QuadraticForm

class QuadraticForm(num_result_qubits=None, quadratic=None, linear=None, offset=None, little_endian=True)

Bases: qiskit.circuit.quantumcircuit.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.

References

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

arXiv:1912.04088

Parameters

num_result_qubits (Optional[int]) – The number of qubits to encode the result. Called $m$ in the class documentation.
quadratic (Union[ndarray, List[List[Union[float, ParameterExpression]]], None]) – A matrix containing the quadratic coefficients, $A$ .
linear (Union[ndarray, List[Union[float, ParameterExpression]], None]) – An array containing the linear coefficients, $b$ .
offset (Union[float, ParameterExpression, None]) – A constant offset, $c$ .
little_endian (bool) – Encode the result in little endianness.

Raises

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.

Methods Defined Here

required_result_qubits

static QuadraticForm.required_result_qubits(quadratic, linear, offset)

Get the number of required result qubits.

Parameters

quadratic (Union[ndarray, List[List[float]]]) – A matrix containing the quadratic coefficients.
linear (Union[ndarray, List[float]]) – An array containing the linear coefficients.
offset (float) – A constant offset.

Return type

int

Returns

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

Attributes

ancillas

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

Return type

List[AncillaQubit]

calibrations

Return calibration dictionary.

The custom pulse definition of a given gate is of the form

{‘gate_name’: {(qubits, params): schedule}}

Return type

dict

clbits

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

Return type

List[Clbit]

data

Return the circuit data (instructions and context).

Returns

a list-like object containing the tuples for the circuit’s data.

Each tuple is in the format (instruction, qargs, cargs), where instruction is an Instruction (or subclass) object, qargs is a list of Qubit objects, and cargs is a list of Clbit objects.

Return type

QuantumCircuitData

extension_lib

Default value: 'include "qelib1.inc";'

global_phase

Return the global phase of the circuit in radians.

Return type

Union[ParameterExpression, float]

header

Default value: 'OPENQASM 2.0;'

instances

Default value: 9

metadata

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 type

dict

num_ancillas

Return the number of ancilla qubits.

Return type

int

num_clbits

Return number of classical bits.

Return type

int

num_parameters

Convenience function to get the number of parameter objects in the circuit.

Return type

int

num_qubits

Return number of qubits.

Return type

int

parameters

Convenience function to get the parameters defined in the parameter table.

Return type

ParameterView

prefix

Default value: 'circuit'

qubits

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

Return type

List[Qubit]

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