We show that the NP-hard quadratic unconstrained binary optimization (QUBO) problem on a graph $G$ can be solved using an adiabatic quantum computer that implements an Ising spin-1/2 Hamiltonian, by reduction through minor-embedding of $G$ in the quantum hardware graph $U$. There are two components to this reduction: embedding and parameter setting. The embedding problem is to find a minor-embedding $G^{emb}$ of a graph $G$ in $U$, which is a subgraph of $U$ such that $G$ can be obtained from $G^{emb}$ by contracting edges. The parameter setting problem is to determine the corresponding parameters, qubit biases and coupler strengths, of the embedded Ising Hamiltonian. In this paper, we focus on the parameter setting problem. As an example, we demonstrate the embedded Ising Hamiltonian for solving the maximum independent set (MIS) problem via adiabatic quantum computation (AQC) using an Ising spin-1/2 system. We close by discussing several related algorithmic problems that need to be investigated in order to facilitate the design of adiabatic algorithms and AQC architectures.
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-min…
Adiabatic quantum programming defines the time-dependent mapping of a quantum algorithm into an underlying hardware or logical fabric. An essential step is embedding problem-specific information into the quantum logical fabric. We present algorithms for embedding arbitrary instances of the adiabatic quantum optimization algorithm into a square lattice of specialized unit cells. These methods extend with fabric growth while scaling linearly in time and quadratically in footprint. We also provide methods for handling hard faults in the logical fabric without invoking approximations to the origin…
A significant challenge in quantum annealing is to map a real-world problem onto a hardware graph of limited connectivity. If the maximum degree of the problem graph exceeds the maximum degree of the hardware graph, one employs minor embedding in which each logical qubit is mapped to a tree of physical qubits. Pairwise interactions between physical qubits in the tree are set to be ferromagnetic with some coupling strength $F<0$. Here we address the question of what value $F$ should take in order to maximise the probability that the annealer finds the correct ground-state of an Ising problem. T…
Quantum annealing provides a promising route for the development of quantum optimization devices, but the usefulness of such devices will be limited in part by the range of implementable problems as dictated by hardware constraints. To overcome constraints imposed by restricted connectivity between qubits, a larger set of interactions can be approximated using minor embedding techniques whereby several physical qubits are used to represent a single logical qubit. However, minor embedding introduces new types of errors due to its approximate nature. We introduce and study quantum annealing corr…
We introduce a framework for mapping NP-Hard problems to adiabatic quantum computing (AQC) architectures that are heavily restricted in both connectivity and dynamic range of couplings, for which minor-embedding -- the standard problem mapping method -- cannot be directly applied. Separating the mapping into two distinct stages, we introduce problem-specific reductions for both quadratic unconstrained binary optimisation (QUBO) and satisfiability (SAT) and develop the subdivision-embedding method that is suitable for directly embedding onto these heavily restricted architectures. The theory un…
A major limitation of current generations of quantum annealers is the sparse connectivity of manufactured qubits in the hardware graph. This technological limitation generated considerable interest, motivating efforts to design efficient and adroit minor-embedding procedures that bypass sparsity constraints. In this paper, starting from a previous equational formulation by Dridi et al. (arXiv:1810.01440), we propose integer programming (IP) techniques for solving the minor-embedding problem. The first approach involves a direct translation from the previous equational formulation to IP, while …
In this work we investigate the capabilities of a hybrid quantum-classical procedure to explore the solution space using the D-Wave $2000Q^{TM}$ Quantum Annealer device. Here we study the ability of the Quantum hardware to solve the Number Partitioning Problem, a well-known NP-Hard optimization model that poses some challenges typical of those encountered in real-world applications. This represents one of the most complex scenario in terms of qubits connectivity and, by increasing the input problem size, we analyse the scaling properties of the quantum-classical workflow. We find remarkable re…
We develop a non-cooperative game-theoretic model for the problem of graph minor-embedding to show that optimal compiling of adiabatic quantum programs in the sense of Nash equilibrium is possible.
We theoretically implement a strategy from quantum computation architectures to simulate Stuart-Landau oscillator dynamics in all-to-all connected networks, also referred to as complete graphs. The technique builds upon the triad structure minor embedding which expands dense graphs of interconnected elements into sparse ones which can potentially be realized in future on-chip solid state technologies with tunable edge weights. As a case study, we reveal that the minor embedding procedure allows simulating the XY model on complete graphs, thus bypassing a severe geometric constraint.
Today, hardware constraints are an important limitation on quantum adiabatic optimization algorithms. Firstly, computational problems must be formulated as quadratic unconstrained binary optimization (QUBO) in the presence of noisy coupling constants. Secondly, the interaction graph of the QUBO must have an effective minor embedding into a two-dimensional nonplanar lattice graph. We describe new strategies for constructing QUBOs for NP-complete/hard combinatorial problems that address both of these challenges. Our results include asymptotically improved embeddings for number partitioning, fill…
We present a heuristic algorithm for finding a graph $H$ as a minor of a graph $G$ that is practical for sparse $G$ and $H$ with hundreds of vertices. We also explain the practical importance of finding graph minors in mapping quadratic pseudo-boolean optimization problems onto an adiabatic quantum annealer.
Minor embedding heuristics have become an indispensable tool for compiling problems in quadratically unconstrained binary optimization (QUBO) into the hardware graphs of quantum and CMOS annealing processors. While recent embedding heuristics have been developed for annealers of moderate size (about 2000 nodes) the size of the latest CMOS annealing processor (with 102,400 nodes) poses entirely new demands on the embedding heuristic. This raises the question, if recent embedding heuristics can maintain meaningful embedding performance on hardware graphs of increasing size. Here, we develop an i…
Using quantum annealing to solve an optimization problem requires minor embeddings of a logic graph into a known hardware graph. In an effort to reduce the complexity of the minor embedding problem, we introduce the minor set cover (MSC) of a known graph G: a subset of graph minors which contain any remaining minor of the graph as a subgraph. Any graph that can be embedded into G will be embeddable into a member of the MSC. Focusing on embedding into the hardware graph of commercially available quantum annealers, we establish the MSC for a particular known virtual hardware, which is a complete…
Adiabatic quantum computing has evolved in recent years from a theoretical field into an immensely practical area, a change partially sparked by D-Wave System's quantum annealing hardware. These multimillion-dollar quantum annealers offer the potential to solve optimization problems millions of times faster than classical heuristics, prompting researchers at Google, NASA and Lockheed Martin to study how these computers can be applied to complex real-world problems such as NASA rover missions. Unfortunately, compiling (embedding) an optimization problem into the annealing hardware is itself a d…
The limited connectivity of current and next-generation quantum annealers motivates the need for efficient graph-minor embedding methods. These methods allow non-native problems to be adapted to the target annealer's architecture. The overhead of the widely used heuristic techniques is quickly proving to be a significant bottleneck for solving real-world applications. To alleviate this difficulty, we propose a systematic and deterministic embedding method, exploiting the structures of both the input graph of the specific problem and the quantum annealer. We focus on the specific case of the Ca…
Adiabatic quantum computation starts from embedding a computational problem into a Hamiltonian whose ground state encodes the solution to the problem. This problem Hamiltonian, $H_{\rm p}$, is normally chosen to be diagonal in the computational basis, that is a product basis for qubits. We point out that $H_{\rm p}$ can be chosen to be non-diagonal in the computational basis. To be more precise, we show how to construct $H_{\rm p}$ in such a way that all its excited states are entangled with respect to the qubit tensor product structure, while the ground state is still of the product form and …
This paper presents an overview of the topology of D-Wave's next-generation quantum processors. It provides examples of minor embeddings and discusses performance of embedding algorithms for the new topology compared to the existing Chimera topology. It also presents some initial performance results for simple, standard Ising model classes of problems.
In order to treat all-to-all connected quadratic binary optimization problems (QUBO) with hardware quantum annealers, an embedding of the original problem is required due to the sparsity of the hardware's topology. Embedding fully-connected graphs - typically found in industrial applications - incurs a quadratic space overhead and thus a significant overhead in the time to solution. Here we investigate this embedding penalty of established planar embedding schemes such as minor embedding on a square lattice, minor embedding on a Chimera graph, and the Lechner-Hauke-Zoller scheme using simulate…
Given quantum hardware that enables sampling from a family of natively implemented Hamiltonians, how well can one use that hardware to sample from a Hamiltonian outside that family? A common approach is to minor embed the desired Hamiltonian in a native Hamiltonian. In Phys. Rev. Research 2, 023020 (2020) it was shown that minor embedding can be detrimental for classical thermal sampling. Here, we generalize these results by considering quantum thermal sampling in the transverse-field Ising model, i.e. sampling a Hamiltonian with non-zero off diagonal terms. To study these systems numerically …
An algebraic method has been developed which allows one to engineer several energy levels including the low-energy subspace of interacting spin systems. By introducing ancillary qubits, this approach allows k-body interactions to be captured exactly using 2-body Hamiltonians. Our method works when all terms in the Hamiltonian share the same basis and has no dependence on perturbation theory or the associated large spectral gap. Our methods allow problem instance solutions to be embedded into the ground energy state of Ising spin systems. Adiabatic evolution might then be used to place a comput…
We present a simple 2d local circuit that implements all-to-all interactions via perturbative gadgets. We find an analytic relation between the values $J_{ij}$ of the desired interaction and the parameters of the 2d circuit, as well as the expression for the error in the quantum spectrum. For the relative error to be a constant $ε$, one requires an energy scale growing as $n^6$ in the number of qubits, or equivalently a control precision up to $ n^{-6}$. Our proof is based on the Schrieffer-Wolff transformation and generalizes to any hardware. In the architectures available today, $5$ digits o…
We extend the family of problems that may be implemented on an adiabatic quantum optimizer (AQO). When a quadratic optimization problem has at least one set of discrete controls and the constraints are linear, we call this a quadratic constrained mixed discrete optimization (QCMDO) problem. QCMDO problems are NP-hard, and no efficient classical algorithm for their solution is known. Included in the class of QCMDO problems are combinatorial optimization problems constrained by a linear partial differential equation (PDE) or system of linear PDEs. An essential complication commonly encountered i…
Adiabatic Quantum Computing (AQC) is an attractive paradigm for solving hard integer polynomial optimization problems. Available hardware restricts the Hamiltonians to be of a structure that allows only pairwise interactions. This requires that the original optimization problem to be first converted -- from its polynomial form -- to a quadratic unconstrained binary optimization (QUBO) problem, which we frame as a problem in algebraic geometry. Additionally, the hardware graph where such a QUBO-Hamiltonian needs to be embedded -- assigning variables of the problem to the qubits of the physical …
We group all known quadratizations of cubic and quartic terms in binary optimization problems into six and seven unique graphs respectively. We then perform a minor embedding of these graphs onto the well-known Chimera graph, and the brand new Pegasus graph. We conclude with recommendations for which gadgets are best to use when aiming to reduce the total number of qubits required to embed a problem.
Optimization or sampling of arbitrary pairwise Ising models, in a quantum annealing protocol of constrained interaction topology, can be enabled by a minor-embedding procedure. The logical problem of interest is transformed to a physical (device programmable) problem, where one binary variable is represented by a logical qubit consisting of multiple physical qubits. In this paper we discuss tuning of this transformation for the cases of clique, biclique, and cubic lattice problems on the D-Wave 2000Q quantum computer. We demonstrate parameter tuning protocols in spin glasses and channel commun…
Designing and optimizing cost functions and energy landscapes is a problem encountered in many fields of science and engineering. These landscapes and cost functions can be embedded and annealed in experimentally controllable spin Hamiltonians. Using an approach based on group theory and symmetries, we examine the embedding of Boolean logic gates into the ground state subspace of such spin systems. We describe parameterized families of diagonal Hamiltonians and symmetry operations which preserve the ground state subspace encoding the truth tables of Boolean formulas. The ground state embedding…
