Reducing multi-qubit interactions in adiabatic quantum computation. Part 1: The “deduc-reduc” method and its application to quantum factorization of numbers Richard Tanburn 1, ∗ 1 Mathematical Institute, Oxford University, OX2 6GG, Oxford, UK. Emile Okada 2, † 2 Department of Mathematics, Cambridge University, CB2 3AP, Cambridge, UK. Nikesh S. Dattani 3,4, ‡ 3 School of Materials Science and Engineering, Nanyang Technological University, 639798, Singapore, and 4 Fukui Institute for Fundamental Chemistry, 606-8103, Kyoto, Japan Adiabatic quantum computing has recently been used to factor 56153 [Dattani & Bryans, arXiv:1411.6758] at room temperature, which is orders of magnitude larger than any number at- tempted yet using Shor’s algorithm (circuit-based quantum computation). However, this number is still vastly smaller than RSA-768 which is the largest RSA number factored thus far on a classi- cal computer. We address a major issue arising in the scaling of adiabatic quantum factorization to much larger numbers. Namely, the existence of many 4-qubit, 3-qubit and 2-qubit interactions in the Hamiltonians. We showcase our method on various examples, one of which shows that we can remove 94% of the 4-qubit interactions and 83% of the 3-qubit interactions in the factorization of a 25- digit number with almost no effort, without adding any auxiliary qubits. Our method is not limited to quantum factoring. Its importance extends to the wider field of discrete optimization. Any CSP (constraint-satisfiability problem), psuedo-boolean optimization problem, or QUBO (quadratic un- constrained Boolean optimization) problem can in principle benefit from the “deduction-reduction” method which we introduce in this paper. We provide an open source code which takes in a Hamilto- nian (or a discrete discrete function which needs to be optimized), and returns a Hamiltonian that has the same unique ground state(s), no new auxiliary variables, and as few multi-qubit (multi-variable) terms as possible with deduc-reduc. PACS numbers: 02.10.Xm, 02.10.Ox, 02.10.De, 02.70.Wz, 03.65.Aa, 03.65.Fd, 03.67.Ac, 03.67.Lx I. INTRODUCTION The quantum algorithm which has generated the most enthusiasm about quantum computing, is Shor’s algorithm for factoring integers [1, 2]. However, de- spite being celebrated for more than 20 years, this algo- rithm has still never been successfully implemented for determining the factors of any integers without using knowledge of the answer to the problem [3, 4]. By using knowledge of the answer to the factorization problem, one can choose a base such that Shor’s algorithm can be implemented with fewer qubits, and by doing this, the algorithm has successfully been implemented for factor- ing 15 [5–9] and 21 [10]. However, at least 8 qubits are needed for genuinely factoring 15 with Shor’s algorithm [3], and the largest number of qubits ever successfully used in the algorithm was 7 [5]. Adiabatic quantum computing (AQC) has succeeded in factoring much larger numbers with far fewer qubits, without any assumptions about the answer to the fac- torization problem [4]. The largest number found so far that has been factored by the room-temperature AQC * richard.tanburn@hertford.ox.ac.uk † eto25@cam.ac.uk ‡ nike.dattani@gmail.com discrete minimization experiment of [11] is 56153 and it only needed 4 qubits [4]. Furthermore, quantum an- nealing can be used on AQC algorithms with up to 2048 qubits, which is enough to factor the 100-digit number RSA-100. Furthermore, quantum annealers have doubled in number of qubits every year between the 4-qubit Ca- lypso machine in 2005 and the 2048-qubit Washington machine in 2015 (a phenomenon analogous to Moore’s law and known as “Rose’s Law”). RSA-220, which is the smallest RSA number that has not yet been factored by any computer (whether quantum or classical), would need only ≈ 5000 qubits to factor successfully with the AQC algorithm of [4], which would be well within the 8192-qubit capacity of a 2017 quantum annealer if Rose’s law continues to hold as it has done for the last 10 years. However, there is a major obstacle holding back such quantum annealers with colossal numbers of qubits, from implementing the AQC factoring algorithm. All such quantum annealing devices reported to date can only implement AQC algorithms which have at most 2-qubit interactions in their Hamiltonian. The algo- rithm which can factor RSA-220 with ≈ 5000 qubits has a Hamiltonian with many 4-qubit and 3-qubit in- teractions. In 2007 Schaller and Schutzhold devised an alternate AQC algorithm for factoring integers, which only has up to 2-qubit interactions in the Hamiltonian,