#  Skipper: Improving the Reach and Fidelity of Quantum Annealers by Skipping
Long Chains

Ramin Ayanzadeh  Georgia Institute of TechnologyAtlantaUSA and  Moinuddin
Qureshi  Georgia Institute of TechnologyAtlantaUSA

###### Abstract.

Quantum Annealers (QAs) operate as single-instruction machines, lacking a SWAP
operation to overcome limited qubit connectivity. Consequently, multiple
physical qubits are _chained_ to form a program qubit with higher
connectivity, resulting in a drastically diminished effective QA capacity by
up to 33x. We observe that in QAs: (a) chain lengths exhibit a power-law
distribution, a few _dominant chains_ holding substantially more qubits than
others; and (b) about 25% of physical qubits remain unused, getting isolated
between these chains. We propose _Skipper_ , a software technique that
enhances the capacity and fidelity of QAs by skipping dominant chains and
substituting their program qubit with two readout results. Using a 5761-qubit
QA, we demonstrate that Skipper can tackle up to 59% (Avg. 28%) larger
problems when eleven chains are skipped. Additionally, Skipper can improve QA
fidelity by up to 44% (Avg. 33%) when cutting five chains (32 runs). Users can
specify up to eleven chain cuts in Skipper, necessitating about 2,000 distinct
quantum executable runs. To mitigate this, we introduce _Skipper-G_ , a greedy
scheme that skips sub-problems less likely to hold the global optimum,
executing a maximum of 23 quantum executables with eleven chain trims.
Skipper-G can boost QA fidelity by up to 41% (Avg. 29%) when cutting five
chains (11 runs).

##  1\. Introduction

Quantum computers (QCs) have the potential to solve certain problems beyond
the capabilities of classical computers (arute2019quantum, ;
villalonga2020establishing, ; preskillNISQ, ; king2021scaling, ; wu2021strong,
). Two main types of QCs exist: digital machines, exemplified by industry
leaders like IBM (IBMQ, ), Google (GoogleAI, ), IonQ (IonQ, ), and Quantinuum
(quantinuum, ), and analog devices such as superconducting _Quantum Annealers_
(_QAs_) developed by D-Wave (D-Wave, ), as well as neutral atom platforms by
QuEra (QuEra, ) and PASQAL (PASQAL, ). Both digital and analog QCs have
polynomial equivalent computing power (aharonov2008adiabatic, ;
albash2018adiabatic, ). For instance, QAs have demonstrated their potential in
tackling real-world applications such as finance (elsokkary2017financial, ),
drug discovery (mulligan2020designing, ), cryptography (peng2019factoring, ;
hu2020quantum, ), Boolean Satisfiability (SAT) (su2016quantum, ;
ayanzadeh2020reinforcement, ; ayanzadeh2018solving, ; ayanzadeh2019sat, ),
planning and scheduling (inoue2021traffic, ; rieffel2015case, ;
venturelli2015quantum, ; tran2016hybrid, ), linear algebra (o2018nonnegative,
), and signal processing (ayanzadeh2019quantum, ; ayanzadeh2020ensemble, ),
extending beyond application-specific acceleration.

While both QC types are accessed via the cloud (AmazonBraKet, ;
MicrosoftAzure, ; D-Wave, ), their operation models and design trade-offs
differ significantly (ayanzadeh2022equal, ). In digital QCs (namely the gate-
based or circuit model quantum computing), as shown in Fig. 1(a), qubits
undergo a scheduled sequence of quantum operations defined by the quantum
algorithm to directly manipulate their states (nielsen2010quantum, ).
Conversely, as shown in Fig. 1(b), analog QCs operate as single-instruction
systems, where the qubit environment is incrementally modified based on the
evolution of a physical system, called _Hamiltonian_ , thereby allowing
natural qubit evolution and indirect state alteration (ayanzadeh2022equal, ;
albash2018adiabatic, ; mcgeoch2020theory, ).

Figure 1.  Operation Model of QCs: (a) Digital QCs execute compiled quantum
circuits. (b) Analog QAs execute the embedded problem Hamiltonian. _Operating
as single-instruction machines, analog QAs do not incorporate quantum
circuits._

Full connectivity of qubits at scale is infeasible. In digital QCs, compilers
introduce SWAP operations to make physical qubits adjacent
(zulehner2018efficient, ; murali2019noise, ; tannu2019not, ). Conversely,
analog QCs cannot apply operations to qubits, thus preventing the use of SWAPs
for qubit routing. Instead, QAs employ _embedding_ (zbinden2020embedding, ;
pelofske20234, ; pelofske2019solving, ; pelofske2022solving, ;
barbosa2021optimizing, ) where multiple physical qubits are _chained_ (or
entangled) to represent a program qubit with higher connectivity, as shown in
Fig. 2(a). Compiling quantum circuits in digital QCs preserves qubit
utilization (1-to-1 mapping between program and physical qubits), however,
embedding in QAs can substantially increase physical qubit utilization
(ayanzadeh2022equal, ). For instance, the 5761-qubit QA can accommodate up to
177 program qubits with all-to-all connectivity, highlighting nearly 33x
reduced _logical capacity_.

(a)  (b)

Figure 2.  (a) Embedding seven program qubits $(Q_{i})$ onto a $5\times 7$
grid of physical qubits. (b) Max embeddable Barabasi–Albert (BA) graphs on a
5761-qubit QA device for different preferential attachment factors (${m}$),
ranging from sparse BA-1 ($m=1$) to highly dense BA-6 ($m=6$) structures.

Given that the hardware graph remains fixed after fabrication, QAs’ logical
capacity is primarily determined by the topology of the problem graph. Real-
world applications typically involve irregular “Power-Law” graphs
(agler2016microbial, ; clauset2016colorado, ; gamermann2019comprehensive, ;
goh2002classification, ; house2015testing, ; mislove2007measurement, ;
pastor2015epidemic, ), and Barabasi–Albert (BA) graphs are widely considered
representative of such real-world graphs (albert2005scale, ;
barabasi1999emergence, ; barabasi2000scale, ; gray2018super, ;
kim2022sparsity, ; lusseau2003emergent, ; wang2019complex, ;
zadorozhnyi2012structural, ; zbinden2020embedding, ). Figure 2(b) illustrates
the largest embeddable BA graphs on a 5761-qubit QA, ranging from sparse (with
attachment factor $m=1$, BA-1) to nearly fully connected (with $m=6$, BA-6)
structures. As $m$ increases linearly, the logical capacity experiences a
superpolynomial reduction, converging to the 177-node fully connected graph.

We observe that chain lengths in QAs follow a “Power-Law” distribution, where
a few _dominant chains_ are significantly longer than most other chains (see
section 4.1 for more information). Moreover, we observe that a significant
portion of physical qubits, nearly 25%, remain unused as they become trapped
in long chains. Furthermore, we observe that long chains can reduce the
fidelity of QAs too. The qubits within a chain might take different values
post-measurement, called _broken chains_. Broken chains can negatively impact
QAs’ reliability, and longer chains are more likely to break.

In this study, we aim to improve the capacity and fidelity of QAs through
eliminating dominant chains, as they account for a substantial portion of
qubit utilization and are the main reason for isolating physical qubits. We
propose _Skipper_ , which _prunes_ these chains by removing their
corresponding program qubits and replacing them with two possible measurement
outcomes: -1 and +1. By eliminating a dominant chain, Skipper accomplishes two
objectives: (a) releasing physical qubits previously used within pruned
chains, and (b) releasing all qubits previously trapped with the pruned chain.
This can enable us to use all released physical qubits to accommodate more
program qubits.

However, identifying and pruning dominant chains is nontrivial. Chains are
formed post-embedding. First, when a (long) chain is eliminated, the remaining
embedding is likely not to be optimum, necessitating re-embedding the new
problem to maximize the reliability of QAs. Embedding itself is nontrivial, as
it can take several hours for problems at scale. Moreover, embedding
techniques are heuristic, and they may fail to find an embedding successfully
for a problem, requiring multiple attempts. Second, pruning the longest chain
can change the position of the second-longest chain when re-embedding the
problem, necessitating an embedding for every pruned chain. To this end,
Skipper adopts a greedy approach to prune $c$ chains by sorting program qubits
based on their degree and removing the top $c$ qubits simultaneously. We
observe that this greedy approach exhibits desirable, near-optimal behavior
for $c\geq 5$ chain cuts.

Importantly, the number of chain cuts in Skipper is user-defined; the system
allows for a maximum of eleven chains to be cut, and this does not scale with
the problem size, offering flexibility within the user’s budget constraints.
Each chain cut bifurcates the search space of the initial problem; therefore,
trimming eleven chains can lead to up to 2048 sub-problems, and Skipper
examines all of them to ensure guaranteed recovery. Our experiments with a
5761-qubit QA by D-Wave demonstrate that Skipper can address up to 59% (Avg.
28.3%) larger problems when up to eleven chains are trimmed. Additionally,
Skipper can significantly enhance QA fidelity by up to 44.4% (Avg. 33.1%),
when trimming up to five chains and running 32 quantum executables.

Skipper is inspired by FrozenQubits (ayanzadeh2023frozenqubits, ). Skipper
enhances both the capacity and fidelity of analog QAs. However, FrozenQubits
has a negligible impact on the capacity of digital QCs, where one program
qubit is represented with one physical qubit. Furthermore, FrozenQubits’
performance diminishes as graph density increases, whereas Skipper effectively
handles graphs ranging from sparse to dense structures.

The quantum cost of Skipper can present affordability challenges for certain
users. We introduce _Skipper-G_ , a greedy approach that bypasses sub-spaces
less likely to include the global optimum. Consequently, it runs at most 23
quantum executables, compared to the 2048 required by Skipper for trimming up
to eleven chains. It is worth noting that Skipper-G is proposed to improve QA
fidelity, with its effect on increasing capacity being negligible. Our
experiments demonstrate that Skipper-G can boost QA fidelity by up to 40.8%
(Avg. 29.2%), with five chain cuts and 11 runs.

Overall, this paper makes the following contributions:

  1. (1)

We show that in QAs, the chain length exhibits a “Power-Law” distribution,
with a few dominant chains having significantly more qubits. Moreover, we
demonstrate that approximately 25% of physical qubits remain unused as they
become trapped within long chains.

  2. (2)

We introduce _Skipper_ that enhances the capacity and reliability of QAs by
cutting dominant chains, thereby addressing up to 59% (Avg. 28.3%) larger
problems and improving QA fidelity by up to 44.4% (Avg. 33.1%), when up to
eleven and five chains are cut, respectively. To our knowledge, Skipper is the
first proposal to simultaneously enhance both the capacity and fidelity of
QAs.

  3. (3)

We demonstrate that the quantum cost of Skipper in enhancing QA fidelity can
be substantially reduced (to only 23 runs, compared to over 2000 runs in
Skipper) by bypassing sub-spaces unlikely to contain optimal solutions.

  4. (4)

We propose _Skipper-G_ , a greedy scheme that enhances QA fidelity by up to
40.8% (Avg. 29.2%), with five chain cuts and only 11 runs (compared to 32 runs
in Skipper).

##  2\. Background and Motivation

###  2.1. Quantum Computers: Digital vs. Analog

QCs fall into two categories: digital and analog. Digital QCs, like those from
IBM and Google, apply precise quantum operations—defined by the quantum
algorithm—to qubits in order to directly manipulate their state
(nielsen2010quantum, ). Conversely, analog QCs, like those from D-Wave and
QuEra, do not directly manipulate the state of qubits. Instead, they apply
precise changes—defined by the quantum program—to the environment in which the
qubits reside, allowing the qubits to evolve and change their states naturally
(albash2018adiabatic, ; ayanzadeh2022equal, ).

###  2.2. Quantum Annealers: Analog Quantum Accelerators

Quantum annealing is a meta-heuristic for tackling optimization problems that
runs on classical computers. _Quantum Annealers_ (_QAs_) are a form of analog
QCs that can sample from the ground state (the configuration with the lowest
energy value) of a physical system, called Hamiltonian (albash2018adiabatic, ;
ayanzadeh2021multi, ; ayanzadeh2022equal, ). QAs by D-Wave are single-
instruction optimization accelerators that can only sample from the ground
state of the following problem Hamiltonian (or Ising model):

(1) $\mathcal{H}_{p}:=\sum_{i}{\mathbf{h}_{i}\mathbf{z}_{i}}+\sum_{I\neq
j}{J_{ij}\mathbf{z}_{i}\mathbf{z}_{j}}$

acting on spin variables $\mathbf{z}_{i}\in{-1,+1}$, where
$\mathbf{h}_{i}\in\mathbb{R}$ and $J_{ij}\in\mathbb{R}$ are linear and
quadratic coefficients, respectively (ayanzadeh2022equal, ).

###  2.3. Operation Model of Single-Instruction QAs

QAs operate as single-instruction computers, and during each execution trial,
they only draw a single sample to approximate the global minimum of (1).
Therefore, we _cast_ real-world problems into Hamiltonians, where $\mathbf{h}$
and ${J}$ are defined in such a way that its global minimum represents the
optimal solution to the problem at hand (albash2018adiabatic, ;
ayanzadeh2022equal, ). The abstract problem Hamiltonian is then _embedded_
into the connectivity map of the QA hardware to generate an executable Quantum
Machine Instruction (QMI) (minorminerGithub, ; cai2014practical, ). Casting
and embedding in QAs are akin to designing and compiling quantum circuits in
digital QCs, respectively (Fig. 1). The QMI is executed for several trials,
and the outcome with the lowest objective value is deemed as the ultimate
result (ayanzadeh2022equal, ).

###  2.4. Anneal Time: Current Technological Barriers

As the energy gap between the global minimum and the adjacent higher state
diminishes linearly, the required annealing time for successful adiabaticity
grows exponentially (albash2018adiabatic, ; das2008colloquium, ), surpassing
the limits of contemporary QAs (yan2022analytical, ). Nonetheless, QAs, akin
to other QCs, are advancing; subsequent generations are expected to bypass
present technological constraints. Specifically, incorporating $XX$ terms into
the time-dependent Hamiltonian can ebb the annealing time scaling from
exponential to linear (nishimori2017exponential, ).

Figure 3.  Embedding example.

###  2.5. Embedding for QAs

The connectivity of QA qubits is sparse, thereby limiting users to only
specify $J_{ij}$ for those qubits that are physically connected. Thus, the
abstract problem Hamiltonian is _embedded_ into QA hardware where a program
qubit ($Q_{i}$) with higher connectivity is represented by multiple physical
qubits ($q_{i}$) called _chain_ (Fig. 3). Satisfying the following conditions
is sufficient to guarantee that both the abstract Hamiltonian and the embedded
Hamiltonian executed on the QA hardware have identical ground states:

  1. (1)

All chains representing program qubits must be a connected component
graph—i.e., there must be a path between any two qubits within a chain.

  2. (2)

There must be at least one connection between chains whose corresponding
program qubits are connected.

  3. (3)

The quadratic coefficient $J_{ij}$ is distributed equally among the couplers
connecting $Q_{i}$ and $Q_{j}$.

  4. (4)

The linear coefficient $\mathbf{h}_{i}$ is distributed equally among all
physical qubits of the corresponding chain.

  5. (5)

Inter-chain quadratic coefficients must be large enough to guarantee that all
qubits within a chain take an identical value—i.e., a very high penalty for
broken chains.

###  2.6. Prior Work Limitations

####  2.6.1. Circuit Cutting in Digital QCs

Circuit cutting techniques, namely CutQC (tang2021cutqc, ), partition quantum
circuits into smaller sub-circuits, enabling larger quantum circuits to be run
on smaller QCs. However, a similar approach is infeasible in the analog
quantum realm because: (a) analog QAs do not incorporate quantum circuits to
cut its wires; and (b) partitioning graphs by edge/node removal is nontrivial
(e.g., highly dense graphs are non-partitionable).

####  2.6.2. Solving Larger Problems on Smalle QAs

Previous methods for solving larger problems on smaller QAs
(pelofske2022solving, ; okada2019improving, ) employ iterative or alternating
approaches involving approximations, leading to reduced reliability as problem
size increases. Additionally, convergence—even to a local optimum—is not
guaranteed with these techniques. Conversely, Skipper explores the entire
search space comprehensively without resorting to approximations, and since it
is not iterative, it does not face convergence issues.

####  2.6.3. Application-Specific Policies

Recent studies have proposed methods for tackling larger instances in various
domains, such as Boolean Satisfiability (SAT) (tan2023hyqsat, ), Max-Clique
(pelofske2023solving, ; pelofske2019solving, ; pelofske2022parallel, ;
pelofske2021decomposition, ), and compressive sensing with matrix uncertainty
(ayanzadeh2019quantum, ; mousavi2019survey, ). However, these techniques are
tailored to their specific applications and cannot be easily adapted to other
domains. In contrast, Skipper is versatile and can be applied to any problem
Hamiltonian. Moreover, reduction to SAT and Max-Clique often leads to a
polynomial increase in program qubits, expanding the problem size.

####  2.6.4. FrozenQubits

Skipper is inspired by FrozenQubits (ayanzadeh2023frozenqubits, ), with both
methods aiming to eliminate high-degree program qubits. While the impact of
FrozenQubits on addressing larger problems in digital QCs is minimal due to
the one-to-one correspondence between program and physical qubits, Skipper, on
the other hand, is capable of solving larger problems on QAs and enhancing QA
fidelity. Moreover, unlike FrozenQubits, whose performance declines with
increasing graph density, Skipper maintains effectiveness across a spectrum of
graph densities, from sparse to dense structures.

###  2.7. Goal of This Paper

Figure 4(a) shows the maximum and average chain lengths for different graph
topologies embedded on a 5761-qubit QA. A few dominant chains contain over
7.9x as many qubits as the average chain lengths. Furthermore, Fig. 4(b)
displays the number of unused qubits when embedding the largest possible
graphs on a 5761-qubit QA for different graph topologies, indicating that more
than 25% of physical qubits remain unutilized, primarily due to dominant
chains.

The severe underutilization of QA qubits, along with utilizing several
physical qubits to represent a single program qubit, severely diminishes the
capacity of QAs by up to 33x. For instance, while current D-Wave QAs boast
over 5,700 qubits, they can accommodate at most 177 program qubits with full
connectivity. The aim of this paper is to enable QAs to tackle larger problems
by pruning dominant chains, while also enhancing the fidelity of the QAs.

(a)  (b)

Figure 4.  Maximum embeddable BA graphs on 5761-qubit QA: (a) Avg and Max
chain lengths, and (b) Number of unutilized qubits.

##  3\. Methodology

###  3.1. Hardware Platform

For our evaluations, we utilize the D-Wave Advantage System (version 6.2),
which features over 5,760 qubits and more than 40,000 couplers, accessed via
the D-Wave Leap cloud service (D-Wave, ). We employ the default annealing time
of 20 microseconds and adhere to the anneal schedule recommended for this
device. Each problem is run for 4,000 trials to comply with the two seconds
maximum job duration limit.

###  3.2. Software Platform

We utilize the _minorminer_ tool (minorminerGithub, ; cai2014practical, ) to
find embeddings for arbitrary problem Hamiltonians on QA hardware. In our
experiments, we set a timeout of 1,000 seconds, a maximum of 20 failed
attempts for improvement, and conduct 20 trials. To program the D-Wave QAs, we
employ the Ocean SDK (dwave_ocean_github, ).

###  3.3. Benchmarking

Although current QAs feature over 5,700 qubits, their single-instruction
operation model limits them to a few hundred program qubits with higher
degrees, which is far below the number of variables required for real-world
applications. Consequently, in this study, we employ synthetic benchmarks
instead of real-world problems. In many real-world applications, graphs often
exhibit a “Power-Law” distribution (agler2016microbial, ; clauset2016colorado,
; gamermann2019comprehensive, ; goh2002classification, ; house2015testing, ;
mislove2007measurement, ; pastor2015epidemic, ), and the _Barabasi–Albert_
(BA) algorithm (albert2005scale, ; barabasi1999emergence, ) is considered
representative of these real-world graph structures
(ayanzadeh2023frozenqubits, ; barabasi2000scale, ; gray2018super, ;
kim2022sparsity, ; lusseau2003emergent, ; wang2019complex, ;
zadorozhnyi2012structural, ; zbinden2020embedding, ). The BA graphs are
generated with a preferential attachment factor $m$, enabling us to vary the
density of the graphs by adjusting $m$—with higher values of $m$ yielding
denser graphs. We generate BA graphs with $m$ values ranging from $m=1$ (BA-1)
to $m=6$ (BA-6) to capture a broad spectrum of topologies, from sparse to
nearly fully connected networks, thus effectively representing the dynamics of
various real-world systems (clauset2016colorado, ). Edge weights are assigned
randomly following a standard normal distribution, which is a common approach
in QA benchmarking (das2008colloquium, ; ayanzadeh2022equal, ;
ayanzadeh2021multi, ).

###  3.4. Figure of merit

In our evaluations, we use the _Energy Residual_ (_ER_) to assess the fidelity
of QA as

(2) $\downarrow\textrm{Energy Residual (ER)}=\left|E_{min}-E_{global}\right|,$

where $E_{global}$ represents the global minimum of the benchmark problem, and
$E_{min}$ corresponds to the best solution obtained by the QA. Ideally, an ER
value closer to zero is desirable as it indicates a solution that closely
aligns with the ground state of the problem Hamiltonian. We conducted
intensive classical computations using state-of-the-art tools
(ayanzadeh_ramin_2021_5142230, ) to determine the global minima of the
benchmarks.

##  4\. Skipper: Skipping Dominant Chains

We introduce _Skipper_ , a software technique to enhance the capacity and
fidelity of QAs through strategically skipping dominant qubit chains.

###  4.1. Key Insights

####  4.1.1. Not All Program Qubits are Equal

In digital QCs, the individual fabrication of physical qubits, such as
superconducting ones, results in inevitable performance variations
(tannu2019not, ). Compilers, therefore, aim to prioritizing high-quality ones
and limit the reliance on those of lower quality (tannu2019not, ;
li2018tackling, ; noiseadaptive, ). However, in analog QCs, our observations
reveal a significant variability at the level of program qubits.

Figure 5(a) shows the histogram of chain lengths (in log-scale) for the BA-3
graph type after embedding onto a 5761-qubit QA device, revealing a _Power-
Law_ distribution with some notably longer _dominant chains_ and a majority of
considerably shorter chains. Figure 5(b) presents the maximum and average
chain lengths as the number of nodes in BA-3 graphs increases, notably
magnifying the variability in chain lengths with the increase in problem size.

These intriguing observations extends beyond the BA-3 graph type, and we
observe it in all benchmark graphs, including BA-1 to BA-6, spanning from
sparse to nearly fully connected graphs. Additionally, we observe the
nonuniformity of chain lengths in regular and fully connected graphs.

(a)  (b)

Figure 5.  (a) Histogram of chain lengths for a 600-node BA-3 graph (log-
scale), indicating a “Power-Law“ distribution of chain lengths. (b) Max and
Avg chain lengths of BA-3 graphs, embedded on a 5761-qubit QA.

####  4.1.2. QA Qubits are significantly Underutilized

We observe that, on average, 25% of physical qubits remain unused as they get
trapped by chains. Additionally, we observe that the dominant chains
significantly contribute to this qubit isolation. This underutilization of QA
qubits, along with utilizing several physical qubits to represent a single
program qubit, severely diminishes the capacity of QAs by up to 33x. For
instance, the 2048-qubit and 5760-qubit QAs by D-Wave can accommodate a
maximum of 64 and 177 fully connected program qubits, respectively.

####  4.1.3. Diminishing Returns with Increased QA Trials

QAs are noisy and prone to errors, leading to a systematic bias during the
execution of quantum programs. This bias causes deviations from the global
optimum, reducing the reliability of QAs (albash2018adiabatic, ;
mcgeoch2020theory, ; ayanzadeh2022equal, ; ayanzadeh2021multi, ). The bias
arises from repeating the same quantum program across multiple iterations,
exposing all trials to a similar noise profile (ayanzadeh2022equal, ).

Figure 6 shows that when the number of trials in QA is increased, the output
distribution reaches saturation. This indicates that the gap between the ideal
solution and the QA does not reduce despite drawing more samples. Moreover,
due to the operation model of QAs as single-instruction computers, strategies
commonly used in gate-based QCs (tannu2019ensemble, ; patel2020veritas, ) to
address this bias are not applicable.

Figure 6.  The Energy Residual (ER) in QAs tends to plateau with an increasing
number of trials, and the global minimum often remains unreachable by QAs.
Figure 7.  Overview of Skipper.

###  4.2. Overview of Skipper

Figure 7 shows the overview of Skipper. Skipper leverages insights into the
distribution of chain lengths and the severe underutilization of qubits in
QAs, employing a strategic approach to prune dominant chains and replace the
corresponding program qubit with two potential measurement outcomes (+1 and
-1). This process involves eliminating each chain, which partitions the search
space into two sub-spaces. Skipper explores all sub-spaces, guaranteeing an
exact recovery of the optimum solutions.

Eliminating a dominant chain accomplishes two significant objectives: firstly,
it frees up physical qubits previously used within pruned chains, and
secondly, it eliminates the isolation of solitary qubits resulting from
dominant chains. As a result, Skipper enables the handling of larger problems
by accommodating a significantly higher number of program qubits.
Additionally, Skipper significantly enhances QA fidelity by substantially
mitigating the impact of dominant chains, a primary factor in compromising QA
reliability. While Skipper utilizes more quantum resources due to the need to
execute $2^{c}$ unique quantum programs for the removal of $c$ chains, it
doesn’t correspondingly enhance QAs (baseline) performance, as demonstrated in
Fig. 6.

###  4.3. Chain Skips: How, Where, and When to Skip?

Figure 8 illustrates the elimination of two chains from a problem with five
variables, creating two and four _independent_ sub-problems, respectively. To
skip a chain, the program qubit is replaced with +1 and -1 (two measurement
outcomes), removing the node and its connected edges from the problem graph.
Unlike digital QCs, where removing one program qubit results in reducing the
physical qubit utilization by one, in QAs, removing one program qubit
liberates all the physical qubits involved in its corresponding chain.

Figure 8.  By replacing $Q_{0}$ with +1 and -1 among five spin variables
(baseline), two sub-problems each with four spin variables are obtained
($c=1$). Fixing $Q_{1}$ in these two sub-problems with +1 and -1 results in
four sub-problems with three spin variables ($c=2$). The same embedding is
utilized for all sub-problems at each level of the binary tree.

Identifying dominant chains to trim in Skipper is nontrivial. In digital QCs,
high-degree program qubits necessitate more CNOT gates, enabling direct
identification prior to circuit compilation. However, in QAs, it is not
feasible to directly recognize program qubits linked to longer chains, thus
requiring embedding techniques to identify them. Furthermore, it is not always
optimal to prune the dominant chain. In Fig. 9(a), the dominant chain is
$Q_{0}$ and consists of ten physical qubits. Pruning $Q_{0}$ (Fig. 9(b)),
liberates all ten physical qubits, leaving the other chains intact. However,
as shown in Fig. 9(c), removing $Q_{2}$ and re-embedding the problem not only
releases the five physical qubits associated with $Q_{2}$ but also effectively
reduces $Q_{0}$ to a singleton chain, totaling fourteen qubits released.

Skipper adopts a greedy approach to prune $c$ chains by sorting program qubits
based on their degree and removing the top $c$ qubits simultaneously. The
removal of a single program qubit can have a substantial impact on other
chains, as shown in Fig. 9(c). In the context of irregular graphs that often
follow the Power-Law distribution in real-world applications, this greedy
approach exhibits a desirable, near-optimal behavior for $c\geq 5$ chain cuts.

Figure 9.  (a) Embedding of five program qubits on a grid. (b) Freeing ten
qubits by pruning the dominant chain $Q_{0}$. (c) Fourteen qubits freed by
pruning $Q_{2}$.

###  4.4. Skip Count: A Cost-Performance Tradeoff

Skipper permits users to trim up to eleven chains. Each chain skipped
bifurcates the search space; therefore, trimming up to eleven chains can lead
to a maximum of 2048 sub-problems. Skipper executes all corresponding QMIs to
ensure exact solution recovery. However, the nontrivial embedding process and
the need to execute up to 2048 embeddings can create a bottleneck for Skipper.
Fortunately, the identical structure of all sub-problems at the $c$-th level
in the binary tree enables sharing the same embedding across them (Fig. 8).

###  4.5. Unembedding: Remediating Broken Chains

The sub-problem resulting from cutting chains consists of $n-c$ program
qubits. However, after embedding, the problem executed on the QA hardware
encompasses $N$ physical qubits, where $c\ll n\ll N$. As a result, the QA
produces outcomes as $N$-bit strings, with each program qubit collectively
represented by multiple bits in a chain. Therefore, Skipper _unembeds_ these
outcome samples, converting them back into the space of program qubits.

Ideally, all physical qubits within a chain should have identical values in a
given QA sample. The value of the associated program qubit is then determined
by observing any one of the physical qubits within it (e.g., program qubit
$Q_{0}$ in Fig. 10). However, QAs are inherently open systems, as interactions
with the environment are unavoidable in QCs, and the annealing process tends
to be diabatic since truly adiabatic processes are often unfeasible
(ayanzadeh2021multi, ). As a result, qubits within a chain can take different
values, an issue known as _broken chains_ (grant2022benchmarking, ;
pelofske2020advanced, ; king2014algorithm, ; barbosa2021optimizing, ).

To remediate broken chains, Skipper employ the _majority voting_ approach. For
instance, in Fig. 10, although $Q_{1}$ exhibits a broken chain with varying
qubit values, the unembedding process assigns a value of -1, reflecting the
majority of -1 values within the chain (4 versus 1). However, not all chains
have an odd length, and forcing the embedding to produce odd chain lengths is
nontrivial. Unembedding even length chains with mostly identical qubit values
(e.g., $Q_{2}$ in Fig. 10) is not challenging, as majority voting can
effectively determine the value of the program qubit. However, as demonstrated
by $Q_{3}$ in Fig. 10, a chain of even length can contain an equal number of
-1 and +1 values, referred to as _balanced chains_ , a condition where
majority voting fails. Skipper manages balanced chains by counting them and
implementing distinct strategies based on their quantity. For problems with
fewer than ten balanced chains, Skipper discards their qubit values and uses a
brute-force approach (with up to 1024 possible configurations), selecting the
configuration that yields the lowest energy value. If the number of balanced
chains exceeds ten, Skipper randomly assigns values to the corresponding
program qubits. When a broken chain occurs, Skipper can optionally apply
Single-Qubit Correction (SQC) (ayanzadeh2021multi, ; ayanzadeh2022equal, )
postprocessing to maintain a feasible solution for the original problem.

Figure 10.  Unembedding examples

###  4.6. Decoding Sub-Problem Results

After unembedding, each sample will encompass $n-c$ bits, while the original
problem includes $n$ variables. The decoding process reintroduces the values
of the $c$ pruned program qubits, which were fixed during the sub-problem
formulation by assigning fixed values to these variables.

###  4.7. Postprocessing

Theoretically, QAs sample from a Boltzmann distribution, exponentially
favoring lower energy values, and thus should locate the global optimum in few
attempts. However, like other QCs, QAs are vulnerable to noise and various
error sources that degrade their fidelity. To enhance the reliability of QAs,
we can optionally apply postprocessing heuristics to the resulting samples
(ayanzadeh2021multi, ).

###  4.8. Deriving the Final Output

In Skipper, all sub-problems are executed independently, each one
corresponding to a separate sub-space of the primary problem. Consequently, in
Skipper, the sample with the lowest energy or objective value is deemed as the
ultimate output, with the originating sub-space of this global optimum being
of no consequence.

###  4.9. Overhead of Skipper

Let ${c}$ represent the number of skipped chains, $e$ denote the edges in the
problem graph, $r$ symbolize the number of trials on the QA, while $n$ and $N$
correspond to the number of program and physical qubits, respectively.

Quantum overhead: Skipper allows for up to eleven chain cuts, necessitating
the execution of at most 2048 distinct quantum executables, each running
independently.

Classical overhead: We separate the embedding overhead of Skipper from all
other classical pre/post-processing modules, as the embedding is the primary
factor influencing the end-to-end runtime of the proposed techniques in this
paper (refer to section 8). Given the fact that $c\ll n\ll N\ll r$, Skipper
demonstrates a classical time complexity of
$O\left(2^{c}\left(rN+c\right)\right)$. This is representative of the
unembedding and decoding processes for outcome samples of sub-problems.

Embedding overhead: In Skipper, all sub-problems of the binary tree at the
$c$-th level share a single embedding, leading to $O(1)$ embedding complexity.
Note that we assume all sub-problems are executed on the same QA hardware or
that all devices have the same working graph topology, allowing them to share
the embedding.

Memory utilization: The memory utilization in Skipper scales according to
$O(rN2^{c})$.

##  5\. Skipper Evaluation Results

We evaluate Skipper using Barabasi–Albert (BA) graphs (barabasi1999emergence,
) with different preferential attachment factor values: $m=1$ (BA-1) to $m=6$
(BA-6).

###  5.1. Solving Larger Problems

####  5.1.1. Impact on Chain Length

Figure 11(a) illustrates that increasing the number of chain cuts ($c$) in
Skipper leads to a reduction in the average chain length of the embeddings.
Figure 11(b) demonstrates that Skipper decreases the mean chain length by up
to 1.32x (with an average of 1.22x) when cutting up to eleven chains.

Figure 12(a) shows that the maximum chain length of the embeddings decreases
as $c$ in Skipper increases. Figure 12(b) shows that cutting up to $c=11$
chains in Skipper reduces the maximum chain length by up to 9.14x (average
1.86x). Our observations indicate that long chains are the primary
contributing factor to the underutilization of physical qubits.

(a)  (b)

Figure 11.  Relative Avg. chain length in Skipper compared to baseline (lower
is better). (a) Relative Avg. chain length for different graphs as cut size
($c$) increase. (b) Overall relative mean chain lengths for up to 11 chain
cuts.

(a)  (b)

Figure 12.  Relative Max chain length in Skipper compared to the baseline
(lower is better). (a) Relative Max chain length for different graphs as $c$
increases. (b) Overall relative max chain lengths for up to 11 chain cuts.

####  5.1.2. Impact on Qubit Utilization

Figure 13(a) displays the average and maximum number of physical qubits when
up to eleven chains are pruned. In Fig. 13(b), Skipper reduces
underutilization of QA qubits by up to 57% (average 22.14%) with up to eleven
trimmed chains.

(a)  (b)

Figure 13.  (a) Utilization of Physical Qubits in Skipper across Different
Graph Types. (b) Relative Number of Unused Physical Qubits in Skipper for up
to 11 Chain Cuts, Compared to the Baseline. Lower is better.

####  5.1.3. Impact on Capacity of QAs

The QA capacity to accommodate specific graph types, from BA-1 to BA-6, is
determined by the largest number of program qubits of each type that can be
embedded on the QA. Figure 14(a) shows that QA capacity in Skipper improves
with increasing $c$ across various graph topologies.

Figure 14(b) demonstrates that Skipper enables the embedding of larger
problems onto current QAs, with an increase of up to 59.61% (average 28.26%).
It is important to note that this growth in the number of program qubits
necessitates a substantial increase in the number of physical qubits, as one
program qubit is represented by multiple physical qubits.

Skipper’s performance remains consistent regardless of the increasing density
of problem graphs (from BA2 to BA6).

(a)  (b)

Figure 14.  Relative QA capacity in Skipper compared to baseline. (a) Relative
capacity for different graphs as cuts increase. (b) Overall relative capacity
for up to 11 chain cuts. Higher is better.

###  5.2. Boosting QA Reliability

In addition to enhancing QA capacity, Skipper can be employed to improve the
reliability of QAs.

####  5.2.1. Impact on Embedding Quality

QAs do not incorporate circuits, thus precluding the use of the Probability of
Successful Trials metric commonly employed to assess compilation quality in
digital QCs (ayanzadeh2023frozenqubits, ; alam2020circuit, ; nishio, ;
tannu2022hammer, ). Prior studies suggest that embeddings with similar chain
lengths can produce better solutions (boothby2016fast, ;
venturelli2015quantum, ; rieffel2015case, ; choi2008minor, ). Figure 15(a)
demonstrates that trimming up to eleven chains in Skipper reduces the average
variance in chain lengths by 2.93x (up to 70.19x).

(a)  (b)

Figure 15.  (a) Relative variance of chain lengths and (b) relative embedding
time in Skipper compared to the baseline when trimming up to eleven chains.
Lower is better.

####  5.2.2. Impact on Embedding Time

Figure 15(b) demonstrates that pruning up to eleven chains in Skipper leads to
a significant reduction in embedding time, with a maximum improvement of
17.13x (average improvement of 7.12x).

####  5.2.3. Impact on Fidelity

Figure 16(a) shows that as Skipper skips more chains, the Energy Residual (ER)
decreases, indicating a progressive approach towards the global optimum.
Additionally, Fig. 16(b) demonstrates a significant reduction in ER by up to
44.4% (average 33.08%), when up to five chains are cut using Skipper, compared
to the baseline.

(a)  (b)

Figure 16.  Relative Energy Residual (ER) in Skipper compared to baseline
(lower is better). (a) Relative ER as $c$ increase. (b) Overall relative ER
for up to five chain cuts.

##  6\. Skipper-G: A Marcovian Approach

We propose _Skipper-G_ , a greedy scheme that reduces the quantum cost of
Skipper by skipping the examination of sub-problems unlikely to contain the
global optimum. However, this strategy entails a trade-off: Skipper-G achieves
marginally lower fidelity gains compared to Skipper and is ineffective for
enhancing QA capacity.

###  6.1. Insight: Not ALL Sub-Spaces Include Global Optimum

Skipper employs a Breadth-First Search (BFS) strategy to examine sub-problems,
as depicted in Fig. 17(a). Trimming each chain bifurcates the search space,
with skipping $c$ chains resulting in a binary tree of depth $c$. To ensure
successful recovery, Skipper evaluates all leaf nodes, running a separate QMI
for each sub-space at the tree’s last level. Notably, Skipper does not examine
intermediate nodes (or sub-spaces) since all chains are trimmed
simultaneously.

Users define the number of chain cuts in Skipper, with the option to skip up
to eleven chains based on their budgetary constraints. For instance, if a user
opts for the maximum allowable eleven cuts, Skipper must run 1024 QMIs when
all linear coefficients are zero (ayanzadeh2023frozenqubits, ), and up to 2048
QMIs otherwise. Nonetheless, these sub-problems are independent, allowing for
parallel execution by Skipper. Notably, Skipper’s overall runtime remains
comparable to the baseline, attributed to the significantly reduced embedding
time, as detailed in Section 5.2. However, the quantum costs incurred on QCs
are substantially higher than those on classical platforms, which may present
affordability issues for some users.

Not every sub-space contains the global optimum. Leveraging this insight, we
introduce _Skipper-G_ (_greedy Skipper_), which reduces the quantum cost of
Skipper by adopting a Depth-First Search (DFS) strategy (Fig. 17(b)), to
bypass sub-spaces unlikely to include the global optimum. When pruning the
maximum of eleven chains, Skipper-G executes 23 QMIs, in contrast to Skipper’s
potential 2048 QMIs.

Figure 17.  (a) Skipper utilizes a Breadth-First Search (BFS) strategy,
examining all leaf nodes (intermediate nodes are not examined). (b) Skipper-G
adopts a Depth-First Search (DFS) strategy, examining only two nodes at each
level of the binary tree (including intermediate nodes). Example: With $c=11$
chain cuts, Skipper and Skipper-G execute at most 2048 and 23 QMIs,
respectively.

###  6.2. How Skipper-G Work?

Figure 18 illustrates the overview of the Skipper-G scheme. In Skipper-G,
similar to Skipper, users can determine the number of chain cuts, with the
possibility of skipping up to eleven chains, depending on their budget
constraints. However, unlike Skipper where all chains are cut simultaneously,
Skipper-G employs an iterative approach, cutting one chain in each iteration.
As illustrated in Fig. 17(b), Skipper-G initiates by setting the root node
(i.e., the baseline with no chains cut) as the current node and executing the
corresponding quantum program. For each chain cut, Skipper-G performs the
following steps:

  1. (1)

In the current node (problem), the dominant chain is trimmed by setting its
corresponding program qubit to either +1 or -1, resulting in two child nodes.
If the current node at level $c$ has the index $x$, then its left and right
children at level $c+1$ will have indices $2x$ and $2x+1$, respectively (e.g.,
node $x=1$ at the third level leads to nodes 2 and 3 in Fig. 17(b)).

  2. (2)

The quantum programs corresponding to the children are executed on the QA.

  3. (3)

The best offspring is set as the current node.

###  6.3. Branch and Bound Criteria

When evaluating a node in Skipper-G, a quantum program is executed on a QA
device for multiple trials. Each trial produces an outcome with an associated
objective value. The assessment of node quality in Skipper-G is based on the
following feature (lower is better):

(3) $\downarrow
f(Z)=\left|\frac{1}{\text{E}_{\text{min}}\times\text{EV}}\right|,$

where $Z$ denotes the set of obtained samples, and $\text{E}_{\text{min}}$ and
EV represent the minimum and the expected value of the energy values in $Z$,
respectively. The lower the value of $f$, the greater the likelihood that a
child includes the global optimum in its corresponding subspace during the
traversal of the associated binary tree. This feature balances the best sample
with the overall quality of all samples, reducing the likelihood of getting
trapped in local optima.

###  6.4. Overhead of Skipper-G

Skipper-G is capable of trimming up to eleven chains, which necessitates a
maximum of 23 distinct QMI executions. Although Skipper-G examines two nodes
at each level of the binary tree, these nodes, due to their identical
structures, can utilize a single embedding. Consequently, Skipper-G requires
$c$ embeddings for $c$ chain cuts. Nonetheless, since these embeddings can be
executed in parallel, the embedding time for Skipper-G remains similar to that
of the baseline, as the root node’s embedding is expected to be more time-
consuming than the embeddings of the smaller intermediate nodes.

Figure 18.  Overview of Skipper-G.

###  6.5. Evaluation Results

Figure 19(a) illustrates the ER for various chain cut counts in Skipper-G,
showing that progressively trimming more dominant chains leads to a decrease
in the ER, approaching the global minimum. Additionally, Fig 19(b) reveals
that pruning up to five chains in Skipper-G can reduce the gap between the
global optimum and the best QA sample by as much as 40.75% (Avg. 29.19%)
compared to the baseline.

Skipper marginally outperforms Skipper-G, achieving a 3.89% greater reduction
in ER, albeit at the expense of significantly higher quantum resource
utilization. Skipper-G includes the baseline at the root of the binary tree,
ensuring it performs no worse than the baseline.

(a)  (b)

Figure 19.  Relative ER for five chain cuts compared to the baseline (lower is
better). (a) Skipper-G: Relative ER with increasing $C$. (b) Overall Relative
ER (lower is better) for Skipper vs. Skipper-G.

##  7\. Skipper and Skipper-G in Classical Realm

Unfortunately, neither Skipper nor Skipper-G can be utilized to enhance the
fidelity of optimization techniques used in classical realm. In the classical
domain, the hardness of optimization problems depends on the number of
variables and the graph topology. For instance, while planar graphs are
tractable (dei2006exact, ; hadlock1975finding, ) in classical realm, neither
regular nor power-law graphs become planar simply by eliminating a few nodes.
Additionally, eliminating ten nodes from a 1000-node graph results in sub-
graphs with 990 nodes, which typically remain intractable in the classical
realm.

Similarly, Skipper is not suitable for tackling larger problems in the
classical realm. Its primary goal is to address the sparse connectivity of
qubits, a key factor limiting the capacity of QAs. However, the full
connectivity of classical bits does not present a similar limitation.

##  8\. Workflow Analysis

The runtime of quantum programs is mainly determined by queuing delays,
execution modes through cloud services (which vary across providers), and
embedding time, rather than the execution time on quantum hardware
(microseconds to milliseconds). To offer a holistic examination of the runtime
between the proposed techniques and the baseline, we employ the following
analytical model:

(4)
$T=T_{\text{emb}}+N_{\text{QMI}}\left({T_{\text{queue}}+T_{\text{QMI}}+T_{\text{net}}}\right)+T_{\text{classical}},$

where $T_{\text{emb}}$ is the embedding time, $N_{\text{QMI}}$ is the number
of quantum executables, $T_{\text{queue}}$ is the job wait time,
$T_{\text{QMI}}$ is the QMI execution time, $T_{\text{net}}$ is the network
delay, and $T_{\text{classical}}$ is the classical pre/post-processing time.

For $r$ trials, $T_{QMI}=t_{p}+\Delta+r\times t_{s}$, where $t_{p}$ is the raw
signal preparation time, $\Delta$ is the 10ms QA initialization time, and
$t_{s}$ is the single annealing/readout time. Given that D-Wave limits
$T_{QMI}$ to two seconds, we assume $T_{QMI}=2$ in all cases. We assume one
second for $T_{\text{net}}$ for each job.

We assume a baseline embedding time of 30 minutes, decreasing proportionally
with skipped chains (as discussed in section 5.2). For example, pruning ten
chains reduces the embedding time to three minutes. All embeddings can be
computed in parallel, making $T_{\text{emb}}$ in Skipper-G the maximum time
for individual embeddings. Additionally, we allocate one second each for pre-
and post-processing.

We examine two access scenarios: _shared_ and _dedicated_ , with one and zero-
second queuing times, respectively. Figure 20 compares the end-to-end runtime
of the baseline and our proposed techniques with $c=11$, resulting in up to
1024 QA runs. Skipper shows significantly greater advantages over others in
the dedicated access mode.

Figure 20.  Overall Runtime comparison.

##  9\. Related Work

Prior studies can be broadly classified into two categories: (a) techniques
for solving larger problems on QAs, which are relevant primarily to Skipper;
and (b) approaches for enhancing QA fidelity, which are considered related
work to both Skipper and Skipper-G.

Prior research on solving larger problems with smaller QAs
(pelofske2022solving, ; okada2019improving, ) are iterative schemes, which
tend to lose reliability as problem size increases due to reliance on
approximations. Conversely, Skipper explores the entire search space without
resorting to approximations. Recent studies have introduced schemes for
addressing larger instances of Boolean Satisfiability (SAT) (tan2023hyqsat, ),
Max-Clique (pelofske2023solving, ; pelofske2019solving, ;
pelofske2022parallel, ; pelofske2021decomposition, ), and compressive sensing
with matrix uncertainty (ayanzadeh2019quantum, ) problems. However, these
methods are specific to their respective applications and are not transferable
to other domains, whereas Skipper is versatile and applicable to any
application.

Policies for improving the fidelity of QAs can be classified as: (a)
Preprocessing (pelofske2019optimizing, ; ayanzadeh2022equal, ;
ayanzadeh2020reinforcement, ), modifying QMIs before submission; (b)
Postprocessing (ayanzadeh2021multi, ), enhancing outcomes using heuristics;
(c) Hybrid strategies (ayanzadeh2022quantum, ; ayanzadeh2020leveraging, ),
combining heuristics and QAs for reliability; (d) Logical analog qubits
(jordan2006error, ; sarovar2013error, ; young2013error, ; pudenz2014error, ;
vinci2015quantum, ; venturelli2015quantum, ; vinci2016nested, ;
matsuura2016mean, ; mishra2016performance, ; matsuura2017quantum, ;
vinci2018scalable, ; pearson2019analog, ; matsuura2019nested, ;
mohseni2021error, ), spreading qubit information over multiple physical
qubits; and ensembling policies (ayanzadeh2022equal, ;
ayanzadeh2020reinforcement, ; ayanzadeh2020ensemble, ), subjecting the quantum
program to different noise profiles to suppress the bias. These proposals are
orthogonal to Skipper and Skipper-G and can effectively boost the reliability
of our proposed techniques.

Skipper is inspired by FrozenQubits (ayanzadeh2023frozenqubits, ) in digital
QCs. While FrozenQubits enhances fidelity of optimization applications in
digital QCs, Skipper excels in addressing larger problems and enhancing QA
fidelity. More importantly, while FrozenQubits’ performance diminishes with
increased problem graph density, Skipper and Skipper-G maintain their
performance, demonstrating the effectiveness of our proposal in handling
sparse to dense graphs.

##  10\. Conclusion

We propose _Skipper_ , a software scheme designed to enhance the capacity and
fidelity of QAs. Observing that chain lengths in QAs follow a “Power-Law”
distribution, with a few _dominant chains_ containing significantly more
qubits than others, Skipper prunes these chains. This approach replaces their
corresponding program qubits with two possible measurement outcomes, freeing
all qubits in the dominant chains and an additional 25% of isolated qubits
previously entrapped in chains. Our experiments on a 5761-qubit QA by D-Wave
show that Skipper allows QAs to solve problems up to 59% larger (Avg. 28.3%)
when up to eleven chains are skipped. Additionally, by removing five chains,
Skipper substantially improves QA fidelity by up to 44.4% (Avg. 33.1%).

The number of chain cuts in Skipper is user-defined; users can trim up to
eleven chains, which necessitates running an average of 1024 (and up to 2048)
distinct quantum executables. However, this may lead to affordability concerns
for some users. To mitigate this, we introduce _Skipper-G_ , a greedy scheme
that prioritizes examining sub-spaces more likely to contain the global
optimum. When up to eleven chains are pruned, Skipper-G runs a maximum of 23
quantum executables. Our experiments show that Skipper-G enhances QA fidelity
by up to 40.8% (Avg. 29.2%), requiring only 11 quantum executable runs for up
to five chain cuts, compared to Skipper’s 32 runs.

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#

Ben-Gurion University of the Negev
The Faculty of Natural Sciences
Department of Mathematics

Poles of degenerate Eisenstein series and Siegel-Weil identities for
exceptional split groups

Thesis Submitted in Partial Fulfillment of the Requirements for the Master of
Sciences Degree

By: Hezi Halawi

Under the Supervision of: Dr Nadya Gurevich

Beer Sheva, August 2016

Ben-Gurion University of the Negev
The Faculty of Natural Sciences
Department of Mathematics

Poles of degenerate Eisenstein series and Siegel-Weil identities for
exceptional split groups

Thesis Submitted in Partial Fulfillment of the Requirements for the Master of
Sciences Degree

By: Hezi Halawi

Under the Supervision of: Dr Nadya Gurevich

Signature of Student:  Date:
Signature of Supervisor:  Date:
Signature of Chairperson of the Committee for Graduate Studies:  Date:

Beer Sheva, August 2016

## Abstract

Let $G$ be a linear split algebraic group. The degenerate Eisenstein series
associated to a maximal parabolic subgroup $E_{P}(f^{0},s,g)$ with the
spherical section $f^{0}$ is studied in the first part of the thesis. In this
part, we study the poles of $E_{P}(f^{0},s,g)$ in the region
$\operatorname{Re}s>0$. We determine when the leading term in the Laurent
expansion of $E_{P}(f^{0},s,g)$ around $s=s_{0}$ is square integrable. The
second part is devoted to finding identities between the leading terms of
various Eisenstein series at different points. We present an algorithm to find
this data and implement it on SAGE. While both parts can be applied to a
general algebraic group, we restrict ourself to the case where $G$ is split
exceptional group of type $F_{4},E_{6},E_{7}$, and obtain new results.

###### Contents

  1. Abstract


About split quaternion algebras over quadratic fields and symbol algebras of
degree $n$

Diana SAVIN

Abstract. In this paper we determine sufficient conditions for a quaternion
algebra to split over a quadratic field. In the last section of the paper, we
find a class of non-split symbol algebras of degree $n$ (where $n$ is a
positive integer, $n\geq 3$) over a $p-$ adic field or over a cyclotomic
field.

Key Words: quaternion algebras, symbol algebras; quadratic fields, cyclotomic
fields; Kummer fields; $p-$ adic fields

2010 AMS Subject Classification: 11R18, 11R37, 11A41, 11R04, 11R52, 11S15,
11F85.

1\. Introduction

Let $K$ be a field with char$K\neq 2.$ Let $K^{\ast}=K\backslash\\{0\\},$
$a,b$ $\in K^{\ast}.$ The quaternion algebra $H_{K}\left(a,b\right)$ is the
$K$-algebra with $K$-basis $\left\\{1;e_{1};e_{2};e_{3}\right\\}$ satisfying
the relations: $e^{2}_{1}=a,$ $e^{2}_{2}=b,$ $e_{3}=e_{1}\cdot
e_{2}=-e_{2}\cdot e_{1.}$
Let $n$ be an arbitrary positive integer, $n\geq 3$ and let $\xi$ be a
primitive $n$-th root of unity. Let $K$ be a field with char$K\neq 2,$ char$K$
does not divide $n$ and $\xi$$\in$ $K.$ Let $a,b$ $\in K^{\ast}$ and let $A$
be the algebra over $K$ generated by elements $x$ and $y$ where

$x^{n}=a,y^{n}=b,yx=\xi xy.$

This algebra is called a symbol algebra  and it is denoted by
$\left(\frac{a,~{}b}{K,\xi}\right).$ For $n=2,$ we obtain the quaternion
algebra. Quaternion algebras and symbol algebras are central simple algebras
of dimension $n^{2}$ over $K$, non-commutative, but associative algebras (see
[Mil; 08]).

In this article we find sufficient conditions for a quaternion algebra to
split over a quadratic field. In the paper [Sa; 16] we found a class of
division quaternion algebra over the quadratic field
$\mathbb{Q}\left(i\right)$ ($i^{2}=-1$), respectively a class of division
symbol algebra over the cyclotomic field $\mathbb{Q}\left(\xi\right),$ where
$\xi$ is a primitive root of order $q$ (prime) of unity. In the last section
of this article we generalize these results for symbol algebras of degree
$n\geq 3,$ not necessarily prime.

2\. Preliminaries

We recall some results of the theory of cyclotomic fields, Kummer fields and
$p-$ adic fields, associative algebras, which will be used in our paper.

Let $n$ be an integer, $n\geq 3$ and let $K$ be a field of characteristic
prime to $n$ in which $x^{n}-1$ splits; and let $\xi$ be a primitive $n$ th
root of unity. The following lemma (which can be found in [Ca, Fr; 67]) gives
information about certain extension of $K.$

Lemma 2.1. If $a$ is a non-zero element of $K,$ there is a well-defined normal
extension $K\left(\sqrt[n]{a}\right),$ the splitting field of $x^{n}-a.$ If
$\alpha$ is a root of $x^{n}=a,$ there is a map of the Galois group
$G\left(K\left(\sqrt[n]{a}\right)/K\right)$ into $K^{*}$ given by
$\sigma\longmapsto\sigma\left(\alpha\right)/\alpha;$ in particular, if $a$ is
of order $n$ in $K^{*}/\left(K^{*}\right)^{n},$ the Galois group is cyclic and
can be generated by $\sigma$ with $\sigma\left(\alpha\right)=\xi\alpha.$
Moreover, the discriminant of $K\left(\sqrt[n]{a}\right)$ over $K$ divides
$n^{n}\cdot a^{n-1};$ $p$ is unramified if $p$ $\nmid$ $na.$

Let $A\neq 0$ be a central simple algebra over a field $K.$ We recall that if
$A$ is a finite-dimensional algebra, then $A$ is a division algebra if and
only if $A$ is without zero divisors ($x\neq 0,y\neq 0\Rightarrow xy\neq 0$).
$A$ is called split by $K$ if $A$ is isomorphic with a matrix algebra over
$K.$ If $K\subset L$ is a fields extension, we recall that $A$ is called split
by $L$ if $A\otimes_{K}L$ is a matrix algebra over $L$. The Brauer group
(Br($K$), $\cdot$) of $K$ is Br($K$)$=\left\\{\left[A\right]|A\ is\ a\
central\ simple\ K-\ algebra\right\\},$ where, two classes of central simple
algebras are equals $\left[A\right]=\left[B\right]$ if and only if there are
two positive integers $r$ and $s$ such that
$A\otimes_{K}M_{r}\left(K\right)\cong B\otimes_{K}M_{s}\left(K\right).$ The
group operation in Br($K$) is : ”$\cdot$”:
Br($K$)$\times$Br($K$)$\longrightarrow$Br($K$),
$\left[A\right]\cdot\left[B\right]=\left[A\otimes_{K}B\right],$ for
$\left(\forall\right)$ $\left[A\right],\left[B\right]$$\in$Br($K$) (see [Mil;
08], [Ko; 00]). A result due Albert-Brauer-Hasse-Noether says that for any
number field $K,$ the following sequence is exact:

$0\longrightarrow
Br\left(K\right)\longrightarrow\oplus_{v}Br\left(K_{v}\right)\longrightarrow\mathbb{Q}/\mathbb{Z}\longrightarrow
0$

Remark 2.1. ([Led; 05]). Let $n$ be a positive integer, $n\geq 3$ and let
$\xi$ be a primitive $n$-th root of unity. Let $K$ be a field such that
$\xi$$\in$$K,$ $a,b\in$$K^{*}$. If $n$ is prime, then the symbol algebra
$\left(\frac{a,~{}b}{K,\xi}\right)$ is either split either a division algebra.

Theorem 2.1. ([Lin; 12]) (Albert-Brauer-Hasse-Noether). Let $H_{F}$ be a
quaternion algebra over a number field $F$ and let $K$ be a quadratic field
extension of $F.$ Then there is an embedding of $K$ into $H_{F}$ if and only
if no prime of $F$ which ramifies in $H_{F}$ splits in $K.$

Proposition 2.1. ([Ki, Vo; 10]). Let $F$ be a number field and let $K$ be a
field containing $F.$ Let $H_{F}$ be a quaternion algebra over $F.$ Let
$H_{K}=H_{F}\otimes_{F}K$ be a quaternion algebra over  $K.$ If $[K:F]=2,$
then $K$ splits $H_{F}$ if and only if there exists an $F$-embedding
$K\hookrightarrow H_{F}.$

3\. Quaternion algebras which split over quadratic fields

Let $p,q$ be two odd prime integers, $p\neq q.$ If a quaternion algebra
$H\left(p,q\right)$ splits over $\mathbb{Q},$ of course it splits over each
algebraic number fields. It is known that if $K$ is an algebraic number field
such that $[K:\mathbb{Q}]$ is odd and $\alpha,\beta$$\in$$\mathbb{Q}^{*},$
then the quaternion algebra $H_{K}\left(\alpha,\beta\right)$ splits if and
only if the quaternion algebra $H_{\mathbb{Q}}\left(\alpha,\beta\right)$
splits (see [Lam; 04]). But, when $[K:\mathbb{Q}]$ is even there are
quaternion algebras $H\left(\alpha,\beta\right)$ which does not split over
$\mathbb{Q},$ but its split over $K.$ For example, the quaternion algebra
$H\left(11,47\right)$ does not split over $\mathbb{Q},$ but it splits over the
quadratic field $\mathbb{Q}\left(i\right)$ (where $i^{2}=-1$).
We want to determine sufficient conditions for a quaternion algebra
$H\left(p,q\right)$ to split over a quadratic field
$K=\mathbb{Q}\left(\sqrt{d}\right).$ Let $\mathcal{O}_{K}$ be the ring of
integers of $K.$ Since $p$ and $q$ lie in $\mathbb{Q},$ the problem whether
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(p,q\right)$ is split reduces to
whether $H_{\mathbb{Q}}\left(p,q\right)$ splits under scalar extension to
$\mathbb{Q}\left(\sqrt{d}\right).$
It is known that, for each prime positive integer
$Br\left(\mathbb{Q}_{p}\right)\cong\mathbb{Q}/\mathbb{Z}$ (the isomorphism is
$inv_{p}:Br\left(\mathbb{Q}_{p}\right)\rightarrow\mathbb{Q}/\mathbb{Z}$) and
for $p=\infty,$ $Br\left(R\right)\cong\mathbb{Z}/2\mathbb{Z}.$
We obtain sufficient conditions for a quaternion algebra $H\left(p,q\right)$
to split over a quadratic field.

Theorem 3.1. Let $d\neq 0,1$ be a free squares integer, $d\not\equiv 1$ (mod
$8$) and let $p,q$ be two prime integers, $q\geq 3,$ $p\neq q.$ Let
$\mathcal{O}_{K}$ be the ring of integers of the quadratic field
$K=\mathbb{Q}\left(\sqrt{d}\right)$ and $\Delta_{K}$ be the discriminant of
$K.$ Then, we have:
i) if $p\geq 3$ and the Legendre symbols
$\left(\frac{\Delta_{K}}{p}\right)\neq 1,$
$\left(\frac{\Delta_{K}}{q}\right)\neq 1,$ then, the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(p,q\right)$ splits;
ii) if $p=2$ and the Legendre symbol $\left(\frac{\Delta_{K}}{q}\right)\neq
1,$ then, the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(2,q\right)$ splits.

Proof. i) Applying Albert-Brauer-Hasse-Noether theorem, we obtain the
following description of the Brauer group of $\mathbb{Q}$ and of the Brauer
group of the quadratic field $\mathbb{Q}\left(\sqrt{d}\right).$

${0}$${Br\left(\mathbb{Q}\right)}$${\oplus_{p}Br\left(\mathbb{Q}_{p}\right)\cong\left(\oplus_{p}\mathbb{Q}/\mathbb{Z}\right)\oplus\mathbb{Z}/2\mathbb{Z}}$${\mathbb{Q}/\mathbb{Z}}$${0}$${0}$${Br\left(\mathbb{Q}\left(\sqrt{d}\right)\right)}$${\oplus_{P}Br\left(\mathbb{Q}\left(\sqrt{d}\right)_{P}\right)\cong\left(\oplus_{P}\mathbb{Q}/\mathbb{Z}\right)}$${\mathbb{Q}/\mathbb{Z}}$${0}$$\scriptstyle{\oplus_{p}\varphi_{p}\oplus
0}$

where $\varphi_{p}$ is the multiplication by $2$ when there is single
$P$$\in$Spec$\left(\mathcal{O}_{K}\right)$ above the ideal $p\mathbb{Z}$ i.e.
$p\mathbb{Z}$ is inert in $\mathcal{O}_{K}$ or $p\mathbb{Z}$ is ramified in
$\mathcal{O}_{K},$ respectively $\varphi_{p}$ is the diagonal map
$\mathbb{Q}/\mathbb{Z}\rightarrow\mathbb{Q}/\mathbb{Z}\oplus\mathbb{Q}/\mathbb{Z}$
if there are two primes $P,$ $P^{{}^{\prime}}$ of $\mathcal{O}_{K}$ above
$p\mathbb{Z}$ i.e. $p\mathbb{Z}$ is totally split in $\mathcal{O}_{K}.$ Using
this description we determine sufficient conditions for a quaternion algebra
$H\left(p,q\right)$ to split over a quadratic field
$K=\mathbb{Q}\left(\sqrt{d}\right).$

It is known that $\Delta_{K}=d$ (if $d\equiv 1$ (mod $4$)) or $\Delta_{K}=4d$
(if $d\equiv 2,3$ (mod $4$)). Since $\left(\frac{\Delta_{K}}{p}\right)\neq 1,$
$\left(\frac{\Delta_{K}}{q}\right)\neq 1$ it results
$\left(\frac{d}{p}\right)=-1$ or $\left(\frac{d}{p}\right)=0,$ respectively
$\left(\frac{d}{q}\right)=-1$ or $\left(\frac{d}{q}\right)=0.$ Applying the
theorem of decomposition of a prime integer $p$ in the ring of integers of a
quadratic field (see for example [Ire, Ros; 90], p. 190), it results that $p$
is ramified in $\mathcal{O}_{K}$ or $p$ is inert in $\mathcal{O}_{K},$
respectively $q$ is ramified in $\mathcal{O}_{K}$ or $q$ is inert in
$\mathcal{O}_{K}.$ So, $p$ and $q$ do not split in $K.$
Let $\Delta$ denote the discriminant of the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(p,q\right).$
It is known that a prime positive integer $p^{{}^{\prime}}$ ramifies in
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(p,q\right)$ if
$p^{{}^{\prime}}|2\Delta$ ([Ko], [Ko; 00]). This implies
$p^{{}^{\prime}}|2pq.$
Since $d\not\equiv 1$ (mod $8$) and the decomposition of $2$ in
$\mathcal{O}_{K}$ (see [Ire, Ros; 90], p. 190), it results that $2$ does not
split in $K.$
From the previously proved and applying Theorem 2.1 and Proposition 2.1, it
results that the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(p,q\right)$ splits.
ii) Let $p^{{}^{\prime}}$ be a prime positive integer which ramifies in
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(2,q\right).$ In this case the
condition $p^{{}^{\prime}}|2\Delta$ implies $p^{{}^{\prime}}|2q.$ With similar
reasoning as i) we get that the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(2,q\right)$ splits.

Remark 3.1. The conditions $\left(\frac{\Delta_{K}}{p}\right)\neq 1,$
$\left(\frac{\Delta_{K}}{q}\right)\neq 1$ from Theorem 3.1 are not necessary
for the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(q,p\right)$ splits. For example, if
$d=-1,$ the conditions $\left(\frac{\Delta_{K}}{p}\right)\neq 1,$
$\left(\frac{\Delta_{K}}{q}\right)\neq 1$ are equivalent to
$p$$\equiv$$q$$\equiv$$3$(mod $4$). We consider the quaternion algebra
$H_{\mathbb{Q}\left(i\right)}\left(5,29\right),$ so $p=5$$\equiv$$1$(mod $4$)
and $q=29$$\equiv$$1$(mod $4$). Doing some calculations in software MAGMA, we
obtain that the algebra $H_{\mathbb{Q}\left(i\right)}\left(5,29\right)$
splits. Analogous, for $p=5$$\equiv$$1$(mod $4$) and $q=19$$\equiv$$3$(mod
$4$), we obtain that the algebra
$H_{\mathbb{Q}\left(i\right)}\left(5,19\right)$ splits. Another example: if
$d=3,$ $p=7,q=47.$ We have $\left(\frac{\Delta_{K}}{p}\right)\neq 1,$ but
$\left(\frac{\Delta_{K}}{q}\right)=1.$ However the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{3}\right)}\left(7,47\right)$ splits. Another remark
is that the quaternion algebra $H_{\mathbb{Q}}\left(7,47\right)$ does not
split.

We wonder what happens with a quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(p,q\right)$ from Theorem 3.1 when
instead of $p$ or $q$ we consider an arbitrary integer $\alpha.$ Immediately
we obtain the following result:

Corollary 3.1. Let $d\neq 0,1$ be a free squares integer, $d\not\equiv 1$ (mod
$8$) and let $\alpha$ be an integer and $p$ be an odd prime integer. Let
$\mathcal{O}_{K}$ be the ring of integers of the quadratic field
$K=\mathbb{Q}\left(\sqrt{d}\right)$ and $\Delta_{K}$ be the discriminant of
$K.$ If the Legendre symbols $\left(\frac{\Delta_{K}}{p}\right)\neq 1,$
$\left(\frac{\Delta_{K}}{q}\right)\neq 1,$ for each odd prime divisor $q$ of
$\alpha$ then, the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(\alpha,p\right)$ splits.

Proof. We want to determine the primes $p^{{}^{\prime}}$ which ramifies in
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(\alpha,p\right),$ i.e the primes
$p^{{}^{\prime}}$ with the property $p^{{}^{\prime}}|2\Delta.$ This implies
$p^{{}^{\prime}}|2\alpha\cdot p.$ Since $\left(\frac{\Delta_{K}}{p}\right)\neq
1,$ $\left(\frac{\Delta_{K}}{q}\right)\neq 1,$ for each odd prime divisor $q$
of $\alpha,$ using a reasoning similar with that of Theorem 3.1, we get that
such primes does not exist, so the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(\alpha,p\right)$ splits.

4\. Symbol algebras of degree $n$

In the paper [Sa; 16] we found a class of division quaternion algebras over
the quadratic field $\mathbb{Q}\left(i\right)$ ([Sa; 16], Th. 3.1) and a class
of division symbol algebras of degree $q$ (where $q$ is an odd prime positive
integer) over a $p$\- adic field or over a cyclotomic field ([Sa; 16], Th.
3.2). Here we generalize theorem 3.2 from [Sa; 16], when $A$ is a symbol
algebra over the $n$-th cyclotomic field, where $n$ is a positive integer,
$n\geq 3.$

Theorem 4.1. Let $n$ be a positive integer, $n\geq 3,$ $p$ be a prime positive
integer such that $p\equiv 1$ (mod $n$), $\xi$ be a primitive root of order
$n$ of unity and let $K=\mathbb{Q}\left(\xi\right)$ be the $n$ th cyclotomic
field. Then there is an integer $\alpha$ not divisible by $p,$ $\alpha$ is not
a $l$ power residue modulo $p,$ for $\left(\forall\right)$
$l$$\in$$\mathbb{N},$ $l|n$ and for every such an $\alpha$, we have:
i) if $A$ is the symbol algebra $A=\left(\frac{\alpha,p}{K,\xi}\right),$ then
$A\otimes_{K}\mathbb{Q}_{p}$ is a non-split algebra over $\mathbb{Q}_{p};$
ii) the symbol algebra $A$ is a non-split algebra over $K.\vskip 3.0pt plus
1.0pt minus 1.0pt$

Proof. i) Let be the homomorphism
$f:\mathbb{F}_{p}^{\ast}\rightarrow\mathbb{F}_{p}^{\ast},$
$f\left(x\right)=x^{n}.$ Since $p\equiv 1$ (mod $n$), it results
$Ker\left(f\right)=\left\\{x\in{F}_{p}^{\ast}|x^{n}\equiv 1\ (mod\
p)\right\\}$ is non -trivial, so $f$ is not injective. So, $f$ is not
surjective. It results that there exists $\overline{\alpha}$ (in
$\mathbb{F}_{p}^{\ast},$) which does not belongs to
$\left(\mathbb{F}_{p}^{\ast}\right)^{n}.$ Let $\beta$ be an $n$ th root of
$\alpha$ (modulo $p$). Since $\alpha$ is not a $l$ power residue modulo $p,$
for $\left(\forall\right)$ $l$$\in$$\mathbb{N},$ $l|n,$ it results that the
extension of fields
$\mathbb{F}_{p}\left(\overline{\beta}\right)/\mathbb{F}_{p}$ is a cyclic
extension of degree $n.$ Applying a consequence of Hensel’s lemma (see for
example [Al,Go;99]) and the fact that $p\equiv 1$ (mod $n$), it results that
$\mathbb{Q}_{p}$ contains the $n$-th roots of the unity, therefore
$\mathbb{Q}\left(\xi\right)\subset\mathbb{Q}_{p}.$ Let the symbol algebra
$A\otimes_{K}\mathbb{Q}_{p}=\left(\frac{\alpha,p}{\mathbb{Q}_{p},\xi}\right).$
Applying Lemma 2.1, it results that the extension
$\mathbb{Q}_{p}\left(\sqrt[n]{\alpha}\right)/\mathbb{Q}_{p}$ is a cyclic
unramified extension of degree $n,$ therefore a norm of an element from this
extension can be a positive power of $p,$ but cannot be $p.$ According to a
criterion for splitting of the symbol algebras (see Corollary 4.7.7, p. 112
from [Gi, Sz; 06]), it results that
$\left(\frac{\alpha,p}{\mathbb{Q}_{p},\xi}\right)$ is a non-split algebra.
ii) Applying i) and the fact that $K\subset\mathbb{Q}_{p},$ it results that
$A$ is a non-split algebra.

Remark 4.1. Although Theorem 4.1 is the generalization of Theorem 3.2 from
[Sa; 16] for symbol algebras of degree $n,$ there are some differences between
these two theorems, namely:
\- One of the conditions of the hypothesis of Theorem 3.2 from [Sa; 16] is:
$\alpha$ is not a $q$ power residue modulo $p.$ With a similar condition in
the hypothesis of Theorem 4.1, namely: $\alpha$ is not a $n$ power residue
modulo $p,$ Theorem 4.1 does not work. We give an example in this regard: let
$p=7,$ $n=6,$ $\alpha=2.$ $2$ is not a $6$ power residue modulo $7,$ but $2$
is a quadratic residue modulo $7.$ Let $\beta$ be an $6$ th root of $\alpha$
(modulo $7$). We obtain that the polynomial $Y^{6}-\overline{2}$ is not
irreducible in $\mathbb{F}_{7}\left[Y\right].$ We have
$Y^{6}-\overline{2}=\left(Y^{3}-\overline{3}\right)\cdot\left(Y^{3}+\overline{3}\right)$
(in $\mathbb{F}_{7}\left[Y\right]$). So, the extension of fields
$\mathbb{F}_{7}\subset\mathbb{F}_{7}\left(\overline{\beta}\right)$ has not the
degree $n=6.$ For this reason, in the hypothesis of Theorem 4.1 we put the
condition: $\alpha$ is not a $l$ power residue modulo $p,$ for
$\left(\forall\right)$ $l$$\in$$\mathbb{N},$ $l|n;$
\- In Theorem 3.2 from [Sa; 16] we proved that $A\otimes_{K}\mathbb{Q}_{p}$ is
a non-split symbol algebra over $\mathbb{Q}_{p}$ (respectively $A$ is a non-
split symbol algebra over $K$) and applying Remark 2.1. this is equivalent to
$A$ is a division symbol algebra over $\mathbb{Q}_{p}$ (respectively $A$ is a
division symbol algebra over $K$). But, Remark 2.1 holds iff $n$ is prime. For
this reason, the conclusion of Theorem 4.1 is: $A$ is a non-split symbol
algebra over $\mathbb{Q}_{p}$ (respectively $A$ is a non-split symbol algebra
over $K$).

Conclusions. In the last section of the paper, we found a class of non-split
symbol algebras of degree $n$ (where $n$ is a positive integer, $n\geq 3$)
over a $p-$ adic field, respectively over a cyclotomic field. In a further
research we intend to improve Theorem 4.1 from this paper, for to find a class
of division symbol algebras of degree $n$ (where $n$$\in$$\mathbb{N}^{*}$,
$n\geq 3$) over a cyclotomic field.

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Diana SAVIN,

Faculty of Mathematics and Computer Science, Ovidius University,

Constanta 900527, Bd. Mamaia no.124, România

Email<EMAIL_ADDRESS><EMAIL_ADDRESS>