Quantum Annealing for Prime Factorization (1804.02733v2)

Abstract: We have developed a framework to convert an arbitrary integer factorization problem to an executable Ising model by first writing it as an optimization function and then transforming the k-bit coupling ($k\geq 3$) terms to quadratic terms using ancillary variables. The method is efficient and uses $\mathcal{O}(\text{log}^2(N))$ binary variables (qubits) for finding the factors of integer $N$. The method was tested using the D-Wave 2000Q for finding an embedding and determining the prime factors for a given composite number. As examples, we present quantum annealing results for factoring 15, 143, 59989, and 376289 using 4, 12, 59, and 94 logical qubits respectively. The method is general and could be used to factor larger numbers

Quantum Annealing for Prime Factorization

The paper "Quantum Annealing for Prime Factorization" presents a method for converting the integer factorization problem into an Ising model that can be solved using quantum annealing. It outlines a framework designed to efficiently factorize integers by leveraging quantum processing. The prime factorization problem, vital to cryptographic systems like RSA, traditionally requires exponential time operations when using classical algorithms. The proposed method claims a complexity of $\mathcal{O}(\text{log}^2(N))$ binary variables for factoring an integer $N$. This is a significant reduction compared to classical approaches.

Methodology

The proposed method involves converting the factorization problem into an optimization problem and mapping this to an Ising Hamiltonian. This allows the integer factorization problem to be visualized as determining the ground state of the corresponding Hamiltonian. Utilizing ancillary variables, the model transforms k-bit coupling terms to quadratic terms. This results in more efficient use of quantum resources by utilizing a polynomial number of ancillary variables.

Quantum annealing, the technique employed for optimization, progresses the system from an initial Hamiltonian with a known ground state to the complex Hamiltonian encoding the solution. The ground state is identified in which the integer factors of the Hamiltonian correspond to the factors of the original integer $N$. This is akin to an adiabatic computation model.

Results

The method was tested using a D-Wave 2000Q quantum annealer. The authors demonstrated the ability to factorize integers such as 15, 143, 59989, and 376289, requiring 4, 12, 59, and 94 logical qubits respectively. The experiment established that the quantum annealing approach could represent the factorization problem within the hardware's constraints, including limitations related to qubit connectivity and control precision.

The encoding used for D-Wave’s chimera hardware graph maintained the energy minimization goal, emphasizing practical constraints like minor embedding and parameter settings. The results reflected that as the number of qubits and their degree of connectivity increase, larger numbers could be factored. The method also suggests a path toward the factorization of large semi-prime numbers utilized in cryptographic secure keys, contingent on the advancement of quantum annealing technology.

Implications and Future Work

The practical implication of this work lies in its potential to change the landscape of cryptography. By efficiently discovering integer factors of large semi-primes, this could compromise traditional cryptosystems like RSA if quantum annealers with sufficient qubits become widely available. Theoretically, the proposed algorithm offers insight into reducing qubit requirements in quantum computation.

An outstanding challenge remains the determination of the spectral gap's complexity in quantum annealing, as it relates to execution efficiency and scalability. The undecidability of the spectral gap problem poses questions on long-term practical uses. Further developments in quantum hardware and embedding strategies could enhance the applicability of this method.

Ultimately, this paper paves the way for further exploration into quantum-based factorization solutions, presenting a potential quantum bridge from theoretical concepts into practical, scalable realities as quantum technologies mature.