We introduce SamBa-GQW, a novel quantum algorithm for solving binary combinatorial optimization problems of arbitrary degree with no use of any classical optimizer. The algorithm is based on a continuous-time quantum walk on the solution space represented as a graph. The walker explores the solution space to find its way to vertices that minimize the cost function of the optimization problem. The key novelty of our algorithm is an offline classical sampling protocol that gives information about the spectrum of the problem Hamiltonian. Then, the extracted information is used to guide the walker to high quality solutions via a quantum walk with a time-dependent hopping rate. We investigate the performance of SamBa-GQW on several quadratic problems, namely MaxCut, maximum independent set, portfolio optimization, and higher-order polynomial problems such as LABS, MAX- k -SAT and a quartic reformulation of the travelling salesperson problem. We empirically demonstrate that SamBa-GQW finds high quality approximate solutions on problems up to a size of n = 30 qubits by only sampling poly( n ) states among 2 n possible decisions. Furthermore, SamBa-GQW compares in par with classically optimized variational approaches, such as the variational guided quantum walk and QAOA (the latter when run on deep circuits). This places Samba-GQW as a promising heuristic to tackle combinatorial problems beyond the NISQ regime.
We introduce SamBa-GQW, a novel quantum algorithm for solving binary combinatorial optimization problems of arbitrary degree with no use of any classical optimizer. The algorithm is based on a continuous-time quantum walk on the solution space represented as a graph. The walker explores the solution space to find its way to vertices that minimize the cost function of the optimization problem. The key novelty of our algorithm is an offline classical sampling protocol that gives information about the spectrum of the problem Hamiltonian. Then, the extracted information is used to guide the walker to high quality solutions via a quantum walk with a time-dependent hopping rate. We investigate the performance of SamBa-GQW on several quadratic problems, namely MaxCut, maximum independent set, portfolio optimization, and higher-order polynomial problems such as LABS, MAX- k -SAT and a quartic reformulation of the travelling salesperson problem. We empirically demonstrate that SamBa-GQW finds high quality approximate solutions on problems up to a size of n = 30 qubits by only sampling poly( n ) states among 2 n possible decisions. Furthermore, SamBa-GQW compares in par with classically optimized variational approaches, such as the variational guided quantum walk and QAOA (the latter when run on deep circuits). This places Samba-GQW as a promising heuristic to tackle combinatorial problems beyond the NISQ regime.
Considerable effort has been made recently in the development of heuristic quantum algorithms for solving combinatorial optimization problems. Meanwhile, these problems have been studied extensively in classical computing for decades. In this paper, we explore a natural approach to leveraging existing classical techniques to enhance quantum optimization. Specifically, we run a classical algorithm to find an approximate solution and then use a quantum circuit to search its "neighborhood" for higher-quality solutions. We propose the Classically-Boosted Quantum Optimization Algorithm (CBQOA) that is based on this idea and can solve a wide range of combinatorial optimization problems, including all unconstrained problems and many important constrained problems such as Max Bisection, Maximum Independent Set, Minimum Vertex Cover, Portfolio Optimization, Traveling Salesperson and so on. A crucial component of this algorithm is an efficiently-implementable continuous-time quantum walk (CTQW) on a properly-constructed graph that connects the feasible solutions. CBQOA utilizes this CTQW and the output of an efficient classical procedure to create a suitable superposition of the feasible solutions which is then processed in certain way. This algorithm has the merits that it solves constrained problems without modifying their cost functions, confines the evolution of the quantum state to the feasible subspace, and does not rely on efficient indexing of the feasible solutions. We demonstrate the applications of CBQOA to Max 3SAT and Max Bisection, and provide empirical evidence that it outperforms previous approaches on these problems.
This paper introduces a non-variational quantum algorithm designed to solve a wide range of combinatorial optimisation problems, including constrained and non-binary problems. The algorithm leverages an engineered interference process achieved through repeated application of two unitaries; one inducing phase-shifts dependent on objective function values, and the other mixing phase-shifted probability amplitudes via a continuous-time quantum walk (CTQW) on a problem-specific graph. The algorithm's versatility is demonstrated through its application to various problems, namely those for which solutions are characterised by either a vector of binary variables, a vector of non-binary integer variables, or permutations (a vector of integer variables without repetition). An efficient quantum circuit implementation of the CTQW for each of these problem types is also discussed. A penalty function approach for constrained problems is also introduced, including a method for optimising the penalty function. The algorithm's performance is demonstrated through numerical simulation for randomly generated instances of the following problems (and problem sizes): weighted maxcut (18 vertices), maximum independent set (18 vertices), k-means clustering (12 datapoints, 3 clusters), capacitated facility location (12 customers, 3 facility locations), and the quadratic assignment problem (9 locations). For each problem instance, the algorithm finds a globally optimal solution with a small number of iterations.
Abstract Recent research on quantum walk-based optimization has shown theoretical potential as an alternative to variational algorithms, avoiding local minima entrapment or barren plateaus by utilizing physically motivated structure rather than unconstrained parameter search. Despite this, the Guided Quantum Walk (GQW) has never been used for portfolio optimization or tested on existing noisy quantum hardware. In this paper we propose a hybrid quantum-classical solver for portfolio optimization that integrates Guided Quantum Walks (GQW) with domain-specific knowledge elicited from real Indian equity data. The core technical contributions are threefold. First, we implement a parameterized GQW circuit in Qiskit (v2.2) with a B\'{e}zier-curve-parameterized hopping rate schedule, mapping the portfolio selection problem directly to the GQW Hilbert space via an Ising Hamiltonian encoding. Second, we introduce a problem-aware warm-start strategy that initializes the energy scale parameters from classical sampling of the feasible portfolio subspace, bypassing the high-energy exploration phase that characterizes naive initialization. Third, we incorporate Conditional Value-at-Risk (CVaR) aggregation into the quantum walk measurement to improve solution space sampling, as well as a hardware-adaptive pipeline that adjusts concentration to per-backend circuit fidelity for IBM Heron devices.\\Our simulation results demonstrate near-optimal approximation ratios exceeding 0.99 for problem sizes up to N=20 on a noiseless setting with CVaR concentration ((α∈{0.1,0.25})), significantly outperforming standard energy expectation values ((α=1.0)). Hardware validation on IBM Heron R2 processors, confirms that parameters optimized on noiseless simulation transfer effectively to real devices without re-optimization, with median approximation ratios above 0.85 across all backends for N=12 drawn from live NIFTY 50 market data. This constitutes the first end-to-end GQW implementation for financial portfolio optimization using IBM quantum hardware, and its application as a scalable quantum solver for real-world financial decision support systems.
This paper introduces in detail a non-variational quantum algorithm designed to solve a wide range of combinatorial optimisation problems, including constrained problems and problems with non-binary variables. The algorithm returns optimal and near-optimal solutions from repeated preparation and measurement of an amplified state. The amplified state is prepared via repeated application of two unitaries; one which phase-shifts solution states dependent on objective function values, and the other which mixes phase-shifted probability amplitudes via a continuous-time quantum walk (CTQW) on a problem-specific mixing graph. The general interference process responsible for amplifying optimal solutions is derived in part from statistical analysis of objective function values as distributed over the mixing graph. The algorithm's versatility is demonstrated through its application to various problems: weighted maxcut, k-means clustering, quadratic assignment, maximum independent set and capacitated facility location. In all cases, efficient circuit implementations of the CTQWs are discussed. A penalty function approach for constrained problems is also introduced, including a method for optimising the penalty function. For each of the considered problems, the algorithm's performance is simulated for a randomly generated problem instance, and in each case, the amplified state produces a globally optimal solution within a small number of iterations.
We utilize the theory of local amplitude transfer (LAT) to gain insights into quantum walks (QWs) and quantum annealing (QA) beyond the adiabatic theorem. By representing the eigenspace of the problem Hamiltonian as a hypercube graph, we demonstrate that probability amplitude traverses the search space through a series of local Rabi oscillations. We argue that the amplitude movement can be systematically guided towards the ground state using a time-dependent hopping rate based solely on the problem's energy spectrum. Building upon these insights, we extend the concept of multistage QW by introducing the guided quantum walk (GQW) as a bridge between QW-like and QA-like procedures. We assess the performance of the GQW on exact cover, traveling salesperson, and garden optimization problems with 9 to 30 qubits. Our results provide evidence for the existence of optimal annealing schedules, beyond the requirement of adiabatic time evolutions. These schedules might be capable of solving large-scale combinatorial optimization problems within evolution times that scale linearly in the problem size. Published by the American Physical Society 2024
This thesis explores quantum computing for combinatorial optimization through the Traveling Salesman Problem (TSP), which aims to find a minimum-cost Hamiltonian cycle visiting each city exactly once. Using Qiskit, we implement the Quantum Approximate Optimization Algorithm (QAOA) on both simulators and quantum hardware, and compare its performance with that of classical exact optimization via the Gurobi solver. TSP instances are encoded as Quadratic Unconstrained Binary Optimization (QUBO) problems derived from the Miller–Tucker–Zemlin formulation, with degree and subtour constraints embedded using quadratic penalties. Results indicate that QAOA consistently generates feasible tours, but classical branch-and-bound solvers outperform it in both solution quality and runtime. On Noisy Intermediate-Scale Quantum (NISQ) devices, the optimality gap grows with problem size, reflecting sensitivity to variational parameter tuning, penalty scaling, and hardware noise. Among classical optimizers for QAOA, COBYLA offers the most effective tradeoff between 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.