Search algorithms based on quantum walks have emerged as a promising approach to solve computational problems across various domains, including combinatorial optimization, and cryptography. Stating a generic search problem in terms of a (quantum) search over a graph makes the efficiency of the algorithmic method depend on the structure of the graph itself. In this work, we propose a complete implementation of a quantum walk search on Johnson graphs, speeding up the solution of the subset-sum problem. We provide a detailed design of each sub-circuit, quantifying their cost in terms of gate number, depth, and width. We compare our solution against a Grover quantum search, showing a reduction of the T-count and T-depth for practically solvable problems. The proposed design provides a building block for the construction of efficient quantum search algorithms that can be modelled on Johnson graphs, filling the gap with the existing theoretical complexity analyses.
Search algorithms based on quantum walks have emerged as a promising approach to solve computational problems across various domains, including combinatorial optimization, and cryptography. Stating a generic search problem in terms of a (quantum) search over a graph makes the efficiency of the algorithmic method depend on the structure of the graph itself. In this work, we propose a complete implementation of a quantum walk search on Johnson graphs, speeding up the solution of the subset-sum problem. We provide a detailed design of each sub-circuit, quantifying their cost in terms of gate number, depth, and width. We compare our solution against a Grover quantum search, showing a reduction of the T-count and T-depth for practically solvable problems. The proposed design provides a building block for the construction of efficient quantum search algorithms that can be modelled on Johnson graphs, filling the gap with the existing theoretical complexity analyses.
Quantum walk-based search algorithms have demonstrated an asymptotic quadratic speedup compared to classical search methods. Formulating a generic search problem as a (quantum) search over a graph makes the efficiency of the algorithm to be closely dependent on the properties of the graph itself. In this work, we present a complete implementation of a quantum walk search procedure on Johnson graphs, speeding up the solution of the Subset Sum Problem, a well-known computational problem with applications in resource allocation, scheduling, and cryptanalysis. We provide a detailed design of each sub-circuit, quantifying their costs in terms of gate count, circuit depth, and qubit width, and exhibit all figures of merit as a function of the problem parameters. Our approach includes two distinct implementations: one that minimizes qubit usage and another focused on optimizing circuit depth. We compare our solutions against the unstructured Grover based quantum search algorithm, demonstrating a reduction of the cost on terms of T-count and T-depth, for practically solvable instances. The proposed design serves as a foundational building block for the development of efficient quantum search algorithms that can be modeled on Johnson graphs, bridging the gap with existing theoretical complexity analyses.
The 𝑘-clique problem, which involves identifying complete subgraphs of size 𝑘 within a graph, is a fundamental challenge in combinatorial optimization with applications in network analysis, bioinformatics, and cryptography.As the number of nodes 𝑛 grows, classical algorithms become computationally intractable, motivating the exploration of quantum computing to address these limitations.This work introduces two quantum algorithms for solving the 𝑘-clique problem: one based on Quantum Amplitude Amplification (QAA) and another leveraging Quantum Walks (QW) over the Johnson graph 𝐽 (𝑛, 𝑘).By exploiting the regular structure of the Johnson graph, the QW approach achieves the same quadratic speedup of the QAA approach over classical algorithms, while remaining applicable to arbitrary undirected, unweighted graphs.We provide detailed quantum circuit designs for both approaches, analyzing their performance in terms of qubit count, gate count, and circuit depth under the NOT-CNOT-Toffoli + arbitrary rotations and Clifford + T gate sets.Compared to state-of-the-art quantum algorithms, our methods achieve significant improvements in scaling, particularly for dense graphs, with speedups ranging from 2 5 to 2 20 .
We investigate the implementation of an oracle for the Subset Sum problem for quantum search using Grover's algorithm. Our work concerns reducing the number of qubits, gates, and multi-controlled gates required by the oracle. We describe the compilation of a Subset Sum instance into a quantum oracle, using a Python library we developed for Qiskit and have published in GitHub. We then present techniques to conserve qubits and gates along with experiments showing their effectiveness on random instances of Subset Sum. These techniques include moving from fixed to varying-width arithmetic, using partial sums of a set's integers to determine specific integer widths, and sorting the set to obtain provably the most efficient partial sums. We present a new method for computing bit-string comparisons that avoids arbitrarily large multiple-control gates, and we introduce a simple modification to the oracle that allows for approximate solutions to the Subset Sum problem via Grover search.
The field of quantum computing has emerged as a powerful tool for solving and optimizing combinatorial optimization problems. To solve many real-world problems with many variables and possible solutions for constraint satisfaction and optimization problems, the required number of qubits of scalable hardware for quantum computing is the bottleneck in the current generation of quantum computers. In this dissertation, we will demonstrate advanced, scalable building blocks for the quantum search algorithms that have been implemented in Grover's search algorithm and the quantum walk algorithm. The scalable building blocks are used to reduce the required number of qubits in the design. The proposed architecture effectively scales and optimizes the number of qubits needed to solve large problems with a limited number of qubits. Thus, scaling and optimizing the number of qubits that can be accommodated in quantum algorithm design directly reflect on performance. Also, accuracy is a key performance metric related to how accurately one can measure quantum states. The search space of quantum search algorithms is traditionally created by using the Hadamard operator to create superposition. However, creating superpositions for problems that do not need all superposition states decreases the accuracy of the measured states. We present an efficient quantum circuit design that the user has control over to create the subspace superposition states for the search space as needed. Using only the subspace states as superposition states of the search space will increase the rate of correct solutions. In this dissertation, we will present the implementation of practical problems for Grover's search algorithm and quantum walk algorithm in logic design, logic puzzles, and machine learning problems such as SAT, MAX-SAT, XOR-SAT, and like SAT problems in EDA, and mining frequent patterns for association rule mining.
The subset-sum problem, a foundational NP-hard problem in theoretical computer science, serves as a critical building block for cryptographic constructions. This work introduces novel classical and quantum heuristic algorithms for the random subset-sum problem at density d=1, where exactly one solution is expected. Classically, we propose the first algorithm based on a ternary tree representation structure, inspired by recent advances in lattice-based cryptanalysis. Through numerical optimization, our method achieves a time complexity of 𝒪˜20.2400n and space complexity of 𝒪˜20.2221n, improving upon the previous best classical heuristic result of 𝒪˜20.2830n. In the quantum setting, we develop a corresponding algorithm by integrating the classical ternary representation technique with a quantum walk search framework. The optimized quantum algorithm attains a time and space complexity of 𝒪˜20.1843n, surpassing the prior state-of-the-art quantum heuristic of 𝒪˜20.2182n. Furthermore, we apply our algorithms to information set decoding in code-based cryptography. For half-distance decoding, our classical algorithm improves the time complexity to 𝒪˜20.0453n, surpassing the previous best of 𝒪˜20.0465n. For full-distance decoding, we achieve a quantum complexity of 𝒪˜20.058326n, advancing beyond the prior best quantum result of 𝒪˜20.058696n. These findings demonstrate the broad applicability and efficiency of our ternary representation technique across both classical and quantum computational models.
The K-SAT problem is a fundamental challenge in computational theory, widely used in artificial intelligence, cryptography, and optimization. Classical algorithms struggle with exponential time complexity, making them inefficient for large-scale instances. In this research, we propose an efficient quantum computing approach based on Grover’s algorithm to enhance the search process for satisfiable solutions in K-SAT problems. Our implementation leverages quantum superposition and amplitude amplification to explore multiple possible solutions simultaneously, reducing the search complexity from O (2ⁿ) in classical methods to O(√2ⁿ) in quantum computing. We design a quantum oracle that encodes the K-SAT clauses and integrates it into Grover’s iterative search framework. The performance is evaluated through Qiskit simulations, demonstrating a significant improvement in search efficiency compared to classical brute-force techniques. The results highlight the potential of quantum algorithms in solving complex combinatorial problems with enhanced speed and accuracy. This study contributes to the development of quantum-accelerated optimization methods, paving the way for real-world applications in machine learning, cryptanalysis, and large-scale data processing.