ising_to_qubo(h, J, offset=0.0)[source]

Convert an Ising problem to a QUBO problem.

Map an Ising model defined on spins (variables with {-1, +1} values) to quadratic unconstrained binary optimization (QUBO) formulation \(x' Q x\) defined over binary variables (0 or 1 values), where the linear term is contained along the diagonal of Q. Return matrix Q that defines the model as well as the offset in energy between the two problem formulations:

\[s' J s + h' s = offset + x' Q x\]

See qubo_to_ising() for the inverse function.

  • h (dict[variable, bias]) – Linear biases as a dict of the form {v: bias, …}, where keys are variables of the model and values are biases.

  • J (dict[(variable, variable), bias]) – Quadratic biases as a dict of the form {(u, v): bias, …}, where keys are 2-tuples of variables of the model and values are quadratic biases associated with the pair of variables (the interaction).

  • offset (numeric, optional, default=0) – Constant offset to be applied to the energy. Default 0.


A 2-tuple containing:

dict: QUBO coefficients.

float: New energy offset.

Return type

(dict, float)


This example converts an Ising problem of two variables that have positive biases of value 1 and are positively coupled with an interaction of value 1 to a QUBO problem and prints the resulting energy offset.

>>> h = {1: 1, 2: 1}
>>> J = {(1, 2): 1}
>>> dimod.ising_to_qubo(h, J, 0.5)[1]