# Error-disturbance uncertainty relations in a superconducting quantum
processor

Tingrui Dong<EMAIL_ADDRESS>Research Institute of Electrical
Communication, Tohoku University, Sendai 980-8577, Japan  Department of
Electronic Engineering, Graduate School of Engineering, Tohoku University,
Sendai 980-8579, Japan  Soyoung Baek<EMAIL_ADDRESS>Research Institute
of Electrical Communication, Tohoku University, Sendai 980-8577, Japan
Fumihiro Kaneda  Department of Physics, Graduate School of Science, Tohoku
University, Sendai 980-8579 Japan  Keiichi Edamatsu  Research Institute of
Electrical Communication, Tohoku University, Sendai 980-8577, Japan

###### Abstract

We experimentally test the error-disturbance uncertainty relation (EDR) in
generalized, variable strength measurements of superconducting qubits on a
NISQ processor. Making use of sequential weak measurements that keeps the
initial signal state practically unchanged prior to the main measurement, we
demonstrate that the Heisenberg EDR is violated, yet the Ozawa and Branciard
EDRs are valid throughout the range of measurement strengths from no
measurement to projection measurement. Our results verify that universal EDRs
are valid even in a noisy quantum processor and will stimulate research on
measurement-based quantum information and communication protocols using a NISQ
processor.

††preprint: APS/123-QED

##  I Introduction

The error-disturbance uncertainty relation (EDR) is a cornerstone of quantum
mechanics since it describes the fundamental limitations on the achievable
accuracy of quantum measurement. Furthermore, with the fast development of
quantum technologies such as quantum metrology [1], quantum communications
[2], and quantum computations [3], it has become more crucial for us to
understand the ultimate quantum limitations in the attainable accuracy of
quantum measurement.

In 1927, Heisenberg [4] formulated that any measurement of the particle’s
position $x$ with the error $\epsilon(x)$ causes the disturbance $\eta(p)$ on
momentum $p$ satisfying $\epsilon(x)\eta(p)\geq\hbar/2$. The generalized form
of Heisenberg’s EDR for an arbitrary pair of observables $A$ and $B$ is given
by

$\epsilon(A)\eta(B)\geq C,$ (1)

where the lower bound $C$ is determined by the commutation relation between
$A$ and $B$, i.e., $C=|\langle[\,A,B]\,\rangle|/2$, $[\,A,B]\,=AB-BA$, and
$\langle...\rangle$ stands for the mean value in a given state.

In 2003, Ozawa [5] proposed an alternative EDR that is theoretically proven to
be universally valid

$\epsilon(A)\sigma(B)+\sigma(A)\epsilon(B)+\epsilon(A)\eta(B)\geq C,$ (2)

where $\sigma(A)=\sqrt{\langle A^{2}\rangle-\langle A\rangle^{2}}$ is the
standard deviation. Ozawa’s relation has two additional correlation terms, the
presence of which allows the error-disturbance product $\epsilon(x)\eta(p)$ to
be below the lower bound of Eq. (1), implying the violation of Heisenberg’s
EDR. Note that there are alternative approaches for the measurement
uncertainty relations based on different definitions of error and disturbance
[6, 7, 8, 9, 10, 11]. For example, Busch et al. [12] defined measurement error
based on the RMS distance of the distributions between the original and the
measurement observables and derived EDR for the qubit case.

Most recently, Branciard [13] has improved the Ozawa’s EDR and obtained even
tighter EDR such as

$\displaystyle\left[\epsilon(A)^{2}\sigma(B)^{2}+\sigma(A)^{2}\eta(B)^{2}\right.$
(3)
$\displaystyle\left.+2\epsilon(A)\eta(B)\sqrt{\sigma(A)^{2}\sigma(B)^{2}-C^{2}}\right]^{1/2}\geq
C.$

Due to its fundamental and practical importance, EDRs have been investigated
in various quantum systems such as polarized neutrons [14, 15], photons [16,
17, 18, 19, 20], and trapped ions [21]. Experimental tests of EDRs commonly
proved that Heisenberg’s EDR can be violated in some systems, yet Ozawa’s and
Branciard’s relations are always validated.

Meanwhile, recent noisy intermediate-scale quantum (NISQ) technology is
regarded as a step toward more powerful quantum technologies to be developed
in the future [22]. Although NISQ devices are not capable of fault-tolerant
operations, NISQ technology gives us new tools for exploring the physics of
many entangled particles by allowing consecutive entangling operations between
neighboring qubits. Proof-of-principle demonstrations of new quantum protocols
using a NISQ processor also proves its prospect as a testbed of new quantum
tasks [23, 24, 25].

In this paper, we focus on availability of a NISQ processor as a quantum
system to verify quantum measurement protocols. To prove it, we experimentally
test the EDRs in generalized, variable strength measurements of
superconducting qubits on the IBM Quantum (IBM-Q) processor [26]. Making use
of sequential weak measurements that keeps the initial signal state
practically unchanged prior to the main measurement, we demonstrate that the
Heisenberg EDR is violated, yet the Ozawa and Branciard EDRs are valid
throughout the range of the measurement strength. To the best of our
knowledge, this is the first investigations of EDRs in a NISQ processor and
our results verify that universal EDRs are valid even in noisy quantum
processor. We anticipated that our study will stimulate research on various
quantum measurement protocols using a quantum processor.

##  II Scheme

Figure 1: Schematic diagram to test EDRs using the weak-probe method for
single-qubit observables $A$ and $B$. Weak probes indirectly measure $A$ and
$B$ with a weak measurement strength, denoted as weak probe $A$ (WPA) and weak
probe $B$ (WPB), prior to the main measurement operated by the measurement
apparatus (MA).We use sufficiently weak measurement strength for the WP that
causes very little disturbance on the system state.

The error and disturbance used to derive EDRs are defined from indirect
measurement model, depicted as a “measurement apparatus” (MA) in Fig. 1. In
that model, measurement of the system state is implemented by the interaction
that (partially) correlates system and meter states, denoted by $U$, followed
by the measurement of the meter states. Using stronger interaction, one can
extract more information from the system state at the cost of a larger
disturbance of the system state. From the measurement model, the error and
disturbance are defined as [5]

$\displaystyle\epsilon(A)$ $\displaystyle=\langle[U^{\dagger}(I\otimes
M)U-A\otimes I)]^{2}\rangle^{1/2},$ (4) $\displaystyle\eta(B)$
$\displaystyle=\langle[U^{\dagger}(B\otimes I)U-B\otimes
I)]^{2}\rangle^{1/2},$

where the average is taken over the system-meter composite state on input. $U$
is a unitary operator that provides interaction between the system and meter
states, and $M$ is the meter observable. The definition of $\epsilon(A)$ is
uniquely derived from the classical notion of root-mean-square error if
$U^{\dagger}(I\otimes M)U$ and $A\otimes I$ commute, and otherwise, it is
considered as a natural quantization of the notion of classical root-mean-
square error [20]. The definition of $\eta(B)$ is derived analogously.

For experimental tests of EDRs, we employ the “weak-probe method”[27, 28]
which has been used in the successful investigation of EDRs on single-photon
polarizations by our co-authors [20]. Fig. 1 shows the schematic diagram to
test EDRs using the weak-probe method for single-qubit observables $A$ and
$B$. Weak probes indirectly measure $A$ and $B$ with a weak measurement
strength, denoted as weak probe $A$ (WPA) and weak probe B (WPB), prior to the
main measurement operated by the measurement apparatus (MA). When the
measurement strength is sufficiently small, the system state is sent to the MA
without being disturbed by the WP. As Lund and Wiseman [27], and Ozawa [28]
pointed out, the error (disturbance) defined by Eq. (4) is given by the “weak-
valued root-mean-square difference” between measurement outcomes of the
WPA(WPB) and the MA (postmeasurement of B).

$\displaystyle\epsilon(A)^{2}$
$\displaystyle=\sum_{i,f}(a_{i}-a_{f})^{2}P_{wv}(a_{i},a_{f}),$ (5)
$\displaystyle\eta(B)^{2}$
$\displaystyle=\sum_{i,f}(b_{i}-b_{f})^{2}P_{wv}(b_{i},b_{f}),$

where $P_{wv}(a_{i},a_{f})$ is the weak-valued joint probability distribution
taking the outcomes $a_{i}$ in the WPA and $a_{f}$ in the MA. As described
later, we can experimentally estimate $P_{wv}(a_{i},a_{f})$, and thus
$\epsilon(A)$, by evaluating the probability distribution $P(a_{i},a_{f})$
that we take the outcomes $a_{i}$ and $a_{f}$. Similarly, $\eta(B)$ is given
by $P_{wv}(b_{i},b_{f})$ by taking outcomes $b_{i}$ in the WPB and $b_{f}$ in
the postmeasurement of $B$.

In this paper, we report the experimental test of EDRs for a superconducting
qubit, placed in a NISQ processor using the weak-probe method. One of the
advantages of using superconducting qubits over photonic qubits [20] is that
it’s easier to create interactions between multiple qubits. Therefore we can
assign independent qubits for system, probe $A$, probe $B$, meter and all
measurement outcomes to derive error and disturbance can be obtained
simultaneously from a single configuration of the quantum circuit. Note that
WPA and WPB were applied only one at a time in Ref. [20] since same probe
qubit (encoded on single photon’s path) is shared for both WPA and WPB to
measure the system (polarization) state of the same photon.

##  III Experiment

Figure 2: (a) Quantum circuit model of testing EDRs for single-qubit
observables $A=Z$ and $B=X$. Our implementation is separated into the state
preparation, weak probe $Z$ (WPZ), weak probe $X$ (WPX), measurement apparatus
(MA), and post $X$ measurement. The system qubit prepared in
$|R\rangle=(|0\rangle-i|1\rangle)\sqrt{2}$, the eigenstate of $Y$, is
indirectly measured by WPZ, WPX, MA, and Post $X$ measurement is followed. The
error (disturbance) is given by the “weak-valued root-mean-square difference”
between measurement outcomes of the WPZ and the MA (WPX and post $X$
measurement). In the experiment, the measurement strength of both WP is set to
$\cos\theta_{w}=0.05$ and the measurement strength of the MA, $\cos\theta$,
was varied from 0 (no measurement) to 1 (projection measurement). (b)
Illustration of qubit assignment on the 7-qubit IBM-Q Falcon processor for the
experimental test of EDRs. The system qubit interact independently with their
neighbors (Probe $Z$, Probe $X$, Meter) in turn. (c) Calibration profile
(averaged over 4 qubits) provided by the IBM-Q processor.

Our implementation is based on the quantum circuit depicted in Fig. 2(a). We
take the system observable to be measured as $A=Z$ and $B=X$, where $X$, $Y$,
and $Z$ denote the Pauli matrices, and $\\{|0\rangle,|1\rangle\\}$ are
eigenbasis of $Z$ with the eigenvalues of {1, –1}. We define weak probe
observables in WPZ and WPX as $Z_{i}$ and $X_{i}$, respectively. Meter
observable in MA and post $X$ measurement observable for the system are
defined as $Z_{f}$ and $X_{f}$, respectively. We then use the following
notation as the measurement outcomes: $a_{i,f}=z_{i,f}=\pm 1$ and
$b_{i,f}=x_{i,f}=\pm 1$.

We employ three cascaded circuits as WPZ, WPX, and MA. All three circuits work
in the same manner. In the MA, meter qubit initialized to $|0\rangle$ is
rotated by
$R_{y}(\theta)=\exp(-i\frac{\theta}{2}Y)=\begin{pmatrix}\cos\frac{\theta}{2}&-\sin\frac{\theta}{2}\\\
\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{pmatrix}$, where
$0\leq\theta\leq\frac{\pi}{2}$. Then, the meter qubit is subjected to a
controlled-NOT (CNOT) operation with the system qubit. The positive operator
valued measure (POVM) elements corresponding to the outcomes of $z_{f}=\pm 1$
are

$\displaystyle\Pi_{z_{f}=\pm 1}=\frac{1}{2}(I\pm(\cos\theta)Z).$ (6)

Here $\cos\theta$ is the measurement strength of the MA. By changing
$\cos\theta$ from 0 to 1, $\Pi_{z_{f}=\pm 1}$ change from identity (no
measurement) to projector (strong measurement). The WP works in exactly the
same manner as the MA except that the measurement strength of the WP is
$\cos\theta_{w}$. In order to keep the WP’s measurement strength sufficiently
weak, $\theta_{w}$ should be close to $\pi/2$. In addition, two Hadamard gates
($H$) are inserted to the system qubit before and after the CNOT in the WPX,
where weak measurement for $X$ is taken.

To take the most stringent test of the EDRs, we chose the system state as
$|R\rangle=(|0\rangle-i|1\rangle)\sqrt{2}$, an eigenstate of $Y$, so that the
rhs of the EDRs become the maximum value in the qubit measurement:
$C=|\langle[Z,X]\rangle|/2=|\langle Y\rangle|=1$. To prepare the system state
in $|R\rangle$, as shown in the preparation part of Fig. 2(a), the system
qubit initialized to $|0\rangle$ is rotated by a $R_{x}(\theta=\pi/2)$ gate,
where
$R_{x}(\theta)=\exp(-i\frac{\theta}{2}X)=\begin{pmatrix}\cos\frac{\theta}{2}&-i\sin\frac{\theta}{2}\\\
-i\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{pmatrix}$. A meter for MA and
weak probes for WPZ and WPX are initialized to $|0\rangle$, interact with
system qubit in sequence, and finally measured in $\\{|0\rangle,|1\rangle\\}$
basis.

In the experiment, the measurement strength for both WPZ and WPX was set to
$\cos\theta_{w}=0.05$ that produced a very small disturbance in the initial
state. We expect the rhs of EDRs, $C=4/(3+\cos 2\theta_{w})-1=0.995$, which is
closed to the ideal value 1. Then, the signal photon was subjected to the MA
and post $X$ measurement, consisting of a Hadamard gate and projection on $Z$
basis, where measurement basis is switched from $Z$ to $X$. From Eq. (5) and
the expression of weak-valued joint probability distribution [27], the error
and disturbance are given by

$\displaystyle\epsilon(Z)^{2}$
$\displaystyle=2\left(1-\frac{1}{\cos\theta_{w}/2}\sum_{z_{i},z_{f}}z_{i}z_{f}P(z_{i},z_{f})\right),$
(7) $\displaystyle\eta(X)^{2}$
$\displaystyle=2\left(1-\frac{1}{\cos\theta_{w}/2}\sum_{x_{i},x_{f}}x_{i}x_{f}P(x_{i},x_{f})\right),$

where the subscript $z_{i}$, $z_{f}$, $x_{i}$, $x_{f}$ denotes the outcomes of
the WPZ, MA, WPX, and post $X$ measurement, respectively. By employing this
quantum circuit, both the error and disturbance can be experimentally
quantified simultaneously.

Experiments were conducted on the 7-qubit IBM Quantum Falcon Processor
(IBM_perth) [26] under the open-source framework Qiskit. Fig. 2(b) illustrates
the qubit assignments for the EDR test. Each node (circle) represents a
superconducting qubit and CNOT operations can be performed between two qubits
connected by a line. In our implementation, qubit 1 surrounded by three
neighboring qubits is assigned as the system qubit and qubit 0, 2, 3 are
assigned as probe $Z$, probe $X$, and meter, respectively. As our EDR test
presented in Fig. 2(a) only requires 4 qubits, qubit 4, 5, and 6 were not
used. Note that we can apply CNOT operations between any two qubits of 7
qubits (even between non-neighboring qubits) after consecutive swap operations
which would introduce additional gate errors.

##  IV Results

Figure 3: Experimentally obtained error $\epsilon(Z)$ (blue circles) and
disturbance $\eta(X)$ (red squares) as functions of the measurement strength
$\cos\theta$. Solid curves are the theoretically calculated error and
disturbance for perfect implementation of the quantum circuit presented in
Fig. 2. Classically simulated error (green triangles) and disturbance (purple
diamonds) are obtained from IBM open-source framework Qiskit after noise in
the quantum processor is taken into account. See text for details.

To evaluate $\epsilon(Z)$ and $\eta(X)$ using Eq. (7), we experimentally
obtain $P(z_{i},z_{f})$ and $P(x_{i},x_{f})$ by executing the designed quantum
circuit in Fig. 2(a) on the IBM-Q processor 100,000 times and analyzing the
statistics of the measurement outcomes $\\{z_{i},x_{i},z_{f},x_{f}\\}$. The
quantities $\epsilon(Z)$ and $\eta(X)$ averaged for 10 repeated measurements,
$10^{5}\times 10=10^{6}$ measurements in total, are shown in Fig. 3 throughout
the range of the measurement strength from no measurement ($\cos\theta=0$) to
projective measurement ($\cos\theta=1$). The error bars represent the rms
error of 10 repeated measurements. The experiment runs approximately in 40 min
in total. Solid curves are the theoretically calculated error (blue) and
disturbance (red) for perfect implementation of the quantum circuit presented
in Fig. 2(a). Fig. 3 also shows the simulated error (green triangles) and
disturbance (purple diamonds), obtained from IBM open-source framework Qiskit
after the calibration data of the quantum processor is taken into account. The
calibration data contains full information characterizing the quantum
processor such as gate error rates and coherence times. Table in Fig. 2(c)
lists the calibration data (averaged over four qubits) that are expected to
have a major impact on the simulation.

We clearly see the trade-off relation between the error and disturbance: as
the measurement strength increases, $\epsilon(Z)$ decreases while $\eta(X)$
increases. The presence of unavoidable noise in the quantum processor causes
deviation of both the simulation results and the experimental data from the
ideal theoretical expectations. Particularly, when the measurement strength is
gradually reduced to $\cos\theta=0$, ideally we expect no disturbance, i.e.,
$\eta(X)=0$, but the measured disturbance is saturated around $\eta(X)=0.6$
due to intrinsic noisy property of the NISQ device. In the same way, for the
region measurement strength reach to $\cos\theta=1$, the measurement error
$\epsilon(Z)$ is ended up around $0.4$ instead of 0. The experimentally
measured error and disturbance not perfectly but closely follow the
theoretically simulated error and disturbance in which the calibration data is
taken into account. The deviation from the simulation might originate from
additional experimental imperfections and limitations of simulations that
cannot account for all real-time noise of the quantum processor.

Figure 4: Comparison of EDRs’ lower bounds in the error–disturbance plot. Blue
(solid) curve: the Heisenberg bound in Eq. (1). Red (short dashed) curve: the
Ozawa bound in Eq. (2). Purple (long dashed) curve: the Branciard bound in Eq.
(3). Green (dotted-dashed) curve: the stronger Branciard bound in Eq. (8).
Black circles: experimental data shown in Fig. 3. Red squares: experimental
data obtained in EDR test for polarization measurement of single photons [20].
The lower-left side of each bound is the forbidden region by the corresponding
EDR. Each bound was calculated for C=1.

In Fig. 4, we plot the predicted lower bounds of the EDRs in Eqs. (1), (2),
and (3), together with the experimental data. We also plot the stronger
Branciard EDR (green) that is applicable to the case (including ours) where
the system and meter observables are both $\pm 1$ valued and $\langle
A\rangle=\langle B\rangle=0$ (hence $\sigma(A)=\sigma(B)=1$) [13]

$\left[\tilde{\epsilon}(A)^{2}+\tilde{\eta}(B)^{2}+2\tilde{\epsilon}(A)\tilde{\eta}(B)\sqrt{1-C^{2}}\right]^{1/2}\geq
C,$ (8)

where $\tilde{\epsilon}=\epsilon\sqrt{1-\epsilon^{2}/4}$ and
$\tilde{\eta}=\eta\sqrt{1-\eta^{2}/4}$. Under the Heisenberg EDR, the error or
disturbance must be infinite when the other goes to zero, while other EDRs
allow a finite error or disturbance even when the other is zero. The lower-
left side of each bound is the forbidden region by the corresponding EDR. Each
bound was calculated for $C=1$.

We clearly see that the experimental data (black circles) is present in the
lower-left side of Heisenberg EDR and right-upper side of other EDRs.
Therefore, our experimental results demonstrate the clear violation of the
Heisenberg EDR, while the Ozawa and Branciard EDRs are always satisfied
throughout the full range of measurement strength. We see that the Branciard
EDRs are stronger than the Ozawa EDR. Our experimental data were closer to the
stronger Branciard bound (dotted-dashed curve) given in Eq. (8), which could
be saturated by ideal experiments [20], than the other EDR bounds. Note that
we need to keep sufficiently weak measurement strengths for WPZ and WPX
($\cos\theta_{w}=0.05$ in our case) to observe validity of Ozawa and Branciard
EDRs.

For the comparison, we also plot experimental data (red squares) obtained in
the investigations of EDRs for polarization measurement of single photons
[20]. We can see that our data measured for superconducting qubits is further
deviated from the stronger Branciard bound than the measurement data for
single photons. Our data also shows slightly larger error bars compared to the
photonic implementation with a comparable sampling number due to intrinsic
noisy property of NISQ device. By comparing the two data sets, we can see that
the NISQ device introduces additional noise and errors along with the benefit
of introducing multi-qubit interactions easily. This noise will introduce
imperfect control over qubits and will place limitations on realizing the
designed quantum measurement protocols in the quantum processor. Therefore,
developing and leveraging tools for characterizing noise and error, mitigating
them, and verifying quantum processing will be crucial in implementing the
measurement protocols on quantum devices and for further applications [29,
30].

##  V Conclusion

We have experimentally tested the Heisenberg, Ozawa, and Branciard EDRs in
generalized, variable strength measurements of superconducting qubits on a
NISQ processor for the first time. Making use of sequential weak measurements
that keeps the initial signal state practically unchanged prior to the main
measurement, we demonstrate that the Heisenberg EDR is violated, yet the Ozawa
and Branciard EDRs are valid throughout the range of measurement strengths
from no measurement to projection measurement. In particular, our results were
closer to the stronger Branciard bound (dotted-dashed curve) given in Eq. (8),
than the other EDR bounds, similar to what was seen in the photonic
implementation in [20]. Such experimental investigation of the EDRs will be of
demanded importance not only in understanding fundamentals of physical
measurement but also in developing novel measurement protocols for quantum
information and communications.

It is important to acknowledge that the presence of intrinsic noise of NISQ
system causes deviations between the experimental results and the theoretical
expectation, which cannot be fully compensated for even with extensive
sampling to reduce the statistical error. Nevertheless, the NISQ processor
proves its prospect as a testbed of new quantum tasks, by performing proof-of-
principle demonstration of quantum measurement protocols and providing results
that tend to follow the theoretical predictions. Particularly, the capability
of introducing multi-qubit entangling operations of NISQ system would allow
experimental investigations of various multi-qubit measurement protocols [31,
32, 33]. By addressing and exploring the existing challenges posed by noise
and errors in NISQ systems [29, 30], we can further enhance the fidelity and
reliability of quantum experiments and pave the way for future advancements in
quantum technologies.

###### Acknowledgements.

This research was supported by MEXT Quantum Leap Flagship Program (MEXT
Q-LEAP) Grant No. JPMXS0118067581. T.D. was supported by JST, the
establishment of university fellowships towards the creation of science
technology innovation Grant No. JPMJFS2102. We acknowledge the use of IBM
Quantum services for this work. The views expressed are those of the authors,
and do not reflect the official policy or position of IBM or the IBM Quantum
team. T.D. and S.B. contributed equally to this work.

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# End-to-End Graph-Sequential Representation Learning for Accurate
Recommendations

Vladimir Baikalov<EMAIL_ADDRESS>0009-0009-4864-2305 Skolkovo
Institute of Science and TechnologyNobel St. 3, Skolkovo Innovation
CenterMoscowRussia143025 and  Evgeny Frolov<EMAIL_ADDRESS>0000-0003-3679-5311 Artificial Intelligence Research Institute
(AIRI)MoscowRussia Skolkovo Institute of Science and TechnologyNobel St. 3,
Skolkovo Innovation CenterMoscowRussia143025

(2024)

###### Abstract.

Recent recommender system advancements have focused on developing sequence-
based and graph-based approaches. Both approaches proved useful in modeling
intricate relationships within behavioral data, leading to promising outcomes
in personalized ranking and next-item recommendation tasks while maintaining
good scalability. However, they capture very different signals from data.
While the former approach represents users directly through ordered
interactions with recent items, the latter aims to capture indirect
dependencies across the interactions graph. This paper presents a novel multi-
representational learning framework exploiting these two paradigms’ synergies.
Our empirical evaluation on several datasets demonstrates that mutual training
of sequential and graph components with the proposed framework significantly
improves recommendations performance.

Recommender Systems, Sequential Learning, Graph Neural Networks, Contrastive
Learning

††journalyear: 2024††copyright: acmlicensed††conference: Companion Proceedings
of the ACM Web Conference 2024; May 13–17, 2024; Singapore,
Singapore††booktitle: Companion Proceedings of the ACM Web Conference 2024
(WWW ’24 Companion), May 13–17, 2024, Singapore, Singapore††doi:
10.1145/3589335.3651499††isbn: 979-8-4007-0172-6/24/05††ccs: Information
systems Recommender systems

Figure 1. The proposed MRGSRec model consists of nine blocks, colored in
yellow and red: (i) Item Embedding Layer and (ii) User Embedding Layer that
encodes input item and user IDs. (iii) Sequential Encoder and (iv) Graph
Encoder that aggregate information leveraging different data representations.
(v) Fusing Layer capable of mixing representations from both encoders. (vi)
Local Objective, (vii) Global Objective, (viii) Fusing Objective, and (ix)
Contrastive Objective used for model training

MRGSRec model overview

##  1\. Introduction

The primary objective of recommender systems is to offer users personalized
recommendations aligned with their interests by leveraging past engagements
between users and items. In recent years, deep learning-based solutions have
emerged as dominant players in the evolution of recommender systems, causing
the rapid development of diverse techniques and methods (Zhang et al., 2019;
Gao et al., 2023). These solutions can construct accurate and expressive
representations of users and items and address various recommendations tasks
(Kang and McAuley, 2018; Sun et al., 2019; He et al., 2020). In most cases,
input data for such models may be represented as a user-specific sequence of
recently interacted items. Leveraging only this information, however, has its
limitations: concerned with individual user interactions, this approach
captures patterns only for direct dependencies, which may lack sensitivity to
a broader context provided by additional interdependencies across entities. On
the contrary, Graph Neural Networks (GNNs) (Wang et al., 2019; He et al.,
2020) provide a powerful way of explicitly modeling information through
indirect dependencies, but adapting them to sequential setup can be
cumbersome. Each approach therefore addresses the same problem of learning
user preferences by leveraging patterns from independent data representations.
To our knowledge, only one study has attempted to integrate both
representations within a single model (Wang et al., 2022). However, it was
constrained by a sequential encoder architecture. In contrast, we aim to
design a general approach, allowing the useage of various versions of
sequential and graph-based components, constrained only by encoder signatures.
Our contributions are as follows:

  * •

We propose MRGSRec (Multi-Representation Graph-based and Sequence-based
approach for Recommendations), novel architecture design that integrates
graph-based and sequential representations into an end-to-end framework;

  * •

We present a scheme for knowledge exchange between different representations
based on contrastive learning;

  * •

We assemble a composite loss function that maximizes the cumulative effect of
all components of the architecture.

##  2\. Related work

We categorize existing models into three types: sequence-based, graph-based,
and multi-representational (models that integrate both categories into one
algorithm).

#### Sequential Recommendations

Transformer architectures (Vaswani et al., 2023) have proven to be effective
across many domains (Brown et al., 2020; Dosovitskiy et al., 2021; Carion et
al., 2020), including their utility in sequence modeling. Their unique
properties have contributed to their success in various applications,
including recommender systems, leading to the development of top-performing
methods. Examples include SASRec (Kang and McAuley, 2018) and BERT4Rec (Sun et
al., 2019) demonstrating state-of-the-art performance. The effectiveness of
these attention-based architectures in modeling sequential data sparked many
further improvements and adaptations aimed at enhancing recommendations
quality (Xie et al., 2021; Qiu et al., 2022; Klenitskiy and Vasilev, 2023).
Notably, the usage of cross-entropy loss for SASRec architecture (SASRec+)
(Klenitskiy and Vasilev, 2023) has demonstrated superior performance compared
to other techniques.

#### Graph-based Recommendations

An alternative approach to match user preferences involves using graph
representations from user-item interactions. Graph Convolution Networks
(GCNs), notably NGCF (Wang et al., 2019) and LightGCN (He et al., 2020), have
been recognized as highly effective solutions within this domain, with NGCF
being foundational and LightGCN serving as both an ancestor and an
enhancement.

#### Multi-Representational Recommendations

Both previous ideas have their limitations: sequential recommendations excel
at extracting patterns from user-specific interactions but overlook broader
interactions. Conversely, GCNs grasp the indirect interdependencies but
struggle to leverage them within user-specific history as precise as
sequential models. To the best of our knowledge, only one approach, namely
MCLSR (Wang et al., 2022) attempts to bridge this gap by using both paradigms,
however, the fusing between different representations is used only during the
training process by propagating patterns from indirect interdependencies into
item representations, while during the inference only sequential part is
leveraged. Also, MCLSR uses an architecture with limited capacity for the
sequential part of the model. MRGSRec enhances the idea from MCLSR by
utilizing both sequence-based and graph-based representations during the whole
training and inference process and providing flexible choices of different
encoders for both sequence-based and graph-based parts by having only
signature constraints. By combining various architectures, this design offers
an end-to-end solution.

##  3\. Proposed approach

This section presents the problem statement for the considered recommendation
scenario and explains in detail the components of the MRGSRec framework,
providing insights into their roles in enhancing model quality. Model overview
is shown in Fig. 1.

We consider a standard next-item prediction task. The goal is to establish
relations between a set $\mathcal{U}$ of $M$ users and a set $\mathcal{I}$ of
$N$ items in a sequential manner, i.e., for a given user $u\in\mathcal{U}$
described by a sequence of the previously interacted items
$S_{u}=(i^{u}_{1},\,i^{u}_{2},\,\dots,\,i^{u}_{|S_{u}|})$,
$i^{u}_{k}\in\mathcal{I}$, the task is to predict the next item
$i^{u}_{|S_{u}|+1}$ the user is likely to interact with. Below we outline the
components of our solution.

###  3.1. Embedding layers

Given a user ID $u$ and an interacted sequence $S_{u}$ of item IDs, we keep
only the last $c$ interactions from $S_{u}$. We then convert the user and
remaining item IDs into embeddings using learnable embedding layers
$U_{e}:\mathbb{N}\rightarrow\mathbb{R}^{d}$ and
$I_{e}:\mathbb{N}\rightarrow\mathbb{R}^{d}$ for user ID and item IDs
respectively. This process produces a user embedding
$e^{u}=U_{e}(u)\in\mathbb{R}^{d}$ and a sequence of item embeddings
$\left(e^{u}_{1},e^{u}_{2},\dots,e^{u}_{c}\right)=E^{u}\in\mathbb{R}^{c\times
d}$, where $e^{u}_{k}=I_{e}(i^{u}_{k})$ and $d$ denotes the embedding size. To
enhance item representation with position-specific information, we add
learnable positional embeddings similar to existing approaches (Kang and
McAuley, 2018; Sun et al., 2019; Xie et al., 2021; Qiu et al., 2022).

###  3.2. Sequential Encoder

To capture the direct interactions, we utilize a sequential encoder block
within the MRGSRec framework. This block is tasked with producing two
representations: an aggregated _local behavioral representation_ and _local
enriched embeddings_ for all input items, with dimensions $\mathbb{R}^{d}$ and
$\mathbb{R}^{c\times d}$ respectively. Without integrating user-specific data,
the encoder output lacks personalized context, producing user-agnostic output.
We address this limitation by concatenating $e^{u}$ and $E^{u}$ before
applying the encoder, allowing the sequential encoder to enrich each item
representation with user-specific context. As specified in the equation above,
then we extract _local behavioral representation_ $e_{l}^{u}$ and _local
enriched embeddings_ $E_{l}^{u}$:

(1)
$e_{l}^{u}\,\|\,E_{l}^{u}=\text{SequentialEncoder}\left(e^{u}\,\|\,E^{u}\right),$

where SequentialEncoder is an encoder part from the transformer architecture
(Vaswani et al., 2023), that processes an input matrix of size $(c+1)\times d$
to produce an enriched representation of the same size.

###  3.3. Graph Encoder

To incorporate the ability to capture indirect interdependencies, we propose
employing GCNs, a method used in various prior works (Gao et al., 2023; Wang
et al., 2019; He et al., 2020). Within the MRGSRec framework, applying a graph
encoder requires similar constraints as discussed in the sequential encoder
section. Using GCNs involves introducing two input matrices: the initial
representation matrix for all users and items
$E^{(0)}_{g}\in\mathbb{R}^{(M+N)\times d}$ and the graph adjacency matrix
$A\in\mathbb{R}^{(M+N)\times(M+N)}$. To construct $E^{(0)}_{g}$, we utilize
embeddings for all users and items from the previously defined embedding
layers $U_{e}$ and $I_{e}$ by concatenating them. For adjacency matrix
initialization, we follow the approach from prior works (Wang et al., 2019; He
et al., 2020). Initially, we introduce the interaction matrix
$R\in\mathbb{R}^{M\times N}$, in which elements $r_{ij}$ are set to 1 when
there is an interaction between user $i$ and item $j$, and 0 otherwise. Using
matrix $R$, we can define matrix $A$ in the block notation:

(2) $A=\begin{pmatrix}0&R\\\ R^{\top}&0\\\ \end{pmatrix}.$

To apply graph encoder, we use $E^{(0)}_{g}$ and $A$ in the recursive
equation:

(3)
$E^{(k)}_{g}=\text{GraphEncoder}\left(E_{g},\,A\right)=D^{-\frac{1}{2}}\,A\,D^{-\frac{1}{2}}\,E^{(k-1)}_{g}\,,$

where $A$ denotes the graph’s adjacency matrix,
$D_{ii}=\sum_{j=1}^{M+N}{a_{ij}}$ stands for the diagonal degree matrix of
$A$, $k$ represents the number of graph encoder layers, and $E^{(k)}_{g}$
signifies the output of the graph encoder, which provides enriched
representations for all users and items. To obtain the _global behavioral
representation_ $e^{u}_{g}\in\mathbb{R}^{d\times 1}$ and _global enriched
embeddings_ $E^{u}_{g}\in\mathbb{R}^{c\times d}$ following the graph encoder’s
execution, we derive the final representation from $E^{(k)}_{g}$ using the IDs
of given user $u$ and each item in $S_{u}$.

###  3.4. Fusion Layer

After processing input data with both sequential and graph encoders, we obtain
pairs of embeddings $e^{u}_{l}$ and $e^{u}_{g}$ for local and global
behavioral representations, respectively. To integrate information from both
encoders into our final representation, we utilize a shallow feed-forward
block as described in Eq. 4:

(4) $e^{u}_{f}=W_{2}\,\text{ReLU}\left(W_{1}(e^{u}_{l}\|e^{u}_{g})\right),$

where $e^{u}_{f}\in\mathbb{R}^{d}$ denotes the fused behavioral
representation. $W_{1}\in\mathbb{R}^{4d\times 2d}$ and
$W_{2}\in\mathbb{R}^{d\times 4d}$ are learnable projection matrices, and
$\|:\mathbb{R}^{d},\,\mathbb{R}^{d}\rightarrow\mathbb{R}^{2d}$ represents the
concatenation function.

###  3.5. Loss Functions

The essential component of MRGSRec, enabling end-to-end usage of both
contexts, lies in its loss functions. Inspired by the cross-domain recommender
systems (Xu et al., 2023), we construct objectives for every intermediate step
within our framework. For context-specific behavior representations, we use
local and global objectives $\mathcal{L}_{l}$ and $\mathcal{L}_{g}$ which are
the same as the downstream loss functions from the chosen encoder architecture
(e.g., next item prediction from SASRec (Kang and McAuley, 2018) for
sequential encoder and BPR-loss plus regularization loss from LightGCN (He et
al., 2020) for graph encoder). Additionally, we introduce an auxiliary
objective $\mathcal{L}_{f}$ based on fused behavioral representation:

(5)
$\mathcal{L}_{f}=\sum_{u\in\mathcal{U}}{-\log\frac{\exp^{(e^{u}_{f})^{\top}I_{e}(i^{u}_{p})}}{\sum_{i_{n}\in\text{Sample}(u)}{\exp^{(e^{u}_{f})^{\top}I_{e}(i_{n})}}+\exp^{(e^{u}_{f})^{\top}I_{e}(i^{u}_{p})}}},$

where $i^{u}_{p}$ is the next interacted item for user $u$, and $i_{n}$ is a
randomly selected negative item ID outside of $S_{u}$. Also, this
representation is used for computing item relevance scores by using the dot-
product between this embedding and item embeddings taken from $I_{e}$.

Finally, we introduce contrastive loss $\mathcal{L}_{c}$ between local and
global enriched item representations $E^{u}_{l}$ and $E^{u}_{g}$, aiding the
model to make a distinction between direct and indirect dependencies:

(6)
$\mathcal{L}_{c}=\sum_{u\in\mathcal{U}}{\sum_{i=1}^{c}{-\log\frac{\exp^{(E^{u}_{l})^{\top}_{i}(E^{u}_{g})_{i}}}{\sum\limits_{j=1}^{c}{\exp^{(E^{u}_{l})^{\top}_{i}(E^{u}_{g})^{\top}_{j}}}}}},$

where $(E^{u}_{g})_{i}$ and $(E^{u}_{l})_{i}$ – global and local enriched
representations of $i$-th item in the sequence respectively.

The overall objective is given as follows:

(7)
$\mathcal{L}(\theta)=\alpha\mathcal{L}_{l}+\beta\mathcal{L}_{g}+\gamma\mathcal{L}_{f}+\delta\mathcal{L}_{c},$

where $\alpha,\,\beta,\,\gamma,\,\delta$ are model hyper-parameters
determining each objective’s influence on the overall model objective.

##  4\. Experimental setup

#### Datasets

We experiment on four benchmark datasets: _Amazon-Beauty_ , _Amazon-Clothing_
, _Amazon-Sports_ , and _MovieLens-1M_. Data preprocessing follows standard
procedure: treating all interactions as positives, filtering out users and
items with fewer than _5_ interactions, and sorting user sequences by
interactions timestamp. Table 1 summarizes statistics after preprocessing.

Table 1. Statistics of the experimental datasets Dataset | #Items | #Users | #Interactions | Avg. length
---|---|---|---|---
Beauty | 12,101 | 22,363 | 198,502 | 8.876
Clothing | 23,033 | 39,387 | 278,677 | 7.075
Sports | 18,357 | 35,598 | 296,337 | 8.325
ML-1M | 3,706 | 6,040 | 1,000,209 | 165.598

#### Evaluation setting

We adopt the leave-one-out strategy as in prior research (Kang and McAuley,
2018; Sun et al., 2019) for train-validation-test splits: the most recent item
is designated as the test target, the second most recent as the validation
target (used for best hyper-parameters search), and the remaining items used
for training. To avoid unintended biases, we do not perform target item
subsampling and rank ground truth items against the entire item catalog. Model
performance is assessed using _HR@n_ and _NDCG@n_ metrics with $n=5,\,10$.

Table 2. Experimental results on benchmark datasets Dataset | Metric | SASRec | BERT4Rec | SASRec+ | NGCF | LightGCN | MCLSR | MRGSRec | Uplift (%)
---|---|---|---|---|---|---|---|---|---
Beauty | HR@5 | 0.0352 | 0.0182 | 0.0426 | 0.0130 | 0.0154 | 0.0371 | 0.0441 | 3.52
HR@10 | 0.0611 | 0.0412 | 0.0663 | 0.0215 | 0.0253 | 0.0614 | 0.0681 | 2.71
NDCG@5 | 0.0214 | 0.0183 | 0.0289 | 0.0070 | 0.0079 | 0.0212 | 0.0301 | 4.15
NDCG@10 | 0.0273 | 0.0254 | 0.0343 | 0.0094 | 0.0098 | 0.0275 | 0.0349 | 1.75
Clothing | HR@5 | 0.0159 | 0.0137 | 0.0207 | 0.0077 | 0.0089 | 0.0168 | 0.0221 | 6.76
HR@10 | 0.0271 | 0.0206 | 0.0339 | 0.0133 | 0.0156 | 0.0287 | 0.0352 | 3.83
NDCG@5 | 0.0090 | 0.0078 | 0.0114 | 0.0041 | 0.0043 | 0.0093 | 0.0121 | 6.15
NDCG@10 | 0.0116 | 0.0105 | 0.0155 | 0.0059 | 0.0071 | 0.0117 | 0.0162 | 4.52
Sports | HR@5 | 0.0216 | 0.0178 | 0.0275 | 0.0115 | 0.0133 | 0.0233 | 0.0289 | 5.09
HR@10 | 0.0324 | 0.0322 | 0.0417 | 0.0155 | 0.0179 | 0.0318 | 0.0434 | 4.08
NDCG@5 | 0.0135 | 0.0102 | 0.0176 | 0.0068 | 0.0072 | 0.0134 | 0.0186 | 5.68
NDCG@10 | 0.0167 | 0.0153 | 0.0218 | 0.0073 | 0.0080 | 0.0168 | 0.0227 | 4.12
ML-1M | HR@5 | 0.1067 | 0.0745 | 0.1244 | 0.0439 | 0.0485 | 0.1077 | 0.1312 | 5.47
HR@10 | 0.1804 | 0.1333 | 0.2133 | 0.0864 | 0.0921 | 0.1813 | 0.2248 | 5.39
NDCG@5 | 0.0639 | 0.0439 | 0.0761 | 0.0286 | 0.0307 | 0.0634 | 0.0829 | 8.94
NDCG@10 | 0.0918 | 0.0623 | 0.1065 | 0.0477 | 0.0549 | 0.0921 | 0.1134 | 6.48

#### Baseline Algorithms

We compare MRGSRec with baselines from both sequence-based and graph-based
approaches. These models include SASRec, BERT4Rec, and SASRec+ for sequence-
based, and NGCF with LightGCN for graph-based models. Additionally, we
evaluate MRGSRec against MCLSR, an approach integrating both data
representations. During this comparison, MRGSRec implementation incorporates
the sequential encoder and loss function from SASRec+, and the graph encoder
with objective function from LightGCN to explore its boost upon these strong
baselines.

##  5\. Results and Discussion

Table 2 reports the overall performance compared with baselines. The obtained
results highlight a more pronounced relative uplift for $n=5$ compared to
$n=10$. Although there is also an uplift for $n=20$ (not reported here for
brevity), it is observed to be of a lesser extent. We conclude that the model
excels at promoting positive items to the top of recommendation lists, but
revealing additional positive examples undetected by other models remains
challenging.

Our analysis shows that the ML-1M dataset, with a higher density of
interactions, demonstrates the most significant relative improvement in the
NDCG metric. This suggests that our model may enhance performance when dealing
with more interactions, possibly due to graph models performing better for
frequently consumed items. This observation could explain the comparatively
lower uplift metric values observed for the Beauty dataset.

In this paper, we have presented a novel end-to-end framework for leveraging
sequence-based and graph-based data representations into a single model to
improve the representation of user behaviors. The proposed framework, MRGSRec,
constructs two independent representations from sequence-based and graph-based
encoders that can capture different patterns in the input data. Our
experiments on classical benchmarks demonstrate the effectiveness of our
approach against diverse families of state-of-the-art methods.

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