Challenging combinatorial optimization problems are ubiquitous in science and engineering. Several quantum methods for optimization have recently been developed, in different settings including both exact and approximate solvers. Addressing this field of research, this manuscript has three distinct purposes. First, we present an intuitive method for synthesizing and analyzing discrete ( i . e . , integer-based) optimization problems, wherein the problem and corresponding algorithmic primitives are expressed using a discrete quantum intermediate representation (DQIR) that is encoding-independent. This compact representation often allows for more efficient problem compilation, automated analyses of different encoding choices, easier interpretability, more complex runtime procedures, and richer programmability, as compared to previous approaches, which we demonstrate with a number of examples. Second, we perform numerical studies comparing several qubit encodings; the results exhibit a number of preliminary trends that help guide the choice of encoding for a particular set of hardware and a particular problem and algorithm. Our study includes problems related to graph coloring, the traveling salesperson problem, factory/machine scheduling, financial portfolio rebalancing, and integer linear programming. Third, we design low-depth graph-derived partial mixers (GDPMs) up to 16-level quantum variables, demonstrating that compact (binary) encodings are more amenable to QAOA than previously understood. We expect this toolkit of programming abstractions and low-level building blocks to aid in designing quantum algorithms for discrete combinatorial problems.
Challenging combinatorial optimization problems are ubiquitous in science and engineering. Several quantum methods for optimization have recently been developed, in different settings including both exact and approximate solvers. Addressing this field of research, this manuscript has three distinct purposes. First, we present an intuitive method for synthesizing and analyzing discrete ( i . e . , integer-based) optimization problems, wherein the problem and corresponding algorithmic primitives are expressed using a discrete quantum intermediate representation (DQIR) that is encoding-independent. This compact representation often allows for more efficient problem compilation, automated analyses of different encoding choices, easier interpretability, more complex runtime procedures, and richer programmability, as compared to previous approaches, which we demonstrate with a number of examples. Second, we perform numerical studies comparing several qubit encodings; the results exhibit a number of preliminary trends that help guide the choice of encoding for a particular set of hardware and a particular problem and algorithm. Our study includes problems related to graph coloring, the traveling salesperson problem, factory/machine scheduling, financial portfolio rebalancing, and integer linear programming. Third, we design low-depth graph-derived partial mixers (GDPMs) up to 16-level quantum variables, demonstrating that compact (binary) encodings are more amenable to QAOA than previously understood. We expect this toolkit of programming abstractions and low-level building blocks to aid in designing quantum algorithms for discrete combinatorial problems.
The Quantum Approximate Optimization Algorithm (QAOA) and its derived variants are widely in use for approximating combinatorial optimization problem instances on gate-based Noisy Intermediate Scale Quantum (NISQ) computers. Commonly, circuits required for QAOA are constructed by first reformulating a given problem as a Quadratic Unconstrained Binary Optimization (QUBO) problem. It is then straightforward to synthesize a QAOA circuit from QUBO equations. In this work, we illustrate a more qubit-efficient circuit construction for combinatorial optimization problems by the example of the Traveling Salesperson Problem (TSP). Conventionally, the qubit encoding in QAOA for the TSP describes a tour using a sequence of nodes, where each node is written as a 1-hot binary vector. We propose to encode TSP tours by selecting edges included in the tour. Removing certain redundancies, the number of required qubits can be reduced by a linear factor compared to the aforementioned conventional encoding. We examined implementations of both QAOA encoding variants in terms of their approximation quality and runtime. Our experiments show that for small instances results are just as accurate using our proposed encoding, whereas the number of required classical optimizer iterations increases only slightly.
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.
This paper proposes a quasi-binary encoding based algorithm for solving a specific quadratic optimization models with discrete variables, in the quantum approximate optimization algorithm (QAOA) framework. The quadratic optimization model has three constraints: 1. Discrete constraint, the variables are required to be integers. 2. Bound constraint, each variable is required to be greater than or equal to an integer and less than or equal to another integer. 3. Sum constraint, the sum of all variables should be a given integer. To solve this optimization model, we use quasi-binary encoding to encode the variables. For an integer variable with upper bound Ui and lower bound Li, this encoding method can use at most 2log2(Ui−Li+1) qubits to encode the variable. Moreover, we design a mixing operator specifically for this encoding to satisfy the hard constraint model. In the hard constraint model, the quantum state always satisfies the constraints during the evolution, and no penalty term is needed in the objective function. In other parts of the QAOA framework, we also incorporate ideas such as CVaR-QAOA and parameter scheduling methods into our QAOA algorithm. In the financial field, by introducing precision, portfolio optimization problems can be reduced to the above model. We will use portfolio optimization cases for numerical simulation. We design an iterative method to solve the problem of coarse precision caused by insufficient qubits of the simulators or quantum computers. This iterative method can refine the precision by multiple few-qubit experiments.
This study explores quantum computing for combinatorial optimization, focusing on the Traveling Salesman Problem (TSP). Using Qiskit, the Quantum Approximate Optimization Algorithm (QAOA) is implemented on simulators and quantum hardware, and compared with classical optimization using Gurobi. TSP instances are encoded as Quadratic Unconstrained Binary Optimization (QUBO) problems derived from the Miller–Tucker–Zemlin formulation, with quadratic penalties enforcing degree and subtour constraints. Results indicate that QAOA consistently generates feasible tours, but classical branch-and-bound methods achieve better solution quality and runtime. On Noisy Intermediate-Scale Quantum (NISQ) devices, performance degrades with problem size due to sensitivity to parameter tuning, penalty scaling, and hardware noise. Among the tested classical optimizers, COBYLA provides the best balance of efficiency and noise tolerance. Overall, this work establishes a reproducible quantum-classical workflow, provides performance baselines, and identifies critical factors such as encoding design, optimizer choice, and error mitigation for improving near-term quantum heuristic performance.
Abstract Combinatorial optimization problems over graphs, such as the traveling salesman problem, longest path problem, and maximum independent set problem, are well-known for being computationally costly, some even NP-hard problems. In this paper, we propose a general quantum algorithm framework searching for approximate solutions to combinatorial optimization problems with linear objective functions. Our framework provides APIs (application programming interfaces) that enable developers to encode weighted graph structures onto quantum circuits and utilize variational algorithms to generate approximate solutions. One key advantage of our framework is that it allows developers to design new graph algorithms for the graph problem represented as linear combinations of edge weights without requiring expertise in quantum programming. Besides, it only uses a logarithmic level of quantum bit scale, making our framework work on quantum computers with limited physical resources. Our experimental results demonstrate that our framework can provide good approximations for the traveling salesman problem compared to current quantum algorithm.
Combinatorial Optimization problems play an important role in various domains such as logis- tics, life science or the financial sector. As their complexity in general grows exponentially with their problem size, finding the optimal solution classically becomes infeasible very fast. This raises the central question of whether quantum computers are in principle suitable for solving combinatorial optimization problems better and faster than classical computers. In order to find answers to that question, one needs to study quantum algorithms designed for those problems under ideal conditions (i.e., simulations, which are however limited due to exponential scaling) to investigate their potential, but one also needs real quantum hardware to test its viability and evaluate whether practical advantages are reasonable. In this regard, we consider variational quantum algorithms, especially the Quantum Approximate Optimization Algorithm (QAOA) consisting of a problem and a mixer Hamiltonian with freely selectable parameters, originally proposed as variational hybrid ansatz with classical optimization. This algorithm structure of- fers flexible and relatively shallow circuits, which makes it suitable for current NISQ hardware. However, optimizing the variational circuit parameters can be an NP-hard problem at worst. In this thesis, we study different versions of QAOA, especially with a focus on limitations, perfor- mance and potential advantages opposite classical methods. The first part of the thesis concerns the investigation of various aspects regarding the imple- mentation of the variational QAOA protocol such as: the incorporation of a problem constraint either by adding a suitable penalty term of by choosing a suitable mixer (that restricts the search space), the scaling of the problem Hamiltonian in comparison to the mixer Hamiltonian, the choice of the initial values for the classical optimization of the circuit parameters, and finally the choice of the function to be minimized (i.e., CVaR vs. expectation value). We apply this proto- col to both the portfolio optimization and the feature selection and evaluate the performance. In order to improve QAOA, we consider the warm-start QAOA for different problem instances and analyze its improved performance with regard to involved quantum effects. For this purpose, we propose a classical pre-processing method and demonstrate that the warm-start results can be reproduced or even exceeded by this method. The second part addresses the issue that the standard variational form requires a high number of circuit parameters that have to be optimized at a sufficiently large depth, which is a major limitationforachievingpotentialscalingadvantagescomparedtoclassicalmethods. Toovercome this limitation, we investigate and further develop fixed schedules for QAOA, which only rely on two free parameters. We start with introducing the continuous adiabatic theorem as it motivates the form of QAOA and show its limitations, which we address by applying a discrete version of the theorem. With our findings, a new version of QAOA based on linear-ramps (LR-QAOA) is introduced and further developed. In order to derive suitable parameter values for this schedule, we propose a method based on extrapolation from smaller towards larger problem sizes. With our results obtained on a noiseless emulator, we evaluate the quantum runtime scaling and discuss whether a scaling advantage can be achieved compared to classical methods.