# FourierChecking¶

class FourierChecking(f, g)[source]

Bases: QuantumCircuit

Fourier checking circuit.

The circuit for the Fourier checking algorithm, introduced in [1], involves a layer of Hadamards, the function $$f$$, another layer of Hadamards, the function $$g$$, followed by a final layer of Hadamards. The functions $$f$$ and $$g$$ are classical functions realized as phase oracles (diagonal operators with {-1, 1} on the diagonal).

The probability of observing the all-zeros string is $$p(f,g)$$. The algorithm solves the promise Fourier checking problem, which decides if f is correlated with the Fourier transform of g, by testing if $$p(f,g) <= 0.01$$ or $$p(f,g) >= 0.05$$, promised that one or the other of these is true.

The functions $$f$$ and $$g$$ are currently implemented from their truth tables but could be represented concisely and implemented efficiently for special classes of functions.

Fourier checking is a special case of $$k$$-fold forrelation [2].

Reference:

[1] S. Aaronson, BQP and the Polynomial Hierarchy, 2009 (Section 3.2). arXiv:0910.4698

[2] S. Aaronson, A. Ambainis, Forrelation: a problem that optimally separates quantum from classical computing, 2014. arXiv:1411.5729

Create Fourier checking circuit.

Parameters
• f (List[int]) -- truth table for f, length 2**n list of {1,-1}.

• g (List[int]) -- truth table for g, length 2**n list of {1,-1}.

Raises

CircuitError -- if the inputs f and g are not valid.

Reference Circuit:

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 CircuitInstructions for each instruction.

Return type

QuantumCircuitData

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

Return the global phase of the circuit in radians.

Return type

Union[ParameterExpression, float]

instances = 2330

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

The number of parameter objects in the circuit.

Return type

int

num_qubits

Return number of qubits.

Return type

int

op_start_times

Return a list of operation start times.

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

Return type

List[int]

Returns

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

Raises

AttributeError -- When circuit is not scheduled.

parameters

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.

Examples

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 unituitive 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
ParameterView([
ParameterVectorElement(x[0]), ParameterVectorElement(x[1]),
ParameterVectorElement(x[2]), ParameterVectorElement(x[3]),
..., ParameterVectorElement(x[11])
])

Return type

ParameterView

Returns

The sorted Parameter objects in the circuit.

prefix = 'circuit'
qubits

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

Return type

List[Qubit]