# Information Harvesting for Far-Field Wireless Power Transfer

Mehmet C. Ilter1, Risto Wichman1, Mikko Saily2 and Jyri Hämäläinen3
1Department of Signal Processing and Acoustics, Aalto University, 02150 Espoo,
Finland  2Network and Architecture Group, Nokia Bell Labs, 02610 Espoo,
Finland  3Department of Communications and Networking, Aalto University, 02150
Espoo, Finland

###### Abstract

Considering ubiquitous connectivity and advanced information processing
capability, huge amount of low-power IoT devices are deployed nowadays and the
maintenance of those devices which includes firmware/software updates and
recharging the units has become a bottleneck for IoT systems. For addressing
limited battery constraints, wireless power transfer is a promising approach
such that it does not require any physical link between energy harvester and
power transfer. Furthermore, combining wireless power transfer with
information transmission has become more appealing. In the systems that apply
radio signals the wireless power transfer has become a popular trend to
harvest RF-radiated energy from received information signal in IoT devices.
For those systems, design frameworks mainly deal with the trade-off between
information capacity and energy harvesting efficiency. Therein various
signaling design frameworks have been proposed for different system
preferences between power and information. In addition to this, protecting the
information part from potential eavesdropping activity in a service area
introduces security considerations for those systems. In this paper, we
propose a novel concept, Information Harvesting, for wireless power transfer
systems. It introduces a novel protocol design from opposite perspective
compared to the existing studies. Particularly, Information Harvesting aims to
transmit information through existing wireless power transfer mechanism
without interfering/interrupting power transfer. To do so, only intended
information receivers are able to decode the transmitted information after
they initiate an information transfer process at wireless power transmitter
while energy harvesters continue energy harvesting procedure without noticing
this procedure. Considering the diversity of IoT networks and the availability
of wireless power transfer infrastructure, proposed Information Harvesting
principles may turn out to a pivotal methodology especially for the cases
where a large number of IoT devices require the software/firmware updates
along with periodical battery recharging needs.

###### Index Terms:

Wireless power and information transfer, information harvesting, Internet of
Things, energy harvesting.

##  I Introduction

The Internet of Things (IoT) creates the potential for new generation industry
use cases and smart cities. With uniquely identifiable devices capable of
communicating through wireless environment, billions of devices are able to
sense and interact with everything and everyone. The vision of IoT is set to
be a communication platform for broadcast and point-to-point communication.
From other aspect, wireless power transfer enables the next stage in the
current consumer electronics revolution, including battery-less sensors,
passive RF identification (RFID), passive wireless sensors, and machine-to-
machine solutions. The origin of wireless power transmitted by using RF
spectrum was pioneered by Tesla and it is based on radiative energy transfer
through a medium (i.e. air) with the help of antennas [1].

There are two types of power transfer: near-field power and far-field power
transfer. The first one depends on inductive/capacitive coupling between
energy harvester and power transmitter so its range is less than a meter. The
latter one exploits radio frequency transmission and due to propagation
characteristics of RF signals, it can be used over longer ranges [2]. For
instance, prototype RF-based energy harvesting circuits operating below 1 GHz
are capable of harvesting microwatts to milliwatts of power over the range of
10 m with similar transmit power of a Wi-Fi router [3].

The main objective of radiative power transfer system is to maximize the
harvested DC power subject to a given transmit power constraint, in other
words, enhancing the end-to-end power transfer efficiency. Designing an
efficient rectenna, the unit that consists of a rectifier and an antenna, to
maximize RF-to-DC conversion has been the traditional line of research in
earlier power transfer literature [4]. Furthermore, signal design for wireless
power mechanism is getting more popular thanks to the availability of wider
range of optimization techniques. For instance, different from grid-based and
lattice-based constellations in conventional communication systems where lower
peak-to-average power ratio is typically desired, real Gaussian signals, flash
signaling and linear frequency-modulated signals are the preferred
constellations for energy harvesting mechanisms since higher PAPR values
result in higher DC voltage by a rectifier leading to better end-to-end power
efficiency [4] at the cost of higher complexity and power consumption in the
transmitter.

Rather than radiating only energy into a particular service area, combining
RF-based power transfer with simultaneous information transmission has become
appealing with the recently introduced simultaneous wireless information and
power transfer (SWIPT) concept. In SWIPT systems, the power and information
components of the transmitting signal are separable by using the energy domain
(power splitting), the time domain (time splitting) and the space domain
(antenna splitting) [5]. For those systems, the receiver architecture is
generally optimized for increasing total harvested energy while decoding
information with minimum power consumption at the same time without
considerable loss in achievable data rates. Therefore, a trade-off exists
between information transfer and energy transfer in such systems.

In practice, sensitivities of information receivers and energy harvesters are
quite different. Specifically, the minimum requirement for the received power
at an information receiver is around –60 dBm, while that of an energy
harvester is –10 dBm. For instance under Friis assumption, 2.4 GHz frequency
introduces a decay of 40 dB decay per meter. Therefore, energy harvester is
usually located closer to wireless power transceiver than information
receivers. However, this setting may create a potential eavesdropping activity
where receiver close by energy harvester can act as a malicious device who
exploits better received signal statistics resulting in a high risk of
eavesdropping and information interception. Similar phenomenon is also
observed in power domain-non-orthogonal multiple access systems where
successive interference cancellation techniques are implemented in the nearby
users in order to eliminate other users’ symbols thanks to having better
channel conditions in the process of extracting its own intended signal from
superposed transmitting symbol. Therefore, nearby users have an advantage to
decode far user information as well and there is a need to combat against
eavesdropping risk of nearby users by applying physical layer security
schemes.

Physical layer (PHY)-security techniques which mainly turn the randomness in
wireless channels into an advantage by exploiting the physical aspects of
communication channels between the nodes provide a powerful tool to achieve
secure communication. Although encryption mechanism is quite useful for
protecting ongoing data transmission in many cases, considering the diversity
in protocols and networks in 5G and beyond, those mechanism might not be
suitable for all available cases. Even with limited computational capability,
it was shown that the PHY-security is able to provide secure communications
for a massive number of IoT devices [6].

Due to huge amount of devices, IoT networks require frequent software updates
which include security updates, bug fixes, and software extensions. From this
perspective, there is considerable amount of data transmission procedure
occurring periodically. For those activities, encryption techniques can
contribute up to 30% of total energy consumption [7]. Furthermore, most IoT
devices/sensors are simple nodes having limited computational capabilities so
the complexity of encryption-decryption procedures are beyond the capabilities
of those devices. It is claimed that no encryption is required in
integrated/tiered IoT devices [8] and SigFox protocol [9]. Those trends can be
easily verified by checking the statistics about existing IoT traffic showing
that the big portion of IoT traffic is unencrypted [10]. In addition, even
encrypted data might not be enough if malicious users have enough time to
analyze the traffic patterns to obtain the information. Therefore
additional/complimentary solution is required.

To tackle these challenges this paper introduces Information Harvesting (IH)
in wireless power transfer mechanism, which proposes data communication
ability on top of existing power transfer procedure; in other words, wireless
power transfer is partnered with a information transfer protocol which allows
secure transfer. The pivotal point of the proposed protocol is that the power
receivers are not aware of information broadcasting and it is required to use
a transmitter entity for the purpose of transmitting broadcasting packets
without disturbing ongoing power transfer. From this perspective, IH can
dissolve wireless power transfer and information broadcasting in one
mechanism. This mechanism can be exploited through updating low complexity
devices without any encryption (or low level of encryption for necessary
cases) by utilizing existing wireless power transceiver unit originally
deployed for charging power receivers.

The rest of the paper is organized as follows. In Section II, the system
components of IH mechanism are introduced. Then, the protocol details of IH
are represented in Section III. Then, Section IV introduces two different
practical realizations, which are spatial modulation and analog beamforming,
respectively. Besides, some additional techniques which increase the physical
layer security of the proposed protocol are also described therein. Section V
represents the feasibility of spatial modulation based-information harvesting
with simulation results generated for an indoor environment. Section VI
concludes the paper.

##  II System Components

The system components of IH are illustrated in Fig. 1. Herein, wireless power
transceiver (WPT) serves as a module supplying power transfer to a portable
power receiver device(s), energy harvester (EH) and also providing information
transfer by utilizing information seeding to a portable information receiver
device, information receiver (IR). The IR contains receive antenna chain where
single/multiple receive antenna(s) is(are) deployed and the energy harvester
consists of a receive antenna chain, a battery charging unit and a battery.
The key motivation herein is to design  information seeding and information
harvesting cycles so that the EHs are not able to detect the on-going
information transfer and in case of this activity detected by EHs, they are
not able to decode the embedded information.

Figure 1: The overall diagram of Information Harvesting system components.

From this point of view, information seeding protocol lies on creating
variations in transmitter entity in WPT with respect to information stream
transmitted to IR. By doing so, broadcasting channels between WPT and IR can
be configured by exploiting existing wireless power transfer mechanism. Those
changes in the chosen transmitter entity cannot be sensed in existing EHs so
broadcasted information can be only decoded at target IRs. At this point, the
procedures handled by IR can be referred to information harvesting which
includes sending request for broadcasting to WPT via request for information
(RFI) signalling, carrying out channel estimation mechanism and detecting the
variations in the transmitter entity from the existing power transmitting
signal. Note that the EH and IR are not necessarily different devices so an EH
can act as an information receiver. The system components of IH can be
described as follows:

  * •

WPT senses power request/information request from portable power
receiver/portable information receiver through identification and
configuration phase. Accordingly, there are two modes for wireless power
transceiver, which are power transfer mode and information seeding mode.
During information seeding, the process of using different transmission entity
in order to send the information to the wireless information receiver is
handled.

  * •

IR contains single/multiple receive antenna(s). Through this/them, the
radiated wireless power transfer signal generated by the wireless power
transceiver is received, then; the harvesting procedure starts with estimating
corresponding transmission entity which was the source of the information.

  * •

EH consists of receive antenna chain, battery charging unit and battery.
Herein, battery charging unit is configured to establish a link between the
receive antenna chain and battery, wherein the manner in which power is
transferred from wireless power transceiver is controlled in accordance with
the parameters and/or state information assigned by the power management unit.

###  II-A Improving PHY security:Dynamic activation pattern and artificial
noise generation

In order to prevent passive attacks (silent eavesdropper and eavesdropping
activities), dynamic transmit entity mapping [11] can be applied in order to
prevent decoding the index of transmitting entities over the air by existing
attacker. For dynamic transmit entity index, dynamic transmit entity
activation pattern can be added in WPT. Those updates on transmit entity index
can be done periodically after certain number of information frames based on
the arrangements between WPT and IR.

Another PHY security improvement can be obtained from artificial noise
generation in full-duplex information receivers. In this scheme, information
receiver can use its own sources in order to mask information transmission
from wireless power transceiver. To do so, when Information receiver sends its
RFI signalling, it can start to generate artificial noise signal. By this way,
it will be more difficult for any existing attackers to trace the transmit
entity index. Since artificial noise is generated by information receiver, the
information receiver can first use a physical layer self-interference
cancellation method, subtracting its own generated artificial noise signal
from received signal, then information harvesting phase can start.

Now, the details of the proposed protocol in information seeding and
information harvesting phases will be described in the next section.

##  III Protocol Design

As mentioned above, WPT initially serves a module supplying power transfer to
a portable EHs and can provide information transfer to portable information
receiver device(s) once information request from the IRs is received. In order
to distinguish the cycles of power transfer and information transfer at WPT,
IH protocol should have pre-signaling, identification and configuration
phases.

For wireless power transfer mode when there is only EH in a service area, the
identification and configuration phases are based on establishing wireless
power transfer link between WPT and EH illustrated in Fig. 2 with blue blocks.
After the wireless power transfer request is initiated by EH, WPT can sense
this request, and the identification of the EH and its configuration is
exchanged between them. Once it is completed, WPT unit turns into power
transfer mode. Any failure in this configuration can be detected by a control
error mechanism and the configuration starts from the beginning if any error
is detected during power transfer period.

Figure 2: The detailed procedures between WPT-EH and WPT-IR during wireless
power and information transfer phases.

In case of IR enters the area, the changes in the existing power transfer
mechanism procedure is highlighted with green blocks in Fig. 2. Basically, the
arrival of IR triggers two different cycles; information seeding and
information harvesting. The first one appears in WPT while the latter is
executed in IR. At the beginning, information harvesting starts with sensing
ongoing wireless power transfer mechanism in the service area. Once it is
sensed by IR, the channel estimation process between WPT-IR links are
initiated and the parameters will be attached to RFI signal based on the
channel estimation process. For instance, this process can determine the
required data rate in the IR so the requirement for how frequent a chosen
transmit entity is varied at WPT during information seeding can be sent
through the RFI signals. Then, extra PHY security enhancement described in
Section II.A can be added based on the IR preference.

Following those, the IR sends its information request to the power transmitter
and this request is sensed through the receiver at the WPT. After sensing this
information request, the communications controller in the transmitter becomes
active and the feature of a chosen transmission entity which was determined by
RF chain activation pattern is updated with respect to information bits. Then,
those changes imposed to the chosen transmission entity are estimated to
extract the information embedded on wireless power signals while ongoing power
transmission configuration is not affected by this procedure.

##  IV Spatial modulation-based IH

Spatial modulation (SM) technique was proposed in [12] where a subset of
active transmit antennas out of multiple transmit antennas can be used in
order to increase spectral efficiency. To do so, extra information bits are
typically conveyed through the index of active transmit antenna(s), in
addition to conventional modulated symbols. We recall that SM was proposed as
a potential NR candidate for 5G networks [13] and its derivatives, index
modulation and media-based modulation, are still popular [14]. The original
version of SM only considered a single active transmit antenna over fading
channels; then, generalized SM was introduced in [15] where a group of active
transmit antennas are considered. Furthermore, there are some examples of SM
proposed for non-fading/low-rank channels beyond 5G systems, i.e. mmWave
channnels [5]. In case of exploiting spatial modulation technique, information
seeding is based on the process of using different transmit antennas at
wireless power transceiver and estimating the transmit antenna in information
receiver.

In the power transfer mode, antenna activation pattern selector is inactive
since there is no information receiver in the range. When the information
receiver appears in the range, it transmits Reguest for Information (RFI)
message. Once RFI is received in wireless power transmitting unit, antenna
activation pattern selector becomes active, so wireless charging power
transmitter turns into information seeding mode. RFI signaling can also
determine how fast active transmit antenna pattern varies. Recently, space-
time mapping has been proposed in [16] where the period of antenna activation
was extended to multiple symbol periods along with marginal increment in
detection complexity. Then, the new transmit antenna(s) can be active in the
next frame based on information bits block. During the period of information
seeding mode ON, information receiver aims to detect the index of active
transmit antenna or indexes of group of active antennas. The overall procedure
is summarized in Fig. 3.

In the case of using analog beamforming for information seeding phase, in
addition to the existing power signals transmitted through EH along the
selected beam pairs, the activated transmit beam indices, when correctly
identified at the IR, can be used to convey additional information.

Figure 3: Spatial modulation-based Information Harvesting. (upper-left) Power
transfer mode. (upper-right) IR-initiated request for information signalling.
(bottom-left) Information seeding mechanism. (bottom-right) Information
harvesting mechanism. TABLE I: The simulation-related parameters

Parameter List | Values
---|---
Total transmitted power at the WPT | 10 dB
Total active transmit antenna at the WPT | 64
The distance of information receiver |  $\displaystyle\rm d_{\rm info}=20$ m
The distance of energy harvester |  $\displaystyle\rm d_{\rm energy}=1$ m
Antenna spacing | $\displaystyle\lambda/2$
Operation frequency |  $\displaystyle 2$ GHz
Array antenna gain |  $\displaystyle 15$ dBi
Path loss model |  $\displaystyle 128.1+37.6\log_{10}(d)$, d[km]
Angular offset’s standard deviation | $\displaystyle 2^{\circ}$
Log-normal shadowing’s standard deviation |  $\displaystyle 8$ dB
Subcarrier bandwidth |  $\displaystyle 15$ kHz
Obstacle cover ratio | $\displaystyle 0-0.9$
Obstacle radius | [0.3, 0.6]
Obstacle height | [5, 25]

##  V Proof of Concept: Spatial Modulation based Information Harvesting

In this section, the feasibility of Information Harvesting will be
investigated through wireless power transceiver which utilizes spatial
modulation mechanism. To do so, the received signal differences at information
receiver and energy harvester will be investigated.

In the wireless power transceiver, $\displaystyle N_{t}$ transmit antennas
sharing a common RF chain are employed and a group of transmit antennas are
active during wireless power transmission as illustrated in Fig. 3. In order
to align with multi-antenna broadcast systems, the constant envelope power
transfer signals considered in [17] are used and a set of phase shifters for
each transmit antenna is deployed. The spatial correlation between transmit
antennas is modeled by the standard Kronecker correlation model which is
widely used in many studies.

The characteristics of the channels for WPT-EH and WPT-IR will vary based on
spatial features (near-distance and far-distance, line-of-sight and non-line-
of-sight) and the WPT-IR shows more variations due to larger distance from
WPT. In order to reflect a more realistic scenario, the obstacles are modeled
by using a homogeneous Poisson point process (PPP) in the calculation of line-
of-sight and non-line-of-sight probabilities for the locations of EH and IR by
using obstacle coverage ratio (OCR). Specifically, the obstacle density is
determined by the average obstacle area and total area covered by the
obstacles and the obstacles are modeled as cylinders that are spatially
distributed according to a homogeneous PPP for an indoor area.

The goal is to show the feasibility of sending information through the changes
in active transmit antenna index where energy harvesters do not notice any
considerable changes in received signals while information receiver can detect
those changes in information harvesting cycle. To do so, the energy harvester
is placed closer with respect to information receiver along with similar
obstacle coverage ratio. The list of the simulation parameters are given in
Table I.

Figure 4: The received signal variations and LoS probabilities in EH over
different distances and the OCR values.

The LoS probabilities with respect to different distances are plotted in Fig.
4. As seen from the curves, higher OCR values lead lower chance of getting LoS
link in any distance. Now, in order to see the effect of the variations in
active transmit antenna indices at wireless power transceiver, two different
distances are considered for energy harvester; 1 m and 20 m. Fig. 4
illustrates the mean received powers harvested in energy harvester at those
distances with different number of active transmit antennas out of
$\displaystyle 64$ antennas at the power transmitter. Those average power
values are obtained from $\displaystyle 1000$ different realizations where the
positions of WPT and EH and the number of active transmit antennas are fixed.
As seen from the figure, changing the transmit antenna creates negligible
differences in the harvested power levels and these variations result from
different power transfer signal for each realization.

The indexes of active transmit antennas during wireless power transmission
period can be detected so the extra information for information receivers can
be decoded without noticing any change in energy harvesters. The upper bound
for spectral efficiency achieved using spatial modulation-based IH is
illustrated in Fig. 5. Therein, different number of active transmit antennas
is considered out of $\displaystyle 64$ total transmit antenna at the WPT. For
higher number of active antennas, total combinations for information seeding
is increasing until half of the antennas turn active resulting in higher
spectral efficiency values at the IR. Higher OCR and longer distance result in
more variations in non-line-sight channel coefficient so it leads to higher
spectral efficiencies in those cases. Interestingly, when no obstacle exists
in the environment, all channels between the WPT and IR turns into roughly
identical LOS channels so spectral efficiency is zero. Note that Fig. 5
reflects the maximum performance of IH when using the spatial modulation-based
IH. Values can be reduced in practical implementation based on different
service area and device capabilities.

Figure 5: Upper bound for achievable spectral efficiency at IR with respect to
different numbers of active transmit antennas are used at WPT.

##  VI Conclusion

We described a method and protocol for data communications where wireless
power transfer is enhanced with a protocol to allow secure information
broadcasting to devices which are in their information harvesting cycle. The
key motivation is the design of information seeding and information harvesting
mechanisms where energy harvesters are not able to detect the ongoing
information broadcasting since harvested energy is not affected by information
seeding mechanism occurring at the wireless power transceiver. From this
aspect, Information Harvesting can be utilized towards to the direction of
updating IoT devices by exploiting available wireless power transfer
infrastructure in the near future. In addition, Information Harvesting is a
potential candidate of 3GPP Rel-18+ study items where the reduced capability
devices may be energy harvesting devices with the radio access and
communication capability, thus operating on the same carrier frequencies with
information harvesting devices. These devices are expected to follow a
protocol allowing both energy harvesting and information harvesting cycles.

##  VII Acknowledgment

This study has been supported by the Academy of Finland (grant number 334000).

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# On the energy barrier of hypergraph product codes

Guangqi Zhao  Centre for Engineered Quantum Systems, School of Physics,
University of Sydney, Sydney, NSW 2006, Australia<EMAIL_ADDRESS>Andrew C. Doherty  Centre for Engineered Quantum Systems, School of Physics,
University of Sydney, Sydney, NSW 2006, Australia  Isaac H. Kim  Department of
Computer Science, University of California, Davis, CA 95616, USA

###### Abstract

A macroscopic energy barrier is a necessary condition for self-correcting
quantum memory. In this paper, we prove tight bounds on the energy barrier
applicable to any quantum code obtained from the hypergraph product of two
classical codes. If the underlying classical codes are low-density parity-
check codes (LDPC), the energy barrier of the quantum code is shown to be the
minimum energy barrier of the underlying classical codes (and their
transposes) up to an additive $O(1)$ constant.

_Introduction_.\- Quantum computers are capable of solving problems that are
likely intractable for classical computers, such as factoring of large
integers Shor (1994) and simulation of quantum systems Feynman (1982); Lloyd
(1996). However, realistic quantum computers are noisy. In order to build a
useful quantum computer capable of carrying out such computational tasks,
quantum error correction is likely necessary.

Since the discovery of the first quantum error-correcting code Shor (1996),
much progress has been made towards finding better codes. While the leading
approach to scalable quantum error correction has been the surface code Kitaev
(2003) for nearly 20 years, lately there has been a surge of interest in using
quantum low-density parity check (LDPC) codes. This recent interest is in part
due to Gottesman, who showed that with quantum LDPC codes, one can achieve a
constant overhead for fault-tolerant quantum computation Gottesman (2014). A
well-known approach to construct such codes is the hypergraph product
construction Tillich and Zemor (2014). More recent studies led to the
development of other families of quantum LDPC codes with improved parameters
Freedman  _et al._ (2002); Evra  _et al._ ; Kaufman and Tessler (2021);
Hastings  _et al._ (2021); Panteleev and Kalachev (2022a); Breuckmann and
Eberhardt (2021a); Panteleev and Kalachev (2022b); Leverrier and Zemor (2022);
Dinur  _et al._ (2023). Moreover, recent studies suggest that there are viable
fault-tolerant quantum computing architectures that can host such codes
Tremblay  _et al._ (2022); Bravyi  _et al._ (2024); Xu  _et al._ (2023).

One challenge in using these codes lies in the decoding. While there are
general methods such as the BP+OSD decoder Panteleev and Kalachev (2021), it
is a priori not obvious how well such a general-purpose decoder would work for
a given code. One attractive approach is to use codes that have an extensive
energy barrier. For such codes, a simple process that iteratively lowers the
energy (quantified in terms of the number of stabilizers violated) is a viable
decoder candidate. Alternatively, such a model can be viewed as a candidate
for self-correcting quantum memory, protecting quantum information by the
macroscopic energy barrier. (We note that the energy barrier is not a
sufficient condition for building a self-correcting quantum memory Haah
(2011); Michnicki (2014); Siva and Yoshida (2017), though it is nevertheless a
necessary condition.)

Such approaches work well for four-dimensional (4D) toric code Alicki  _et
al._ (2010) and the quantum expander code Leverrier  _et al._ (2015); Fawzi
_et al._ (2020), both of which have extensive energy barriers. Unfortunately,
rigorous bounds on the energy barrier are hard to come by and often derived
for specific codes (or family of codes) Bravyi and Haah (2011); Michnicki
(2014); Leverrier  _et al._ (2015); Williamson and Baspin (2023); Lin  _et
al._ (2023).

In this paper, we prove a tight bound on the energy barrier for the hypergraph
product codes Tillich and Zemor (2014), defined in terms of the energy barrier
of the underlying classical codes. The hypergraph product is defined in terms
of the parity check matrices (or equivalently, the Tanner graphs) of two
classical codes. While there are codes with better parameters Hastings  _et
al._ (2021); Breuckmann and Eberhardt (2021a); Panteleev and Kalachev (2022a,
b), the hypergraph product remains a flexible framework for constructing
quantum LDPC codes with many advantages, such as the variety of decoders
Leverrier  _et al._ (2015); Fawzi  _et al._ (2020); Panteleev and Kalachev
(2021); Roffe  _et al._ (2020); Grospellier  _et al._ (2021), logical gates
Krishna and Poulin (2021); Cohen  _et al._ (2022); Quintavalle  _et al._
(2023), and distance-preserving syndrome extraction circuit Tremblay  _et al._
(2022); Manes and Claes (2023).

Now we describe our main result. Without loss of generality, consider a
hypergraph product of two classical codes, defined in terms of the parity
check matrices $H_{1}$ and $H_{2}$. We denote the parity check matrix of the
resulting quantum code as $H_{(H_{1},H_{2})}$. For both the quantum and the
classical code, we denote the energy barrier as $\Delta(H)$, where $H$ can be
a parity check matrix of the quantum or the classical code. We prove that
under a modest condition,

$\Delta(H_{(H_{1},H_{2})})=\min(\Delta(H_{1}),\Delta(H_{2}),\Delta(H_{1}^{T}),\Delta(H_{2}^{T})),$
(1)

where $H^{T}$ is the transpose of $H$. Eq. (1) holds if the energy barriers of
the classical codes are larger than or equal to a certain sparsity parameter
of the code. Importantly, if the underlying codes are LDPC, then the sparsity
parameter is $O(1)$. Therefore, Eq. (1) holds if the energy barriers of
$H_{1},H_{2},H_{1}^{T},$ and $H_{2}^{T}$ grow as the code size grows. If this
condition is not satisfied, the energy barrier is bounded by a constant, a
fact that follows trivially from the definition of the hypergraph product
code.

The proof of Eq. (1) is based on two results. First, if two logical operators
of a quantum LDPC code are equivalent up to a stabilizer, their energy
barriers differ only by a constant related to the code’s sparsity parameters
$w_{c},w_{q}$ [Theorem 1]. Thanks to this result, proving the energy barrier
for a given code reduces to proving the energy barrier for any complete set of
logical operators, which can be a much smaller set than the set of all logical
operators. We then identify a set of logical operators for which the exact
energy barrier can be determined in terms of the energy barriers of the
underlying classical codes. Together, these two results imply Eq. (1).

_Energy barrier of codes_.\- We first present a formal definition of the
energy barrier for stabilizer codes Bravyi and Terhal (2009). Let
$\mathcal{C}$ be the code subspace of a stabilizer code, defined in terms of
the stabilizer group $S$. The code subspace can be viewed as the ground state
subspace of a Hamiltonian of the form $\hat{H}=\sum_{i=1}^{m}(I-s_{i})/2$,
where $\\{s_{1},\ldots,s_{m}\\}\subset S$ is a set of generators. Note that
the energy barrier depends on the choice of the generating set. For an
operator $P$ in the Pauli group $\mathcal{P}$, the energy of the state
$P|\psi\rangle$ is given by
$\bra{\psi}P^{\dagger}\hat{H}P\ket{\psi}=\epsilon(P)$. Here $\ket{\psi}$ is
any ground state with $\bra{\psi}\hat{H}\ket{\psi}=0$, and $\epsilon(P)$ is
the energy cost of $P$. This is the number of $s_{i}$s that anticommute with
$P$, i.e., $\epsilon(P)=|\\{i:s_{i}P=-Ps_{i}\\}|$. Equivalently, one may
define the energy barrier in terms of the parity check matrix $H$ of the
stabilizer code. Let $v(P)$ be the binary (bit-string) representation of a
Pauli $P$:

$\epsilon(P)=\text{wt}(Hv(P)).$ (2)

A sequence $P_{0},P_{1},\ldots,P_{t}$ from the Pauli group $\mathcal{P}$ forms
a _path_ from $P_{0}$ to $P_{t}$ if for each index $i$, the operators $P_{i}$
and $P_{i+1}$ differ at no more than one qubit. The notation $w(P_{0},P_{t})$
represents the collection of all such paths from $P_{0}$ to $P_{t}$. For $r\in
w(P_{0},P_{t})$, $\epsilon_{\max}(r)$ denotes the highest energy along path
$r$, i.e., $\epsilon_{\max}(r)=\max_{P_{i}\in r}\epsilon(P_{i})$, as the
energy barrier of $E$ along the path $r$.

The minimum energy associated with a Pauli $P$ is the smallest value of
$\epsilon_{\text{max}}$ across all possible paths from $0$ to $P$. This is the
energy barrier of $P$, denoted as $\Delta(P)$:

$\displaystyle\Delta(P)=\min_{r\in w(I,P)}\epsilon_{\max}(r).$ (3)

The energy barrier of the quantum code is the minimum energy barrier over the
set of nontrivial logical operators.

######  Definition 1.

Let $S$ be a stabilizer group and $L(S)$ be the set of nontrivial logical
operators. The energy barrier is

$\displaystyle\Delta(H):=\min_{\ell\in L(S)}\Delta(\ell).$ (4)

This is the smallest energy the environment has to overcome to enact a logical
operation on the encoded qubit. We can similarly define the energy barrier of
classical codes by only considering the path formed by Pauli-$X$s. We shall
denote this energy barrier also as $\Delta(H)$, where $H$ in this case is the
parity check matrix of the classical code.

_Quantum LDPC codes and their energy barriers_.\- A quantum LDPC code is a
stabilizer code with a sparse parity-check matrix; see Babar  _et al._ (2015);
Breuckmann and Eberhardt (2021b) for recent reviews. The sparsity parameters
are $w_{c}$ and $w_{q}$, which are the maximum row and column weights of the
parity check matrix, respectively. These represent the maximum weight amongst
all the checks and the maximum number of checks associated with a single bit.
A code is LDPC if $w_{c},w_{q}=O(1)$.

Here, we prove a property that holds true for any quantum LDPC code. It states
that the energy barrier of two logical operators equivalent under stabilizers
are equal, provided that at least one of their energy barriers is larger than
or equal to $w_{c}w_{q}$. For quantum LDPC codes, $w_{c}w_{q}=O(1)$.
Therefore, if the energy barrier is $\Omega(1)$ for any given non-trivial
logical operator, it must also be the same energy barrier for any equivalent
logical operator.

We first prove the following bound.

######  Lemma 1.

Let $\mathcal{C}$ be a quantum code with sparsity parameters $(w_{c},w_{q})$.
For any stabilizer $s\in S$,

$\Delta(s)\leq w_{c}w_{q}.$ (5)

###### Proof.

Without loss of generality, any stabilizer $s$ can be expressed as a product
$s=s_{1}\cdots s_{m}$, where $s_{1},\ldots,s_{m}$ are the stabilizer
generators. Consider a path that applies $s_{i}$ in sequence, from $i=1$ to
$m$. For each $s_{i}$, we envision applying a (sub)sequence of Paulis in the
support of $s_{i}$. This subsequence has a length of at most $w_{c}$, and each
Pauli in the sequence affects at most $w_{q}$ stabilizers. Therefore, the
highest energy attained within this subsequence is at most $w_{c}w_{q}$. Once
$s_{i}$ is applied, the energy cost becomes zero. Therefore, $\Delta(s)\leq
w_{c}w_{q}$.

∎

######  Theorem 1.

Let $\mathcal{C}$ be a $(w_{c},w_{q})$ quantum LDPC code. For any logical
operator $L$ and stabilizer $s\in S$,

$\Delta(Ls)\leq\max(\Delta(L),w_{c}w_{q}).$ (6)

###### Proof.

Without loss of generality, consider a path $r=(\ell_{1},\ldots,\ell_{N})\in
w(0,L)$. We can consider a new path $r^{\prime}\in w(0,Ls)$ by appending $r$
with a sequence $\delta_{r}\in w(0,s)$, forming
$(\ell_{1},\ldots,\ell_{N},\ell_{1}^{\prime},\ldots,\ell_{M}^{\prime})$ where
$\delta_{r}=(\ell_{1}^{\prime},\ldots,\ell_{M}^{\prime})$. By assumption,

$\epsilon_{\max}(r^{\prime})=\max(\epsilon_{\max}(r),\Delta(s)).$ (7)

According to Lemma 1, $\Delta(s)\leq w_{c}w_{q}$. It implies that for any
given path $r$, there exists a path $r^{\prime}$ such that
$\epsilon_{\max}(r^{\prime})\leq\max(\epsilon_{\max}(r),w_{c}w_{q})$. Choose
$r$ such that $\Delta(L)=\epsilon_{\max}(r)$. Since
$\Delta(Ls)\leq\epsilon_{\max}(r^{\prime})$, we have
$\Delta(Ls)\leq\max(\Delta(L),w_{c}w_{q})$. ∎

We remark that using the same logic, one can deduce
$\Delta(L)=\Delta(Lss)\leq\max(\Delta(Ls),w_{c}w_{q})$. Consequently, if
$\Delta(L)\geq w_{c}w_{q}$ or $\Delta(Ls)\geq w_{c}w_{q},$
$\Delta(L)=\Delta(Ls)$. Therefore, to determine the asymptotic scaling of the
energy barrier of a quantum LDPC code, it suffices to consider the energy
barrier of any _fixed_ complete set of logical operators. Once an energy
barrier is obtained for such a set, the energy of all the other logical
operators is also essentially determined, thanks to Theorem 1.

However, it is a priori not obvious how to choose such a set. Below, we will
solve this problem for a large family of quantum codes known as the hypergraph
product code Tillich and Zemor (2014).

_Hypergraph product code and its logical operators_.\- Hypergraph product
codes are CSS codes formed from two classical linear codes. Without loss of
generality, let $H_{1}$ and $H_{2}$ be $r_{1}\times n_{1}$ and $r_{2}\times
n_{2}$ parity check matrices, respectively. The parity-check matrix of the
hypergraph product code becomes Tillich and Zemor (2014):

$\displaystyle H_{X}$ $\displaystyle=$
$\displaystyle\left(H_{1}\otimes\mathbf{I}_{n_{2}}\ \mathbf{I}_{r_{1}}\otimes
H_{2}^{T}\right),$ (8) $\displaystyle H_{Z}$ $\displaystyle=$
$\displaystyle\left(\mathbf{I}_{n_{1}}\otimes H_{2}\
H_{1}^{T}\otimes\mathbf{I}_{r_{2}}\right).$

Because $H_{X}H_{Z}^{T}=0$, these two parity check matrices define a CSS code,
with a quantum parity check matrix

$H_{(H_{1},H_{2})}=\left(\begin{array}[]{cc}H_{X}&0\\\
0&H_{Z}\end{array}\right).$ (9)

The classical codes defined by $H_{1},H_{2},H_{1}^{T},\text{ and }H_{2}^{T}$
form the parent classical codes. Without loss of generality, we will assume
that $H_{i}$ and $H_{i}^{T}$ define codes with parameters
$[n_{i},k_{i},d_{i}]$ and $[r_{i},k_{i}^{T},d_{i}^{T}]$, for $i=1,2$. Under
this assumption, the quantum code is a
$\left[[n_{1}n_{2}+r_{1}r_{2},k_{1}k_{2}+k_{1}^{T}k_{2}^{T},\min(d_{1},d_{2},d_{1}^{T},d_{2}^{T})]\right]$
code Tillich and Zemor (2014).

We now introduce a particularly useful set of logical operators, which we
refer to as the _canonical_ logical operators Quintavalle and Campbell (2022);
Manes and Claes (2023). For the $Z$-type logical operators, consider the
following operator

$\begin{pmatrix}\sum_{k,j}\lambda_{kj}\overline{x}_{k}\otimes y_{j}\\\
\sum_{\ell,m}\kappa_{\ell m}a_{\ell}\otimes\overline{b}_{m}\end{pmatrix},$
(10)

where (i) $H_{1}\overline{x}_{i}=H_{2}^{T}\overline{b}_{m}=0$ and (ii)
$y_{j}\not\in\text{Im}(H_{2}^{T})$ and $a_{\ell}\not\in\text{Im}(H_{1})$ are
unit vectors. We note that $j\in\\{1,\ldots,k_{2}\\}$ and
$\ell\in\\{1,\ldots,k_{1}^{T}\\}$ and that the set of logical operators
expressible in this form is complete Quintavalle and Campbell (2022); Manes
and Claes (2023), which means all $k_{1}k_{2}+k_{1}^{T}k_{2}^{T}$ logical
operators are provided. A canonical logical operator is _elementary_ if only
one of the coefficients, either $\lambda_{kj}$ or $\kappa_{\ell m}$, equals
one. A similar form of canonical $X$-type logical operators can also be
constructed. Because our discussion below can be applied to such operators
with little change, we omit the discussion about the $X$-type logical
operators.

Let $\mathcal{G}_{1}(V_{1},C_{1})$ and $\mathcal{G}_{2}(V_{2},C_{2})$ be the
Tanner graphs of codes defined by $H_{1}$ and $H_{2}$, respectively. Here,
$V_{1}$ and $V_{2}$ represent the set of bits, and $C_{1}$ and $C_{2}$ denote
the set of checks. We use $\\{v_{1}^{i}:i\in\\{1,2,\cdots n_{1}\\}\\}$ (resp.
$\\{v_{2}^{j}:j\in\\{1,2,\cdots n_{2}\\}\\}$) to refer to bit vertices in
$V_{1}$ (resp. $V_{2}$), and also as length $n_{1}$ (resp. $n_{2}$) unit
vectors with the $i$th (resp. $j$th) entry as $1$. The hypergraph product
$\mathcal{G}_{1}\times\mathcal{G}_{2}$ is a bipartite graph with vertex set
$V\cup C$, where $V=V_{1}\otimes V_{2}\cup C_{1}\otimes C_{2}$ is the qubit
set and $C=C_{1}\otimes V_{2}\cup V_{1}\otimes C_{2}$ is the stabilizer set.

The set of qubits can be partitioned into two subsets, $V_{1}\otimes V_{2}$
and $C_{1}\times C_{2}$. For $H_{X}$, $H_{1}\otimes\mathbf{I}_{n_{2}}$ acts on
$V_{1}\otimes V_{2}$ and $\mathbf{I}_{r_{1}}\otimes H_{2}^{T}$ acts on
$C_{1}\otimes C_{2}$. Moreover, the subset $V_{1}\otimes V_{2}$ can be further
partitioned into $n_{2}$ subsets $\\{V_{1}\otimes v_{2}^{1},V_{1}\otimes
v_{2}^{2},\cdots,V_{1}\otimes v_{2}^{n_{2}}\\}$, where $V_{1}\otimes
v_{2}^{k}:=\\{v\otimes v_{2}^{k}:v\in V_{1}\\}$ and
$V_{2}=\\{v_{2}^{1},\ldots,v_{2}^{n_{2}}\\}$ [Fig. 1].

Thus a $Z$-type Pauli operator can be expressed as a bit-string
$z=\left(z^{(1)},z^{(2)}\right)^{T}$, where $z^{(1)}$ is supported on the
qubit subset $V_{1}\otimes V_{2}$ with vector space
$\mathbb{F}_{2}^{n_{1}}\otimes\mathbb{F}_{2}^{n_{2}}$, and $z^{(2)}$ is
supported on the qubit set $C_{1}\otimes C_{2}$ with vector space
$\mathbb{F}_{2}^{r_{1}}\otimes\mathbb{F}_{2}^{r_{2}}$. Because $z^{(1)}$ and
$z^{(2)}$ act on a tensor product of two vector spaces, one can view them also
as a matrix. (For instance, $v_{i}\otimes u_{j}$ for unit vectors $v_{i}$ and
$u_{j}$ can be viewed as a matrix whose entry is $1$ on the $i$’th row and the
$j$’th column and zero elsewhere.) We call this procedure as vector reshaping,
and explain a few basic facts in the Appendix. After the reshaping, $z^{(1)}$
and $z^{(2)}$ become $Z^{(1)}$ and $Z^{(2)}$ respectively. Here, $Z^{(1)}$ is
an $n_{1}\times n_{2}$ matrix, and the entry $Z^{(1)}_{i,j}$ is supported on
qubit $v_{1}^{i}\otimes v_{2}^{j}$. Moreover, the $j$th column $Z^{(1)}_{j}$
is supported on qubits $\\{v_{1}^{1}\otimes v_{2}^{j},v_{1}^{2}\otimes
v_{2}^{j},\cdots,v_{1}^{n_{1}}\otimes v_{2}^{j}\\}$, which we refer to as
$V_{1}\otimes v_{2}^{j}$. Similarly, the subset $C_{1}\otimes C_{2}$ can be
partitioned into $r_{1}$ rows, and $i$th row is supported on qubit subset
$c_{1}^{i}\otimes C_{2}$ (defined similarly).

An elementary canonical logical-$Z$ operator, which is in the form
$\left(\bar{x}_{k}\otimes y_{j},0_{r_{1}r_{2}}\right)^{T}$ (where
$0_{r_{1}r_{2}}$ is the $r_{1}r_{2}$-dimensional zero vector), is supported on
the subset $V_{1}\otimes v_{2}^{j}$ if $y_{j}=v_{2}^{j}$. In the matrix form,
it is supported on the $j$th column of $Z^{(1)}$.

Figure 1: Graphical illustration the hypergraph product structure: (a) The
qubit subset $V_{1}\otimes V_{2}$ is partitioned into $\\{V_{1}\otimes
v_{2}^{1},V_{1}\otimes v_{2}^{2},\ldots,V_{1}\otimes v_{2}^{n_{2}}\\}$, where
$\\{v_{2}^{1},v_{2}^{2},\ldots,v^{n_{2}}_{2}\\}\in V_{2}$. (b) The qubit
subset $C_{1}\otimes C_{2}$ is divided into $\\{c^{1}_{1}\otimes
C_{2},c^{2}_{1}\otimes C_{2},\ldots,c^{r_{1}}_{1}\otimes C_{2}\\}$, with
$\\{c_{1}^{1},c^{2}_{1},\cdots,c^{r_{1}}_{1}\\}\in C_{1}$. An elementary
canonical logical operator $(\bar{x}_{k}\otimes y_{j},0_{r_{1}r_{2}})^{T}$
resides on the qubit subset $V_{1}\otimes v^{j}_{2}$ if $y_{j}=v_{2}^{j}$.
Similarly, the logical operator
$(0_{n_{1}n_{2}},a_{\ell}\otimes\bar{b}_{m})^{T}$ is localized on the subset
$c^{i}_{1}\otimes C_{2}$ if $c^{i}_{1}=a_{\ell}$.

It would be instructive to discuss an example of the canonical logical
operators. It is well-known that Kitaev’s toric code Kitaev (2003) can be
viewed as two copies of 1D repetition code Tillich and Zemor (2014). In this
context, the canonical logical operators are the string operators of minimal
lengths [Fig. 2].

Figure 2: The hypergraph product construction of the Toric code from two
repetition codes (periodic boundary conditions). (a) Delineates the two
subsets of qubits, $V_{1}\otimes V_{2}$ and $C_{1}\otimes C_{2}$. (b) Show the
logical $X$ operator (blue circles) and the logical-$Z$ operator (red circles)
within the subset $V_{1}\otimes V_{2}$. (c) Showcasing the logical $X$ (blue
circles) and $Z$ (red circles) operators within the subset $C_{1}\otimes
C_{2}$.

_Energy barrier of hypergraph product code._ \- We now study relationship
between the energy barrier of hypergraph product codes and their underlying
classical codes. We first upper bound the energy barrier of the hypergraph
product code in terms of the energy barrier of the classical codes. We then
prove a matching lower bound, establishing their equivalence.

_(1) Upper bound_.\- The upper bound on the energy barrier can be obtained by
considering a specific path from the identity to some logical operator. By
choosing this logical operator to be a canonical logical operator, we obtain
an upper bound. First, consider an elementary canonical logical operator of
the form of $(\bar{x}_{k}\otimes y_{j},0_{r_{1}r_{2}})^{T}$, where
$0_{r_{1}r_{2}}$ is the $r_{1}r_{2}$-dimensional zero vector. We will consider
a path consisting of the operators of the form of $(x_{i}\otimes y_{j})^{T}$
from $i=1$ to $i=N$, interpolating between $x_{1}=(0,\ldots,0)^{T}$ to
$x_{N}=\overline{x}_{k}$. The energy at the $i$th step is precisely
$\mathrm{wt}(H_{1}x_{i})$, which is the energy of the classical code $H_{1}$
evaluated with respect to $x_{i}$. Because $x_{N}=\overline{x}_{k}$ is a
codeword of $H_{1}$, the energy barrier along this path is at most
$\Delta(H_{1})$.

Similarly, we can consider a path of the form of
$(0_{n_{1}n_{2}},a_{\ell}\otimes b_{i})$ from $i=1$ to $i=N$, interpolating
between $b_{1}=(0,\ldots,0)^{T}$ to $b_{N}=\overline{b}_{m}$, where
$\overline{b}_{m}$ is a coreword of $H_{2}^{T}$. This yields an energy barrier
upper bound of $\Delta(H_{2}^{T})$. Together, we obtain an energy upper bound
of $\min(\Delta(H_{1}),\Delta(H_{2}^{T}))$ for the canonical logical-$Z$
operators. A similar argument can be applied to the canonical logical $X$
operators, yielding an upper bound of $\min(\Delta(H_{2}),\Delta(H_{1}^{T}))$.

_(2) Lower bound_.\- We now prove a matching lower bound for the energy
barrier of the canonical logical-$Z$ operators.

######  Proposition 1.

For any nontrivial canonical logical-$Z$ operator $L$,

$\Delta(L)\geq\min(\Delta(H_{1}),\Delta(H_{2}^{T})).$ (11)

We introduce two lemmas to prove this proposition. We use the following
convention in the proof. A path $r=\left\\{P_{0},P_{1},\ldots,P_{F}\right\\}$
is said to be supported on $U\subseteq V$ if the support of all $P_{i}$ in $r$
is included in $U$.

######  Lemma 2.

For any elementary canonical logical-$Z$ operator $L$ supported on
$V_{1}\otimes v^{\alpha}_{2}$ (resp. $c_{1}^{\beta}\otimes C_{2}$), its energy
barrier is attained by a path supported on $V_{1}\otimes v^{\alpha}_{2}$
(resp. $c^{\beta}_{2}\otimes C_{2}$).

######  Lemma 3.

For any nontrivial canonical logical-$Z$ operator $L$, the energy barrier
$\Delta(L)$ is greater than or equal to the minimum energy barrier of the
elementary canonical logical-$Z$ operators.

Consider an elementary canonical logical-$Z$ operator $L$ supported on
$V_{1}\otimes v_{2}^{\alpha}$. Suppose $\Delta(L)$ is given by a path $r$. The
main idea behind the proof of Lemma 2 is to deform a general path $r$ into a
new path $r^{\prime}$, supported on $V_{1}\otimes v_{2}^{\alpha}$.
Importantly, we show that the energy barrier of $r^{\prime}$ is no greater
than that of the original path $r$. A similar argument can be applied to prove
Lemma 3; see the Appendix for the proofs.

Let $L=(\bar{x}_{k}\otimes v_{2}^{j},0_{r_{1}r_{2}})^{T}$ be an elementary
canonical logical-$Z$ operator, supported on $V_{1}\otimes v_{2}^{j}$. Lemma 2
implies there is a path $r$ supported on $V_{1}\otimes v_{2}^{j}$ that attains
the energy barrier of $\Delta(L)$.

Using such a path, we can prove a lower bound on the energy barrier of $L$.
Without loss of generality, let $r=(P_{0},P_{1},\cdots,P_{F})$ with $P_{0}=I$
and $P_{F}=L$. Because this path is supported on $V_{1}\otimes v_{2}^{j}$, in
the binary representation, each $P_{i}$ becomes $u_{i}\otimes v_{2}^{j}$ for
some $u_{i}\in V_{1}$. In particular, at the $i$’th step, the energy is
$\text{wt}(H_{1}u_{i})$, which is exactly the energy of the classical code
with respect to $u_{i}\in V_{1}$. Because $u_{1}=(0,\ldots,0)$ and
$u_{F}=\bar{x}_{k}$ for some codeword $\bar{x}_{k}$ of $H_{1}$, the path
$(u_{0},\ldots,u_{F})$ must have an energy barrier of at least
$\Delta(H_{1})$. Similarly, for any elementary canonical logical-$Z$ operator
$L$ on subset $c_{1}^{\beta}\otimes C_{2}$, one can show that
$\Delta\left(L\right)\geq\Delta\left(H_{2}^{T}\right)$. Then Lemma 3
imeidately implies Proposition 1.

_(3) Energy barrier_.\- Since the lower bound and the upper bound match, the
energy barrier of the canonical logical-$Z$ operators is precisely
$\min(\Delta(H_{1}),\Delta(H_{2}^{T}))$. Similarly, the energy barrier of the
canonical logical-$X$ operators can be shown to be
$\min(\Delta(H_{2}),\Delta(H_{1}^{T}))$. Moreover, due to Theorem 1, the
energy barrier of any logical operator can be bounded in terms of the
canonical logical operators’ energy barrier and the code’s sparsity parameter.
Note that this conclusion applies to the hypergraph product of _any_ two
classical codes.

If the code is LDPC and the energy barrier is $\Omega(1)$, we conclude that
the energy barrier of the quantum code is exactly equal to
$\min(\Delta(H_{1}),\Delta(H_{2}),\Delta(H_{1}^{T}),\Delta(H_{2}^{T}))$.

######  Theorem 2.

Let $\Delta(H)$ be the energy barrier of the hypergraph product code obtained
from parity check matrices $H_{1}$ and $H_{2}$ of $O(1)$ row and column
weight. If
$\Delta(H_{1}),\Delta(H_{2}),\Delta(H_{1}^{T}),\Delta(H_{2}^{T})=\Omega(1)$,

$\Delta(H)=\min(\Delta(H_{1}),\Delta(H_{2}),\Delta(H_{1}^{T}),\Delta(H_{2}^{T})).$
(12)

More precisely, for any $\left(w_{c},w_{q}\right)$ quantum code derived from a
hypergraph product, if the parent classical codes’ energy barriers exceed
$w_{c}w_{q}$, the quantum code’s energy barrier matches the minimum energy
barrier of these parent codes. This can be used to prove tight bounds on the
energy barrier. For instance, a classical code whose Tanner graph is a
bipartite expander graph has an extensive macroscopic energy barrier. Thus our
result implies that the quantum expander code Leverrier  _et al._ (2015);
Fawzi  _et al._ (2020) also has an extensive energy barrier. While this is
already known Fawzi  _et al._ (2020), note that this argument does not rely on
the expansion property of quantum expander code.

_Outlook_.\- We proved a tight bound on the energy barrier of the hypergraph
product code, determined in terms of the energy barriers of the underlying
classical codes. While it was expected that the energy barrier of the quantum
code is related to its counterpart Rakovszky and Khemani (2024), we provided a
first rigorous proof of this statement [Eq. (1)] to our knowledge. Looking
forward, it would be interesting to study the energy barrier of the codes such
as the homological product Bravyi and Hastings (2014), balanced product codes
Breuckmann and Eberhardt (2021a) and lifted product codes Panteleev and
Kalachev (2022a). The generalized bicycle code Kovalev and Pryadko (2013);
Panteleev and Kalachev (2022a), due to its simple structure, is another
natural candidate to explore. Understanding how our proof technique can be
generalized to these setups and how to design efficient decoders that can
leverage the energy barrier is left for future work.

_Acknowledgement_ \- We thank Niko Breuckmann, Sergey Bravyi, Earl Campbell,
Jeongwan Haah, Vedika Khemani, and Anthony Leverrier for helpful discussions.
GZ acknowledges the financial support from Sydney Quantum Academy. This work
was supported by the Australian Research Council Centre of Excellence for
Engineered Quantum Systems (EQUS, CE170100009). IK acknowledges support from
NSF Grant QCIS-FF 2013562.

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##  I Appendix

Throughout the appendix, we are always working in the field $\mathbb{F}_{2}$.
Thus, all addition operations are modulo $2$ except for the computation of the
weight of vectors or matrices, i.e., the function $\mathrm{wt}(\cdot)$.

###  I.1 Vector reshaping

Consider a basis $\mathcal{B}$ of the vector space
$\mathbb{F}_{2}^{n_{1}}\otimes\mathbb{F}_{2}^{n_{2}}$:

$\displaystyle\mathcal{B}=\left\\{a_{i}\otimes b_{j}\mid
i=1,\ldots,n_{1}\text{ and }j=1,\ldots,n_{2}\right\\}.$ (13)

Then any vector $v\in\mathbb{F}_{2}^{n_{1}}\otimes\mathbb{F}_{2}^{n_{2}}$ can
be written as

$\displaystyle v=\sum_{a_{i}\otimes
b_{j}\in\mathcal{B}}v_{ij}\left(a_{i}\otimes b_{j}\right)$ (14)

for some $v_{ij}\in\mathbb{F}_{2}$. We call the $n_{1}\times n_{2}$ matrix $V$
with entries $v_{ij}$ the _reshaping_ of the vector $v$. By this definition,
if $A,B$ are respectively $m_{1}\times n_{1}$ and $m_{2}\times n_{2}$
matrices, then

$\displaystyle(A\otimes B)v\Rightarrow AVB^{T}.$ (15)

Define $\mathrm{wt}(M)$ as the number of ones in the vector (or matrix) $M$.
We have $\mathrm{wt}((A\otimes B)v)=\mathrm{wt}(AVB^{T})$.

###  I.2 Proof of Lemma 2

Recall that a $Z$-type Pauli error of the hypergraph product code can be
expressed as a bit-string $z=\left(z^{(1)},z^{(2)}\right)^{T}$. The
corresponding energy $\epsilon(z)=\text{wt}(H_{X}z)$ is

$\epsilon(z)=\text{wt}\left((H_{1}\otimes I_{n_{2}})z^{(1)}+(I_{r_{1}}\otimes
H_{2}^{T})z^{(2)}\right).$ (16)

By applying vector reshaping, the energy can be written as

$\epsilon(z)=\text{wt}\left(H_{1}Z^{(1)}+Z^{(2)}H_{2}\right),$ (17)

where $Z^{(1)}$ and $Z^{(2)}$ are the matrices reshaped from $z^{(1)}$ and
$z^{(2)}$, respectively. The $j$th column of $Z^{(1)}$ is supported on qubit
subset $V_{1}\otimes v_{2}^{j}$.

Using Eq. (17), we aim to prove a lower bound on the energy $\epsilon(z)$
[Lemma 4]. To that end, we shall use the following convention. Let $L_{c}$ be
a nontrivial codeword of $H_{2}$. The set of columns of $Z^{(1)}$ associated
with the nonzero entries of $L_{c}$ will play an important role. We define the
index set of such columns as $\mathcal{C}(L_{c})$:

$\mathcal{C}(L_{c}):=\\{j:(L_{c})_{j}=1\\}.$ (18)

Given a codeword $L_{c}$ of $H_{2}$, one can construct a vector $z^{1,s}$ from
$Z^{(1)}$ by summing all the columns in the set $\mathcal{C}(L_{c})$. More
formally,

$z^{1,s}_{i}=\sum_{j:j\in\mathcal{C}(L_{c})}Z^{(1)}_{ij},$ (19)

where the addition is modulo $2$. For example, if $L_{c}=110100$, we would sum
columns $1,2$, and $4$. Using Eq. (19), we can deform an arbitrary path to a
path consisting of Paulis only supported on subset $V_{1}\otimes v_{2}^{k}$
for some $k$.111The precise choice of $k$ does not matter; any
$k\in\mathcal{C}(L_{c})$ would suffice. In particular, we can prove an
inequality between the energy of the original Pauli and the deformed Pauli,
proved in Lemma 4.

######  Lemma 4.

$\mathrm{wt}(H_{1}z^{1,s})\leqslant\mathrm{wt}\left(H_{1}Z^{(1)}+Z^{(2)}H_{2}\right).$
(20)

###### Proof.

We prove this by contradiction. Consider the row spaces of $H_{1}z^{1,s}$ and
$H_{1}Z^{(1)}+Z^{(2)}H_{2}$. If
$\mathrm{wt}(H_{1}z^{1,s})>\mathrm{wt}\left(H_{1}Z^{(1)}+Z^{(2)}H_{2}\right)$,
then there exists a row $j$ such that

$\displaystyle\mathrm{wt}((H_{1}z^{1,s})_{\text{row
}j})>\mathrm{wt}\left((H_{1}Z^{(1)}+Z^{(2)}H_{2})_{\text{row }j}\right).$ (21)

We will prove that Eq. (21) cannot be satisfied, thereby proving the claim.

Without loss of generality, consider the $j$’th row. Since $H_{1}z^{1,s}$ is a
$r_{1}\times 1$ vector, $\mathrm{wt}((H_{1}z^{1,s})_{\text{row }j})$ must be
either 0 or 1. If $\mathrm{wt}((H_{1}z^{1,s})_{\text{row }j})=0$, Eq. (21)
cannot be satisfied. Therefore, we consider the
$\mathrm{wt}((H_{1}z^{1,s})_{\text{row }j})=1$ case.

If $\mathrm{wt}((H_{1}z^{1,s})_{\text{row }j})=1$, $(H_{1}Z^{(1)})_{\text{row
}j}$ must contain an odd number of ones on the columns in the set
$\mathcal{C}(L_{c})$. Otherwise, we would have had
$\mathrm{wt}((H_{1}z^{1,s})_{\text{row }j})=0$, which is a contradiction. On
the other hand, we remark that $(Z^{(2)}H_{2})_{\text{row }j}$ consists of an
even number of ones on the column set $\mathcal{C}(L_{c})$. To see why, note
that each row of $Z^{(2)}H_{2}$ is a linear combination of the checks in
$H_{2}$. Because $L_{c}$ is a codeword of $H_{2}$,
$(Z^{(2)}H_{2})_{\text{row},j}L_{c}=0$. Therefore, in the
$\mathrm{wt}((H_{1}z^{1,s})_{\text{row }j})=1$ case, the number of ones in
$H_{1}Z^{(1)}+Z^{(2)}H_{2}$ on the $j$’th row and the columns in
$\mathcal{C}(L_{c})$ is odd.

Thus we conclude $\mathrm{wt}\left((H_{1}Z^{(1)}+Z^{(2)}H_{2})_{\text{row
}j}\right)\geq 1$. As such, Eq. (21) cannot be satisfied. This completes the
proof. ∎

Now we are in a position to prove Lemma 2. We do so by identifying a path
$r^{\prime}=\left\\{P_{0}^{\prime},P_{1}^{\prime},\cdots,P_{F}^{\prime}\right\\}$
that is only supported on $V_{1}\otimes v_{2}^{\alpha}$ while ensuring that
$\epsilon_{\max}\left(r^{\prime}\right)\leqslant\epsilon_{\max}(r)$. Lemma 4
suggests a way to deform the path $r$ to the one supported on $V_{1}\otimes
v^{\alpha}_{2}$.

Without loss of generality, let $r=\left\\{P_{0},P_{1},\cdots,P_{F}\right\\}$
be a path that $\Delta(L)=\epsilon_{\max}(r)$, with $P_{0}=I$ and $P_{F}=L$.
We consider a Pauli $P_{i}$ in the path $r$. It will be convenient to work in
its binary representation, written as $(p^{(1)},p^{(2)})^{T}$, where $p^{(1)}$
and $p^{(2)}$ represent the Paulis supported on $V_{1}\otimes V_{2}$ and
$C_{1}\otimes C_{2}$, respectively. First, we remove all the Paulis supported
on $C_{1}\otimes C_{2}$ by setting $p^{(2)}$ as the zero vector. Next, we
apply the following transformations to $p^{(1)}$. We reshape $p^{(1)}$ and
refer to the reshaped matrix as $P^{(1)}$. Let $L_{c}$ be a codeword of
$H_{2}$ such that $(L_{c})_{\alpha}=1$. Consider a set of columns in $P^{(1)}$
corresponding to the index set $\mathcal{C}(L_{c})$. Denoting each column as
$u_{k}$, where $k\in\mathcal{C}(L_{c})$, we update the column $u_{\alpha}$ in
the following way:

$u_{\alpha}\to
u_{\alpha}+\sum_{k\in\mathcal{C}(L_{c})\setminus\\{\alpha\\}}u_{k}.$ (22)

Afterwards, the other columns of $P^{(1)}$ are set to the zero vector. This
yields the deformed Pauli operator $P_{i}^{\prime}$.

By construction, the resulting $P_{i}^{\prime}$ is supported on $V_{1}\otimes
v^{\alpha}_{2}$. Note that $r^{\prime}$ is a valid path because
$\mathrm{wt}(P_{i}^{\prime}P_{i+1}^{\prime})\leq 1$ for every $i$. Also,
because $P_{F}^{\prime}=P_{F}=L$, $r^{\prime}$ is still a path for $L$.
Moreover, because
$\epsilon\left(P_{i}^{\prime}\right)\leqslant\epsilon\left(P_{i}\right)$ for
all $i$ [Lemma 4], we have
$\epsilon_{\max}\left(r^{\prime}\right)\leqslant\epsilon_{\max}(r)$. Both
$r^{\prime}$ and $r$ are paths for $L$, by definition
$\epsilon_{\max}\left(r^{\prime}\right)\geqslant\epsilon_{\max}(r)$, we
conclude $\epsilon_{\max}\left(r^{\prime}\right)=\epsilon_{\max}(r)$. Thus, by
deforming $r$, we obtained a new path $r^{\prime}$ supported on $V_{1}\otimes
v^{\alpha}_{2}$ that yields the energy barrier $\Delta(L)$.

This argument can be applied to prove similar lower bounds for logical
operators on $c^{\beta}_{1}\otimes C_{2}$. To conclude, for any elementary
canonical logical operator $L$ supported on $V_{1}\otimes v^{\alpha}_{2}$
(resp. $c^{\beta}_{1}\otimes C_{2}$), their energy barrier can be given by a
path supported on $V_{1}\otimes v^{\alpha}_{2}$ (resp. $c^{\beta}_{1}\otimes
C_{2}$).

###  I.3 Proof of Lemma 3

Any nontrivial canonical logical-$Z$ operator $L$ belongs to one of the
following categories:

  * •

Case 1: $L$ is supported solely on the qubit subset $V_{1}\otimes V_{2}$.

  * •

Case 2: $L$ is supported solely on the qubit subset $C_{1}\otimes C_{2}$.

  * •

Case 3: $L$ is supported on both subsets.

We will focus solely on Case 1. Case 2 can be analyzed similarly by
considering subsets $C_{1}\otimes C_{2}$, while case 3 can be treated as Case
1 or 2.

Without loss of generality, let the energy barrier of $L$ be attained by a
path $r=\\{P_{0},P_{1},\cdots,P_{F}\\}$, with $P_{0}=I$ and $P_{F}=L$. Similar
to the approach taken in Lemma 2, we aim to deform the path $r$ to the one
supported on $V_{1}\otimes v_{2}^{k}$ for some $k$, such that the energy
barrier of the deformed path lower bounds that of the $r$.

The deformation works in the same way as in the proof of Lemma 2. We describe
this procedure again for the readers’ convenience. Let $L_{c}$ be a nontrivial
codeword of $H_{2}$ and $\mathcal{C}(L_{c})$ be its corresponding column index
set. We consider a binary representation of a Pauli $P_{i}$, written as
$(p^{(1)},p^{(2)})^{T}$. As in the proof of Lemma 2, we remove the Paulis
supported on $C_{1}\otimes C_{2}$ by setting $p^{(2)}$ as the zero vector.
Next, reshape $p^{(1)}$ into a matrix $P^{(1)}$ and update its columns in the
following way. Choose $\alpha\in\mathcal{C}(L_{c})$. This column is updated as
Eq. (22). The other columns of $P^{(1)}$ are converted to zero vectors.

Thanks to Lemma 4, we obtain a new path
$r^{\prime}=\\{P_{0}^{\prime},P_{1}^{\prime},\cdots,P_{F}^{\prime}\\}$
supported on $V_{1}\otimes v_{\alpha}^{2}$ with the property
$\epsilon_{\max}\left(r^{\prime}\right)\leqslant\epsilon_{\max}\left(r\right)$.
Note that $P_{F}^{\prime}$, in the binary representation, is of the form
$\bar{x}\otimes v_{\alpha}^{2}$, where $\bar{x}$ is a codeword of $H_{1}$.
Therefore, $P_{F}^{\prime}$ is either a nontrivial elementary canonical
logical operator or the identity. In the latter case, $P_{F}^{\prime}$ is the
trivial codeword (zero vector) in the binary representation. Henceforth, we
denote this as $L^{\prime}=P_{F}^{\prime}$.

If $L^{\prime}$ is nontrivial, we can use the relation between the energy
barriers of $L$ and $L^{\prime}$:

$\displaystyle\Delta(L)=\epsilon_{\max}(r)\geqslant\epsilon_{\max}(r^{\prime})\geqslant\Delta(L^{\prime}).$
(23)

Because $L^{\prime}$ is an elementary logical operator, $\Delta(L)$ is greater
or equal to the minimum energy barrier of elementary canonical logical
operators. Thus, if $L^{\prime}$ is nontrivial, the proof follows immediately.

If $L^{\prime}$ is an identity, the above argument does not work. Fortunately,
it turns out that for any $L^{\prime}$, one can choose $L_{c}$ (the codeword
of $H_{2}$ used in the current proof) such that $L^{\prime}$ is not an
identity.

Without loss of generality, consider a canonical logical-$Z$ operator $L$,
expressed as

$L=\begin{pmatrix}\sum_{k,j}\lambda_{kj}\bar{x}_{k}\otimes y_{j}\\\
0_{r_{1}r_{2}}\end{pmatrix},$ (24)

where (i) $H_{1}\bar{x}_{i}=0$ and (ii)
$y_{j}\notin\operatorname{Im}\left(H_{2}^{T}\right)$ are unit vectors. If a
given path ends with $L$, its deformation (using Eq. (22)) yields the
following operator $L^{\prime}$:

$L^{\prime}=\begin{pmatrix}\sum_{k}c_{k}\bar{x}_{k}\otimes y_{\alpha}\\\
0_{r_{1}r_{2}}\end{pmatrix},$ (25)

where $\alpha\in\mathcal{C}(L_{c})$ and $c_{k}$ is defined as

$c_{k}:=\sum_{j\in\mathcal{C}(L_{c})}\lambda_{kj}.$ (26)

Note that $L^{\prime}$ is trivial if and only if $c_{k}=0$ for all $k$.
Therefore, we aim to prove that there exists a choice of $L_{c}$ such that
$c_{k}=1$ for at least one $k$.

Let us prove the contrapositive. Suppose $c_{k}=0$ for all $k$, for any choice
of $L_{c}$. Consider the following vector:

$u_{k}:=\sum_{j}\lambda_{kj}y_{j}.$ (27)

Note that $c_{k}=u_{k}^{T}L_{c}$. By our assumption $c_{k}=0$ for any choice
of $L_{c}$ and so the inner product of $u_{k}$ with any codeword of $H_{2}$
must be zero. On the other hand, $u_{k}$, if it is nonzero, must lie outside
of $\text{Im}(H_{2}^{T})$ by the definition of the $y_{j}$’s. Thus, $u_{k}$ is
not an element of the row space of $H_{2}$. However, this is a contradiction
for the following reason. For a linear code, let $H$ and $G$ be the parity
check matrix and the generator matrix. Then $v^{T}G=0$ if and only if $v$ is a
vector in the row space of $H$. In our setup, if $c_{k}=0$ for any $L_{c}$,
then $u_{k}^{T}G_{2}=0$. This implies that $u_{k}$ must be in the row space of
$H_{2}$, which contradicts the fact that it lies outside of
$\text{Im}(H_{2}^{T})$. To conclude, there must be at least one $k$ such that
$c_{k}=1$. Thus, there is always a choice of $L_{c}$ such that $L^{\prime}$ is
not an identity, thereby proving the claim.

Case 2 can be analyzed similarly to Case 1 by considering rows in the subset
$C_{1}\otimes C_{2}$. For Case 3, one can treat it just as Case 1 or Case 2.
For example, when treating it as Case 1, the logical operator $L$ has a
nontrivial part in the subset $V_{1}\otimes V_{2}$. One can prove there exists
a codeword $L_{c}$ of $H_{2}$, such that after the deformation, the resulting
$L^{\prime}$ is a nontrivial elementary canonical logical operator.