# Source code for qiskit.circuit.library.hidden_linear_function

```
# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2020.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.
"""Hidden Linear Function circuit."""
from typing import Union, List
import numpy as np
from qiskit.circuit.quantumcircuit import QuantumCircuit
from qiskit.circuit.exceptions import CircuitError
[docs]class HiddenLinearFunction(QuantumCircuit):
r"""Circuit to solve the hidden linear function problem.
The 2D Hidden Linear Function problem is determined by a 2D adjacency
matrix A, where only elements that are nearest-neighbor on a grid have
non-zero entries. Each row/column corresponds to one binary variable
:math:`x_i`.
The hidden linear function problem is as follows:
Consider the quadratic form
.. math::
q(x) = \sum_{i,j=1}^{n}{x_i x_j} ~(\mathrm{mod}~ 4)
and restrict :math:`q(x)` onto the nullspace of A. This results in a linear
function.
.. math::
2 \sum_{i=1}^{n}{z_i x_i} ~(\mathrm{mod}~ 4) \forall x \in \mathrm{Ker}(A)
and the goal is to recover this linear function (equivalently a vector
:math:`[z_0, ..., z_{n-1}]`). There can be multiple solutions.
In [1] it is shown that the present circuit solves this problem
on a quantum computer in constant depth, whereas any corresponding
solution on a classical computer would require circuits that grow
logarithmically with :math:`n`. Thus this circuit is an example
of quantum advantage with shallow circuits.
**Reference Circuit:**
.. jupyter-execute::
:hide-code:
from qiskit.circuit.library import HiddenLinearFunction
import qiskit.tools.jupyter
A = [[1, 1, 0], [1, 0, 1], [0, 1, 1]]
circuit = HiddenLinearFunction(A)
%circuit_library_info circuit
**Reference:**
[1] S. Bravyi, D. Gosset, R. Koenig, Quantum Advantage with Shallow Circuits, 2017.
`arXiv:1704.00690 <https://arxiv.org/abs/1704.00690>`_
"""
[docs] def __init__(self, adjacency_matrix: Union[List[List[int]], np.ndarray]) -> None:
"""Create new HLF circuit.
Args:
adjacency_matrix: a symmetric n-by-n list of 0-1 lists.
n will be the number of qubits.
Raises:
CircuitError: If A is not symmetric.
"""
adjacency_matrix = np.asarray(adjacency_matrix)
if not np.allclose(adjacency_matrix, adjacency_matrix.transpose()):
raise CircuitError("The adjacency matrix must be symmetric.")
num_qubits = len(adjacency_matrix)
circuit = QuantumCircuit(num_qubits, name="hlf: %s" % adjacency_matrix)
circuit.h(range(num_qubits))
for i in range(num_qubits):
for j in range(i + 1, num_qubits):
if adjacency_matrix[i][j]:
circuit.cz(i, j)
for i in range(num_qubits):
if adjacency_matrix[i][i]:
circuit.s(i)
circuit.h(range(num_qubits))
super().__init__(*circuit.qregs, name=circuit.name)
self.compose(circuit.to_gate(), qubits=self.qubits, inplace=True)
```