Abstract We present two novel contributions for achieving and assessing quantum advantage in solving difficult optimisation problems, both in theory and foreseeable practice. (1) We introduce the “Quantum Tree Generator” to generate in superposition all feasible solutions of a given 0-1 knapsack instance; combined with amplitude amplification, this identifies optimal solutions. Assuming fully connected logical qubits and comparable quantum clock speed, QTG offers perspectives for runtimes competitive to classical state-of-the-art knapsack solvers for instances with only 100 variables. (2) By introducing a new technique that exploits logging data from a classical solver, we can predict the runtime of our method way beyond the range of existing quantum platforms and simulators, for benchmark instances with up to 600 variables. Under the given assumptions, we demonstrate the QTG’s potential practical quantum advantage for such instances, indicating the promise of an effective approach for hard combinatorial optimisation problems.
Abstract We present two novel contributions for achieving and assessing quantum advantage in solving difficult optimisation problems, both in theory and foreseeable practice. (1) We introduce the “Quantum Tree Generator” to generate in superposition all feasible solutions of a given 0-1 knapsack instance; combined with amplitude amplification, this identifies optimal solutions. Assuming fully connected logical qubits and comparable quantum clock speed, QTG offers perspectives for runtimes competitive to classical state-of-the-art knapsack solvers for instances with only 100 variables. (2) By introducing a new technique that exploits logging data from a classical solver, we can predict the runtime of our method way beyond the range of existing quantum platforms and simulators, for benchmark instances with up to 600 variables. Under the given assumptions, we demonstrate the QTG’s potential practical quantum advantage for such instances, indicating the promise of an effective approach for hard combinatorial optimisation problems.
We present two novel contributions for achieving and assessing quantum advantage in solving difficult optimisation problems, both in theory and foreseeable practice. (1) We introduce the “Quantum Tree Generator” to generate in superposition all feasible solutions of a given 0-1 knapsack instance; combined with amplitude amplification, this identifies optimal solutions. Assuming fully connected logical qubits and comparable quantum clock speed, QTG offers perspectives for runtimes competitive to classical state-of-the-art knapsack solvers for instances with only 100 variables. (2) By introducing a new technique that exploits logging data from a classical solver, we can predict the runtime of our method way beyond the range of existing quantum platforms and simulators, for benchmark instances with up to 600 variables. Under the given assumptions, we demonstrate the QTG’s potential practical quantum advantage for such instances, indicating the promise of an effective approach for hard combinatorial optimisation problems.
Solving combinatorial optimization problems is a promising application area for quantum algorithms in real-world scenarios. In this work, we extend the “Quantum Tree Generator” (QTG), previously proposed for the 0-1 Knapsack Problem, to the 0-1 Quadratic Knapsack Problem (QKP) and the Multidimensional Knapsack Problem (MDKP). The QTG constructs a superposition of all feasible solutions for a given instance and can therefore be utilized as a promising state preparation routine within amplitude amplification to produce high-quality solutions. Previously, QTG-based search was tested on the 0-1 Knapsack Problem, where it demonstrated the potential for practical quantum advantage, once quantum computers with a few hundred logical and fully connected qubits are available. Here, we evaluate the algorithm's performance on QKP and MDKP against the classical solver Gurobi. To facilitate large-scale evaluations, we employ an advanced benchmarking technique that enables runtime predictions for instances with up to 2000 variables for QKP and up to 1500 variables and 100 constraints for MDKP. Our results indicate that QTG-based search can produce high-quality solutions with competitive runtimes for QKP. However, its performance declines for MDKP, highlighting the challenges quantum algorithms face when tackling highly constrained optimization problems.
Combinatorial optimisation is a main topic for real-world applications, where fast solutions are required. Optimisation algorithms are incredibly powerful, but still face bottlenecks for certain problems. In recent years, quantum algorithms have been proposed as a potential candidate for solving such problems. The question that naturally arises is whether they could provide meaningful speed-ups over classical approaches. The study of asymptotic complexity has become central in the development of quantum algorithms. Nevertheless, having a promising asymptotic improvement does not directly translate into good practical performance. This is already well known for classical algorithms, which can be executed and evaluated directly. When it comes to quantum algorithms, the situation is more nuanced, as solving practically relevant instances would require a fully fault-tolerant quantum computer. In this thesis, we address the challenge of assessing the practical performance of a quantum algorithm without access to quantum hardware. Our strategy is to evaluate how these algorithms would perform under idealised fault-tolerant assumptions and to identify parameter regimes in which quantum approaches could potentially become useful. Our goal is to guide algorithmic development long before implementation becomes possible. We first give a general overview of the quantum (sub)routines that serve as building blocks for more advanced algorithms, as well as of the three hybrid benchmarking techniques used throughout this thesis. Then, we elaborate on each method: we start with the algorithmic descriptions, proceed to the methodology, and conclude with their hybrid evaluation on practically relevant datasets. Within the hybrid benchmarking method 1 (gate count analysis), we analytically derive the minimum number of gates that a quantum version of the classical simplex algorithm would require to solve realistic instances. By executing classical counterparts of the quantum subroutines, we infer the execution speed required per gate for the quantum simplex to outperform its classical counterpart. Within the hybrid benchmarking method 2 (query count analysis), we benchmark functional quantum linear solvers by analytically deriving a precise number of query calls made to the same oracles as a function of problem parameters. This provides insight into which algorithm might be more suitable in a given parameter regime. It allows for a direct comparison only between different quantum algorithms and is independent of classical runtime measurements. Within the hybrid benchmarking method 3 (cycle count analysis), we develop a runtime estimation framework tailored to the quantum tree generator–based search for solving 0-1-knapsack problems. We evaluate its performance on large scale instances beyond the reach of general simulation. We compare it with state-of-the-art classical linear programming solvers in terms of processor cycle counts. Hybrid benchmarking provides a fair, systematic benchmarking methodology, ensuring reproducible and relevant comparisons not only between classical and quantum approaches, but also among different quantum algorithms.
This paper introduces a novel approach to the Quantum Approximate Optimization Algorithm (QAOA), specifically tailored to the knapsack problem. We combine the recently proposed quantum tree generator as an efficient state preparation circuit for all feasible solutions to the knapsack problem with the framework of Grover-mixer QAOA to form the first representative of Amplitude Amplification-mixer QAOA (AAMQAOA). On hard benchmark sets with up to 20 knapsack items, we demonstrate our method's improved performance over the current state-of-the-art Copula-QAOA. However, for larger instance sizes, both approaches fail to deliver better outcomes than greedily packing items in descending value-to-weight ratio, at least for the considered circuit depths. For sufficiently high circuit depths, however, we can prove that AAM-QAOA will eventually be able to sample the optimal solution.
Combinatorial optimization problems play a central role in computer science and real-world decision-making, as they aim to find the optimal solution from a large set of possibilities. Among these, the Knapsack Problem is a fundamental example with many practical applications. The Quantum Approximate Optimization Algorithm (QAOA) is a quantum-classical hybrid approach designed to address such problems. However, existing simulation approaches for QAOA applied to the Knapsack Problem do not scale well because they fail to exploit the problem's structural properties. To overcome these limitations, this study exploits the entangling characteristics of circuits in the Knapsack Problem and develops an optimized solver tailored to this setting. The key technical contributions include an entanglement-free circuit simulation method that reduces complexity from exponential to linear for non-entangling circuits, enabling 24,000x speedups for QAOA ansatz circuit simulations, and a two-phase parameter selection strategy that enhances the efficiency of finding optimal Knapsack solutions. Our solver extends the scalability of QAOA simulations from the previous limit of approximately 10 qubits to 100 qubits within a system memory of 256 GB. It achieves performance gains of over 100,000x for solving Knapsack Problems while maintaining high solution quality.
Quantum annealing (QA) has the potential to significantly improve solution quality and reduce time complexity in solving combinatorial optimization problems compared to classical optimization methods. However, due to the limited number of qubits and their connectivity, the QA hardware did not show such an advantage over classical methods in past benchmarking studies. Recent advancements in QA with more than 5000 qubits, enhanced qubit connectivity, and the hybrid architecture promise to realize the quantum advantage. Here, we use a quantum annealer with state-of-the-art techniques and benchmark its performance against classical solvers. To compare their performance, we solve over 50 optimization problem instances represented by large and dense Hamiltonian matrices using quantum and classical solvers. The results demonstrate that a state-of-the-art quantum solver has higher accuracy (~0.013%) and a significantly faster problem-solving time (~6561×) than the best classical solver. Our results highlight the advantages of leveraging QA over classical counterparts, particularly in hybrid configurations, for achieving high accuracy and substantially reduced problem solving time in large-scale real-world optimization problems.