Quantum computing promises solutions to classically difficult and new-found problems through controlling the subtleties of quantum computing. The Quantum Approximate Optimisation Algorithm (QAOA) is a recently proposed quantum algorithm designed to tackle difficult combinatorial optimisation problems utilising both quantum and classical computation. The hybrid nature, generality and typically low gate-depth make it a strong candidate for near-term implementation in quantum computing. Finding the practical limits of the algorithm is currently an open problem. Until now, no tools to facilitate the design and validation of probabilistic quantum optimisation algorithms such as the QAOA on a non-trivial scale exist. Graph similarity is a long standing classically difficult problem withstanding decades of research from academia and industry. Determining the maximal edge overlap between all possible node label permutations is an NP-Complete task and provides an apt measure of graph similarity. We introduce a novel quantum optimisation simulation package facilitating investigation of all constituent components of the QAOA from desktop to cluster scale using graph similarity as an example. Our simulation provides flexibility and performance. We investigate eight classical optimisation methods each at six levels of decomposition. Moreover an encoding for permutation based problems such as graph similarity through edge overlap to the QAOA allows for significant quantum memory savings at the cost of additional operations. This compromise extends into the classical portion of the algorithm as the inclusion of infeasible solutions creates a challenging cost-function landscape. We present performance analysis of our simulation and of the QAOA setting a precedent for investigating and validating numerous other difficult problems to the QAOA as we move towards realising practical quantum computation.
Quantum computing promises solutions to classically difficult and new-found problems through controlling the subtleties of quantum computing. The Quantum Approximate Optimisation Algorithm (QAOA) is a recently proposed quantum algorithm designed to tackle difficult combinatorial optimisation problems utilising both quantum and classical computation. The hybrid nature, generality and typically low gate-depth make it a strong candidate for near-term implementation in quantum computing. Finding the practical limits of the algorithm is currently an open problem. Until now, no tools to facilitate the design and validation of probabilistic quantum optimisation algorithms such as the QAOA on a non-trivial scale exist. Graph similarity is a long standing classically difficult problem withstanding decades of research from academia and industry. Determining the maximal edge overlap between all possible node label permutations is an NP-Complete task and provides an apt measure of graph similarity. We introduce a novel quantum optimisation simulation package facilitating investigation of all constituent components of the QAOA from desktop to cluster scale using graph similarity as an example. Our simulation provides flexibility and performance. We investigate eight classical optimisation methods each at six levels of decomposition. Moreover an encoding for permutation based problems such as graph similarity through edge overlap to the QAOA allows for significant quantum memory savings at the cost of additional operations. This compromise extends into the classical portion of the algorithm as the inclusion of infeasible solutions creates a challenging cost-function landscape. We present performance analysis of our simulation and of the QAOA setting a precedent for investigating and validating numerous other difficult problems to the QAOA as we move towards realising practical quantum computation.
The quantum approximation optimisation algorithm (QAOA) is one of the leading
candidates for testing the applicability of gate-model quantum resources at solving
optimisation problems on small-sized quantum hardware. A combinatorial problem that
has been understudied with QAOA is correlation clustering: given a weighted (+1,-1)
graph, where the edge weights indicate whether two nodes are similar (positive edge
weight) or different (negative edge weight), the task in correlation clustering is to find a
clustering that either maximises agreements, minimises disagreements or a combination
of both. In this thesis, we design Hamiltonian formulations that encode correlation
clustering problems such that they can be solved with QAOA. For all Hamiltonian
formulations we propose circuit implementations and study their complexities. To
benchmark the performances of a basic QAOA algorithm using these formulations, we
use numerical simulations on complete graph data sets. For one of the formulations,
which uses a multi-level approach naturally suitable to qudit systems, we investigate
the performances of several optimisers and introduce heuristic strategies to further
improve its performance. On all instances in our data-sets, which include complete as
well as Erdős–Renyi graphs, the improved algorithm shows competitive performances
for QAOA depth p ≥ 2. We also show that for this algorithm at p = 1 parameters exists
such that it has a performance guarantee of 0.670 on 3-regular graphs.
Abstract The quantum approximate optimisation algorithm (QAOA) is a hybrid quantum–classical algorithm used to approximately solve combinatorial optimisation problems. It involves multiple iterations of a parameterised ansatz that consists of a problem and mixer Hamiltonian, with the parameters being classically optimised. While QAOA can be implemented on near-term quantum hardware, physical limitations such as gate noise, restricted qubit connectivity, and state-preparation-and-measurement (SPAM) errors can limit circuit depth and decrease performance. To address these limitations, this work introduces the eXpressive QAOA (XQAOA), an overparameterised variant of QAOA that assigns more classical parameters to the ansatz to improve its performance at low depths. XQAOA also introduces an additional Pauli-Y component in the mixer Hamiltonian, allowing the mixer to implement arbitrary unitary transformations on each qubit. To benchmark the performance of XQAOA at unit depth, we derive its closed-form expression for the MaxCut problem and compare it to QAOA, Multi-Angle QAOA (MA-QAOA) (Herrman et al 2022 Sci. Rep. 12 6781), a classical-relaxed algorithm, and the state-of-the-art Goemans–Williamson algorithm on a set of unweighted regular graphs with 128 and 256 nodes for degrees ranging from 3 to 10. Our results indicate that at unit depth, XQAOA has benign loss landscapes with local minima concentrated near the global optimum, allowing it to consistently outperform QAOA, MA-QAOA, and the classical-relaxed algorithm on all graph instances and the Goemans–Williamson algorithm on graph instances with degrees greater than 4. Small-scale simulations also reveal that unit-depth XQAOA invariably surpasses both QAOA and MA-QAOA on all tested depths up to five. Additionally, we find an infinite family of graphs for which XQAOA solves MaxCut exactly and analytically show that for some graphs in this family, special cases of XQAOA are capable of achieving a much larger approximation ratio than QAOA. Overall, XQAOA is a more viable choice for variational quantum optimisation on near-term quantum devices, offering competitive performance at low depths.
Quantum computing has emerged as a promising paradigm to tackle computationally intensive problems that classical computers struggle with. In this study, we explore the application of the Edge-based quantum approximate optimization algorithm (QAOA) to the MAX-CUT problem, a well-known combinatorial optimization challenge. MAX-CUT aims to partition the vertices of a graph into two subsets such that the number of edges between the subsets is maximized. We define the edge-based MAX-CUT problem and propose a method for applying QAOA specifically tailored to this formulation. We conduct simulations using IBM’s Qiskit framework, examining both vertex-based and edge-based QAOA implementations across various graph structures. Our results highlight the comparative performance of these approaches in terms of solution quality and computational efficiency. Specifically, we analyze the impact of different graph sizes and edge densities on the complexity and CNOT gate counts of the proposed algorithms. This analysis provides insights into leveraging quantum computing for combinatorial optimization tasks, particularly focusing on the implications for practical applications and future research directions.
The transition from theoretical quantum computing to practical real-world applications heavily relies onhybrid quantum-classical architectures designed for NoisyIntermediate-Scale Quantum (NISQ) devices. Among these approaches, the Quantum Approximate Optimization Algorithm(QAOA) is a prominent strategy for addressing complex combinatorial optimization challenges, such as the NP-hard MaxCut problem. Instead of performing the entire computationon a quantum processor, QAOA utilizes a quantum circuit toevaluate a parameterized cost function, whereas a classical optimizer refines the parameters iteratively. This study presentsa rigorous empirical evaluation comparing the performanceof QAOA against established classical solvers on small-scaleMax-Cut instances. All experiments were implemented andexecuted using Python (Qiskit Aer) on a state-vector simulator.Using Erdős–Rényi random graphs and regular graphs with4 to 20 nodes, we benchmark QAOA at depths p = 2 andp = 3 against simulated annealing, the Goemans–WilliamsonSDP-based algorithm, and a linear-time greedy heuristic. Ourexperimental results demonstrate that low-depth QAOA implementations exhibit severe performance degradation, prohibitive computational overhead (O(2^n) scaling), and low convergencerates compared to mature classical algorithms. At 20 nodes,QAOA (p = 2) achieves only a 0.467 approximation ratio versus0.856 for Goemans–Williamson, with a 210× runtime slowdownover greedy heuristics
The Quantum Approximate Optimisation Algorithm (QAOA) is a widely studied quantum-classical iterative heuristic for combinatorial optimisation. While QAOA targets problems in complexity class NP, the classical optimisation procedure required in every iteration is itself known to be NP-hard. Still, advantage over classical approaches is suspected for certain scenarios, but nature and origin of its computational power are not yet satisfactorily understood. By introducing means of efficiently and accurately approximating the QAOA optimisation landscape from solution space structures, we derive a new algorithmic variant of unit-depth QAOA for two-level Hamiltonians (including all problems in NP): Instead of performing an iterative quantum-classical computation for each input instance, our non-iterative method is based on a quantum circuit that is instance-independent, but problem-specific. It matches or outperforms unit-depth QAOA for key combinatorial problems, despite reduced computational effort. Our approach is based on proving a long-standing conjecture regarding instance-independent structures in QAOA. By ensuring generality, we link existing empirical observations on QAOA parameter clustering to established approaches in theoretical computer science, and provide a sound foundation for understanding the link between structural properties of solution spaces and quantum optimisation.
The main object of study of this work is the Quantum Approximate Optimization Algorithm (QAOA) —the best-known heuristic quantum algorithm for addressing combinatorial optimization problems. Despite its popularity, QAOA poses major challenges. Namely, its performance remains poorly understood, especially in the low depth regime most relevant to near-term quantum computing. Besides, the algorithm is of variational nature, relying on several hyperparameters to be classically trained. In this work, we contribute to addressing these issues by developing techniques to understand the performance of QAOA at low depth, and deducing from these pretrained hyperparameters, thereby eliminating or considerably reducing the need for a classical optimization loop. We first consider applying QAOA to constraint satisfaction problems with the goal of maximizing the algorithm’s approximation ratio (fraction of satisfied constraints). As a particular example, we consider QAOA applied to MAXCUT on non-regular graphs, showing how analytic techniques provide good guess hyperparameters in this setting. Next, we analyze the performance of QAOA on a well-studied constraint satisfaction problem: random k-SAT. In this case, the performance is measured by the probability of finding an assignment satisfying all constraints (“success probability”). To estimate this quantity, we develop new techniques, mapping quantum expectation values of the QAOA circuits to classical spin models. We further show how these techniques can be generalized to a larger family of random constraint satisfaction problems. We conclude the work by two feasibility studies inspired by real-world use cases encountered in collaborations with industrial partners: peptide sampling, and solving flow problems. These problems are naturally phrased by encoding combinatorial variables in qudits, a relatively unexplored setting in the existing literature. We present partial analytic and numerical results for optimal QAOA hyperparameters and the scaling of its success probability in this framework, underlining potential specific difficulties when using QAOA with qudits.