Quantum Annealing for Prime Factorization

Jiang, Shuxian; Britt, Keith A.; McCaskey, Alexander; Humble, Travis S.; Kais, Sabre

doi:10.1038/s41598-018-36058-z

Cited by 150 publications

(132 citation statements)

References 31 publications

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“…We are grateful for discussions with Peter Hannaford, Adolfo del Campo and Richard Warren, who also brought the work [3] to our attention after the completion of this paper. The publication fee for this work is covered by Swinburne University.…”

Section: Discussionmentioning

confidence: 98%

“…In this paper we also consider the factorisation problem in the realm of quantum computation but with adiabatic processes, in complementary addition to the computation with quantum circuits. Also recently, the authors of [3] have considered the problem with quantum annealing. We first reformulate in the next section the factorisation into two integer factors as an optimisation of some corresponding Diophantine polynomials over the integer domain.…”

Section: Introductionmentioning

confidence: 99%

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A factorisation algorithm in Adiabatic Quantum Computation

Kieu

2019

J. Phys. Commun.

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The problem of factorising positive integer N into two integer factors x and y is first reformulated as an optimisation problem over the positive integer domain of either of the Diophantine polynomials Q x y N N xy x x y ,, of each of which the optimal solution is unique with x N y   , and x=1 if and only if N is prime. An algorithm in the context of Adiabatic Quantum Computation is then proposed for the general factorisation problem.Factoring an integer into its prime constituents has attracted much interest since the advance of the RSA publicprivate key encryption [1]. It is suspected that factorisation is NP-intermediate, that is, in the NP class but may be not quite NP-complete. While there does not yet exist any polynomial-time algorithm for the problem on a classical/Turing computer, the discovery of Shor's quantum algorithm with quantum circuits [2] has been one of the main motivations for research into quantum computation and building of quantum computers.In this paper we also consider the factorisation problem in the realm of quantum computation but with adiabatic processes, in complementary addition to the computation with quantum circuits. Also recently, the authors of [3] have considered the problem with quantum annealing. We first reformulate in the next section the factorisation into two integer factors as an optimisation of some corresponding Diophantine polynomials over the integer domain. The optimisation could also be repeatedly applied to any integer having more than two prime factors. Based on this reformulation, we then present an algorithm in the context of AQC (Adiabatic Quantum Computation) for the general factorisation problem. Following that are some numerical illustrations of the algorithm and discussion on the lower bound of the computing time with the help of an energy-time uncertainty relation. The paper is then concluded with some remarks. Factorisation as an optimisation problemWe first consider the problem of factorising a natural integer N into two integer factors x and y. We propose that this problem can be reformulated as an optimisation problem over the integer domain of the following Diophantine polynomial

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Section: Discussionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

A factorisation algorithm in Adiabatic Quantum Computation

Kieu

2019

J. Phys. Commun.

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“…This could allow all-to-all connectivity beyond nearest neighbor coupling without requiring any special encoding [32,33]. Moreover, it should allow the implementation of arbitrary k-body interactions that are usually avoided by introducing ancillary bits to map them into 2-body interactions [34,35].…”

Section: Scopementioning

confidence: 99%

Scalable Emulation of Sign-Problem–Free Hamiltonians with Room-Temperature p -bits

Çamsarı¹,

Chowdhury²,

Datta³

2019

Phys. Rev. Applied

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The growing field of quantum computing is based on the concept of a q-bit which is a delicate superposition of 0 and 1, requiring cryogenic temperatures for its physical realization along with challenging coherent coupling techniques for entangling them. By contrast, a probabilistic bit or a p-bit is a robust classical entity that fluctuates between 0 and 1, and can be implemented at room temperature using present-day technology. Here, we show that a probabilistic coprocessor built out of room temperature p-bits can be used to accelerate simulations of a special class of quantum many-body systems that are sign-problem−free or "stoquastic", leveraging the well-known Suzuki-Trotter decomposition that maps a d-dimensional quantum many body Hamiltonian to a d+1-dimensional classical Hamiltonian. This mapping allows an efficient emulation of a quantum system by classical computers and is commonly used in software to perform Quantum Monte Carlo (QMC) algorithms. By contrast, we show that a compact, embedded MTJ-based coprocessor can serve as a highly efficient hardware-accelerator for such QMC algorithms providing several orders of magnitude improvement in speed compared to optimized CPU implementations. Using realistic device-level SPICE simulations we demonstrate that the correct quantum correlations can be obtained using a classical p-circuit built with existing technology and operating at room temperature. The proposed coprocessor can serve as a tool to study stoquastic quantum many-body systems, overcoming challenges associated with physical quantum annealers. arXiv:1810.07144v4 [quant-ph]

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“…Most of these efforts (e.g. [71,72,73,74,75,76,77,78,79]) have used global embedding, in which an entire Ising model is minor-embedded heuristically [41] or a fixed embedding is used [39,40]. However Su et al [45] used a general place-and-route approach, while Trummer et al [80], Chancellor et al [68], Zaribafiyan et al [40], and Andriyash et al [81] used a placement approach optimized for the specific constraints at hand.…”

Section: Related Workmentioning

confidence: 99%

Solving SAT and MaxSAT with a Quantum Annealer: Foundations and a Preliminary Report

Bian

Chudak

Macready

et al. 2017

Lecture Notes in Computer Science

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Quantum annealers (QAs) are specialized quantum computers that minimize objective functions over discrete variables by physically exploiting quantum effects. Current QA platforms allow for the optimization of quadratic objectives defined over binary variables (qubits), also known as Ising problems. In the last decade, QA systems as implemented by D-Wave have scaled with Moore-like growth. Current architectures provide 2048 sparsely-connected qubits, and continued exponential growth is anticipated, together with increased connectivity.We explore the feasibility of such architectures for solving SAT and MaxSAT problems as QA systems scale. We develop techniques for effectively encoding SAT -and, with some limitations, MaxSAT-into Ising problems compatible with sparse QA architectures. We provide the theoretical foundations for this mapping, and present encoding techniques that combine offline Satisfiability and Optimization Modulo Theories with on-the-fly placement and routing. Preliminary empirical tests on a current generation 2048-qubit D-Wave system support the feasibility of the approach for certain SAT and MaxSAT problems. 1 Superposition is perhaps the best-known and most surprising aspect of quantum physics (e.g. the famous Scrödinger's cat which is both dead and alive prior to observation).

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Quantum Annealing for Prime Factorization

Cited by 150 publications

References 31 publications

A factorisation algorithm in Adiabatic Quantum Computation

A factorisation algorithm in Adiabatic Quantum Computation

Scalable Emulation of Sign-Problem–Free Hamiltonians with Room-Temperature p -bits

Solving SAT and MaxSAT with a Quantum Annealer: Foundations and a Preliminary Report

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