Ising formulations of many NP problems (1302.5843v3)

Abstract: We provide Ising formulations for many NP-complete and NP-hard problems, including all of Karp's 21 NP-complete problems. This collects and extends mappings to the Ising model from partitioning, covering and satisfiability. In each case, the required number of spins is at most cubic in the size of the problem. This work may be useful in designing adiabatic quantum optimization algorithms.

• The paper extends mappings from NP-complete and NP-hard problems to Ising models, offering systematic formulations for use in adiabatic quantum optimization.
• It details various problem classes, including partitioning, covering, and graph coloring, and analyzes spin-scaling and ancillary variables for each mapping.
• The work emphasizes optimizing Hamiltonians to improve spectral gaps and lays a theoretical framework for advancing quantum computing algorithms.

Ising Formulations of Many NP Problems

Andrew Lucas's paper tackles the significant challenge of encoding NP-complete and NP-hard problems into the framework of Ising spin glasses. This manuscript contributes a comprehensive review and extension of methods to map several renowned NP problems onto Ising models, with a keen focus on adiabatic quantum optimization (AQO) algorithms. In this essay, we will explore the essential aspects and implications of these mappings.

Key Concepts and Methodologies

The Ising model, traditionally a foundational model in statistical mechanics, represents spin systems where interactions occur between neighboring spins. Translating combinatorial problems into this model involves formulating an energy function (Hamiltonian) where the ground states correspond to solutions of the original problem. AQO leverages these Hamiltonians to theoretically achieve polynomial-time solutions for certain classes of problems by exploiting quantum mechanical properties.

Lucas's work extends known mappings and introduces new formulations for a broad array of NP problems, including Karp's 21 NP-complete problems. The mappings reviewed and developed in the paper are categorized into several traditional combinatorial problem classes:

1. Partitioning Problems: These involve dividing sets into subsets under specific constraints, exemplified by number partitioning, graph partitioning, and clique problems.
2. Binary Integer Linear Programming (ILP): The paper translates ILP problems into Ising models by enforcing linear constraints via penalizing terms in the Hamiltonian and then optimizing a linear objective function.
3. Covering and Packing Problems: This includes vertex cover, set packing, and satisfiability problems, where subsets are selected to cover or pack a set under given constraints.
4. Problems with Inequalities: Examples include set cover and the knapsack problem, which require encoding constraints involving inequalities.
5. Coloring Problems: These encompass graph coloring and related problems like clique cover, requiring distinct labels for elements under adjacency conditions.
6. Hamiltonian Cycle Problems: It addresses classical problems like the Hamiltonian cycle and traveling salesman problems using permutation matrices in quantum space.
7. Tree Problems: This involves creating minimal spanning trees or Steiner trees with additional constraints like limited vertex degrees.
8. Graph Isomorphisms: A problem of determining whether two graphs are structurally identical, mapped into an Ising model to leverage quantum annealing's potential.

Numerical and Efficiency Considerations

The paper scrutinizes the number of spins required for each mapping—critical for practical implementations on quantum devices. For instance, problems like number partitioning may need spins scaling linearly with problem size, while others like the traveling salesman necessitate quadratic scaling due to the encoding of permutations.

Special attention is given to minimizing the number of auxiliary spins (ancilla bits) introduced to enforce constraints and reduce higher-order interactions down to pairwise interactions. For instance, Lucas employs binary variables to transform polynomial constraints into linear ones, minimizing the complexity of resultant Ising Hamiltonians.

Contradictory Claims and Bold Assertions

One of the bold assertions of the paper is the possibility that AQO might efficiently solve some problem classes where classical algorithms do not perform optimally, though this remains speculative due to the exponential time usually required by AQO algorithms. Exponential time scaling arising from exponentially small energy gaps presents significant physical constraints and has sparked debate about AQO’s practical superiority over classical solutions.

Implications and Future Developments

The implications of Lucas's work are twofold. Practically, it serves as a guide for encoding numerous NP problems on quantum devices, aiding the design of future AQO systems. Theoretically, it opens inquiries into the physical properties of these mappings, such as energy gap sizes and how these affect quantum computational efficiency.

Future research directions might revolve around optimizing these mappings to enhance spectral gaps in Hamiltonians, making these quantum algorithms more practical for real-world problem sizes. Moreover, investigating alternative quantum computation models beyond AQO that can leverage these Ising formulations effectively represents another significant avenue for exploration.

Conclusion

Andrew Lucas's paper offers a meticulous groundwork for the quantum computational solving of NP problems through Ising model formulations, showing both the breadth of application and depth of thought invested in extending this field. While the practical dominance of AQO over classical methods remains unresolved, the theoretical advancements and thorough mappings provided by Lucas are invaluable for the ongoing development of quantum optimization algorithms. This work lays the groundwork for future explorations into more efficient quantum computing paradigms and their potential to address some of the most challenging computational problems.