A case study in programming a quantum annealer for hard operational planning problems

Abstract

We report on a case study in programming an early quantum annealer to attack optimization problems related to operational planning. While a number of studies have looked at the performance of quantum annealers on problems native to their architecture, and others have examined performance of select problems stemming from an application area, ours is one of the first studies of a quantum annealer's performance on parametrized families of hard problems from a practical domain. We explore two different general mappings of planning problems to quadratic unconstrained binary optimization (QUBO) problems, and apply them to two parametrized families of planning problems, navigation-type and scheduling-type. We also examine two more compact, but problemtype specific, mappings to QUBO, one for the navigation-type planning problems and one for the scheduling-type planning problems. We study embedding properties and parameter setting, and examine their effect on the efficiency with which the quantum annealer solves these problems. From these results we derive insights useful for the programming and design of future quantum annealers: problem choice, the mapping used, the properties of the embedding, and the annealing profile all matter, each significantly affecting the performance.

Coherent Ising machines—optical neural networks operating at the quantum limit

“…As α increases, the first nonzero steady state appears at the minimal value of the function 2ϵ N H as shown in Eq. (13). As shown in Fig.…”

“…The Ising coupling J ij and the local field h i take continuous (real) values, the magnitude of which are determined by mapping a given combinatorial optimization problem on the Ising model. 13,21 The three-dimensional Ising model and the two-dimensional Ising model with local fields belong to the NP-hard class in complexity theory. 18 Therefore, you can imagine that many hard problems in the real world can be solved through the Ising model.…”

“…18 Therefore, you can imagine that many hard problems in the real world can be solved through the Ising model. 13,21 In order to implement the cost function described in Eq. (7) as the effective loss of the DOPO network, pairs of DOPOs must be coupled with the coupling constant J ij and also the constant optical field h i must be injected into each DOPO.…”

“…6,7 Sophisticated AQC/QA devices are already under development, [8][9][10][11] but providing dense connectivity between qubits remains a major challenge 12 with serious implications for the efficiency of AQC/QA systems. 13 Networks of degenerate optical parametric oscillators (DOPOs) are an alternative physical system, with an unconventional operating mechanism, [14][15][16][17] for solving the Ising problem, [18][19][20] and by extension many other combinatorial optimization problems. 21 Formally, the N-spin Ising problem is to find the configuration of spins σ i 2 À1; þ1 f g i ¼ 1; ; N ð Þthat minimizes the energy function H ¼ À P 1 i<j N J ij σ i σ j À P 1 i N h i σ i , where the particular problem instance being solved is specified by the N × N matrix J (with elements J ij ) and the N-dimensional vector h (with elements h i ).…”

Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets

“…Problem instances represented by large but incompletely connected input graphs must use embeddings that are both resource efficient and time efficient in order to ensure fast and correct solutions. Examples include dynamic job scheduling [16] and route planning [17], as well as time-dependent fault-detection [18]. Alleviating the classical processing bottleneck while retaining the resource efficiency of the CMR algorithm is therefore an important problem for solving optimization problems with quantum annealing and integrating these quantum processing units into future computing systems [19].…”