Quantum Factoring Algorithm Achieves Space Reduction To, Enabling Practical Implementation and Verification of -Factorization

The challenge of integer factorization, crucial to the security of widely used RSA encryption, receives renewed attention from Wentao Yang of Tsinghua University, Bao Yan from the State Key Laboratory of Mathematical Engineering and Advanced Computing, Muxi Zheng of Tsinghua University and the National University of Singapore, and colleagues. Recent theoretical work by Regev offers a potentially more efficient approach, but demands considerable computational space, hindering practical application. This team addresses this limitation by developing a novel qubit reuse method, inspired by reversible computing, that dramatically reduces the space complexity of Regev’s algorithm, achieving a significant improvement over existing approaches. Through both detailed simulations and a successful experimental implementation on a superconducting computer, they demonstrate the feasibility of Regev-style factoring and provide a valuable blueprint for future advances in quantum algorithms and hardware.

Shor’s Algorithm Factoring of Thirty-Five

This research details a comprehensive implementation of Shor’s algorithm for factoring the number 35, with a strong emphasis on optimizing the quantum circuit for current and near-term quantum hardware. The team focused on reducing the number of qubits and quantum gates required, making the algorithm more feasible to run on limited resources. Key optimizations include partial uncomputation and recompilation, which simplify the circuit by leveraging specific states encountered during computation, alongside optimized circuits for multiplication and squaring. State-specific simplification further reduces circuit size by focusing on the states actually used during the computation. The research highlights the significant reduction in resource requirements achieved through these techniques, paving the way for more practical quantum factoring algorithms.

Reversible Computation Optimizes Factorization Space Complexity

Researchers have addressed the challenge of integer factorization by optimizing Regev’s high-dimensional generalization of Shor’s algorithm. The team identified repeated squaring as a primary source of increased space complexity and engineered a novel qubit reuse method based on intermediate uncomputation, inspired by reversible computing principles. This approach reduced space complexity from O(n3/2) to O(n5/4), significantly improving resource efficiency, and further refinement achieved a space complexity of O(n log n). The team rigorously proved that O(n log n) represents a space lower bound within this framework, demonstrating the optimality of their approach. Numerical simulations analyzed performance characteristics, and the method was validated by designing and compiling quantum circuits to factor 35. A simplified experimental circuit was successfully executed on a superconducting quantum computer, retrieving the factors of 35 and verifying the practical feasibility of the optimized factoring method.

Optimal Quantum Factoring with Logarithmic Space Complexity

Scientists have achieved a significant breakthrough in quantum factoring by optimizing Regev’s algorithm, a high-dimensional generalization of Shor’s algorithm. The research team focused on reducing the substantial space complexity inherent in the original construction. Initial strategies lowered space complexity from O(n3/2) to O(n5/4), and further refinements, inspired by reversible computing principles, ultimately achieved a space complexity of O(n log n). Crucially, the team proved that O(n log n) represents a space lower bound within this framework, demonstrating the optimality of their approach.

The work involved designing and compiling quantum circuits to factor the number 35, verifying the effectiveness of the method through noisy simulations. Results indicate that the optimized algorithm reduces qubit consumption and improves circuit performance in the presence of noise. A simplified circuit was constructed and executed on a superconducting quantum computer, successfully retrieving the factors using lattice-based post-processing, demonstrating the feasibility of Regev’s approach on current quantum hardware.

Logarithmic Space Factoring Algorithm Achieved

This research presents significant advances in the efficiency of factoring algorithms, specifically addressing the computational demands of Regev’s approach. Scientists developed novel qubit reuse methods, inspired by reversible computing, that substantially reduce the space complexity required for these calculations. Initial strategies lowered register consumption while maintaining comparable computational time, and further refinement, employing binary and k-ary recursion, ultimately achieved a space complexity of O(log m), representing a fundamental lower bound within this computational model. These improvements were demonstrated through simulations and, crucially, by successfully compiling and executing a simplified experimental circuit on a superconducting computer. This implementation verified the effectiveness of the new methods in retrieving factors for a test case, demonstrating a pathway towards practical application of Regev-style factoring. The team acknowledges that while space complexity is minimized, time complexity increases with certain strategies, suggesting future work may focus on optimizing the trade-offs between these two factors and exploring variable-arity recursion to achieve near-linear time complexity alongside sublinear space usage.