HiddenLinearFunction

class qiskit.circuit.library.HiddenLinearFunction(adjacency_matrix)[source]

Bases: QuantumCircuit

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 \(x_i\).

The hidden linear function problem is as follows:

Consider the quadratic form

\[q(x) = \sum_{i,j=1}^{n}{x_i x_j} ~(\mathrm{mod}~ 4)\]

and restrict \(q(x)\) onto the nullspace of A. This results in a linear function.

\[2 \sum_{i=1}^{n}{z_i x_i} ~(\mathrm{mod}~ 4) \forall x \in \mathrm{Ker}(A)\]

and the goal is to recover this linear function (equivalently a vector \([z_0, ..., z_{n-1}]\)). 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 \(n\). Thus this circuit is an example of quantum advantage with shallow circuits.

Reference Circuit:

Reference:

[1] S. Bravyi, D. Gosset, R. Koenig, Quantum Advantage with Shallow Circuits, 2017. arXiv:1704.00690

Create new HLF circuit.

Parameters:

adjacency_matrix (List[List[int]] | ndarray) – a symmetric n-by-n list of 0-1 lists. n will be the number of qubits.

Raises:

CircuitError – If A is not symmetric.

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 CircuitInstructions for each instruction.

Return type:

QuantumCircuitData

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

Return the global phase of the current circuit scope in radians.

header = 'OPENQASM 2.0;'
instances = 161
layout

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.

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

The number of parameter objects in the circuit.

num_qubits

Return number of qubits.

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.

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