Large integer factorization is a prominent research challenge, particularly in the context of quantum computing. This holds significant importance, especially in information security that relies on public key cryptosystems. The classical computation of prime factors for an integer has exponential time complexity. Quantum computing offers the potential for significantly faster computational processes compared to classical processors. In this paper, we propose a new quantum algorithm, Shallow Depth Factoring (SDF), to factor a biprime integer. SDF consists of three steps. First, it converts a factoring problem to an optimization problem without an objective function. Then, it uses a Quantum Feasibility Labeling (QFL) method to label every possible solution according to whether it is feasible or infeasible for the optimization problem. Finally, it employs the Variational Quantum Search (VQS) to find all feasible solutions. The SDF utilizes shallow-depth quantum circuits for efficient factorization, with the circuit depth scaling linearly as the integer to be factorized increases. Through minimizing the number of gates in the circuit, the algorithm enhances feasibility and reduces vulnerability to errors.
Large integer factorization is a prominent research challenge, particularly in the context of quantum computing. This holds significant importance, especially in information security that relies on public key cryptosystems. The classical computation of prime factors for an integer has exponential time complexity. Quantum computing offers the potential for significantly faster computational processes compared to classical processors. In this paper, we propose a new quantum algorithm, Shallow Depth Factoring (SDF), to factor a biprime integer. SDF consists of three steps. First, it converts a factoring problem to an optimization problem without an objective function. Then, it uses a Quantum Feasibility Labeling (QFL) method to label every possible solution according to whether it is feasible or infeasible for the optimization problem. Finally, it employs the Variational Quantum Search (VQS) to find all feasible solutions. The SDF utilizes shallow-depth quantum circuits for efficient factorization, with the circuit depth scaling linearly as the integer to be factorized increases. Through minimizing the number of gates in the circuit, the algorithm enhances feasibility and reduces vulnerability to errors.
Abstract With the advent of powerful quantum computers, the quest for more efficient quantum algorithms becomes crucial in attaining quantum supremacy over classical counterparts in the noisy intermediate-scale quantum era. While Grover's search algorithm and its generalization, quantum amplitude amplification, offer quadratic speedup in solving various important scientific problems, their exponential time complexity limits scalability as the quantum circuit depths grow exponentially with the number of qubits. To overcome this challenge, we propose Variational Quantum Search (VQS), a novel algorithm based on variational quantum algorithms and parameterized quantum circuits. We show that a depth-10 Ansatz can amplify the total probability of k (k ≥ 1) good elements, out of 2n elements represented by n + 1 qubits, from k/2n to nearly 1, as verified for n up to 26, and that the maximum depth of quantum circuits in the VQS increases linearly with the number of qubits. Our experimental results have validated the efficacy of VQS and its exponential advantage over Grover's algorithm in circuit depth for up to 26 qubits. We demonstrate that a depth-56 circuit in VQS can replace a depth-270,989 circuit in Grover's algorithm. Envisioning its potential, VQS holds promise to accelerate solutions to critical problems.
Factoring large integers using a quantum computer is an outstanding research problem that can illustrate true quantum advantage over classical computers. Exponential time order is required in order to find the prime factors of an integer by means of classical computation. However, the order can be drastically reduced by converting the factorization problem to an optimization one and solving it using a quantum computer. Recent works involving both theoretical and experimental approaches use Shor's algorithm, adiabatic quantum computation and quantum annealing principles to factorize integers. However, our work makes use of the generalized Grover's algorithm as proposed by Liu, with an optimal version of classical algorithm/analytic algebra. We utilize the phase-matching property of the above algorithm for only amplitude amplification purposes to avoid an inherent phase factor that prevents perfect implementation of the algorithm. Here we experimentally demonstrate the factorization of two bi-primes, 4088459 and 966887 using IBM's 5- and 16-qubit quantum processors, hence making those the largest numbers that have been factorized on a quantum device. Using the above 5-qubit processor, we also realize the factorization of a tri-prime integer 175, which had not been achieved to date. We observe good agreement between experimental and theoretical results with high fidelities. The difficulty of the factorization experiments has been analyzed and it has been concluded that the solution to this problem depends on the level of simplification chosen, not the size of the number factored. In principle, our results can be extended to factorize any multi-prime integer with minimum quantum resources.
Abstract Many important science and engineering problems can be converted into NP-complete problems which are of significant importance in computer science and mathematics. Currently, neither existing classical nor quantum algorithms can solve these problems in polynomial time. To address this difficulty, this paper proposes a quantum feasibility labeling (QFL) algorithm to label all possible solutions to the vertex coloring problem, which is a well-known NP-complete problem. The QFL algorithm converts the vertex coloring problem into the problem of searching an unstructured database where good and bad elements are labeled. The recently proposed variational quantum search (VQS) algorithm was demonstrated to achieve an exponential speedup, in circuit depth, up to 26 qubits in finding good element(s) from an unstructured database. Using the labels and the associated possible solutions as input, the VQS can find all feasible solutions to the vertex coloring problem. The number of qubits and the circuit depth required by the QFL each is a polynomial function of the number of vertices, the number of edges, and the number of colors of a vertex coloring problem. We have implemented the QFL on an IBM Qiskit simulator to solve a 4-colorable 4-vertex 3-edge coloring problem.
We present a compact quantum circuit for factoring a large class of integers, including some whose classical hardness is expected to be equivalent to RSA (but not including RSA integers themselves). Most notably, we factor n-bit integers of the form P2Q with logQ=Θ(na) for a∈(2/3,1) in space and depth sublinear in n (specifically, O~(logQ)) using O~(n) quantum gates; for these integers, no known classical algorithms exploit the relatively small size of Q to run asymptotically faster than general-purpose factoring algorithms. To our knowledge, this is the first polynomial-time circuit to achieve sublinear qubit count for a classically-hard factoring problem. Our circuit builds on the quantum algorithm for squarefree decomposition discovered by Li, Peng, Du, and Suter (Nature Scientific Reports 2012), which relies on computing the Jacobi symbol in quantum superposition. The technical core of our contribution is a new space-efficient quantum algorithm to compute the Jacobi symbol of A mod B, in the regime where B is classical and much larger than A. Our circuit for computing the Jacobi symbol generalizes to related problems such as computing the greatest common divisor and modular inverses, and thus could be of independent interest.
Abstract With powerful quantum computers already built1,2, we need more efficient quantum algorithms to achieve quantum supremacy over classical computers in the noisy intermediate-scale quantum (NISQ) era2,3. Grover's search algorithm4,5 and its generalization, quantum amplitude amplification6, provide quadratic speedup in solving many important scientific problems7. However, they still have exponential time complexity as the depths of their quantum circuits increase exponentially with the number of qubits6,7. To address this problem, we propose a new algorithm, Variational Quantum Search (VQS), which is based on the celebrated variational quantum algorithms8,9 and includes a parameterized quantum circuit, known as Ansatz. We show that a depth-10 Ansatz can amplify the total probability of k (k≥1) good elements, out of 2n elements represented by n+1 qubits, from k/2n to nearly 1, as verified for n up to 26, and that the maximum depth of quantum circuits in the VQS increases linearly with the number of qubits. We demonstrate that a depth-56 circuit in VQS can replace a depth-270,989 circuit in Grover's algorithm, and thus VQS is more suitable for NISQ computers. We envisage our VQS could exponentially speed up the solutions to many important problems, including the NP-complete problems, which is widely considered impossible.
The discrete logarithm problem is a significant challenge in number theory,and due to the difficulty of solving it,classical computers lack efficient algorithms for this task.As a result,the discrete logarithm problem is widely used in public key cryptosystems,and if it were cracked,it would pose a direct threat to the security of these systems.However,with the advent of quantum computing,researchers have begun exploring quantum computers as a potential solution for the discrete logarithm problem.Currently,most quantum algorithms for solving the discrete logarithm problem are based on Shor’s algorithm.However,due to Shor’s inherent limitations,these algorithms often face issues such as large quantum circuit depth,high qubit usage and complex post-processing steps.This makes it difficult for Shor’s algorithm to be implemented on NISQ computers.To address these issues,this paper proposes a novel approach by introducing a variational quantum algorithm for solving the discrete logarithm pro-blem.This algorithm leverages the parallelism of quantum computing to compute the modular exponentiation of parameterized quantum states.It also designs a marked solution circuit that maps valid solutions of the discrete logarithm problem onto auxiliary qubits.Then,a classical optimizer is used to iteratively adjust the parameters within the parameterized quantum circuit,continuously reducing the value of a designed loss function.Finally,the optimized parameters from the classical optimizer are fed into the measurement circuit,where the solution to the discrete logarithm problem can be obtained with high probability.Compared to Shor’s algorithm,the proposed method significantly reduces the required number of qubits and nearly halves the quantum circuit depth.Furthermore,this paper provides a detailed design of the quantum circuit and verifies the correctness of the proposed algorithm using the Qiskit package in Python.