---
title: LinearAmplitudeFunctionGate (v2.0)
description: API reference for qiskit.circuit.library.LinearAmplitudeFunctionGate in qiskit v2.0
source: https://eu-de.quantum.cloud.ibm.com/docs/en/api/qiskit/2.0/qiskit.circuit.library.LinearAmplitudeFunctionGate
---

# LinearAmplitudeFunctionGate

*class* `qiskit.circuit.library.LinearAmplitudeFunctionGate(num_state_qubits, slope, offset, domain, image, rescaling_factor=1, breakpoints=None, label='F')`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/library/arithmetic/linear_amplitude_function.py#L179-L340)

Bases: [`Gate`](/docs/api/qiskit/2.0/qiskit.circuit.Gate "qiskit.circuit.gate.Gate")

A circuit implementing a (piecewise) linear function on qubit amplitudes.

An amplitude function $F$ of a function $f$ is a mapping

$$
F|x\rangle|0\rangle = \sqrt{1 - \hat{f}(x)} |x\rangle|0\rangle + \sqrt{\hat{f}(x)}
|x\rangle|1\rangle.
$$

for a function $\hat{f}: \{ 0, ..., 2^n - 1 \} \rightarrow [0, 1]$, where $|x\rangle$ is a $n$ qubit state.

This circuit implements $F$ for piecewise linear functions $\hat{f}$. In this case, the mapping $F$ can be approximately implemented using a Taylor expansion and linearly controlled Pauli-Y rotations, see \[1, 2] for more detail. This approximation uses a `rescaling_factor` to determine the accuracy of the Taylor expansion.

In general, the function of interest $f$ is defined from some interval $[a,b]$, the `domain` to $[c,d]$, the `image`, instead of $\{ 1, ..., N \}$ to $[0, 1]$. Using an affine transformation we can rescale $f$ to $\hat{f}$:

$$
\hat{f}(x) = \frac{f(\phi(x)) - c}{d - c}
$$

with

$$
\phi(x) = a + \frac{b - a}{2^n - 1} x.
$$

If $f$ is a piecewise linear function on $m$ intervals $[p_{i-1}, p_i], i \in \{1, ..., m\}$ with slopes $\alpha_i$ and offsets $\beta_i$ it can be written as

$$
f(x) = \sum_{i=1}^m 1_{[p_{i-1}, p_i]}(x) (\alpha_i x + \beta_i)
$$

where $1_{[a, b]}$ is an indication function that is 1 if the argument is in the interval $[a, b]$ and otherwise 0. The breakpoints $p_i$ can be specified by the `breakpoints` argument.

**References**

**\[1]: Woerner, S., & Egger, D. J. (2018).**

Quantum Risk Analysis. [arXiv:1806.06893](http://arxiv.org/abs/1806.06893)

**\[2]: Gacon, J., Zoufal, C., & Woerner, S. (2020).**

Quantum-Enhanced Simulation-Based Optimization. [arXiv:2005.10780](http://arxiv.org/abs/2005.10780)

**Parameters**

- **num\_state\_qubits** ([*int*](https://docs.python.org/3/library/functions.html#int)) – The number of qubits used to encode the variable $x$.
- **slope** ([*float*](https://docs.python.org/3/library/functions.html#float)  *|*[*list*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*float*](https://docs.python.org/3/library/functions.html#float)*]*) – The slope of the linear function. Can be a list of slopes if it is a piecewise linear function.
- **offset** ([*float*](https://docs.python.org/3/library/functions.html#float)  *|*[*list*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*float*](https://docs.python.org/3/library/functions.html#float)*]*) – The offset of the linear function. Can be a list of offsets if it is a piecewise linear function.
- **domain** ([*tuple*](https://docs.python.org/3/library/stdtypes.html#tuple)*\[*[*float*](https://docs.python.org/3/library/functions.html#float)*,* [*float*](https://docs.python.org/3/library/functions.html#float)*]*) – The domain of the function as tuple $(x_\min{}, x_\max{})$.
- **image** ([*tuple*](https://docs.python.org/3/library/stdtypes.html#tuple)*\[*[*float*](https://docs.python.org/3/library/functions.html#float)*,* [*float*](https://docs.python.org/3/library/functions.html#float)*]*) – The image of the function as tuple $(f_\min{}, f_\max{})$.
- **rescaling\_factor** ([*float*](https://docs.python.org/3/library/functions.html#float)) – The rescaling factor to adjust the accuracy in the Taylor approximation.
- **breakpoints** ([*list*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*float*](https://docs.python.org/3/library/functions.html#float)*] | None*) – The breakpoints if the function is piecewise linear. If None, the function is not piecewise.
- **label** ([*str*](https://docs.python.org/3/library/stdtypes.html#str)) – A label for the gate.

## Attributes

### base\_class

Get the base class of this instruction. This is guaranteed to be in the inheritance tree of `self`.

The “base class” of an instruction is the lowest class in its inheritance tree that the object should be considered entirely compatible with for \_all\_ circuit applications. This typically means that the subclass is defined purely to offer some sort of programmer convenience over the base class, and the base class is the “true” class for a behavioral perspective. In particular, you should *not* override [`base_class`](#qiskit.circuit.library.LinearAmplitudeFunctionGate.base_class "qiskit.circuit.library.LinearAmplitudeFunctionGate.base_class") if you are defining a custom version of an instruction that will be implemented differently by hardware, such as an alternative measurement strategy, or a version of a parametrized gate with a particular set of parameters for the purposes of distinguishing it in a [`Target`](/docs/api/qiskit/2.0/qiskit.transpiler.Target "qiskit.transpiler.Target") from the full parametrized gate.

This is often exactly equivalent to `type(obj)`, except in the case of singleton instances of standard-library instructions. These singleton instances are special subclasses of their base class, and this property will return that base. For example:

```python
>>> isinstance(XGate(), XGate)
True
>>> type(XGate()) is XGate
False
>>> XGate().base_class is XGate
True
```

In general, you should not rely on the precise class of an instruction; within a given circuit, it is expected that `Instruction.name` should be a more suitable discriminator in most situations.

### decompositions

Get the decompositions of the instruction from the SessionEquivalenceLibrary.

### definition

Return definition in terms of other basic gates.

### label

Return instruction label

### mutable

Is this instance is a mutable unique instance or not.

If this attribute is `False` the gate instance is a shared singleton and is not mutable.

### name

Return the name.

### num\_clbits

Return the number of clbits.

### num\_qubits

Return the number of qubits.

### params

The parameters of this `Instruction`. Ideally these will be gate angles.

## Methods

### add\_decomposition

`add_decomposition(decomposition)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/instruction.py#L321-L326)

Add a decomposition of the instruction to the SessionEquivalenceLibrary.

### broadcast\_arguments

`broadcast_arguments(qargs, cargs)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/gate.py#L188-L245)

Validation and handling of the arguments and its relationship.

For example, `cx([q[0],q[1]], q[2])` means `cx(q[0], q[2]); cx(q[1], q[2])`. This method yields the arguments in the right grouping. In the given example:

```python
in: [[q[0],q[1]], q[2]],[]
outs: [q[0], q[2]], []
      [q[1], q[2]], []
```

The general broadcasting rules are:

> - If len(qargs) == 1:
>
>   ```python
>   [q[0], q[1]] -> [q[0]],[q[1]]
>   ```
>
> - If len(qargs) == 2:
>
>   ```python
>   [[q[0], q[1]], [r[0], r[1]]] -> [q[0], r[0]], [q[1], r[1]]
>   [[q[0]], [r[0], r[1]]]       -> [q[0], r[0]], [q[0], r[1]]
>   [[q[0], q[1]], [r[0]]]       -> [q[0], r[0]], [q[1], r[0]]
>   ```
>
> - If len(qargs) >= 3:
>
>   ```python
>   [q[0], q[1]], [r[0], r[1]],  ...] -> [q[0], r[0], ...], [q[1], r[1], ...]
>   ```

**Parameters**

- **qargs** ([*list*](https://docs.python.org/3/library/stdtypes.html#list)) – List of quantum bit arguments.
- **cargs** ([*list*](https://docs.python.org/3/library/stdtypes.html#list)) – List of classical bit arguments.

**Returns**

A tuple with single arguments.

**Raises**

[**CircuitError**](/docs/api/qiskit/2.0/circuit#qiskit.circuit.CircuitError "qiskit.circuit.CircuitError") – If the input is not valid. For example, the number of arguments does not match the gate expectation.

**Return type**

[*Iterable*](https://docs.python.org/3/library/typing.html#typing.Iterable)\[[tuple](https://docs.python.org/3/library/stdtypes.html#tuple)\[[list](https://docs.python.org/3/library/stdtypes.html#list), [list](https://docs.python.org/3/library/stdtypes.html#list)]]

### control

`control(num_ctrl_qubits=1, label=None, ctrl_state=None, annotated=None)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/gate.py#L105-L145)

Return the controlled version of itself.

Implemented either as a controlled gate (ref. [`ControlledGate`](/docs/api/qiskit/2.0/qiskit.circuit.ControlledGate "qiskit.circuit.ControlledGate")) or as an annotated operation (ref. [`AnnotatedOperation`](/docs/api/qiskit/2.0/qiskit.circuit.AnnotatedOperation "qiskit.circuit.AnnotatedOperation")).

**Parameters**

- **num\_ctrl\_qubits** ([*int*](https://docs.python.org/3/library/functions.html#int)) – number of controls to add to gate (default: `1`)
- **label** ([*str*](https://docs.python.org/3/library/stdtypes.html#str) *| None*) – optional gate label. Ignored if implemented as an annotated operation.
- **ctrl\_state** ([*int*](https://docs.python.org/3/library/functions.html#int)  *|*[*str*](https://docs.python.org/3/library/stdtypes.html#str) *| None*) – the control state in decimal or as a bitstring (e.g. `'111'`). If `None`, use `2**num_ctrl_qubits-1`.
- **annotated** ([*bool*](https://docs.python.org/3/library/functions.html#bool) *| None*) – indicates whether the controlled gate is implemented as an annotated gate. If `None`, this is set to `False` if the controlled gate can directly be constructed, and otherwise set to `True`. This allows defering the construction process in case the synthesis of the controlled gate requires more information (e.g. values of unbound parameters).

**Returns**

Controlled version of the given operation.

**Raises**

[**QiskitError**](/docs/api/qiskit/2.0/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – unrecognized mode or invalid ctrl\_state

### copy

`copy(name=None)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/instruction.py#L431-L446)

Copy of the instruction.

**Parameters**

**name** ([*str*](https://docs.python.org/3/library/stdtypes.html#str)) – name to be given to the copied circuit, if `None` then the name stays the same.

**Returns**

a copy of the current instruction, with the name updated if it was provided

**Return type**

[qiskit.circuit.Instruction](/docs/api/qiskit/2.0/qiskit.circuit.Instruction "qiskit.circuit.Instruction")

### inverse

`inverse(annotated=False)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/instruction.py#L374-L429)

Invert this instruction.

If annotated is False, the inverse instruction is implemented as a fresh instruction with the recursively inverted definition.

If annotated is True, the inverse instruction is implemented as [`AnnotatedOperation`](/docs/api/qiskit/2.0/qiskit.circuit.AnnotatedOperation "qiskit.circuit.AnnotatedOperation"), and corresponds to the given instruction annotated with the “inverse modifier”.

Special instructions inheriting from Instruction can implement their own inverse (e.g. T and Tdg, Barrier, etc.) In particular, they can choose how to handle the argument `annotated` which may include ignoring it and always returning a concrete gate class if the inverse is defined as a standard gate.

**Parameters**

**annotated** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – if set to True the output inverse gate will be returned as [`AnnotatedOperation`](/docs/api/qiskit/2.0/qiskit.circuit.AnnotatedOperation "qiskit.circuit.AnnotatedOperation").

**Returns**

The inverse operation.

**Raises**

[**CircuitError**](/docs/api/qiskit/2.0/circuit#qiskit.circuit.CircuitError "qiskit.circuit.CircuitError") – if the instruction is not composite and an inverse has not been implemented for it.

### is\_parameterized

`is_parameterized()`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/instruction.py#L286-L291)

Return whether the `Instruction` contains [compile-time parameters](/docs/api/qiskit/2.0/circuit#circuit-compile-time-parameters).

### post\_processing

`post_processing(scaled_value)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/library/arithmetic/linear_amplitude_function.py#L321-L340)

Map the function value of the approximated $\hat{f}$ to $f$.

**Parameters**

**scaled\_value** ([*float*](https://docs.python.org/3/library/functions.html#float)) – A function value from the Taylor expansion of $\hat{f}(x)$.

**Returns**

The `scaled_value` mapped back to the domain of $f$, by first inverting the transformation used for the Taylor approximation and then mapping back from $[0, 1]$ to the original domain.

**Return type**

[float](https://docs.python.org/3/library/functions.html#float)

### power

`power(exponent, annotated=False)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/gate.py#L63-L94)

Raise this gate to the power of `exponent`.

Implemented either as a unitary gate (ref. [`UnitaryGate`](/docs/api/qiskit/2.0/qiskit.circuit.library.UnitaryGate "qiskit.circuit.library.UnitaryGate")) or as an annotated operation (ref. [`AnnotatedOperation`](/docs/api/qiskit/2.0/qiskit.circuit.AnnotatedOperation "qiskit.circuit.AnnotatedOperation")). In the case of several standard gates, such as [`RXGate`](/docs/api/qiskit/2.0/qiskit.circuit.library.RXGate "qiskit.circuit.library.RXGate"), when the power of a gate can be expressed in terms of another standard gate that is returned directly.

**Parameters**

- **exponent** ([*float*](https://docs.python.org/3/library/functions.html#float)) – the power to raise the gate to
- **annotated** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – indicates whether the power gate can be implemented as an annotated operation. In the case of several standard gates, such as [`RXGate`](/docs/api/qiskit/2.0/qiskit.circuit.library.RXGate "qiskit.circuit.library.RXGate"), this argument is ignored when the power of a gate can be expressed in terms of another standard gate.

**Returns**

An operation implementing `gate^exponent`

**Raises**

[**CircuitError**](/docs/api/qiskit/2.0/circuit#qiskit.circuit.CircuitError "qiskit.circuit.CircuitError") – If gate is not unitary

### repeat

`repeat(n)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/instruction.py#L494-L524)

Creates an instruction with `self` repeated :math\`n\` times.

**Parameters**

**n** ([*int*](https://docs.python.org/3/library/functions.html#int)) – Number of times to repeat the instruction

**Returns**

Containing the definition.

**Return type**

[qiskit.circuit.Instruction](/docs/api/qiskit/2.0/qiskit.circuit.Instruction "qiskit.circuit.Instruction")

**Raises**

[**CircuitError**](/docs/api/qiskit/2.0/circuit#qiskit.circuit.CircuitError "qiskit.circuit.CircuitError") – If n \< 1.

### reverse\_ops

`reverse_ops()`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/instruction.py#L348-L372)

For a composite instruction, reverse the order of sub-instructions.

This is done by recursively reversing all sub-instructions. It does not invert any gate.

**Returns**

**a new instruction with**

sub-instructions reversed.

**Return type**

[qiskit.circuit.Instruction](/docs/api/qiskit/2.0/qiskit.circuit.Instruction "qiskit.circuit.Instruction")

### soft\_compare

`soft_compare(other)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/instruction.py#L217-L257)

Soft comparison between gates. Their names, number of qubits, and classical bit numbers must match. The number of parameters must match. Each parameter is compared. If one is a ParameterExpression then it is not taken into account.

**Parameters**

**other** (*instruction*) – other instruction.

**Returns**

are self and other equal up to parameter expressions.

**Return type**

[bool](https://docs.python.org/3/library/functions.html#bool)

### to\_matrix

`to_matrix()`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/gate.py#L49-L61)

Return a Numpy.array for the gate unitary matrix.

**Returns**

if the Gate subclass has a matrix definition.

**Return type**

np.ndarray

**Raises**

[**CircuitError**](/docs/api/qiskit/2.0/circuit#qiskit.circuit.CircuitError "qiskit.circuit.CircuitError") – If a Gate subclass does not implement this method an exception will be raised when this base class method is called.

### to\_mutable

`to_mutable()`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/instruction.py#L146-L154)

Return a mutable copy of this gate.

This method will return a new mutable copy of this gate instance. If a singleton instance is being used this will be a new unique instance that can be mutated. If the instance is already mutable it will be a deepcopy of that instance.

### validate\_parameter

`validate_parameter(parameter)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.0/qiskit/circuit/gate.py#L247-L261)

Gate parameters should be int, float, or ParameterExpression