The emergence of Quantum Computing has resulted in a slate of quantum algorithms that can solve a wide variety of algorithmic and computational problems faster than any classical computer. How we can apply these various quantum algorithms to practical problems in computing is still an ongoing area of research. In this work, we present a ''hybrid classical/quantum'' algorithm that uses quantum counting to solve a network verification problem involving the estimation of probabilities. We present a detailed cost analysis of our algorithm, showing that under certain assumptions it outperforms standard verification methods.
The emergence of Quantum Computing has resulted in a slate of quantum algorithms that can solve a wide variety of algorithmic and computational problems faster than any classical computer. How we can apply these various quantum algorithms to practical problems in computing is still an ongoing area of research. In this work, we present a ''hybrid classical/quantum'' algorithm that uses quantum counting to solve a network verification problem involving the estimation of probabilities. We present a detailed cost analysis of our algorithm, showing that under certain assumptions it outperforms standard verification methods.
Network verification, broadly defined as proving the correctness of certain properties resulting from a network's configuration, cannot be efficiently solved on classical hardware via brute force. Prior work has developed a variety of methods that scale by observing a structure in the search space and then evaluating classes induced by that structure. However, even these classification mechanisms have their limitations. In this paper, we consider a radically different approach: applying quantum computing to more efficiently solve network verification problems. We provide an overview of how to map variants of verification problems into unstructured search problems that can be solved via quantum computing with quadratic speedup, making the approach feasible in theory to problems that twice as big in the size of the input. Emerging quantum systems cannot yet tackle problems of practical interest, but rapid advances in hardware and algorithm development make now a great time to start thinking about their application. With this in mind, we explore the limits of scale of the problem for which quantum computing can solve network verification problems as unstructured search.
In recent years, quantum computers and algorithms have made significant progress indicating the prospective importance of quantum computing (QC). Especially combinatorial optimization has gained a lot of attention as an application field for near-term quantum computers, both by using gate-based QC via the Quantum Approximate Optimization Algorithm and by quantum annealing using the Ising model. However, demonstrating an advantage over classical methods in real-world applications remains an active area of research. In this work, we investigate the robustness verification of ReLU networks, which involves solving many-variable mixed-integer programs (MIPs), as a practical application. Classically, complete verification techniques struggle with large networks as the combinatorial space grows exponentially, implying that realistic networks are difficult to be verified by classical methods. To alleviate this issue, we propose to use QC for neural network verification and introduce a hybrid quantum procedure to compute provable certificates. By applying Benders decomposition, we split the MIP into a quadratic unconstrained binary optimization and a linear program which are solved by quantum and classical computers, respectively. We further improve existing hybrid methods based on the Benders decomposition by reducing the overall number of iterations and placing a limit on the maximum number of qubits required. We show that, in a simulated environment, our certificate is sound, and provides bounds on the minimum number of qubits necessary to approximate the problem. Finally, we evaluate our method within simulations and on quantum hardware.
In recent years, quantum computers and algorithms have made significant progress indicating the prospective importance of quantum computing (QC). Especially combinatorial optimization has gained a lot of attention as an application field for near-term quantum computers, both by using gate-based QC via the Quantum Approximate Optimization Algorithm and by quantum annealing using the Ising model. However, demonstrating an advantage over classical methods in real-world applications remains an active area of research. In this work, we investigate the robustness verification of ReLU networks, which involves solving a many-variable mixed-integer programs (MIPs), as a practical application. Classically, complete verification techniques struggle with large networks as the combinatorial space grows exponentially, implying that realistic networks are difficult to be verified by classical methods. To alleviate this issue, we propose to use QC for neural network verification and introduce a hybrid quantum procedure to compute provable certificates. By applying Benders decomposition, we split the MIP into a quadratic unconstrained binary optimization and a linear program which are solved by quantum and classical computers, respectively. We further improve existing hybrid methods based on the Benders decomposition by reducing the overall number of iterations and placing a limit on the maximum number of qubits required. We show that, in a simulated environment, our certificate is sound, and provide bounds on the minimum number of qubits necessary to approximate the problem. Finally, we evaluate our method within simulation and on quantum hardware.
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.
Building a network that is resilient to a component failure is vital. Our access to electricity and telecommunications or the internet of things all hinge on an uninterrupted service provided by a robust network. Calculating the network reliability R is ♯P-complete and intractable to calculate exactly for medium and large networks. Here, we present an explicit, circuit-level implementation of a quantum algorithm that computes R. Our algorithm requires O(EV/ε) gate operations and O(E) qubits, where V and E are the number of nodes and edges in the graph and ε is the uncertainty in the reliability estimation. This constitutes a significant polynomial speedup over the best classical approaches currently known. We further provide quantum gate counts, relevant for both pre-fault-tolerant and fault-tolerant regimes, sufficient to compute R.
Quantum algorithms run on quantum computers are qualitatively different from those that run on classical computers. Quantum computing algorithms can be used for several problems in graph theory. Most of the classical algorithms involve searching over some space for finding the shortest-paths problem between two points in a graph and a minimal weight spanning tree. We modified classical Dijkstra's algorithm and implement quantum search instead of classical search, of which it will lead to more efficient algorithm. Also we proposed the structure for non-classical algorithms and design the various phases of the probabilistic quantum-classical algorithm for classical and quantum parts. Finally, we represent the result of implementing and simulating Dijkstra's algorithm as the probabilistic quantum-classical algorithm.