323 lines
15 KiB
Plaintext
323 lines
15 KiB
Plaintext
---
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title: LinearAmplitudeFunction (latest version)
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description: API reference for qiskit.circuit.library.LinearAmplitudeFunction in the latest version of qiskit
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in_page_toc_min_heading_level: 1
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python_api_type: class
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python_api_name: qiskit.circuit.library.LinearAmplitudeFunction
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---
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# LinearAmplitudeFunction
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<Class id="qiskit.circuit.library.LinearAmplitudeFunction" isDedicatedPage={true} github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/circuit/library/arithmetic/linear_amplitude_function.py#L22-L173" signature="qiskit.circuit.library.LinearAmplitudeFunction(num_state_qubits, slope, offset, domain, image, rescaling_factor=1, breakpoints=None, name='F')" modifiers="class">
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Bases: [`QuantumCircuit`](qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.QuantumCircuit")
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A circuit implementing a (piecewise) linear function on qubit amplitudes.
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An amplitude function $F$ of a function $f$ is a mapping
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$$
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F|x\rangle|0\rangle = \sqrt{1 - \hat{f}(x)} |x\rangle|0\rangle + \sqrt{\hat{f}(x)}
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|x\rangle|1\rangle.
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$$
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for a function $\hat{f}: \{ 0, ..., 2^n - 1 \} \rightarrow [0, 1]$, where $|x\rangle$ is a $n$ qubit state.
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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.
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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 $\hat{f}$:
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$$
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\hat{f}(x) = \frac{f(\phi(x)) - c}{d - c}
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$$
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with
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$$
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\phi(x) = a + \frac{b - a}{2^n - 1} x.
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$$
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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
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$$
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f(x) = \sum_{i=1}^m 1_{[p_{i-1}, p_i]}(x) (\alpha_i x + \beta_i)
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$$
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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.
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**References**
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**\[1]: Woerner, S., & Egger, D. J. (2018).**
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Quantum Risk Analysis. [arXiv:1806.06893](http://arxiv.org/abs/1806.06893)
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**\[2]: Gacon, J., Zoufal, C., & Woerner, S. (2020).**
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Quantum-Enhanced Simulation-Based Optimization. [arXiv:2005.10780](http://arxiv.org/abs/2005.10780)
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**Parameters**
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* **num\_state\_qubits** ([*int*](https://docs.python.org/3/library/functions.html#int "(in Python v3.13)")) – The number of qubits used to encode the variable $x$.
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* **slope** ([*float*](https://docs.python.org/3/library/functions.html#float "(in Python v3.13)") *|*[*list*](https://docs.python.org/3/library/stdtypes.html#list "(in Python v3.13)")*\[*[*float*](https://docs.python.org/3/library/functions.html#float "(in Python v3.13)")*]*) – The slope of the linear function. Can be a list of slopes if it is a piecewise linear function.
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* **offset** ([*float*](https://docs.python.org/3/library/functions.html#float "(in Python v3.13)") *|*[*list*](https://docs.python.org/3/library/stdtypes.html#list "(in Python v3.13)")*\[*[*float*](https://docs.python.org/3/library/functions.html#float "(in Python v3.13)")*]*) – The offset of the linear function. Can be a list of offsets if it is a piecewise linear function.
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* **domain** ([*tuple*](https://docs.python.org/3/library/stdtypes.html#tuple "(in Python v3.13)")*\[*[*float*](https://docs.python.org/3/library/functions.html#float "(in Python v3.13)")*,* [*float*](https://docs.python.org/3/library/functions.html#float "(in Python v3.13)")*]*) – The domain of the function as tuple $(x_\min{}, x_\max{})$.
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* **image** ([*tuple*](https://docs.python.org/3/library/stdtypes.html#tuple "(in Python v3.13)")*\[*[*float*](https://docs.python.org/3/library/functions.html#float "(in Python v3.13)")*,* [*float*](https://docs.python.org/3/library/functions.html#float "(in Python v3.13)")*]*) – The image of the function as tuple $(f_\min{}, f_\max{})$.
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* **rescaling\_factor** ([*float*](https://docs.python.org/3/library/functions.html#float "(in Python v3.13)")) – The rescaling factor to adjust the accuracy in the Taylor approximation.
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* **breakpoints** ([*list*](https://docs.python.org/3/library/stdtypes.html#list "(in Python v3.13)")*\[*[*float*](https://docs.python.org/3/library/functions.html#float "(in Python v3.13)")*] | None*) – The breakpoints if the function is piecewise linear. If None, the function is not piecewise.
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* **name** ([*str*](https://docs.python.org/3/library/stdtypes.html#str "(in Python v3.13)")) – Name of the circuit.
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## Attributes
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### ancillas
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.ancillas">
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A list of `AncillaQubit`s in the order that they were added. You should not mutate this.
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</Attribute>
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### calibrations
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.calibrations">
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Return calibration dictionary.
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The custom pulse definition of a given gate is of the form `{'gate_name': {(qubits, params): schedule}}`
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</Attribute>
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### clbits
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.clbits">
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A list of `Clbit`s in the order that they were added. You should not mutate this.
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</Attribute>
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### data
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.data">
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The circuit data (instructions and context).
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**Returns**
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a list-like object containing the [`CircuitInstruction`](qiskit.circuit.CircuitInstruction "qiskit.circuit.CircuitInstruction")s for each instruction.
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**Return type**
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QuantumCircuitData
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</Attribute>
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### global\_phase
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.global_phase">
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The global phase of the current circuit scope in radians.
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</Attribute>
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### instances
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.instances" attributeValue="178" />
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### layout
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.layout">
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Return any associated layout information about the circuit
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This attribute contains an optional [`TranspileLayout`](qiskit.transpiler.TranspileLayout "qiskit.transpiler.TranspileLayout") object. This is typically set on the output from [`transpile()`](compiler#qiskit.compiler.transpile "qiskit.compiler.transpile") or [`PassManager.run()`](qiskit.transpiler.PassManager#run "qiskit.transpiler.PassManager.run") to retain information about the permutations caused on the input circuit by transpilation.
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There are two types of permutations caused by the [`transpile()`](compiler#qiskit.compiler.transpile "qiskit.compiler.transpile") function, an initial layout which permutes the qubits based on the selected physical qubits on the [`Target`](qiskit.transpiler.Target "qiskit.transpiler.Target"), and a final layout which is an output permutation caused by [`SwapGate`](qiskit.circuit.library.SwapGate "qiskit.circuit.library.SwapGate")s inserted during routing.
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</Attribute>
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### metadata
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.metadata">
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Arbitrary user-defined metadata for the circuit.
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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.
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</Attribute>
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### num\_ancillas
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.num_ancillas">
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Return the number of ancilla qubits.
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</Attribute>
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### num\_captured\_vars
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.num_captured_vars">
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The number of real-time classical variables in the circuit marked as captured from an enclosing scope.
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This is the length of the `iter_captured_vars()` iterable. If this is non-zero, [`num_input_vars`](#qiskit.circuit.library.LinearAmplitudeFunction.num_input_vars "qiskit.circuit.library.LinearAmplitudeFunction.num_input_vars") must be zero.
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</Attribute>
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### num\_clbits
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.num_clbits">
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Return number of classical bits.
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</Attribute>
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### num\_declared\_vars
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.num_declared_vars">
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The number of real-time classical variables in the circuit that are declared by this circuit scope, excluding inputs or captures.
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This is the length of the `iter_declared_vars()` iterable.
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</Attribute>
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### num\_input\_vars
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.num_input_vars">
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The number of real-time classical variables in the circuit marked as circuit inputs.
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This is the length of the `iter_input_vars()` iterable. If this is non-zero, [`num_captured_vars`](#qiskit.circuit.library.LinearAmplitudeFunction.num_captured_vars "qiskit.circuit.library.LinearAmplitudeFunction.num_captured_vars") must be zero.
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</Attribute>
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### num\_parameters
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.num_parameters">
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The number of parameter objects in the circuit.
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</Attribute>
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### num\_qubits
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.num_qubits">
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Return number of qubits.
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</Attribute>
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### num\_vars
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.num_vars">
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The number of real-time classical variables in the circuit.
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This is the length of the `iter_vars()` iterable.
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</Attribute>
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### op\_start\_times
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.op_start_times">
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Return a list of operation start times.
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This attribute is enabled once one of scheduling analysis passes runs on the quantum circuit.
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**Returns**
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List of integers representing instruction start times. The index corresponds to the index of instruction in `QuantumCircuit.data`.
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**Raises**
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[**AttributeError**](https://docs.python.org/3/library/exceptions.html#AttributeError "(in Python v3.13)") – When circuit is not scheduled.
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</Attribute>
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### parameters
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.parameters">
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The parameters defined in the circuit.
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This attribute returns the [`Parameter`](qiskit.circuit.Parameter "qiskit.circuit.Parameter") objects in the circuit sorted alphabetically. Note that parameters instantiated with a [`ParameterVector`](qiskit.circuit.ParameterVector "qiskit.circuit.ParameterVector") are still sorted numerically.
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**Examples**
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The snippet below shows that insertion order of parameters does not matter.
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```python
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>>> from qiskit.circuit import QuantumCircuit, Parameter
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>>> a, b, elephant = Parameter("a"), Parameter("b"), Parameter("elephant")
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>>> circuit = QuantumCircuit(1)
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>>> circuit.rx(b, 0)
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>>> circuit.rz(elephant, 0)
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>>> circuit.ry(a, 0)
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>>> circuit.parameters # sorted alphabetically!
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ParameterView([Parameter(a), Parameter(b), Parameter(elephant)])
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```
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Bear in mind that alphabetical sorting might be unintuitive when it comes to numbers. The literal “10” comes before “2” in strict alphabetical sorting.
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```python
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>>> from qiskit.circuit import QuantumCircuit, Parameter
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>>> angles = [Parameter("angle_1"), Parameter("angle_2"), Parameter("angle_10")]
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>>> circuit = QuantumCircuit(1)
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>>> circuit.u(*angles, 0)
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>>> circuit.draw()
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┌─────────────────────────────┐
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q: ┤ U(angle_1,angle_2,angle_10) ├
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└─────────────────────────────┘
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>>> circuit.parameters
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ParameterView([Parameter(angle_1), Parameter(angle_10), Parameter(angle_2)])
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```
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To respect numerical sorting, a [`ParameterVector`](qiskit.circuit.ParameterVector "qiskit.circuit.ParameterVector") can be used.
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```python
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>>> from qiskit.circuit import QuantumCircuit, Parameter, ParameterVector
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>>> x = ParameterVector("x", 12)
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>>> circuit = QuantumCircuit(1)
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>>> for x_i in x:
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... circuit.rx(x_i, 0)
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>>> circuit.parameters
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ParameterView([
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ParameterVectorElement(x[0]), ParameterVectorElement(x[1]),
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ParameterVectorElement(x[2]), ParameterVectorElement(x[3]),
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..., ParameterVectorElement(x[11])
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])
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```
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**Returns**
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The sorted [`Parameter`](qiskit.circuit.Parameter "qiskit.circuit.Parameter") objects in the circuit.
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</Attribute>
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### prefix
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.prefix" attributeValue="'circuit'" />
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### qubits
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.qubits">
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A list of `Qubit`s in the order that they were added. You should not mutate this.
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</Attribute>
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### name
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.name" attributeTypeHint="str">
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A human-readable name for the circuit.
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</Attribute>
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### qregs
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.qregs" attributeTypeHint="list[QuantumRegister]">
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A list of the `QuantumRegister`s in this circuit. You should not mutate this.
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</Attribute>
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### cregs
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.cregs" attributeTypeHint="list[ClassicalRegister]">
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A list of the `ClassicalRegister`s in this circuit. You should not mutate this.
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</Attribute>
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### duration
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.duration" attributeTypeHint="int | float | None">
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The total duration of the circuit, set by a scheduling transpiler pass. Its unit is specified by [`unit`](#qiskit.circuit.library.LinearAmplitudeFunction.unit "qiskit.circuit.library.LinearAmplitudeFunction.unit").
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</Attribute>
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### unit
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<Attribute id="qiskit.circuit.library.LinearAmplitudeFunction.unit">
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The unit that [`duration`](#qiskit.circuit.library.LinearAmplitudeFunction.duration "qiskit.circuit.library.LinearAmplitudeFunction.duration") is specified in.
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</Attribute>
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## Methods
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### post\_processing
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<Function id="qiskit.circuit.library.LinearAmplitudeFunction.post_processing" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/circuit/library/arithmetic/linear_amplitude_function.py#L154-L173" signature="post_processing(scaled_value)">
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Map the function value of the approximated $\hat{f}$ to $f$.
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**Parameters**
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**scaled\_value** ([*float*](https://docs.python.org/3/library/functions.html#float "(in Python v3.13)")) – A function value from the Taylor expansion of $\hat{f}(x)$.
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**Returns**
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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.
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**Return type**
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[float](https://docs.python.org/3/library/functions.html#float "(in Python v3.13)")
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</Function>
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</Class>
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