We introduce an algorithm to generate (not solve) spin-glass instances with planted solutions of arbitrary size and structure. First, a set of small problem patches with open boundaries is solved either exactly or with a heuristic, and then the individual patches are stitched together to create a large problem with a known planted solution. Because in these problems frustration is typically smaller than in random problems, we first assess the typical computational complexity of the individual patches using population annealing Monte Carlo, and introduce an approach that allows one to fine-tune the typical computational complexity of the patch-planted system. The scaling of the typical computational complexity of these planted instances with various numbers of patches and patch sizes is investigated and compared to random instances. PACS numbers: 75.50.Lk, 75.40.Mg, 05.50.+q, Many optimization problems belong to the NP-hard complexity class, for which it is believed that no algorithms exist to solve them in polynomial time. Spin-glass problems without biases and on nonplanar topologies, such as the EdwardAnderson (EA) model [1], represent a sub-class of the NPhard class. Because spin glasses are the simplest models with both disorder and frustration that fall into the NP-hard class, they represent the ideal model systems to benchmark algorithms, as well as novel computing architectures. A number of heuristics, as well as exhaustive search methods, have been designed and developed to minimize spin-glass Hamiltonians as efficiently as possible. These method also include simulated annealing [2] , parallel tempering Monte Carlo [3] [4] [5] [6] , population annealing Monte Carlo [7] [8] [9] [10] , and genetic algorithms [11, 12] , as well as branch-and-cut [13] algorithms, to name a few. Many of these optimization algorithms use only local updates during the minimization procedure. However, in many cases, the use of cluster algorithms with nonlocal updates can greatly enhance the searching process when the energy landscape has many metastable states with small overlap [14] [15] [16] . In the last two decades, quantum heuristics have been proposed as an alternative to classical heuristics, due to their potential to exploit quantum superposition and quantum tunneling effects. Among quantum approaches, adiabatic quantum optimization (AQO) is widely used [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] and likely the method most amenable to hardware implementations [31] . Current state-of-the-art AQO hardware is manufactured by D-Wave System Inc., whose latest chip allows for the quantum optimization of problems of approximately up to 2000 variables. However, whether AQO can be more efficient than classical algorithms for certain problems is still controversial [32] [33] [34] . Given the importance of comparing optimization techniques across disciplines, it is necessary to have benchmark problems that are (1) representative of the hardness of a typical NP-hard problem, (2) scalable for large systems, and for which (3) the ground state is known a priori. While it is easy to fulfill criteria (1) and (2), it is challenging to have large FIG. 1: Schematic diagram of the patch planting method for a twodimensional lattice. First, the ground state for each (easily solvable) patch (A -D) is computed using free boundary conditions. Then, a ground state configuration of each patch is selected. Edge spins between the patches are represented by empty and full circles. Couplers between the edge spins between adjacent patches are added under the condition that all interactions are satisfied, i.e., if two spins have the same value (i.e., both full or both empty circles), a ferromagnetic coupler (straight blue line) is added. If, however, two adjacent spins are different, then an antiferromagnetic coupler (red wiggly line) is added. The direct product of the ground state of the patches A -D is then the ground state of the large planted system. problems with known solutions. There have been previous approaches to plant solutions for benchmarking purposes. For example, Ref. [35] used an approach based on constraint satisfaction problems. Although these problems are tunable in hardness, there is little control when selecting the coupler values between the individual variables. For analog machines with finite precision, such as the D-Wave quantum annealers, this could be an unnecessary restriction. Other approaches [36] start from a random coupler arXiv:1706.02825v2 [cond-mat.dis-nn]
A recent paper REF introduced the method of patch planting, in which subgraphs with known solution are coupled to each other, satisfying all interactions and thus planting an overall ground state.
Patch-planting spin-glass solution for benchmarking
Phys. Rev. E 96, 023312 (2017)
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