# 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.
- 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.
- 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.
- 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.

- 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 \(J_{i,j}\).

- Minimum gap
- The minimum distance between the ground state and the first excited state throughout any point in the anneal.
- Objective function
- A mathematical expression of the energy of a system as a function of binary variables representing the qubits.
- 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
- 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}- 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.