prepare the ground state of a quantum system. They discovered that the
combination of both methods worked best most of the time and concluded that
the optimal protocol for minimizing the energy of a quantum state under time
constraints is often of the “bang-anneal-bang” form rather than the previously
thought “bang-bang” pulse structure based on Pontryagin’s principle [267].

In communication, Cui et al. [268] applied QAOA to the problem of maximum
likelihood (ML) detection of binary symbols transmitted over a multiple-input
and multiple-output (MIMO) channel. They demonstrated that a QAOA-based ML
detector could approach the performance of a classical ML detector, revealing
the potential for large-scale classical optimization problems to be
effectively addressed on NISQ computers.

In the work of Chandarana et al. [269], a hybrid classical-quantum ansatz was
proposed, which uses counterdiabatic protocols to solve the protein folding
problem on a tetrahedral lattice. The ansatz is inspired by digitized-
counterdiabatic quantum computation, which can accelerate adiabatic quantum
algorithms and compress quantum circuits [79, 80]. The authors applied this
algorithm to various proteins with different numbers of amino acids and
qubits, and it was shown to outperform state-of-the-art quantum algorithms,
such as the original QAOA, in terms of convergence and circuit depth. These
demonstrations were performed on several quantum hardware platforms, including
trapped-ions and superconducting systems, achieving high success probabilities
and efficient ground state convergence. The authors noted the remaining
difficulties for such an approach to solving complex optimization problems
with quantum computers. However, the proposed digitized counterdiabatic
protocol opened an investigation into implementing problem-inspired ansatz to
industrial use cases using NISQ devices.

In computer vision, Li et al. [23] investigated the application of partially
occluded object detection under the broader framework of QUBO. They proposed a
three-tiered improvement approach for a hybrid quantum-classical optimization
for object detection, resulting in significant execution speedup—an over
13-fold speedup was achieved by selecting L-BFGS-B as the classical optimizer.
They also demonstrated that optimally rescheduling gate operations, especially
in deeper circuits, resulted in better circuit fidelity at the third level.
The findings of this study shed light on the potential benefits of QAOA for
object detection tasks.

Very recently, Date et al. [270] proposed using quantum computers to
accelerate training of machine learning models. They formulated three machine
learning problems (linear regression, support vector machine, and balanced
$k$-means clustering) as QUBO problems and suggested solving them with
adiabatic quantum computing. In this context, QAOA can be employed as an
alternative method to find good approximated solutions to such problems. This
could potentially pave the way for using QAOA in the deep learning realm for
neural networks’ training in the future. Some other potential future scopes
from [271] include experimental implementation of non-Gaussian gates,
exploration of quantum advantages in the presence of decoherence, development
of specialized QNNs, investigation of joint architectures, and deeper
exploration of fundamental quantum physics concepts in QNNs.

###  5.3 The Quantum Advantage of QAOA

The QAOA shows promising potential in realizing quantum advantage over
classical algorithms in specific problem instances and under certain
conditions. This approach targets classical optimization problems, providing
increased efficiency and improved performance solutions. However, this
advantage is not yet fully realized due to various challenges, including noise
and hardware limitations in near-term quantum devices and the competitiveness
of state-of-the-art classical solvers.

In terms of computational runtime efficiency, instances where QAOA has
demonstrated superiority include solving the MaxCut problem on dense graphs,
offering exponential acceleration for large Minimum Vertex Cover (MVC)
problems, and achieving a superlinear quantum speedup compared to Simulated
Annealing (SA) for the Maximum Independent Set (MIS) problem on specific graph
instances. Moreover, QAOA has shown potential in other areas, such as
unstructured search problems and Quantum Linear System Problems (QLSP), where
it outperforms various classical and quantum algorithms.

In terms of solution quality, multiple studies comparing QAOA to classical
algorithms on various optimization problems, including MaxCut, Max-$k$XOR, and
other Constraint Satisfaction Problems (CSPs), have revealed that QAOA can
outperform classical algorithms under specific conditions or for certain
problems. For instance, QAOA surpasses classical threshold algorithms for
Max-$k$XOR problems when $k>4$ in the large degree limit. Furthermore,
compared to classical local algorithms, QAOA provides superior solutions for
MaxCut on large-girth random regular graphs at depth $p=11$ and beyond.

Despite these encouraging results, other studies have found that classical
algorithms can still match or surpass QAOA’s performance in many scenarios. A
simple modification of the classical algorithm, the Gaussian wave process, has
achieved a larger improvement over random assignment compared to QAOA1 in the
asymptotic limit as $D\to\infty$ for MaxCut on triangle-free $D$-regular
graphs. There also exists a 2-local classical MaxCut algorithm that
consistently outperforms QAOA2 for all $D$-regular graphs of girth $>5$.
Furthermore, QAOA can encounter limitations in scenarios where it cannot
outperform the best classical algorithm, such as solving the MaxCut problem on
bipartite $D$-regular graphs. It can also face restrictions due to the
locality constraint, which limits its algorithmic performance when the entire
graph is not visible to the algorithm. When applied to problems with the
Overlap Gap Property (OGP), such as the MIS problem on sparse random graphs or
Max-$k$XOR with even $k\geq 4$, QAOA can face obstructions limiting its
performance. Despite these challenges, a potential quantum advantage is still
achievable in some problems, particularly when QAOA surpasses sub-logarithmic
depths, as classical Approximate Message Passing (AMP) algorithms exhibit
suboptimality. Consequently, more research is needed to understand the extent
of QAOA’s capabilities and limitations and identify improvement areas or
alternative optimization approaches.

One area that can be improved to bolster the QAOA’s efficacy is its ansatz
design. Recent advancements in ansatz variants such as ab-QAOA, ADAPT-QAOA,
and QAOAnsatz have significantly improved over the standard QAOA. The ab-QAOA
drastically reduces computation time and achieves faster convergence, with
improvements increasing proportionally with problem size. The ADAPT-QAOA,
applying the principle of shortcuts to adiabaticity, exhibits enhanced
convergence speed and reduces the number of CNOT gates and optimization
parameters by about half, streamlining quantum computation. QAOAnsatz
introduces more flexibility in defining parts of the ansatz, thereby expanding
the range of solvable problems and enabling a larger and potentially more
useful set of states to be represented than possible with the original
formulation. These advancements could accelerate the realization of quantum
advantage in solving combinatorial optimization problems. Other ansatz
variants, such as WS-QAOA, have also shown potential advantages over the
standard QAOA at low depth, critical for implementation on NISQ devices.
Likewise, the FALQON+ algorithm has illustrated a significant increase in
approximation ratio and success probability with a minor increase in circuit
depth and noise degradation, making it suitable for NISQ devices. Furthermore,
FQAOA has demonstrated substantial performance advantages in portfolio
optimization problems by effectively handling constraints. Other approaches
like Quantum Dropout and ST-QAOA have also indicated improvements in
performance for specific types of combinatorial optimization problems.

The performance of QAOA is also strongly tied to its depth (i.e., the number
of layers), as a larger depth implies more parameters to optimize. Finding
optimal parameters often requires polynomial time, making it difficult to
achieve a quantum speedup with QAOA, even at low depths. Additional
complications, such as barren plateaus, aggravate this challenge and
necessitate improved parameter optimization strategies. Therefore, parameter
optimization is crucial for QAOA’s quantum advantage. Various techniques have
been proposed to effectively initialize QAOA parameters, achieving better
results than random initializations. Many works have also shown that when
coupled with diverse optimization strategies such as gradient-based methods,
gradient-free techniques, and certain machine learning approaches, QAOA often
displays rapid convergence to optimal solutions and better solution quality.
Moreover, parameter concentration in certain problem instances and symmetries
can be exploited to increase the efficiency of parameter optimization further.

There remain, however, substantial obstacles to achieving a quantum advantage
in more practical settings, i.e., when running QAOA on real quantum computers.
Challenges related to various noise sources, including local and correlated
errors, present significant hurdles. While strategies to mitigate some of
these errors have been proposed, quantum devices’ error rates need to continue
to improve in order to address issues such as noise-induced barren plateaus.
Therefore, work towards achieving consistent quantum advantage with QAOA
remains an ongoing and active area of research.

Nonetheless, QAOA exhibits strong compatibility with NISQ devices, enhancing
its near-term applicability. Some QAOA variants display resilience to noise,
making them well-suited for NISQ devices. Empirical evidence suggests that
QAOA could effectively manage industry-relevant constrained-optimization
problems on these quantum platforms, efficiently addressing scalability and
compatibility issues. This versatility, combined with the successful
application of QAOA to various real-world problems, underscores its adaptive
nature and a broad range of use cases.

###  5.4 Open Questions and Future Directions

The QAOA represents a noteworthy milestone in quantum computing with its
potential for addressing various optimization problems. Despite significant
progress, numerous research opportunities remain in the realm of QAOA.
Unlocking the full potential of this quantum optimization algorithm may
require a combination of theoretical understanding, empirical studies,
algorithmic improvements, parameter optimization, and a thorough exploration
of entanglement and problem structures, among other factors.

##### Theoretical insights & mathematical frameworks:

Enhancing our theoretical understanding of QAOA, primarily at higher depths,
is fundamental. This includes developing rigorous theoretical frameworks that
provide a comprehensive understanding of QAOA’s behavior, performance, and
inherent limitations compared to classical optimization algorithms. Further
research is needed to understand the interplay between entanglement and
circuit depth, particularly how they collectively impact QAOA’s performance at
higher depths. This will also facilitate the development of problem-specific
algorithms and efficient ansatz designs. Research on how different problem
structures interact with QAOA’s functionality and performance is also
essential. This understanding can offer valuable insights into improving
performance and applicability across diverse problem sets.

Some specific questions to be addressed in this regard include:

  * •

A rigorous proof of QAOA achieving the Parisi value would be an important
result. Basso et al. [176] conjectured that as $p\to\infty$, QAOA would be
able to achieve the Parisi value on random $D$-regular graphs as $D\to\infty$.
If true, this would mean QAOA could also optimally solve the SK model, as well
as the MaxCut problem [182].

  * •

The Overlap Gap Property (OGP) exhibited by certain problem instances has been
shown to pose obstacles for QAOA in achieving optimal solutions. These
challenge arise when QAOA does not see the whole graph (e.g., when operating
at sub-logarithmic depths) [192], as well as when it is able to see the whole
graph (e.g., on fully-connected $k$-spin models) [176]. In comparison, the
classical Approximate Message Passing (AMP) algorithm can provably find
solutions arbitrarily close to the true optima of $k$-spin models in tha
absence of OGP (conjectured for $k=2$), but it also faces suboptimality when
$k\geq 4$ is even. It therefore remains an open question whether QAOA can
achieve a quantum advantage in scenarios involving the OGP.

  * •

For instances with exceedingly small spectral gaps, QAOA can overcome the
adiabatic limitations. Zhou et al. [15] noticed it for small problem sizes,
but how this tendency could scale and allow to solve problems out of reach for
adiabatic evolution is still unknown.

  * •

Rajakumar et al. [76] proposed a method enabling the construction of arbitrary
coupling operations on quantum spin-systems, using global Ising operations and
single qubit bit flips. This method exhibits promising scaling properties,
suggesting potential advantages for dense, unweighted coupling operations like
those in QAOA. However, an optimal operation sequence for arbitrary graphs
remains elusive, likely pointing to an NP-hard problem. Future work may focus
on developing efficient solutions for specific types of graphs, seeking
tighter upper bounds on construction, and formally establishing this problem’s
NP-hard nature. Further research is also needed to explore how the complexity
of compilation affects quantum algorithms when constructing arbitrary
unweighted and weighted coupling graphs using global entangling operations in
quantum spin systems.

  * •

Boulebnane and Montanaro [187] derived analytical bounds on the average
success probability of QAOA on random $k$-SAT in the limit of infinite problem
size. Further research could be done to explore QAOA’s performance on other
Boolean satisfaction problems and instances and investigate the relationship
between QAOA’s success probability and running time.

  * •

Dupont et al. [143] studied the growth and spread of entanglement resulting
from optimized and randomized QAOA circuits, which could have implications for
the simulation of QAOA circuits with tensor network-based methods. Further
research could be conducted on these connections and their impact on
overcoming barren plateaus.

##### Parameter optimization strategies and initialization techniques:

Efficient methods for finding optimal parameters and initializing the
algorithm are crucial for improving QAOA’s performance, especially in noise
and hardware limitations. Strategies include overcoming barren plateaus,
exploiting parameter symmetries, and leveraging machine learning or
reinforcement learning techniques. Furthermore, the process of parameter
initialization significantly influences QAOA’s overall performance. Enhancing
strategies for optimal parameter initialization and understanding their impact
on performance at higher depths are critical future research directions.

In this direction, some promising avenues for future investigation include:

  * •

It has been noticed that for similar graphs, the parameters of the QAOA
ansatze share some similarities [181]. However, it is unclear whether this
phenomenon is due to the small graphs that have been investigated or whether
this is more of a universal property that connects the ansatze to closely
related graphs.

  * •

The performance of BFGS for larger parameter vectors (i.e., for $p\geq 3$) in
QAOA optimization is unclear [128]. It would be interesting to evaluate the
behavior and effectiveness of BFGS for parameter optimization in QAOA circuits
with larger depths.

  * •

How can parameter transferability and reusability be further utilized for
optimizing QAOA performance? Galda et al. [115] provided a theoretical
foundation for parameter transferability in QAOA. However, more research is
needed to understand how this can be applied across a wide range of problem
instances and optimization tasks.

  * •

Explore additional ways to leverage parameter symmetries for more efficient
QAOA performance. Shaydulin et al. [137] investigated the correlation between
QAOA and the inherent symmetries of the target function, using machine
learning techniques to predict the QAOA performance accurately. Additional
research could be conducted to improve the understanding of symmetry in
objective functions, cost Hamiltonians, and QAOA parameters themselves,
leading to more efficient optimization processes.

  * •

Can optimal QAOA parameters be derived analytically? Wang et al. [138] derived
analytical expressions for solving optimal parameters for the level-1 QAOA
applied to MaxCut on general graphs. However, it is unclear whether this
approach can be extended to higher values of p or other problem instances.
Investigate the possibility of deriving analytical expressions for optimal
QAOA parameters in various problem instances and optimization tasks. This
could potentially simplify the search for optimal parameter values and improve
the overall efficiency of the QAOA algorithm.

  * •

The effectiveness of unsupervised machine learning approaches, such as
clustering, in setting QAOA parameters has been demonstrated [19]. However,
how well these methods can generalize to other problems and larger instances
remains to be seen. The same goes for reinforcement learning [133, 100].
Investigate the applicability and generalization of unsupervised learning and
RL methods for QAOA parameter optimization in a broader range of problem
settings and larger problem instances.

  * •

Can better parameter optimization strategies mitigate the challenge of
achieving a quantum speedup with QAOA, even at low depths, due to factors such
as barren plateaus [71, 152]?

##### Understanding performance, benchmarking and comparison with classical
algorithms:

Developing application-specific benchmarking methods and evaluating QAOA’s
performance across various problem instances will help determine its
advantages over classical algorithms and guide researchers in choosing the
most appropriate solver for a given optimization problem. Benchmarking against
classical algorithms will help identify the conditions under which QAOA can
consistently surpass classical benchmarks, which is crucial to harnessing its
full potential. This includes exploring cases where QAOA can offer a quantum
advantage. Empirical studies will also provide practical insights into the
performance of QAOA. Simulator-based studies can shed light on how the
algorithm performs in real-world scenarios. Further empirical investigations
that complement theoretical studies can contribute to a more holistic
understanding of the algorithm. Finally, opportunities exist to adapt or
enhance classical algorithms for quantum optimization. This could lead to
significant advancements in quantum computing, including novel solutions that
outperform existing quantum optimization algorithms.

Along these directions, some areas for further research include:

  * •

Machine Learning (ML) models have been employed to predict QAOA’s performance
and identify situations where QAOA is most likely to outperform classical
algorithms [198, 132]. Based on their findings, Moussa et al. [198] suggested
exploring the potential of graph sparsification to enhance QAOA’s performance.
ML models like these provide valuable insights into the behavior of QAOA and
offer guidance on leveraging its advantages, helping uncover opportunities for
achieving quantum advantage.

  * •

Marwaha and Hadfield [173] found that QAOA starts to outperform the threshold
algorithm when $k>4$. However, this does not rule out the possibility that a
different local tensor algorithm will match or outperform QAOA at larger $k$.
Can QAOA outperform other local tensor algorithms at larger $k$?

  * •

Larkin et al. [156] proposed a new metric for evaluating the runtime
performance of QAOA and developed a method based on it to reduce the execution
time of QAOA for solving MaxCut on 3-regular graphs. The effectiveness and
scalability of this technique on other problems remains to be investigated.
Furthermore, studies on different performance metrics may provide more
insights into how to improve QAOA’s performance across different problems.

  * •

Can the observed quantum speedup in specific problem domains be extended to
more general cases? For instance, the speedup shown by QAOA in the Maximum
Independent Set (MIS) problem on 2D Rydberg atom arrays remains an open
question for general cases [26]

##### Noise and error effects, mitigation techniques, and fault tolerance:

Understanding the impact of various noise types on QAOA’s performance is
essential. Developing noise mitigation strategies, error suppression schemes,
and fault-tolerant techniques will improve the practicality of QAOA in real-
world settings with decoherence and hardware limitations. Specific strategies
to reduce noise in QAOA and other Variational Quantum Algorithms (VQAs), such
as optimizing SWAP networks, reducing the total number of CNOT gates, and
leveraging equivalent circuit averaging, continue to be promising future
directions. Techniques such as symmetry verification have been proposed for
mitigating errors in QAOA. This approach has shown effectiveness in improving
output state fidelity, expected objective function value, and probability of
sampling the optimal solution.

Some more specific potential areas for future research involve:

  * •

According to Campos et al. [216], local coherent dephasing noise can remove
training saturation in layer-wise learning, potentially aiding the
optimization process. However, it is unclear if there are other noise sources
beyond the simple noise model that can play a similar role. Further
investigation is needed to determine this.

  * •

It was observed in numerical simulation that QAOA performance improves as
noise correlation strength increases at fixed local error rates, suggesting a
certain degree of noise resilience [223]. However, further studies are
required to test the generalizability of this result.

  * •

Optimizing SWAP networks can help reduce the negative impact of noise on QAOA
and other VQAs [226]. More research should be done on improving SWAP network
optimization techniques to enhance performance in practical settings.

  * •

Is it possible to mitigate Noise-Induced Barren Plateaus (NIBP) with novel
error mitigation strategies [152]?

  * •

What is the prospect for quantum advantage of QAOA in the present of noise and
errors? For example, a quantitative analysis of the error rates required to
maintain the advantage of QAOA in noiseless scenarios would provide valuable
insights. To further address this question, an empirical demonstration of
quantum advantage of QAOA in real quantum devices is crucial.

##### Hardware-specific challenges and scalability:

Addressing hardware limitations such as connectivity, decoherence, and gate
error rates is vital for scaling up QAOA to solve larger problems. Leveraging
specific hardware to enhance QAOA performance has shown potential. However,
different architectures possess their specific advantages and disadvantages
that need thorough exploration. Investigating QAOA’s performance on different
hardware platforms, including trapped ions, neutral atoms, superconducting
qubits, photonic quantum computers, and the use of higher energy levels
(qudits), will provide a deeper understanding of its capabilities and
limitations. For example, the utilization of higher energy levels beyond
qubits presents an interesting prospect. In photonic quantum computers, qudits
can be more efficiently mapped to the hardware for various problems. Tailoring
algorithms for photonic quantum computing also present many other advantages,
such as potential improvements in terms of cluster state dimension compared to
QAOA gate-based circuit algorithms.

Along these lines, several promising opportunities for further study consist
of the following:

  * •

A systematic investigate the impact of compilation and routing qubits with
swap networks on QAOA’s performance in real hardware is needed. This
additional overhead can significantly impact the performance of the algorithm,
especially when solving problems on graphs that differ from hardware
connectivity [202].

  * •

When using ML models to predict the performance of QAOA [198], adding some
hardware-related features will assist in deciding whether or not to choose
QAOA over classical algorithms on a specific quantum hardware.

  * •

Recent observations showed that a decomposed implementation could
significantly reduce the performance of QAOA. Hence, it would be interesting
to investigate the effect on QAOA by a direct hardware implementation of a
combination of a controlled arbitrary phase and a SWAP gate [254].

  * •

Dlaska et al. [240] concluded that we do not know how experimental
investigations beyond numerical simulations for QAOA will perform using the
innovative four-qubit Rydberg parity gate and parity architecture.

  * •

According to Proietti et al. [74], we should continue developing and tailoring
algorithms for photonic quantum computing, as their MBQC QAOA for photonic
quantum computers showed an up-to 30-fold improvement in terms of cluster
state dimension when compared to QAOA gate-based circuit algorithms.

  * •

Explore improvements in the surface code implementation, such as faster state
distillation, to potentially produce quantum advantages on early fault-
tolerant quantum processors for algorithms offering only quadratic speedups
[153, 154].

##### Alternative optimization approaches and problem-specific algorithms:

Exploring alternative optimization methods and tailoring algorithms to
specific problem structures will undeniably enhance the performance and
applicability of the QAOA across a diverse range of optimization problems. It
is worth investigating strategies such as FALQON, FQAOA, and others.
Furthermore, the concept of shortcuts to adiabaticity has played a crucial
role in enhancing many ansatz designs. As such, exploring this concept could
lead to more efficient algorithms with faster convergence and reduced resource
requirements.

In addition to these alternative optimization approaches, there is uncharted
territory for potential improvements within the existing QAOA framework. This
includes adopting “warm-starting” approaches, which could provide superior
capabilities, unique features, and enhanced practical use. These algorithmic
enhancements could contribute significantly to the ongoing evolution of QAOA,
broadening its scope and effectiveness.

  * •

New ansatz designs: research suggests exploring ma-QAOA to simplify parameter
optimization [77, 93], investigating ab-QAOA for scalability and quantum
advantage in combinatorial optimization problems [80] and leveraging QAOAnsatz
for a broader set of problems [84].

  * •

Herrman [94] investigated the connection between ma-QAOA and Continuous-Time
Quantum Walks (CTQW) on dynamic graphs, showing their equivalence. Research
could be done on investigating well-studied CTQW phenomena, such as hitting
times, to improve our understanding of ma-QAOA and help find optimal
parameters.

  * •

Further research on WS-QAOA [88] to improve the initialization of QAOA
parameters based on the solution to the relaxed QUBO problem, potentially
enhancing performance at low depths, which is particularly important for
implementation on NISQ devices.

  * •

Research methods to deal with hard cases of combinatorial optimization
problems where the energy landscape is rugged, such as the approach proposed
by Wang et al. [91] using selective clause dropout.

  * •

Applying QAOA to various Constraint Satisfaction Problems (CSPs) can deepen
our understanding of the algorithm’s effectiveness and applicability.
Evaluating its performance across these problems, especially those with unique
properties is a valuable future research direction.

##  6 A Practical Guide to QAOA

The Quantum Approximate Optimization Algorithm is a versatile quantum
algorithm that can be used to solve combinatorial optimization problems on
quantum computers. This guide aims to provide researchers with detailed
information on when and how to use QAOA effectively. It answers key questions
such as which QAOA variant or ansatz to use for a specific problem, the
benefits of using QAOA for different problem types, and how to tune and
optimize the parameters of the QAOA ansatz. It also provides guidance to apply
QAOA to a real-world problem practically.

###  6.1 Which QAOA Ansatz Variant Should I Use for My Problem?

QAOA and its variants have been applied to a range of combinatorial
optimization problems and various other problems relating to graph theory,
constrained optimization, linear systems, and unstructured search. Some
specific problems that QAOA and its variants have been successfully applied to
include MaxCut, Traveling Salesman Problem, $k$-Vertex Cover, Discrete
Portfolio Rebalancing, Sherrington-Kirkpatrick (SK) Hamiltonian, $k$-graph
coloring, electric vehicle charging problems, and Max-$k$-Cut.

In light of this, when selecting the appropriate QAOA ansatz variant for a
given problem, it is vital to consider the problem type, hardware constraints,
and the trade-off between specificity and generality. Some QAOA variants seem
particularly suitable for problems with hard constraints that always need to
be satisfied and soft constraints that need to be minimized in their
violations. They effectively solve optimization problems with rugged cost
function landscapes and numerous local minima and have been demonstrated to
sometimes outperform classical algorithms or provide some quantum advantage in
such instances. In addition, QAOA’s performance can benefit from leveraging
problem symmetries, odd cycles, density, and other structural features. QAOA
also seems to work well for hardware grid problems and problems where higher-
depth versions are necessary for achieving satisfactory results. Below we
summarize the key features of some notable QAOA ansatz variants and which
problems they might be good for.

Standard QAOA

This basic version works well for various combinatorial optimization problems
like MaxCut, Minimum Vertex Cover (MVC), Constraint Satisfaction Problems
(CSPs), and Maximum Independent Set (MIS). Use it as a starting point for
problems with an unknown structure or simple instances.

Multi-Angle QAOA (ma-QAOA)

This ansatz introduces new parameters into the circuit so that each element of
the cost and mixer layers has its angle. It achieves better approximation
ratios than vanilla QAOA and may require shallower circuits. It is suitable
for combinatorial problems represented as graphs where a more complex
parameter optimization is acceptable, as it may require shallower circuits but
provides better or equal approximation ratios. The connection between ma-QAOA
and Continuous-Time Quantum Walks (CTQW) on dynamic graphs might help find
optimal parameters. Exploiting the natural symmetries of input graphs can
reduce the number of ma-QAOA parameters by approximately 33% with little to no
impact on the objective function.

QAOA+

This ansatz augments the traditional $p=1$ QAOA with additional multi-
parameter problem-independent layers of parameterized ZZ gates and a layer of
mixer X gates. It is suitable for problems like MaxCut on random regular
graphs and achieves higher approximation ratios than $p=1$ QAOA while keeping
the circuit depth low. The added circuit depth beyond the vanilla QAOA grows
only in the number of qubits used as a set of $2N-1$ parameters for $N$
qubits. QAOA+ outperforms the alternative multi-angle QAOA ansatz in most
cases.

Digitized Counterdiabatic QAOA (DC-QAOA)

This ansatz reduces computational complexity and circuit depth by utilizing
counterdiabatic driving to speed up the optimization process. It is suitable
for problems such as Ising models, classical optimization problems, and
$k$-spin models. DC-QAOA seems to outperform the standard QAOA in all cases
when applied to these problems.

Adaptive Bias QAOA (ab-QAOA)

This ansatz incorporates adaptive bias fields into the mixer operators to
accelerate convergence. The local fields (bias fields) are not optimized but
updated according to a specific prescription. It is suitable for faster
convergence and reduced computation time in combinatorial optimization
problems and has a polynomially shorter computation time than vanilla QAOA.
The improvement in computation time further increases with problem size.

Adaptive Derivative-Assembled Problem-Tailored QAOA (ADAPT-QAOA)

This ansatz iteratively selects the QAOA mixer from a pre-defined pool of
operators, maximizing the gradient of the commutator of the pool operator and
the cost Hamiltonian over the ansatz of the previous step. It was reported to
converge faster than standard QAOA while reducing the number of CNOT gates and
optimization parameters by about 50% each, particularly when entangling gates
are included in the operator pool. The drawback of this method is that the
selection of mixing operators requires additional measurements depending on
the size of the operator pool.

Recursive QAOA (RQAOA)

This non-local variant of QAOA iteratively reduces the size of the problem. It
is designed to address the limitations caused by the $\mathbb{Z}_{2}$ symmetry
of QAOA states and the geometric locality of the ansatz. It has shown
promising results on NISQ devices and applies to combinatorial optimization
problems like MaxCut.

Quantum Alternating Operator Ansatzes (QAOAnsatz)

This ansatz alternates between a more general set of operators, allowing it to
solve a broader range of problems, especially those with hard constraints
defining feasible subspaces and soft constraints to minimize violations.
QAOAnsatz encompasses many variants such as XY mixer, Grover mixer, etc. that
can be tailored to different problems or constraints.

Warm-Starting QAOA (WS-QAOA)

This approach initializes QAOA parameters based on the solution to the relaxed
QUBO problem (continuous variables instead of binary ones). It provides
initial states with the best approximation guarantee available classically in
polynomial time and can be incorporated into the workflow of recursive QAOA.
Suitable for recursive QAOA and provides initial states with the best
approximation guarantee available classically in polynomial time.

Feedback-based ALgorithm for Quantum OptimizatioN (FALQON)

This algorithm uses qubit measurements to assign values to quantum circuit
variational parameters constructively, enabling approximate solutions without
classical optimization. While it was designed for the nearly fault-tolerant
era where deep circuits are feasible, it can also be used as a warm-start
technique for NISQ devices, improving the initialization of standard QAOA.

Fermionic QAOA (FQAOA)

This approach is suitable for solving combinatorial optimization problems with
constraints, utilizing fermion particle number preservation to intrinsically
impose these constraints and using fermion-based driver Hamiltonian for
problem Hamiltonians with constraints.

Spanning Tree QAOA (ST-QAOA)

In this ansatz, an approximate solution is constructed with $r$ rounds of
gates explicitly tailored to the given problem instance using insights from
the classical algorithm in the pre-computation routine. When $r=1$, the ST-
QAOA is guaranteed to match the performance of the classical solver. As the
number of rounds increases, the ST-QAOA approaches the exact solution to the
MaxCut problem. It achieves the same performance guarantee as the classical
algorithm and can outperform vanilla QAOA at low depths.

Ansatz Architecture Search (AAS)

This approach optimizes the QAOA ansatz and variational parameters by
searching the discrete space of quantum circuit architectures near QAOA. It
starts with a greedy search strategy and iteratively removes two-qubit gates
from the best ansatz of the previous level, scored, and the best of them is
selected as the output of this level. This method has shown significant
improvement in the probability of finding low-energy states while using fewer
two-qubit gates.

Hardware-specific ansatz designs

Tailor your ansatz for better performance on specific platforms such as
trapped ions, neutral atoms, superconducting qubits, and photonic quantum
computers.

In the meantime, it is essential to note that while QAOA can be adapted and
extended to various problems, it might not always be the most efficient or
best-suited method for every problem. For example, problems that do not have a
well-defined graph representation, do not fit into the combinatorial
optimization framework, or are not easily represented by Hamiltonians might
prove more difficult to benefit from it. QAOA may also not be a good fit for
certain problems due to performance limitations, locality constraints, and
obstructions caused by specific properties of the problems [83, 162, 165]. For
example, it might not be suitable for problems that require a large depth or
high entanglement due to limitations in quantum hardware and noise-related
issues. Finally, the algorithm may also be less suitable when the graph
topology differs significantly from the quantum hardware’s connectivity, as
this may introduce substantial overhead in compiling and routing qubits.

Since the choice of QAOA variant deeply depends on the specific problem
instance and available quantum hardware, the following guidelines can be
helpful:

  * •

MaxCut problems: Several QAOA variants have been suggested for MaxCut
problems, including QAOA+, Warm-Starting QAOA (WS-QAOA), Recursive QAOA
(RQAOA), Multi-Angle QAOA (ma-QAOA), Adaptive Bias QAOA (ab-QAOA), and
Measurement Based Quantum Computing QAOA (MBQC QAOA). They all seem
particularly well-suited for solving MaxCut problems, with multiple research
papers mentioning their success in achieving a high approximation ratio. QAOA+
improves over the traditional $p=1$ QAOA ansatz using additional layers of
parameterized gates, providing higher approximation ratios. WS-QAOA leverages
the best classical approximation guarantees, while RQAOA iteratively reduces
the problem size, showing promise for NISQ devices. MBQC QAOA demonstrates
improvements when applied to photonic quantum computers. The variant choice
depends on the specific problem instance and available quantum hardware.

  * •

Combinatorial optimization problems with constraints: Adaptive Bias QAOA (ab-
QAOA) and Fermionic QAOA (FQAOA) are suggested for these problems and are
particularly applicable when the constraints are integral to the problem
structure. ab-QAOA accelerates convergence by incorporating adaptive bias
fields, while FQAOA utilizes fermion particle number preservation to impose
constraints intrinsically. Grover Mixer QAOA (GM-QAOA) is suggested for CSPs,
as it uses Grover-like selective phase shifting operators for efficient
optimization and is not susceptible to Trotterization or Hamiltonian
simulation errors. Other approaches might also be suitable; the choice of
optimal QAOA variant depends on the problem’s structure and constraints.

  * •

Ising models: Digitized Counterdiabatic QAOA (DC-QAOA) and Ansatz Architecture
Search (AAS) have been proposed as suitable variants for Ising models. DC-QAOA
reduces computational complexity by leveraging counterdiabatic driving, while
AAS modifies both the QAOA ansatz and parameter selection process to improve
low-energy state identification. DC-QAOA and AAS are not the only possible
approaches for Ising models, but they are highlighted here for their unique
approaches to reducing computational complexity and identifying low-energy
states, respectively.

  * •

Graph-based problems: Recursive QAOA (RQAOA), Spanning Tree QAOA (ST-QAOA),
and Graph Neural Networks (GNNs) initialization are recommended for graph-
based problems. RQAOA iteratively reduces problem size and works well for
graph-based problems, while ST-QAOA approaches the exact solution to the
MaxCut problem as the number of rounds increases. GNNs-based initialization
can speed up inference time across graphs. Adaptive Derivative-Assembled
Problem-Tailored (ADAPT-QAOA) is also a good choice for graph-based problems
because it demonstrates faster convergence than the original QAOA due to
shortcuts to adiabaticity.

  * •

Other problems: QAOA has also shown potential for solving problems such as the
Sherrington-Kirkpatrick (SK) Hamiltonian problems, $k$-graph coloring and
electric vehicle charging problems with global power contraints, Exact Cover
problem, $k$-spin model, and ground-state energy calculations for small
chemistry and material science problems. Quantum Alternating Operator Ansatzes
(QAOAnsatzes) has been proposed as a versatile extension of QAOA that works
well for optimization problems with both hard and soft constraints.

###  6.2 How to Optimize the QAOA Parameters?

Initialization and optimization of QAOA parameters are crucial for achieving
good performance. Different QAOA ansatz designs and hardware platforms will
require different parameter optimization methods. There is no universal
solution for the selection of the parameter optimization method. Nevertheless,
it is possible to extract valuable insights and broad recommendations for
choosing an appropriate method for QAOA applications.

#### Gradient-free methods

Gradient-free methods are, in general, more suitable for problems with
moderate circuit depths and limited computational resources. Gradient-free
methods often employed include BOBYQA, COBYLA, NEWUOA, Nelder-Mead, PRAXIS,
and SBPLX. BOBYQA within the APOSSM framework has shown the best performance
for a fixed number of functional evaluations.

Strengths:

  * •

Computationally efficient

  * •

Requires fewer function evaluations when compared to gradient-based methods

  * •

Some methods, such as SPSA and Powell’s method, are less affected by noise
than other methods

Weaknesses:

  * •

Performance can be challenged with an increasing number of layers

  * •

Some methods, like COBYLA, Nelder-Mead, and Conjugate-Gradient, are
significantly affected by noise

When to use: Gradient-free methods can be appropriate when computational
efficiency is important and you are working with problems with modest circuit
depths. Some gradient-free methods may be preferred when noise levels are low
or moderate.

#### Gradient-based approaches

Gradient-based approaches are more robust to variations in problems and noise
but may require more computational resources. Often employed gradient-based
approaches include Gradient Descent, Stochastic Gradient Descent (SGD), Model
Gradient Descent (MGD), and BFGS.

Strengths:

  * •

Widely used for parameter optimization

  * •

Can handle problems with a smooth objective function

  * •

Can perform well even if the objective function is not smooth with respect to
the error (policy-gradient-based reinforcement learning)

  * •

Some methods (e.g., AMSGrad and BFGS) perform better than gradient-free
methods in the presence of shot-noise

Weaknesses:

  * •

Can be computationally expensive

  * •

Vulnerable to noise in NISQ devices

  * •

May require many measurements for each gradient component

When to use: Gradient-based approaches may be appropriate when the objective
function is smooth and when you require a method that can handle a wide range
of problems. These methods may also be preferred when noise levels are low or
when working with problems requiring a robust optimization process.

#### Machine Learning Approaches

Machine Learning approaches can provide faster convergence and better
optimization results but may require more training instances and computational
resources. Some ML approaches include Gaussian Process Regression (GPR),
Linear Regression (LM), Regression Tree (RTREE), Support Vector Machine
Regression (RSVM), Long Short-Term Memory (LSTM) neural networks, Graph Neural
Networks (GNNs), and Reinforcement Learning (RL).

Strengths:

  * •

Can exploit correlations and patterns among parameters

  * •

Can accelerate QAOA optimization

  * •

Can generalize across different problem instances and graph sizes

Weaknesses:

  * •

Scalability can be an issue, particularly for more complex problems

  * •

May require many training instances for good performance

When to use: Machine learning approaches can be appropriate when working with
problems with correlations or patterns among parameters or when you require a
method that can generalize across different problem instances and graph sizes.
These approaches may also be suitable when working with large datasets or
complex problems requiring advanced optimization techniques.

#### Other considerations

Certain optimization problems can showcase specific attributes such as barren
plateaus, transferability and reusability of parameters, or the existence of
parameter symmetries. These characteristics can be harnessed to enhance the
optimization procedure. For instance, adopting layerwise learning strategies
can aid in bypassing barren plateaus, and exploiting parameter symmetries can
simplify the optimization process.

Some general strategies and ideas include:

Warm-starting QAOA

Initializing QAOA parameters based on the solution to the relaxed QUBO problem
can provide a good starting point for optimization. It provides an advantage
over standard QAOA at low depth, which is particularly important for NISQ
devices.

FALQON’s purely quantum optimization loop

By using FALQON’s measurement-based feedback loop, one can assign values to
quantum circuit variational parameters, improving the initialization of
standard QAOA. It is primarily designed for fault-tolerant quantum devices
that do not yet exist, limiting current applicability.

FQAOA

This method is designed to solve combinatorial optimization problems with
constraints, utilizing fermion particle number preservation to impose these
constraints intrinsically.

AAS

This technique searches the discrete space of quantum circuit architectures
near QAOA to find a better ansatz.

Layerwise learning strategy

Layerwise learning strategies grow the circuit incrementally and update only
subsets of parameters during optimization. This can help to avoid initializing
on a plateau and reduce the probability of creeping onto a plateau during
training. This can improve the optimization process and help QAOA converge to
better solutions.

Parameter Transferability and Reusability

Optimal QAOA parameters can be transferred and reused across different problem
instances based on their local characteristics. This can improve the quality
of the solution and reduce the number of evaluations required to reach it.

Constraint-based optimization

Constraining QAOA circuit parameters to the range $(0,\pi)$ to exploit their
symmetry can result in runtime acceleration of up to 5.5 times.

Parameter regression

Using parameter regression to optimize QAOA parameters can achieve an
acceleration of more than 1.23 times.

Leveraging Parameter Symmetries

Parameter symmetries can simplify the optimization process and eliminate
degeneracies in the parameter space. Exploiting these symmetries can make the
search for optimal QAOA parameters more efficient.

Exploit natural symmetries

Exploiting the natural symmetries of input graphs can reduce the number of
parameters and improve QAOA performance by approximately 33%.

Use Continuous-Time Quantum Walks

Relate ma-QAOA to CTQW on dynamic graphs to leverage well-studied CTQW
phenomena for finding optimal parameters.

Heuristic strategies for parameter initialization

Optimal initialization of QAOA’s parameters can be determined in
$O(\text{poly}(p))$ time, while random initialization would necessitate
$2^{O(p)}$ optimization runs to attain comparable performance.

Hardware-specific optimizations

Consider the unique properties and limitations of your chosen quantum hardware
platform when optimizing QAOA parameters. Depending on the hardware used, this
may involve leveraging connectivity, qudits, global entangling operations, or
higher energy levels.

###  6.3 Practical Considerations for Implementing QAOA

When applying QAOA to a problem, consider the following practical advice:

  * •

Choose an ansatz that suits the problem type and hardware design while
balancing specificity and generality.

  * •

Employ Ansatz Architecture Search (AAS) if you seek to optimize both the QAOA
ansatz and variational parameters. This method uses a greedy search strategy,
exploring the space of quantum circuit architectures to find a more suitable
ansatz for your problem.

  * •

Assess the impact of different mixer designs on your QAOA algorithm and the
trade-offs between improved performance and longer circuit depth.

  * •

Experiment with different parameter initialization and optimization methods,
such as heuristic strategies, parameter fixing, and Graph Neural Networks
(GNNs), to find the best approach for your specific problem type and hardware
platform.

  * •

Select an appropriate optimization algorithm for your problem, considering the
trade-offs of gradient-free methods, gradient-based approaches, and machine
learning techniques.

  * •

Explore the transferability and reusability of optimal parameters across
different problem instances based on local characteristics of subgraphs.

  * •

Leverage parameter symmetries to simplify optimization, eliminate degeneracies
in the parameter space, and improve QAOA performance. Investigate symmetries
in objective functions, cost Hamiltonians, and QAOA parameters themselves.

  * •

To overcome barren plateaus in cost function landscapes, consider employing
layerwise learning strategies, incremental circuit depth growth, or
initializing closer to target parameters.

  * •

When working with large-scale problems or complex optimization landscapes, be
prepared to employ advanced optimization techniques, hybrid quantum-classical
approaches, or specialized ansatz variations to overcome challenges and
achieve desired results.

  * •

To overcome noise and error challenges, consider implementing error mitigation
techniques such as gate count reduction, SWAP network optimization, symmetry
verification, and leveraging hardware-specific features.

## Acknowledgements

We thank Dr. Michele Grossi and Michał Stęchły for their invaluable
discussions and insights that have greatly benefited this project. We would
like to express our gratitude to the Quantum Open Source Foundation (QOSF) for
organizing the mentorship program that enabled the formation of our team. We
acknowledge the use of IBM Quantum services for this work. The views expressed
are those of the authors, and do not reflect the official policy or position
of IBM or the IBM Quantum team. In this paper we used `ibm_oslo`, `ibm_lagos`,
`ibm_nairobi` and `ibm_perth` which are some of the IBM Quantum Falcon r5.11H
devices. The authors disclosed receipt of financial support for the
publication of this article from Quantum Neural Technologies S.A.

##### Author Contributions:

Conceptualization, KB; methodology, KB, AC, RL and AS; software, major
contributions: KB, DB, AC, RL, AS; minor contributions: CC, KP; formal
analysis and investigation, all authors contributed equally; writing—original
draft, and writing—review and editing, all authors contributed equally;
supervision, KB; project administration, KB, AC and RL.

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