# Concepts¶

See the Glossary for short definitions of terminology or learn Ocean concepts here:

Concepts | Related terms |
---|---|

Binary Quadratic Models | BQM, Ising, QUBO |

Constraint Satisfaction | CSP, binary CSP |

Hybrid | quantum-classical hybrid, Leap’s hybrid solvers, hybrid workflows |

Minor-Embedding | embedding, mapping logical variables to physical qubits, chains, chain strength |

QPU Topology | Chimera, Pegasus |

Samplers and Composites | solver |

Solutions | samples, sampleset, probabilistic, energy |

## Glossary¶

- binary quadratic model
- BQM
- A collection of binary-valued variables (variables that can be assigned two values, for example -1, 1) with associated linear and quadratic biases. Sometimes referred to in other tools as a problem. See a fuller description under Binary Quadratic Models.
- Chain
- One or more nodes or qubits in a target graph that represent a single variable in the source graph. See embedding. See a fuller description under Minor-Embedding.
- Chain length
- The number of qubits in a Chain. See a fuller description under Minor-Embedding.
- Chain strength
- Magnitude of the negative quadratic bias applied between variables to form a chain. See a fuller description under Minor-Embedding.
- Chimera
- The D-Wave QPU is a lattice of interconnected qubits. While some qubits connect to others via couplers, the D-Wave QPU is not fully connected. Instead, the qubits interconnect in an architecture known as Chimera. See a fuller description under QPU Topology.
- Complete graph
- Fully connected
- See complete graph. on wikipedia. A fully connected or complete binary quadratic model is one that has interactions between all of its variables.
- Composed sampler
- Samplers that apply pre- and/or post-processing to binary quadratic programs without changing the underlying sampler implementation by layering composite patterns on the sampler. For example, a composed sampler might add spin transformations when sampling from the D-Wave system.
- Composite
- A sampler can be composed. The composite pattern allows layers of pre- and post-processing to be applied to binary quadratic programs without needing to change the underlying sampler implementation. We refer to these layers as “composites”. A composed sampler includes at least one sampler and possibly many composites.
- Embed
- Embedding
- Minor-embed
- Minor-embedding
- The nodes and edges on the graph that represents an objective function translate to the qubits and couplers in Chimera. Each logical qubit, in the graph of the objective function, may be represented by one or more physical qubits. The process of mapping the logical qubits to physical qubits is known as minor embedding. See a fuller description under Minor-Embedding.
- Excited state
- States of a quantum system that have higher energy than the ground state. Such states represent non-optimal solutions for problems represented by an Objective function and infeasible configurations for problems represented by a penalty model.
- Graph
- A collection of nodes and edges. A graph can be derived from a model: a node for each variable and an edge for each pair of variables with a non-zero quadratic bias.
- Ground state
- The lowest-energy state of a quantum-mechanical system and the global minimum of a problem represented by an Objective function.
- Hamiltonian
A classical Hamiltonian is a mathematical description of some physical system in terms of its energies. We can input any particular state of the system, and the Hamiltonian returns the energy for that state. For a quantum system, a Hamiltonian is a function that maps certain states, called

*eigenstates*, to energies. Only when the system is in an eigenstate of the Hamiltonian is its energy well defined and called the*eigenenergy*. When the system is in any other state, its energy is uncertain. For the D-Wave system, the Hamiltonian may be represented as\begin{equation} {\cal H}_{ising} = \underbrace{\frac{A({s})}{2} \left(\sum_i {\hat\sigma_{x}^{(i)}}\right)}_\text{Initial Hamiltonian} + \underbrace{\frac{B({s})}{2} \left(\sum_{i} h_i {\hat\sigma_{z}^{(i)}} + \sum_{i>j} J_{i,j} {\hat\sigma_{z}^{(i)}} {\hat\sigma_{z}^{(j)}}\right)}_\text{Final Hamiltonian} \end{equation}where \({\hat\sigma_{x,z}^{(i)}}\) are Pauli matrices operating on a qubit \(q_i\), and \(h_i\) and \(J_{i,j}\) are the qubit biases and coupling strengths.

- Hardware graph
- See hardware graph. The hardware graph is the physical lattice of interconnected qubits. See also working graph. See a fuller description under QPU Topology.
- Ising
Traditionally used in statistical mechanics. Variables are “spin up” (\(\uparrow\)) and “spin down” (\(\downarrow\)), states that correspond to \(+1\) and \(-1\) values. Relationships between the spins, represented by couplings, are correlations or anti-correlations. The objective function expressed as an Ising model is as follows:

\begin{equation} \text{E}_{ising}(\pmb{s}) = \sum_{i=1}^N h_i s_i + \sum_{i=1}^N \sum_{j=i+1}^N J_{i,j} s_i s_j \end{equation}where the linear coefficients corresponding to qubit biases are \(h_i\), and the quadratic coefficients corresponding to coupling strengths are \(J_{i,j}\). See also Ising Model on Wikipedia.

- Minimum gap
- The minimum distance between the ground state and the first excited state throughout any point in the anneal.
- Model
- A collection of variables with associated linear and
quadratic biases. Sometimes referred to as a
**problem**. - Objective function
- A mathematical expression of the energy of a system as a function of binary variables representing the qubits.
- Pegasus
- The D-Wave QPU is a lattice of interconnected qubits. While some qubits connect to others via couplers, the D-Wave QPU is not fully connected. Instead, the qubits interconnect in an architecture known as Pegasus. See a fuller description under QPU Topology.
- Penalty function
- An algorithm for solving constrained optimization problems. In the context of Ocean tools, penalty functions are typically employed to increase the energy level of a problem’s objective function by penalizing non-valid configurations. See Penalty method on Wikipedia
- Penalty model
- An approach to solving constraint satisfaction problems (CSP) using an Ising model or a QUBO by mapping each individual constraint in the CSP to a ‘small’ Ising model or QUBO.
- QPU
- Quantum processing unit
- QUBO
Quadratic unconstrained binary optimization. QUBO problems are traditionally used in computer science. Variables are TRUE and FALSE, states that correspond to 1 and 0 values. A QUBO problem is defined using an upper-diagonal matrix \(Q\), which is an \(N\) x \(N\) upper-triangular matrix of real weights, and \(x\), a vector of binary variables, as minimizing the function

\begin{equation} f(x) = \sum_{i} {Q_{i,i}}{x_i} + \sum_{i<j} {Q_{i,j}}{x_i}{x_j} \end{equation}where the diagonal terms \(Q_{i,i}\) are the linear coefficients and the nonzero off-diagonal terms are the quadratic coefficients \(Q_{i,j}\). This can be expressed more concisely as

\begin{equation} \min_{{x} \in {\{0,1\}^n}} {x}^{T} {Q}{x}. \end{equation}In scalar notation, the objective function expressed as a QUBO is as follows:

\begin{equation} \text{E}_{qubo}(a_i, b_{i,j}; q_i) = \sum_{i} a_i q_i + \sum_{i<j} b_{i,j} q_i q_j. \end{equation}See also QUBO on Wikipedia.

- Sampler
- Samplers are processes that sample from low energy states of a problem’s objective function, which is a mathematical expression of the energy of a system. A binary quadratic model (BQM) sampler samples from low energy states in models such as those defined by an Ising equation or a QUBO problem and returns an iterable of samples, in order of increasing energy.
- SAPI
- Solver API used by clients to communicate with a solver.
- Solver
- A resource that runs a problem. Some solvers interface to the QPU; others leverage CPU and GPU resources.
- Source
- Source graph
- In the context of embedding, the model or induced graph that we
wish to embed. Sometimes referred to as the
**logical**graph/model. See a fuller description under Minor-Embedding. - Structured sampler
- Samplers that are restricted to sampling only binary quadratic models defined on a specific graph.
- Subgraph
- See subgraph on wikipedia.
- Target
- Target graph
- Embedding attempts to create a target model from a target
graph. The process of embedding takes a source model, derives the source
graph, maps the source graph to the target graph, then derives the target
model. Sometimes referred to as the
**embedded**graph/model. See a fuller description under Minor-Embedding. - Working graph
- In a D-Wave QPU, the set of qubits and couplers that are available for computation is known as the working graph. The yield of a working graph is typically less than 100% of qubits and couplers that are fabricated and physically present in the QPU. See hardware graph.