We introduce a qubit- and gate-efficient higher-order unconstrained binary optimization (HUBO) encoding for graph partitioning problems requiring label-count minimization. This widely applicable class of problems includes minimum graph coloring, minimum \textit{k}-cut, and community detection. To the best of our knowledge, this is the first work to address the optimization versions of these problems in a quantum setting, rather than only their decision counterparts. Our construction encodes each vertex variable in a number of bits logarithmic in the number of available labels---the information-theoretic minimum---and employs a novel lexicographic penalty system that implicitly minimizes partition count without requiring dedicated indicator variables. We derive provably sufficient conditions on all penalty coefficients, including those arising from Rosenberg quadratization, guaranteeing feasibility and optimality of the lowest-energy solution. Analogous conditions are derived for a one-hot encoding to enable controlled comparison. We also show that our encoding reduces two-qubit gate count per layer of the quantum approximate optimization algorithm (QAOA). Benchmarking on a quantum annealer demonstrates that our logarithmic encoding greatly improves solution quality and time-to-solution for minimum graph coloring relative to one-hot encoding, with greater advantage as problem size increases.
We introduce a qubit- and gate-efficient higher-order unconstrained binary optimization (HUBO) encoding for graph partitioning problems requiring label-count minimization. This widely applicable class of problems includes minimum graph coloring, minimum \textit{k}-cut, and community detection. To the best of our knowledge, this is the first work to address the optimization versions of these problems in a quantum setting, rather than only their decision counterparts. Our construction encodes each vertex variable in a number of bits logarithmic in the number of available labels---the information-theoretic minimum---and employs a novel lexicographic penalty system that implicitly minimizes partition count without requiring dedicated indicator variables. We derive provably sufficient conditions on all penalty coefficients, including those arising from Rosenberg quadratization, guaranteeing feasibility and optimality of the lowest-energy solution. Analogous conditions are derived for a one-hot encoding to enable controlled comparison. We also show that our encoding reduces two-qubit gate count per layer of the quantum approximate optimization algorithm (QAOA). Benchmarking on a quantum annealer demonstrates that our logarithmic encoding greatly improves solution quality and time-to-solution for minimum graph coloring relative to one-hot encoding, with greater advantage as problem size increases.
Quantum approaches to combinatorial optimization problems (COPs) are often limited by the resource demands of Quadratic Unconstrained Binary Optimization (QUBO) encodings, which enlarge circuits through penalty terms and increase qubit and gate counts. We show that Higher-Order Unconstrained Binary Optimization (HUBO) enables a more resource-efficient formulation. Our method systematically constructs HUBO Hamiltonians and, compared to a QUBO formulation in benchmarks on Gate Assignment (GAP), Maximum k-Colorable Subgraph (MkCS), and Integer Programming (IP) problems, significantly reduces qubit requirements and decreases total CNOT gate counts by at least 89.6% for all tested instances. These results highlight HUBO as a practical alternative for quantum optimization on near-term devices. To promote adoption, we release an open-source Python library that automates HUBO model construction, extends beyond the examples presented in this work, and broadens access to resource-efficient quantum optimization.
Combinatorial optimization problems are typically formulated using Quadratic Unconstrained Binary Optimization (QUBO), where constraints are enforced through penalty terms that introduce auxiliary variables and rapidly increase Hamiltonian complexity, limiting scalability on near term quantum devices. In this work, we systematically study Quadratic Unconstrained D-ary Optimization (QUDO) as an alternative formulation in which decision variables are encoded directly in higher dimensional Hilbert spaces. We demonstrate that QUDO naturally captures structural constraints across a range of problem classes, including the Traveling Salesman Problem, two variants of the Vehicle Routing Problem, graph coloring, job scheduling, and Max-K-Cut, without the need for extensive penalty constructions. Using a qudit-level implementation of the Quantum Approximate Optimization Algorithm (qudit QAOA), we benchmark these formulations against their binary QUBO counterparts and exact classical solutions. Our study show consistently improved approximation ratios and substantially reduced computational overhead at comparable circuit depths, highlighting QUDO as a scalable and expressive representation for quantum combinatorial optimization.
Quantum computing holds the potential to efficiently solve certain problems believed to be intractable for classical computers. Realizing this potential, however, requires substantial physical resources, including large qubit counts, sufficient connectivity, and long coherence times. As current quantum devices are still far from this regime, the efficient compilation of quantum algorithms into hardware-compatible instructions remains a central challenge in quantum computing, especially for near-term applications. A representative example of such applications is combinatorial optimization, where exponentially large search spaces motivate quantum algorithms that leverage interference to approximate solutions more efficiently than classical heuristics, even at modest system sizes. Two prominent paradigms in this context are quantum annealing and the Quantum Approximate Optimization Algorithm (QAOA): Quantum annealing realizes optimization via continuous-time quantum dynamics implemented on analog hardware, while QAOA is a variational, gate-based approach that builds parameterized quantum circuits from problem-specific Hamiltonians. Both paradigms require compilation strategies tailored to the constraints of their respective hardware architectures. Most existing compilation strategies are problem-agnostic and therefore broadly applicable, yet often suboptimal for specific tasks. This thesis uses the graph coloring problem as a case study to investigate a problem-driven approach to quantum compilation, where structural features of the problem class are exploited to design tailored compilation schemes that offer improved efficiency and scalability for the targeted application. To address hardware connectivity constraints, the presented approaches adopt the parity framework, which encodes long-range interactions into single-qubit terms. This thesis develops two novel problem-driven compilation methods for the graph coloring problem, the Bandwidth-Optimized LHZ layout and the 2D-Twine circuit. The former leverages bandwidth minimization to optimize parity encodings for analog quantum annealers, while the latter constructs QAOA unitaries for gate-based optimization using parallelized Parity Twine circuits. Both approaches are benchmarked against general-purpose methods and demonstrate substantial improvements in qubit usage or circuit complexity. These results illustrate the advantages of incorporating problem structure into the compilation process and provide a proof-of-concept for the effectiveness of problem-driven compilation strategies.
Quantum computers are devices, which allow more efficient solutions of problems as compared to their classical counterparts.As the timeline to developing a quantum-error corrected computer is unclear, the quantum computing community has dedicated much attention to developing algorithms for currently available noisy intermediate-scale quantum computers (NISQ).Thus far, within NISQ, optimization problems are one of the most commonly studied and are quite often tackled with the quantum approximate optimization algorithm (QAOA).This algorithm is best known for computing graph partitions with a maximal separation of edges (MaxCut), but can easily calculate other problems related to graphs.Here, I present a novel quantum optimization algorithm, which uses exponentially less qubits as compared to the QAOA while requiring a significantly reduced number of quantum operations to solve the MaxCut problem.Such an improved performance allowed me to partition graphs with 32 nodes on publicly available 5 qubit gate-based quantum computers without any preprocessing such as division of the graph into smaller subgraphs.These results represent a 40% increase in graph size as compared to state-of-art experiments on gate-based quantum computers such as Google Sycamore.The obtained lower bound is 54.9% on the solution for actual hardware benchmarks and 77.6% on ideal simulators of quantum computers.Furthermore, large-scale optimization problems represented by graphs of a 128 nodes are tackled with simulators of quantum computers, again without any predivision into smaller subproblems and a lower solution bound of 67.9% is achieved.The study presented here paves way to using powerful genetic optimizer in synergy with quantum computers.
Hypergraph partitioning is a fundamental optimization problem with applications in data management and other domains involving higher-order relations. In this paper, we study balanced hypergraph partitioning from the perspective of quantum optimization. We formalize balanced k-way hypergraph partitioning with general hyperedge cut functions, and derive corresponding binary optimization formulations targeted at quantum optimization methods in both the two-way and multi-way settings. Our discussion highlights which cut functions admit Quadratic Unconstrained Binary Optimization (QUBO) encodings and which instead lead to higher-order binary objectives or rational forms. As a preliminary empirical validation, we focus on balanced two-way partitioning with the all-or-nothing cut on 3-uniform hypergraphs, where a direct QUBO is available, and evaluate simulated Quantum Approximate Optimization Algorithm (QAOA) and Simulated Annealing (SA) on small instances against exact solutions. The results show that the formulation is effective on small hypergraphs and that the balance-penalty weight plays a critical role in trading off cut quality and balance.
Quantum computers are devices which allow the solution of problems unsolvable to their classical counterparts. As an error-corrected quantum computer is still a decade away the quantum computing community has dedicated much attention to developing algorithms for currently available Noisy Intermediate-Scale Quantum computers (NISQ). Thus far, within NISQ, optimization problems are one of the most commonly studied and are exclusively tackled with the Quantum Approximate Optimization Algorithm (QAOA). This algorithm predominantly computes graph partitions with a maximal separation of edges (MaxCut), but can also be modified to calculate other properties of graphs. Here, I present a novel quantum optimization algorithm which uses exponentially less qubits as compared to the QAOA while requiring a significantly reduced number of quantum operations to solve the MaxCut problem. Such an improved performance allowed me to partition a graphs 32 nodes on publicly available 5 qubit gate-based solid state quantum devices without any preprocessing such as division of the graph into smaller subgraphs. This results represent a 40% increase in graph size as compared to state-of-art experiments on solid state devices such as Google Sycamore. The obtained lower bound is 54.9% on the solution for actual hardware benchmarks and 77.6% on ideal quantum simulators. Furthermore, large-scale optimization problems represented by graphs of a 128 nodes are tackled with quantum simulators, again without any pre-division into smaller subproblems and a lower solution bound of 67.9% is achieved.