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class qiskit.circuit.library.HiddenLinearFunction(adjacency_matrix)

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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 xix_i.

The hidden linear function problem is as follows:

Consider the quadratic form

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

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

2i=1nzixi (mod 4)xKer(A)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 [z0,...,zn1][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 nn. Thus this circuit is an example of quantum advantage with shallow circuits.

Reference Circuit:



[1] S. Bravyi, D. Gosset, R. Koenig, Quantum Advantage with Shallow Circuits, 2017. arXiv:1704.00690(opens in a new tab)

Create new HLF circuit.


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


CircuitError – If A is not symmetric.



A list of AncillaQubits in the order that they were added. You should not mutate this.


Return calibration dictionary.

The custom pulse definition of a given gate is of the form {'gate_name': {(qubits, params): schedule}}


A list of Clbits in the order that they were added. You should not mutate this.


The circuit data (instructions and context).


a list-like object containing the CircuitInstructions for each instruction.

Return type



The global phase of the current circuit scope in radians.


Default value: 200


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 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.


Arbitrary user-defined metadata for the circuit.

Qiskit will not examine the content of this mapping, but it will pass it through the transpiler and reattach it to the output, so you can track your own metadata.


Return the number of ancilla qubits.


The number of real-time classical variables in the circuit marked as captured from an enclosing scope.

This is the length of the iter_captured_vars() iterable. If this is non-zero, num_input_vars must be zero.


Return number of classical bits.


The number of real-time classical variables in the circuit that are declared by this circuit scope, excluding inputs or captures.

This is the length of the iter_declared_vars() iterable.


The number of real-time classical variables in the circuit marked as circuit inputs.

This is the length of the iter_input_vars() iterable. If this is non-zero, num_captured_vars must be zero.


The number of parameter objects in the circuit.


Return number of qubits.


The number of real-time classical variables in the circuit.

This is the length of the iter_vars() iterable.


Return a list of operation start times.

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


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


AttributeError(opens in a new tab) – When circuit is not scheduled.


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.


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


The sorted Parameter objects in the circuit.


Default value: 'circuit'


A list of Qubits in the order that they were added. You should not mutate this.


Type: str

A human-readable name for the circuit.


Type: list[QuantumRegister]

A list of the QuantumRegisters in this circuit. You should not mutate this.


Type: list[ClassicalRegister]

A list of the ClassicalRegisters in this circuit. You should not mutate this.


Type: int | float | None

The total duration of the circuit, set by a scheduling transpiler pass. Its unit is specified by unit.


The unit that duration is specified in.

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