Factoring integers with sublinear resources on a superconducting quantum processor (2212.12372v1)

Abstract: Shor's algorithm has seriously challenged information security based on public key cryptosystems. However, to break the widely used RSA-2048 scheme, one needs millions of physical qubits, which is far beyond current technical capabilities. Here, we report a universal quantum algorithm for integer factorization by combining the classical lattice reduction with a quantum approximate optimization algorithm (QAOA). The number of qubits required is O(logN/loglog N), which is sublinear in the bit length of the integer $N$, making it the most qubit-saving factorization algorithm to date. We demonstrate the algorithm experimentally by factoring integers up to 48 bits with 10 superconducting qubits, the largest integer factored on a quantum device. We estimate that a quantum circuit with 372 physical qubits and a depth of thousands is necessary to challenge RSA-2048 using our algorithm. Our study shows great promise in expediting the application of current noisy quantum computers, and paves the way to factor large integers of realistic cryptographic significance.

• The paper introduces a novel hybrid approach combining classical Schnorr’s lattice algorithm with QAOA to achieve sublinear qubit scaling for integer factorization.
• It demonstrates experimental success by factoring up to 48-bit integers with only 10 superconducting qubits, achieving high quantum state fidelity.
• Resource estimates indicate that challenging RSA-2048 requires 372 physical qubits and moderate circuit depth, paving the way for practical cryptanalysis.

Insights into Factoring Integers with Sublinear Resources on a Superconducting Quantum Processor

The paper "Factoring integers with sublinear resources on a superconducting quantum processor" presents a novel approach to integer factorization, an essential problem underpinning modern cryptographic systems like RSA-2048. This research addresses the limitations of current quantum capabilities by proposing a sublinear quantum algorithm for integer factorization involving the hybrid utilization of classical lattice reduction techniques and Quantum Approximate Optimization Algorithms (QAOA) on a superconducting quantum processor.

Summary of the Algorithm and Results

The core advancement lies in combining classical Schnorr's lattice-based algorithm with QAOA to optimize the integer factorization process. Through this methodology, the paper demonstrates an innovative quantum-classical hybrid that reduces the qubit requirements to $O(\text{log N/\text{loglog} N)$, which is sublinear concerning the bit length of the integer $N$. This method substantially diminishes qubit requirements compared to existing algorithms such as Shor's algorithm, making it highly efficient in qubit resource usage.

Experimentally, the team successfully factored integers up to 48 bits using just 10 superconducting qubits, notably the largest general integer factored on a quantum device to date. The implementation involved a superconducting quantum processor with a carefully designed QAOA circuit. The approach resulted in an impressive level of quantum state fidelity relative to theoretically derived distributions, indicating robust optimization performance even in the noisy intermediate-scale quantum (NISQ) era.

Quantum Resource Estimation

One of the key contributions of this work is the estimation of quantum resources necessary to challenge RSA-2048. The method requires 372 physical qubits and a circuit depth in the field of a few thousand, even in a simplified 1D-chain topology. This estimate underscores the pragmatism in advancing quantum computing capabilities towards realistically tackling RSA encryption standards, aligning with projections for near-future quantum device capabilities.

Theoretical and Practical Implications

From a theoretical standpoint, this research advances the understanding of variational quantum algorithms' capability to solve classically challenging problems by leveraging quantum optimization to effectively refine classical computational techniques. Practically, the hybrid model sets a trajectory for how current quantum systems can extend their applicability to meaningful cryptographic challenges well before fully fault-tolerant quantum computers become available.

Potential Future Directions

The paper implicitly introduces several avenues for further research. Enhancements in fault-tolerant operations and optimizing QAOA efficiency can further improve this method. Exploring additional lattice reduction algorithms that can benefit from quantum optimization may expand the applicability and efficiency of quantum-assisted factorizations. Moreover, with the continued evolution in quantum hardware fidelity and connectivity, reducing the circuit depth and investigating alternative quantum system architectures could refine the process even further.

Conclusion

This work presents a promising approach to efficiently factor large integers, crucial for advancing quantum cryptanalysis potential. Its sublinear-resource algorithm for quantum processors marks a significant step toward practically challenging established cryptographic protocols like RSA. Future developments building upon this hybrid classical-quantum approach may soon enable quantum computers to address real-world cryptographic problems, signaling a shift in how information security systems are conceptualized and developed.