho}_{0}(t)=r\text{Tr}_{\mathcal{R}_{i}}[\mathcal{U}(\tau)\varrho(t)\mathcal{U}^{\dagger}(\tau)-\varrho(t)]$
(25)

for the reduced dynamics of PQS is derived from
$\dot{\varrho}_{0}(t)=(\varrho_{0}(t+\delta t)-\varrho_{0}(t))/\delta t$.
Rewriting Eq.(20) as $\mathcal{U}(\tau)\simeq{\bf{1}}-U_{1}(\tau)-U_{2}(\tau)$
with $U_{1}(\tau)=\text{i}\tau U$ and $U_{2}(\tau)=\tau^{2}U^{2}/2$, and
inserting into Eq.(25), one obtains

$\displaystyle\dot{\varrho}_{0}(t)=$ $\displaystyle
r\text{Tr}_{\mathcal{R}_{i}}[U_{1}(\tau)\varrho(t)U_{1}^{\dagger}(\tau)-U_{1}(\tau)\varrho(t)$
$\displaystyle-
U_{2}(\tau)\varrho(t)-\varrho(t)U_{1}^{\dagger}(\tau)-\varrho(t)U_{2}^{\dagger}(\tau)].$
(26)

Here, we have disregarded terms that are higher than second order in $\tau$,
such as $U_{2}(\tau)\varrho(t)U_{2}^{\dagger}(\tau)\propto\tau^{4}$ or
$U_{1}(\tau)\varrho(t)U_{2}^{\dagger}(\tau)\propto\tau^{3}$. The explicit form
of the master equation for PQS is obtained by tracing out the degrees of
freedom in the information environment and utilizing the linear and cyclic
properties of the trace operation.

Following the recipe above, the obtained master equation for the proposed
model reads

$\dot{\varrho}_{0}=-\text{i}r\tau\left[\sum_{i}^{N}g_{i}\left(\langle\sigma_{i}^{-}\rangle
S_{0}^{+}+\langle\sigma_{i}^{+}\rangle S_{0}^{-}\right),\varrho\right]\\\
+\frac{r\tau^{2}}{2}\sum_{i=1}^{N}g_{i}^{2}\left(\langle\sigma_{i}^{+}\sigma_{i}^{-}\rangle\mathcal{L}[S_{0}^{+}]+\langle\sigma_{i}^{-}\sigma_{i}^{+}\rangle\mathcal{L}[S_{0}^{-}]\right)\\\
+2r\tau^{2}\sum_{i<j}^{N^{\prime}}g_{i}g_{j}\left(\langle\sigma_{i}^{+}\rangle\langle\sigma_{j}^{+}\rangle\mathcal{L}_{s}[S_{0}^{-}]+\langle\sigma_{i}^{-}\rangle\langle\sigma_{j}^{-}\rangle\mathcal{L}_{s}[S_{0}^{+}]\right).$
(27)

The above expression provides the mathematical proof of the transfer of
information from the reservoirs to PQS through the calculation of average
values obtained with Pauli operators. The relationship between these average
values and the density matrix elements of the information reservoirs, which
represent the input quantum information, is related to the relevant
parametrizations as

$\displaystyle\mathcal{\rho}_{\mathcal{R}_{i}}$
$\displaystyle=\begin{bmatrix}\frac{1+\cos\theta_{i}}{2}&\frac{e^{-i\phi_{i}}}{2}\sin\theta_{i}\\\
\frac{e^{i\phi_{i}}}{2}\sin\theta_{i}&\frac{1-\cos\theta_{i}}{2}\end{bmatrix}:=\begin{bmatrix}\langle\sigma_{i}^{+}\sigma_{i}^{-}\rangle&\langle\sigma_{i}^{-}\rangle\\\
\langle\sigma_{i}^{+}\rangle&\langle\sigma_{i}^{-}\sigma_{i}^{+}\rangle\end{bmatrix}$
(28)

where $\mathcal{\rho}_{\mathcal{R}_{i}}$ is the $i$th information reservoir
density matrix.

The density matrix representation of the initial PQS state can be stated as

$\varrho_{0}=\frac{1}{3}\begin{bmatrix}P_{11}&C_{12}&C_{13}\\\
C_{21}&P_{22}&C_{23}\\\
C_{31}&C_{32}&P_{33}\end{bmatrix}\equiv\ket{PQS}_{0~{}0}\bra{PQS}$ (29)

where
$\ket{PQS}_{0}=\frac{1}{\sqrt{3}}(c_{1}\ket{e_{1}}+c_{2}\ket{e_{2}}+c_{3}\ket{g})$
with $\bra{e_{1}}=(1\quad 0\quad 0)$, $\bra{e_{2}}=(0\quad 1\quad 0)$, and
$\bra{e_{3}}=(0\quad 0\quad 1)$ being the dual form of the eigenbasis of
$S_{z}$ for $J=1$. Note that $P_{nn}=c_{n}c_{n}^{*}$ and
$C_{nm}=c_{n}c_{m}^{*}$. Following these notations, the corresponding Bloch
equations read

$\displaystyle\langle{\dot{S_{x}}}\rangle=$
$\displaystyle\frac{1}{3\sqrt{2}}\left(2\gamma^{-}_{3}C_{21}-\gamma^{+}_{2}C_{21}+2\gamma^{+}_{3}C_{12}-\gamma^{-}_{2}C_{23}+2\gamma^{-}_{3}C_{32}\right)$
$\displaystyle+\frac{1}{3\sqrt{2}}\left(-\gamma^{-}_{2}C_{32}+2\gamma^{+}_{3}C_{23}\right)+\frac{{\langle
S_{z}\rangle}_{0}}{\sqrt{2}}\left(\gamma^{-}_{1}-\gamma^{+}_{1}\right)$ (30)
$\displaystyle\langle{\dot{S_{y}}}\rangle=$
$\displaystyle\frac{\text{i}}{3\sqrt{2}}\left(2\gamma^{-}_{3}C_{21}+\gamma^{+}_{2}C_{21}-2\gamma^{+}_{3}C_{12}-\gamma^{-}_{2}C_{23}+2\gamma^{-}_{3}C_{32}\right)$
$\displaystyle+\frac{i}{3\sqrt{2}}\left(\gamma^{-}_{2}C_{32}-2\gamma^{+}_{3}C_{23}\right)+i\frac{{\langle
S_{z}\rangle}_{0}}{\sqrt{2}}\left(\gamma^{-}_{1}+\gamma^{+}_{1}\right)$ (31)
$\displaystyle\langle\dot{S_{z}}\rangle=$
$\displaystyle\gamma^{+}_{1}C_{12}-\gamma^{-}_{1}P_{22}+2\gamma^{+}_{2}\left(P_{22}+P_{33}\right)-2\gamma^{-}_{2}\left(P_{11}+P_{22}\right)$
$\displaystyle-\gamma^{-}_{1}C_{32}+i\gamma^{+}_{1}C_{23}).$ (32)

As previously stated, the presented model gives the binary classification
decision in a steady state. Therefore, the steady state solutions of the
master equation and the Bloch equations should be obtained. The steady state
of the PQS density matrix is expressed in Eq. 14 as the solution of
$\dot{\varrho}_{0}=0$. Solutions of the Bloch equations in the steady state
found to be as

$\displaystyle\langle S_{z}\rangle^{ss}=3{\langle S_{z}\rangle}_{0}$
$\displaystyle=\frac{(\sum_{i=1}^{N}g^{2}_{i}p_{e_{i}})^{2}-(\sum_{i=1}^{N}g^{2}_{i}p_{g_{i}})^{2}}{(\sum_{i=1}^{N}g^{2}_{i}p_{e_{i}})^{2}+(\sum_{i=1}^{N}g^{2}_{i}p_{g_{i}})^{2}+\sum_{i,j=1}^{N}g^{2}_{i}g^{2}_{j}p_{e_{i}}p_{g_{j}}}$
$\displaystyle\langle
S_{z}\rangle^{ss}=\frac{g_{1}^{4}\langle\sigma_{z}^{1}\rangle+g_{2}^{4}\langle\sigma_{z}^{2}\rangle+2g_{1}^{2}g_{2}^{2}(p_{e_{1}}p_{e_{2}}-p_{g_{1}}p_{g_{2}})}{g_{1}^{4}(1-p_{e_{1}}p_{g_{1}})+g_{2}^{4}(1-p_{e_{2}}p_{g_{2}})+g_{1}^{2}g_{2}^{2}(1+p_{g_{1}}p_{g_{2}})}$
(33) $\displaystyle\langle S_{y}\rangle^{ss}=\text{i}\frac{3\langle
S_{z}\rangle_{0}}{\sqrt{2}}\left(\gamma^{-}_{1}+\gamma^{+}_{1}\right)$ (34)
$\displaystyle\langle S_{x}\rangle^{ss}=\frac{3\langle
S_{z}\rangle_{0}}{\sqrt{2}}\left(\gamma^{-}_{1}-\gamma^{+}_{1}\right)$ (35)

where ${\langle S_{z}\rangle}_{0}=\frac{1}{3}(P_{11}-P_{33})$, and $p_{e_{i}}$
and $p_{g_{i}}$ are, the diagonal elements of the reservoir units’ density
matrix.

The expressions for the steady magnetization and the Bloch solutions of PQS
can be obtained using the same methods by considering higher values of $J$ in
Eq. 20. Unfortunately, the expressions obtained with higher spin values are
too complicated to be generalized for a general $J$ value, so we will only
share the graph we obtained for the cost function using the analytical results
we obtained up to $J=5/2$. However, as stated in the main text, higher values
of $J$ for PQS result in a stronger non-linear response in the steady state
and provide a faster decrease in the cost function, which will be further
discussed below.

Figure 6:  Cost function minimization against iteration steps depending on
different values of $J$. The spin angular quantum numbers are $J=1/2$ for the
blue solid line, $J=1$ for the orange dotted line, $J=3/2$ for the green dash-
dotted line, and $J=5/2$ for the red dashed line. For the both of them the
initial values of the reservoirs are $\langle\sigma_{z}^{1}\rangle=0.95$,
$\langle\sigma_{z}^{2}\rangle=-0.11$, $g_{1}=0.001$, $g_{2}=0.04$ and $\langle
S_{z}^{0}\rangle_{des}^{ss}=0.42$, respectively.

##  Appendix C Derivation of the cost function

In this section, we analyze the derivation of the cost function using Eqs.
(16), (17) and (18). To start, we extend Eq. (18) by taking the partial
derivative with respect to $g_{i}$ as

$\frac{\partial C}{\partial{g}_{i}}=(\langle
S_{z}^{0}\rangle_{des}^{ss}-\langle
S_{z}^{0}\rangle_{act}^{ss})(-\frac{\partial\langle
S_{z}^{0}\rangle_{act}^{ss}}{\partial{g}_{i}}).$ (36)

It’s important to note that in the current example, the total angular momentum
of the probe system is $J=1$ and PQS is connected to two information
reservoirs. As a result, we use the steady magnetization, represented by $A$,
as the actual value of the cost function as

$\displaystyle A$ $\displaystyle=\langle S_{z}^{0}\rangle_{act}^{ss}$
$\displaystyle=\frac{g_{1}^{4}\langle\sigma_{z}^{1}\rangle+g_{2}^{4}\langle\sigma_{z}^{2}\rangle+2g_{1}^{2}g_{2}^{2}(p_{e_{1}}p_{e_{2}}-p_{g_{1}}p_{g_{2}})}{g_{1}^{4}(1-p_{e_{1}}p_{g_{1}})+g_{2}^{4}(1-p_{e_{2}}p_{g_{2}})+g_{1}^{2}g_{2}^{2}(1+p_{g_{1}}p_{g_{2}})}.$
(37)

Here, $\langle\sigma_{z}^{i}\rangle=p_{e_{i}}-p_{g_{i}}$. For simplicity, we
denote the numerator in eq. (C) as $X(g_{1},g_{2})$ and the denominator as
$Y(g_{1},g_{2})$. According to the recipe to derive the cost function, the
partial derivatives with respect to $g_{1}$ and $g_{2}$ separately obtained as

$\displaystyle\frac{\partial A}{\partial{g}_{1}}=\frac{\frac{\partial
X(g_{1},g_{2})}{\partial{g}_{1}}Y(g_{1},g_{2})-\frac{\partial
Y(g_{1},g_{2})}{\partial{g}_{1}}X(g_{1},g_{2})}{{(Y(g_{1},g_{2}))}^{2}}$
$\displaystyle\frac{\partial A}{\partial{g}_{2}}=\frac{\frac{\partial
X(g_{1},g_{2})}{\partial{g}_{2}}Y(g_{1},g_{2})-\frac{\partial
Y(g_{1},g_{2})}{\partial{g}_{2}}X(g_{1},g_{2})}{{(Y(g_{1},g_{2}))}^{2}}$ (38)

In our example, the desired magnetization $\langle
S_{z}^{0}\rangle_{des}^{ss}=0.4$ is an arbitrary value in the cost function.
By substituting Eqs. (C) and (C) into Eq. (36), and Eq. 18 into Eq. 17 we
obtain the explicit form of the cost function and the coupling strengths
update expressions as

$\displaystyle(g_{1})_{k+1}=(g_{1})_{k}+\delta(g_{1})_{k}$
$\displaystyle(g_{2})_{k+1}=(g_{2})_{k}+\delta(g_{2})_{k}.$ (39)

In the main text, we compared the cost function in Fig.4 for $J=1/2$ and
$J=1$, and in Fig.6, we have compared it for higher values of $J$. Since
analytical expressions become cumbersome for higher values of $J$, we only
provide the graphical results as shown Fig. 6. The obtained results show that
for the same learning rate values, changing only the $J$ value rapidly affects
the convergence in the iteration.

## Acknowledgments

The authors acknowledge support from the Scientific and Technological Research
Council of Turkey (TÜBİTAK-Grant No. 120F353). The authors also wish to extend
special thanks to the Cognitive Systems Lab in the Department of Electrical
Engineering providing the atmosphere for motivational and stimulating
discussions.

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|  Ufuk Korkmaz received the BSc, MSc and PhD degrees from the Ondokuz Mayıs
University (OMU), Physics Department, in 2006, 2010 and 2014 respectively. He
worked as a Postdoctoral Researcher at Istanbul Technical University (ITU) in
2018-2019. He starting researches as Post-Doc in Istanbul Technical University
(ITU) in 2021. His research interests are IR and UV spectroscopy, X-ray single
crystal diffraction, Understanding the nature of H bonds in supramolecular
structure, Quantum Mechanics and Quantum information theory.
---|---
|  Deniz Türkpençe received M.S., and Ph.D. degrees in Atomic Physics from
from Ondokuz Mayıs University in 2007 and 2013. He worked as a visiting
student researcher at Dortmund Technical University for 1 year with a YÖK
scholarship to conduct research and examination abroad related to his doctoral
thesis. He worked as a Research Assistant at Koç University in April 2014 and
took part in an international project. He completed her studies at Koç
University in 2016. He worked as a postdoctoral researcher in the Cognitive
Systems Laboratory of the Electrical Engineering Department of Istanbul
Technical University between 2017-2018. He started to work as a Lecturer in
the Department of Electrical Engineering at Istanbul Technical University in
March 2018. In November 2019, he was awarded the title of Associate Professor
by the Turkey Inter-university Board. Currently he is a faculty member in
Electric Electronic Faculty of Istanbul Technical University.
---|---