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.
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.
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.
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.
Numerous conceptually important quantum algorithms rely on a blackbox device known as an oracle, which is typically difficult to construct without knowing the answer to the problem that the algorithm is intended to solve.A notable example is Grover's search algorithm.Here we propose a Grover search for solutions to a class of NP-complete decision problems known as subset sum problems, including the special case of number partitioning.Each problem instance is encoded in the couplings of a set of qubits to a central spin or boson, which enables a realization of the oracle without knowledge of the solution.The algorithm provides a quantum speedup across a known phase transition in the computational complexity of the partition problem, and we identify signatures of the phase transition in the simulated performance.Whereas the naive implementation of our algorithm requires a spectral resolution that scales exponentially with system size for NP-complete problems, we also present a recursive algorithm that enables scalability.We propose and analyze implementation schemes with cold atoms, including Rydberg-atom and cavity-QED platforms.
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.
Several prominent quantum computing algorithms—including Grover’s search algorithm and Shor’s algorithm for finding the prime factorization of an integer—employ subcircuits termed ‘oracles’ that embed a specific instance of a mathematical function into a corresponding bijective function that is then realized as a quantum circuit representation. Designing oracles, and particularly, designing them to be optimized for a particular use case, can be a non-trivial task. For example, the challenge of implementing quantum circuits in the current era of NISQ-based quantum computers generally dictates that they should be designed with a minimal number of qubits, as larger qubit counts increase the likelihood that computations will fail due to one or more of the qubits decohering. However, some quantum circuits require that function domain values be preserved, which can preclude using the minimal number of qubits in the oracle circuit. Thus, quantum oracles must be designed with a particular application in mind. In this work, we present two methods for automatic quantum oracle synthesis. One of these methods uses a minimal number of qubits, while the other preserves the function domain values while also minimizing the overall required number of qubits. For each method, we describe known quantum circuit use cases, and illustrate implementation using an automated quantum compilation and optimization tool to synthesize oracles for a set of benchmark functions; we can then compare the methods with metrics including required qubit count and quantum circuit complexity.
The Subset Sum Problem is a member of the NP-complete class, so no known polynomial time algorithm exists for it. Although there are polynomial time approximations and heuristics, these are not always acceptable, yet exact-solution algorithms are unfeasible for large input. Quantum computation offers new insights for not only the Subset Sum Problem but also the entire NP-complete class; most notably, Grover's quantum algorithm for an unstructured database search can be tailored to identify solutions to problems within mathematics and computer science. This paper discusses the physical and conceptual feasibility of quantum computation and demonstrates the utility of quantum search by analyzing the time complexities of the classical dynamic programming algorithm and Grover's algorithm in solving the Subset Sum Problem, evincing the implications this has on the NP-complete class in general.