This paper addresses the problem of finding the densest
k-vertex subgraph in an arbitrary graph. This problem is NP-hard and has important applications in social network analysis, fraud detection, recommendation systems, and bioinformatics. We propose two quantum approaches to solve this problem: reduction to Quadratic Unconstrained Binary Optimization (QUBO) and using Grover's quantum search algorithm. For the latter approach, we present an explicit gate-based oracle circuit utilizing Dicke states and Quantum Fourier Transform for edge counting. Numerical simulations demonstrate a quadratic speedup over classical Brute-force search.
Quadratic Unconstrained Binary Optimization (QUBO) is a combinatorial optimization to find an optimal binary solution vector that minimizes the energy value defined by a quadratic formula of binary variables in the vector. The graph isomorphism problem is one of the applications of combinatorial optimization, and is not known to be solvable in polynomial time. The problem is also not known to be NP-complete. We propose four QUBO formulations for graph isomorphism problem instances. We also propose four QUBO formulations for induced subgraph isomorphism problem instances and two QUBO formulations for subgraph isomorphism problem instances, which are NP-complete problems. We solve QUBO instances defined by our QUBO formulations, using known QUBO solvers: Gurobi optimizer, Fixstars Amplify AE and OpenJij with SA.
We study a variant of the subgraph isomorphism problem that is of high interest to the quantum computing community. Our results give an algorithm to perform pattern matching in quantum circuits for many patterns simultaneously, independently of the number of patterns. After a pre-computation step in which the patterns are compiled into a decision tree, the running time is linear in the size of the input quantum circuit. More generally, we consider connected port graphs, in which every edge
e incident to
v has a label
Lv(e) unique in
v. Jiang and Bunke showed that the subgraph isomorphism problem
H⊆G for such graphs can be solved in time
O(∣V(G)∣⋅∣V(H)∣). We show that if in addition the graphs are directed acyclic, then the subgraph isomorphism problem can be solved for an unbounded number of patterns simultaneously. We enumerate all
m pattern matches in time
O(P)P+3/2⋅∣V(G)∣+O(m), where
P is the number of vertices of the largest pattern. In the case of quantum circuits, we can express the bound obtained in terms of the maximum number of qubits
N and depth
δ of the patterns :
O(N)N+1/2⋅δlogδ⋅∣V(G)∣+O(m).
Quantum Approximate Optimization Algorithm (QAOA) and Quantum Annealing are prominent approaches for solving combinatorial optimization problems, such as those formulated as Quadratic Unconstrained Binary Optimization (QUBO). These algorithms aim to minimize the objective function
xTQx, where
Q is a QUBO matrix. However, the number of two-qubit CNOT gates in QAOA circuits and the complexity of problem embeddings in Quantum Annealing scale linearly with the number of non-zero couplings in
Q, contributing to significant computational and error-related challenges. To address this, we introduce the concept of \textit{semi-symmetries} in QUBO matrices and propose an algorithm for identifying and factoring these symmetries into ancilla qubits. \textit{Semi-symmetries} frequently arise in optimization problems such as \textit{Maximum Clique}, \textit{Hamilton Cycles}, \textit{Graph Coloring}, and \textit{Graph Isomorphism}. We theoretically demonstrate that the modified QUBO matrix
Qmod retains the same energy spectrum as the original
Q. Experimental evaluations on the aforementioned problems show that our algorithm reduces the number of couplings and QAOA circuit depth by up to
45%. For Quantum Annealing, these reductions also lead to sparser problem embeddings, shorter qubit chains and better performance. This work highlights the utility of exploiting QUBO matrix structure to optimize quantum algorithms, advancing their scalability and practical applicability to real-world combinatorial problems.
Quantum physics has the potential to revolutionize computer science by building what is called a quantum computer. Quantum computers can perform tasks that are impossible with Turing-machine based classical computers (e.g., generating random sequences of bits) and can outperform classical computers (to different degrees) in a variety of classical computation problems. Several different models of quantum computing exist. Based on the Adiabatic Theorem in physics, Adiabatic Quantum Computing is a novel computing framework that is drastically different from the more traditional quantum logic gate model. Traditionally, quantum computing has only been tested on devices with a very small number of qubits, severely limiting their uses in practice. In recent years however, advances in engineering and physics have allowed the construction for (relatively) large machines called quantum annealers that do what is known as a specific type of adiabatic quantum computation called quantum annealing. Solving classical computation problems with the quantum annealing framework is not an easy task and there are many obstacles to overcome if we want to achieve any kind of speedups over classical algorithms. In this thesis, we present quantum annealing solutions to several classical problems including the Hamiltonian Cycle Problem, Mixed Dominating Set Problem, Graph Isomorphism Problem, Densest k-Subgraph Problem and the Maximum Weighted Independent Set Problem. Experimental results of these quantum solutions on DWave quantum annealers are also reported. We also look at some of the bottlenecks in the quantum annealing framework and propose several novel approach to mitigate their effects on the efficiency of the quantum solutions.