Imaginary-time evolution has been shown to be a promising framework for tackling combinatorial optimization problems on quantum hardware. In this work, we propose a classical quantum-inspired strategy for solving combinatorial optimization problems with integer-valued decision variables by encoding decision variables into multi-level quantum states known as qudits. This method results in a reduced number of decision variables compared to binary formulations while inherently incorporating single-association constraints. Efficient classical simulation is enabled by constraining the system to remain in a product state throughout optimization. The qudit states are optimized by applying a sequence of unitary operators that iteratively approximate the dynamics of imaginary time evolution. Unlike previous studies, we propose a gradient-based method of adaptively choosing the Hermitian operators used to generate the state evolution at each optimization step, as a means to improve the convergence properties of the algorithm. The proposed algorithm demonstrates promising results on Min-d-Cut problem with constraints, outperforming Gurobi on penalized constraint formulation, particularly for larger values of d.
Imaginary-time evolution has been shown to be a promising framework for tackling combinatorial optimization problems on quantum hardware. In this work, we propose a classical quantum-inspired strategy for solving combinatorial optimization problems with integer-valued decision variables by encoding decision variables into multi-level quantum states known as qudits. This method results in a reduced number of decision variables compared to binary formulations while inherently incorporating single-association constraints. Efficient classical simulation is enabled by constraining the system to remain in a product state throughout optimization. The qudit states are optimized by applying a sequence of unitary operators that iteratively approximate the dynamics of imaginary time evolution. Unlike previous studies, we propose a gradient-based method of adaptively choosing the Hermitian operators used to generate the state evolution at each optimization step, as a means to improve the convergence properties of the algorithm. The proposed algorithm demonstrates promising results on Min-d-Cut problem with constraints, outperforming Gurobi on penalized constraint formulation, particularly for larger values of d.
Solving combinatorial optimization problems using variational quantum algorithms (VQAs) has emerged as a promising research direction. Since the introduction of the Quantum Approximate Optimization Algorithm (QAOA), numerous variants have been proposed to enhance its performance. QAOA was later extended to the Quantum Alternating Operator Ansatz (QAOA+), which generalizes the initial state, phase-separation operator, and mixer to address constrained problems without relying on the standard Quadratic Unconstrained Binary Optimization (QUBO) formulation. However, QAOA+ often requires additional ancilla qubits and a large number of multi-controlled Toffoli gates to prepare the superposition of feasible states, resulting in deep circuits that are challenging for near-term quantum devices. Furthermore, VQAs are generally hindered by issues such as barren plateaus and suboptimal local minima. Recently, Quantum Imaginary Time Evolution (QITE), a groundstate preparation algorithm, has been explored as an alternative to QAOA and its variants. QITE has demonstrated improved performance in quantum chemistry problems and has been applied to unconstrained combinatorial problems such as MaxCut. In this work, we apply the variational form of QITE (VarQITE) to solve the Multiple Knapsack Problem (MKP), a constrained problem, using a Max-Cut-tailored ansatz. To the best of our knowledge, this is the first attempt to address constrained optimization using VarQITE. We show that VarQITE achieves significantly lower mean optimality gaps compared to QAOA and other conventional methods. Moreover, we demonstrate that scaling the Hamiltonian coefficients can further reduce optimization costs and accelerate convergence.
Abstract We propose a hybrid quantum–classical algorithm for solving quadratic unconstrained binary optimization (QUBO) problems using an imaginary time evolution-mimicking circuit (ITEMC). The circuit parameters are optimized to mimic short-time ITE, using only single- and two-qubit expectation values. This significantly reduces the measurement overhead by avoiding full energy evaluation. By updating the initial state based on results from last step iteratively, the algorithm quickly converges to the low-energy solutions. With a pre-sorting step that optimizes quantum gate ordering based on QUBO coefficients, the convergence is further improved. Our classical simulations achieve approximation ratios above 99 % up to 150 qubits. Furthermore, the linear scaling of entanglement entropy with system size suggests that the circuit is challenging to simulate classically using tensor networks. We also demonstrate hardware runs on IBM’s device for 40, 60, and 80 qubits, and obtain solutions compatible with that from simulated annealing.
Algorithms based on non-unitary evolution have attracted much interest for ground state preparation on quantum computers. One recently proposed method makes use of ancilla qubits and controlled unitary operators to implement weak measurements related to imaginary-time evolution. In this work we specialize and extend this approach to the setting of combinatorial optimization. We first generalize the algorithm from exact to approximate optimization, taking advantage of several properties unique to classical problems. In particular we show how to select parameters such that the success probability of each measurement step is bounded away from 1/2. We then show how to adapt our paradigm to the setting of constrained optimization for a number of important classes of hard problem constraints. For this we compare and contrast both penalty-based and feasibility-preserving approaches, elucidating the significant advantages of the latter approach. Our approach is general and may be applied to easy-to-prepare initial states as a standalone algorithm, or deployed as a quantum postprocessing stage to improve performance of a given parameterized quantum circuit. We then propose a more sophisticated variant of our algorithm that adaptively applies a mixing operator or not, based on the measurement outcomes seen so far, as to speeds up the algorithm and helps the system evolution avoid slowing down or getting stuck suboptimally. In particular, we show that mixing operators from the quantum alternating operator ansatz can be imported directly, both for the necessary eigenstate scrambling operator and for initial state preparation, and discuss quantum resource tradeoffs.
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.
Approximate combinatorial optimization is a promising use case for quantum computers. The quantum optimization algorithms often employ a fixed ansatz that evolves an unbiased initial state towards states with better values of the optimand, then samples the states to determine an approximately optimal solution. However, promising alternative approaches have considered ``warm-start" and sampling-based methods that instead begin from the best known solution, which can be directly optimized with the quantum computer and updated as new information becomes available, potentially outperforming the fixed ansätze. Here we use these ideas to design a nonvariational quantum algorithm for combinatorial optimization. At each step the algorithm begins with a state superposed around the best known solution, then drives it to lower energy using quantum imaginary time evolution. These nonvariational, initial-state-dependent circuits are determined using analytic equations that are evaluated using only a conventional computer. After implementing the circuits, the state is sampled, potentially obtaining a new best-known solution to use as the initial state at the next iteration. Using simulations of the algorithm solving MaxCut on 3-regular graphs with 30 or fewer vertices and a shot budget of 100 total shots, the approach obtains median solutions within 95\% of the global optimum and finds optimal solutions in 11\% or more of cases, significantly outperforming random and simplified classical search procedures. We discuss several future directions.
Integer programming (IP) is an NP-hard combinatorial optimization problem that is widely used to represent a diverse set of real-world problems spanning multiple fields, such as finance, engineering, logistics, and operations research. It is a hard problem to solve using classical algorithms, as its complexity increases with problem size. Most quantum algorithms for solving IP are highly resource-inefficient because they encode integers into qubits. In [1], the issue of resource inefficiency was addressed by mapping integer variables to qudits. However, [1] has limited practical value due to a lack of scalability to multiple qudits to encode larger problems. In this work, by extending upon the ideas of [1], a circuit-based scalable quantum algorithm is presented using multiple interacting qudits for which we show a quantum speed-up for solving the integer programming problem. The quantum algorithm consists of a distillation function that efficiently separates the feasible from the infeasible regions, a phase-amplitude encoding for the cost function, and a quantum phase estimation coupled with a multi-controlled single-qubit rotation for optimization. We prove that our algorithm yields the optimal solution with the maximum measurement probability. The time complexity for the quantum algorithm is shown to be O(rP(mn2logd+dn/2/Nys1/2+nεQPE−1)) for a problem with the number of variables n taking d integer values, satisfying m constraints with a precision of εQPE and rP repetitions required for reaching a success probability P. Compared to the classical time complexity of brute force O(dn) and the classical exact algorithm O((logn)O(n)), it incurs a reduction of ∼dn/2 in the time complexity (for a polynomial number of repetitions and precision requirements) for solving a general polynomial IP problem.