Prime factorization using quantum annealing and computational algebraic geometry

We investigate prime factorization from two perspectives: quantum annealing and computational algebraic geometry, specifically Gr\"obner bases. We present a novel scalable algorithm which combines the two approaches and leads to the factorization of all bi-primes up to just over $200 \, 000$, the largest number factored to date using a quantum processor.

In the above table, M stands for the bi-prime, n is the number of variables in the QUBO problem, emTry is the number of block trials of the sapiFindEmbedding routine, idC is the total number of identified couplers, prC is the total number of problem couplers, #qubits is the total number of (physical) qubits, jRatio is the ratio , and rTime is the chip run time in seconds.
For Gröbner bases computation, we have used J. C. Faugère FGb C library available at http://www-polsys.lip6.fr/ jcf/. The calculations were performed on 2.5 GHz Intel Core i5 processor with 16 GB RAM. This hardware constrained the cutoff to be around 8 and around 4 for (s p + s q + 1) − 2ndcutoff. The percentage R% of the obtained reduction is reported in the main text (Cell Algorithm and Column Algorithm tables) and is around 13% for the Cell Algorithm and 35% for the Column Algorithm. We could not achieve more reduction (for the bi-prime numbers reported) using this hardware.

Continuous optimization problems for the requirements (ii-iii)
In Results, we describe how a positive quadratic polynomial H + i j can be extracted using Gröbner bases. Here we provide the details of the calculation.
The second requirement (ii) is equivalent to each of the following linear polynomials being greater than zero: For the third requirement (iii), the first choice for the objective function f : The solution is a

Basic description of the quantum annealing processor
Here we introduce the quantum annealing concept that ultimately solves a general Ising (quadratic unconstrained binary optimization, or "QUBO") problem, then talk about the important topic of embedding a QUBO problem into the specific quantum annealer (the D-Wave 2X processor).
Quantum annealing (QA), along with the D-Wave processors, have been the focus of much research. We refer the interested reader to [2][3][4][5][6][7][8][9] . QA is a paradigm designed to find the ground state of systems of interacting spins represented by a time-evolving Hamiltonian: The parameters h i and J i j encode the particular QUBO problem P into its Ising formulation. QA is performed by first setting ∆ E , which results in a ground state into which the spins can be easily initialized. Then ∆ is slowly reduced and E is increased until E ∆. At this point the system is dominated by H P , which encodes the optimization problem. Thus, the ground state represents the solution to the optimization problem.
An embedding is the mapping of the nodes of an input graph to the nodes of the destination graph. The graph representing the problem's QUBO matrix needs to be embedded into the actual physical qubits on the processor in order for it to solve 2/3 the QUBO problem. The specific existing connectivity pattern of qubits in the D-Wave chip is called the Chimera graph. Embedding an input graph (a QUBO problem graph) into the hardware graph (the Chimera graph) is in general NP-hard ( 10 ). Figure 1-right shows an embedding of the (column algorithm) QUBO corresponding to the bi-prime M = 200 099 into the Chimera graph of the D-Wave 2X chip consisting of a 12 by 12 lattice of 4 by 4 bipartite blocks. The Chimera graph is structured so that the vertical and horizontal couplers in its lattice are connected only to either side of each bipartite block. Each node in this graph represents one qubit and each edge represents a coupling between two qubits. Adjacent nodes in the Chimera graph can be grouped together to form new effective (i.e., logical) nodes, creating nodes of a higher degree. Such a grouping is performed on the processor by setting the coupler between two qubits to a large negative value, forcing two Ising spins to align such that the two qubits end up with the same values. These effective qubits are expected to behave identically and remain in the same binary state at the time of measurement. The act of grouping adjacent qubits (hence forming new effective qubits) is called chain creation or chain identification.