184 lines
6.8 KiB
Plaintext
184 lines
6.8 KiB
Plaintext
---
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title: QuadraticForm
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description: API reference for qiskit.circuit.library.QuadraticForm
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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.QuadraticForm
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---
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# QuadraticForm
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<Class id="qiskit.circuit.library.QuadraticForm" isDedicatedPage={true} github="https://github.com/qiskit/qiskit/tree/stable/0.18/qiskit/circuit/library/arithmetic/quadratic_form.py" signature="QuadraticForm(num_result_qubits=None, quadratic=None, linear=None, offset=None, little_endian=True)" modifiers="class">
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Bases: `qiskit.circuit.quantumcircuit.QuantumCircuit`
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Implements a quadratic form on binary variables encoded in qubit registers.
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A quadratic form on binary variables is a quadratic function $Q$ acting on a binary variable of $n$ bits, $x = x_0 ... x_{n-1}$. For an integer matrix $A$, an integer vector $b$ and an integer $c$ the function can be written as
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$$
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Q(x) = x^T A x + x^T b + c
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$$
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If $A$, $b$ or $c$ contain scalar values, this circuit computes only an approximation of the quadratic form.
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Provided with $m$ qubits to encode the value, this circuit computes $Q(x) \mod 2^m$ in \[two’s complement]\([https://stackoverflow.com/questions/1049722/what-is-2s-complement](https://stackoverflow.com/questions/1049722/what-is-2s-complement)) representation.
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$$
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|x\rangle_n |0\rangle_m \mapsto |x\rangle_n |(Q(x) + 2^m) \mod 2^m \rangle_m
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$$
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Since we use two’s complement e.g. the value of $Q(x) = 3$ requires 2 bits to represent the value and 1 bit for the sign: 3 = ‘011’ where the first 0 indicates a positive value. On the other hand, $Q(x) = -3$ would be -3 = ‘101’, where the first 1 indicates a negative value and 01 is the two’s complement of 3.
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If the value of $Q(x)$ is too large to be represented with m qubits, the resulting bitstring is $(Q(x) + 2^m) \mod 2^m)$.
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The implementation of this circuit is discussed in \[1], Fig. 6.
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**References**
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**\[1]: Gilliam et al., Grover Adaptive Search for Constrained Polynomial Binary Optimization.**
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[arXiv:1912.04088](https://arxiv.org/pdf/1912.04088.pdf)
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**Parameters**
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* **num\_result\_qubits** (`Optional`\[`int`]) – The number of qubits to encode the result. Called $m$ in the class documentation.
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* **quadratic** (`Union`\[`ndarray`, `List`\[`List`\[`Union`\[`float`, `ParameterExpression`]]], `None`]) – A matrix containing the quadratic coefficients, $A$.
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* **linear** (`Union`\[`ndarray`, `List`\[`Union`\[`float`, `ParameterExpression`]], `None`]) – An array containing the linear coefficients, $b$.
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* **offset** (`Union`\[`float`, `ParameterExpression`, `None`]) – A constant offset, $c$.
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* **little\_endian** (`bool`) – Encode the result in little endianness.
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**Raises**
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* **ValueError** – If `linear` and `quadratic` have mismatching sizes.
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* **ValueError** – If `num_result_qubits` is unspecified but cannot be determined because some values of the quadratic form are parameterized.
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## Methods Defined Here
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### required\_result\_qubits
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<Function id="qiskit.circuit.library.QuadraticForm.required_result_qubits" signature="QuadraticForm.required_result_qubits(quadratic, linear, offset)" modifiers="static">
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Get the number of required result qubits.
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**Parameters**
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* **quadratic** (`Union`\[`ndarray`, `List`\[`List`\[`float`]]]) – A matrix containing the quadratic coefficients.
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* **linear** (`Union`\[`ndarray`, `List`\[`float`]]) – An array containing the linear coefficients.
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* **offset** (`float`) – A constant offset.
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**Return type**
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`int`
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**Returns**
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The number of qubits needed to represent the value of the quadratic form in twos complement.
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</Function>
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## Attributes
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### ancillas
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<Attribute id="qiskit.circuit.library.QuadraticForm.ancillas">
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Returns a list of ancilla bits in the order that the registers were added.
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</Attribute>
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### calibrations
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<Attribute id="qiskit.circuit.library.QuadraticForm.calibrations">
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Return calibration dictionary.
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**The custom pulse definition of a given gate is of the form**
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\{‘gate\_name’: \{(qubits, params): schedule}}
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</Attribute>
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### clbits
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<Attribute id="qiskit.circuit.library.QuadraticForm.clbits">
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Returns a list of classical bits in the order that the registers were added.
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</Attribute>
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### data
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<Attribute id="qiskit.circuit.library.QuadraticForm.data">
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Return the circuit data (instructions and context).
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**Returns**
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a list-like object containing the tuples for the circuit’s data.
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Each tuple is in the format `(instruction, qargs, cargs)`, where instruction is an Instruction (or subclass) object, qargs is a list of Qubit objects, and cargs is a list of Clbit objects.
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**Return type**
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QuantumCircuitData
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</Attribute>
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### extension\_lib
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<Attribute id="qiskit.circuit.library.QuadraticForm.extension_lib" attributeValue="'include "qelib1.inc";'" />
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### global\_phase
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<Attribute id="qiskit.circuit.library.QuadraticForm.global_phase">
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Return the global phase of the circuit in radians.
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</Attribute>
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### header
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<Attribute id="qiskit.circuit.library.QuadraticForm.header" attributeValue="'OPENQASM 2.0;'" />
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### instances
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<Attribute id="qiskit.circuit.library.QuadraticForm.instances" attributeValue="16" />
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### metadata
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<Attribute id="qiskit.circuit.library.QuadraticForm.metadata">
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The user provided metadata associated with the circuit
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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.
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</Attribute>
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### num\_ancillas
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<Attribute id="qiskit.circuit.library.QuadraticForm.num_ancillas">
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Return the number of ancilla qubits.
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</Attribute>
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### num\_clbits
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<Attribute id="qiskit.circuit.library.QuadraticForm.num_clbits">
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Return number of classical bits.
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</Attribute>
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### num\_parameters
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<Attribute id="qiskit.circuit.library.QuadraticForm.num_parameters">
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Convenience function to get 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.QuadraticForm.num_qubits">
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Return number of qubits.
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</Attribute>
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### parameters
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<Attribute id="qiskit.circuit.library.QuadraticForm.parameters">
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Convenience function to get the parameters defined in the parameter table.
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</Attribute>
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### prefix
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<Attribute id="qiskit.circuit.library.QuadraticForm.prefix" attributeValue="'circuit'" />
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### qubits
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<Attribute id="qiskit.circuit.library.QuadraticForm.qubits">
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Returns a list of quantum bits in the order that the registers were added.
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</Attribute>
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</Class>
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