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

Get the number of required result qubits.

Attributes

ancillas

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

calibrations

Return calibration dictionary.

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

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

clbits

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

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 = 'include "qelib1.inc";'

global_phase

Return the global phase of the circuit in radians.

header = 'OPENQASM 2.0;'

instances = 16

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.

num_ancillas

Return the number of ancilla qubits.

num_clbits

Return number of classical bits.

num_parameters

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

num_qubits

Return number of qubits.

parameters

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

prefix = 'circuit'

qubits

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