class qiskit.circuit.library.QuadraticForm(num_result_qubits=None, quadratic=None, linear=None, offset=None, little_endian=True)[source]#

Bases: QuantumCircuit

Implements a quadratic form on binary variables encoded in qubit registers.

A quadratic form on binary variables is a quadratic function \(Q\) acting on a binary variable of \(n\) bits, \(x = x_0 ... x_{n-1}\). For an integer matrix \(A\), an integer vector \(b\) and an integer \(c\) the function can be written as

\[Q(x) = x^T A x + x^T b + c\]

If \(A\), \(b\) or \(c\) contain scalar values, this circuit computes only an approximation of the quadratic form.

Provided with \(m\) qubits to encode the value, this circuit computes \(Q(x) \mod 2^m\) in [two’s complement](https://stackoverflow.com/questions/1049722/what-is-2s-complement) representation.

\[|x\rangle_n |0\rangle_m \mapsto |x\rangle_n |(Q(x) + 2^m) \mod 2^m \rangle_m\]

Since we use two’s complement e.g. the value of \(Q(x) = 3\) requires 2 bits to represent the value and 1 bit for the sign: 3 = ‘011’ where the first 0 indicates a positive value. On the other hand, \(Q(x) = -3\) would be -3 = ‘101’, where the first 1 indicates a negative value and 01 is the two’s complement of 3.

If the value of \(Q(x)\) is too large to be represented with m qubits, the resulting bitstring is \((Q(x) + 2^m) \mod 2^m)\).

The implementation of this circuit is discussed in [1], Fig. 6.


[1]: Gilliam et al., Grover Adaptive Search for Constrained Polynomial Binary Optimization.


  • ValueError – If linear and quadratic have mismatching sizes.

  • ValueError – If num_result_qubits is unspecified but cannot be determined because some values of the quadratic form are parameterized.



Returns a list of ancilla bits in the order that the registers were added.


Return calibration dictionary.

The custom pulse definition of a given gate is of the form {'gate_name': {(qubits, params): schedule}}


Returns a list of classical bits in the order that the registers were added.


Return the circuit data (instructions and context).


a list-like object containing the CircuitInstructions for each instruction.

Return type:


extension_lib = 'include "qelib1.inc";'#

Return the global phase of the circuit in radians.

header = 'OPENQASM 2.0;'#
instances = 422#

Return any associated layout information about the circuit

This attribute contains an optional TranspileLayout object. This is typically set on the output from transpile() or PassManager.run() to retain information about the permutations caused on the input circuit by transpilation.

There are two types of permutations caused by the transpile() function, an initial layout which permutes the qubits based on the selected physical qubits on the Target, and a final layout which is an output permutation caused by SwapGates inserted during routing.


The user provided metadata associated with the circuit.

The metadata for the circuit is a user provided dict of metadata for the circuit. It will not be used to influence the execution or operation of the circuit, but it is expected to be passed between all transforms of the circuit (ie transpilation) and that providers will associate any circuit metadata with the results it returns from execution of that circuit.


Return the number of ancilla qubits.


Return number of classical bits.


The number of parameter objects in the circuit.


Return number of qubits.


Return a list of operation start times.

This attribute is enabled once one of scheduling analysis passes runs on the quantum circuit.


List of integers representing instruction start times. The index corresponds to the index of instruction in QuantumCircuit.data.


AttributeError – When circuit is not scheduled.


The parameters defined in the circuit.

This attribute returns the Parameter objects in the circuit sorted alphabetically. Note that parameters instantiated with a ParameterVector are still sorted numerically.


The snippet below shows that insertion order of parameters does not matter.

>>> from qiskit.circuit import QuantumCircuit, Parameter
>>> a, b, elephant = Parameter("a"), Parameter("b"), Parameter("elephant")
>>> circuit = QuantumCircuit(1)
>>> circuit.rx(b, 0)
>>> circuit.rz(elephant, 0)
>>> circuit.ry(a, 0)
>>> circuit.parameters  # sorted alphabetically!
ParameterView([Parameter(a), Parameter(b), Parameter(elephant)])

Bear in mind that alphabetical sorting might be unintuitive when it comes to numbers. The literal “10” comes before “2” in strict alphabetical sorting.

>>> from qiskit.circuit import QuantumCircuit, Parameter
>>> angles = [Parameter("angle_1"), Parameter("angle_2"), Parameter("angle_10")]
>>> circuit = QuantumCircuit(1)
>>> circuit.u(*angles, 0)
>>> circuit.draw()
q: ┤ U(angle_1,angle_2,angle_10) ├
>>> circuit.parameters
ParameterView([Parameter(angle_1), Parameter(angle_10), Parameter(angle_2)])

To respect numerical sorting, a ParameterVector can be used.

>>> from qiskit.circuit import QuantumCircuit, Parameter, ParameterVector
>>> x = ParameterVector("x", 12)
>>> circuit = QuantumCircuit(1)
>>> for x_i in x:
...     circuit.rx(x_i, 0)
>>> circuit.parameters
    ParameterVectorElement(x[0]), ParameterVectorElement(x[1]),
    ParameterVectorElement(x[2]), ParameterVectorElement(x[3]),
    ..., ParameterVectorElement(x[11])

The sorted Parameter objects in the circuit.

prefix = 'circuit'#

Returns a list of quantum bits in the order that the registers were added.


static required_result_qubits(quadratic, linear, offset)[source]#

Get the number of required result qubits.

  • quadratic (ndarray | List[List[float]]) – A matrix containing the quadratic coefficients.

  • linear (ndarray | List[float]) – An array containing the linear coefficients.

  • offset (float) – A constant offset.


The number of qubits needed to represent the value of the quadratic form in twos complement.

Return type: