Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design (1001.3116v2)

Abstract: In [Choi08], we introduced the notion of minor-embedding in adiabatic quantum optimization. A minor-embedding of a graph G in a quantum hardware graph U is a subgraph of U such that G can be obtained from it by contracting edges. In this paper, we describe the intertwined adiabatic quantum architecture design problem, which is to construct a hardware graph U that satisfies all known physical constraints and, at the same time, permits an efficient minor-embedding algorithm. We illustrate an optimal complete-graph-minor hardware graph. Given a family F of graphs, a (host) graph U is called F-minor-universal if for each graph G in F, U contains a minor-embedding of G. The problem for designing a F-minor-universal hardware graph U_{sparse} in which F consists of a family of sparse graphs (e.g., bounded degree graphs) is open.

• The paper introduces the TRIAD construction for minor-embedding complete graphs, optimizing qubit and coupler usage in quantum hardware design.
• The study leverages a triangular layout with degree-bounding to enhance the scalability of embedding NP-hard Ising Hamiltonians.
• The proposed design offers a sparsely connected, high treewidth graph framework that could significantly improve adiabatic quantum algorithms.

Overview of Minor-Embedding in Adiabatic Quantum Computation: II. Minor-Universal Graph Design

The paper "Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design" by Vicky Choi presents an exploration into optimizing quantum hardware design for adiabatic quantum computation (AQC) focusing on minor-embedding in graph-theory contexts. The paper advances the understanding of how quantum computational problems, specifically those represented in Ising Hamiltonians, can be efficiently solved via carefully structured quantum hardware graphs.

Adiabatic Quantum Computation and Ising Hamiltonians

Adiabatic Quantum Computation, established as a potent computational paradigm by Farhi et al., utilizes the gradual transformation of Hamiltonians to reach a solution to computational problems. The paper concentrates on the Ising Hamiltonian, representing computational challenges in a quadratic unconstrained binary optimization framework, highlighting its NP-hard nature. The author's work builds on leveraging the principles of AQC to achieve efficient problem-solving by embedding problem graphs into quantum hardware graphs.

Minor-Embedding and Optimization

The concept of minor-embedding in graph theory allows problems mapped on a graph $G$ to be implemented on a quantum hardware graph $U$ by capturing $G$ as a minor of $U$. This paper extends previous research by providing insights into the design of hardware graphs optimal for such embeddings, considering physical constraints like qubit degree-bounding and edge length limits inherent in superconducting architectures like those employed by D-Wave Systems.

Triangular Layout and \textit{TRIAD} Construction

A significant contribution of the paper is the construction of a triangular minor-universal layout, known as the \textit{TRIAD}, optimized for embedding complete graphs $K_n$. Through this construction, each vertex in $K_n$ is represented by a chain of virtual vertices, offering a solution to embed complete graphs while adhering to degree constraints. The scalability of this layout ensures efficient representation with minimal qubits and couplers, surpassing the demands of generic embedding of sparse graphs.

Implications and Future Considerations

By proposing a hardware graph design that is both sparse and possesses a large treewidth, the paper suggests avenues for a more efficient realization of adiabatic quantum algorithms. The notion of minor-universal graphs offers a richly connected yet sparse framework that could potentially enable efficient minor-embeddings of a broad class of challenging computational problems. The research identifies significant gaps in existing expander constructions, stressing the requirement for geometric expanders that satisfy necessary hardware constraints.

The implications are clear: advances in quantum processor layouts can drastically enhance the performance of quantum optimization techniques. Future developments might focus on refining graph decompositions and achieving geometric expanders tailored for specific problem classes, enhancing the scalability and efficiency of quantum computations further.

Choi's paper thus plays a critical role in the ongoing efforts to adapt hardware architectures, paving the way for broader applicability of AQC in tackling classically intractable problems more effectively. As this field evolves, the ideas presented here might serve as a foundation upon which future quantum computational frameworks are constructed.