# QuadraticForm

`qiskit.circuit.library.QuadraticForm(num_result_qubits=None, quadratic=None, linear=None, offset=None, little_endian=True)`

Bases: `QuantumCircuit`

Implements a quadratic form on binary variables encoded in qubit registers.

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

If $A$, $b$ or $c$ contain scalar values, this circuit computes only an approximation of the quadratic form.

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 (opens in a new tab)) representation.

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.

If the value of $Q(x)$ is too large to be represented with m qubits, the resulting bitstring is $(Q(x) + 2^m) \mod 2^m)$.

The implementation of this circuit is discussed in [1], Fig. 6.

## References

**[1]: Gilliam et al., Grover Adaptive Search for Constrained Polynomial Binary Optimization.**

arXiv:1912.04088 (opens in a new tab)

**Parameters**

**num_result_qubits**(*int*(opens in a new tab)*| None*) – The number of qubits to encode the result. Called $m$ in the class documentation.**quadratic**(*ndarray*(opens in a new tab)*|**List*(opens in a new tab)*[**List*(opens in a new tab)*[**float*(opens in a new tab)*|**ParameterExpression**]] | None*) – A matrix containing the quadratic coefficients, $A$.**linear**(*ndarray*(opens in a new tab)*|**List*(opens in a new tab)*[**float*(opens in a new tab)*|**ParameterExpression**] | None*) – An array containing the linear coefficients, $b$.**offset**(*ParameterExpression**|**float*(opens in a new tab)*| None*) – A constant offset, $c$.**little_endian**(*bool*(opens in a new tab)) – Encode the result in little endianness.

**Raises**

**ValueError**(opens in a new tab) – If`linear`

and`quadratic`

have mismatching sizes.**ValueError**(opens in a new tab) – If`num_result_qubits`

is unspecified but cannot be determined because some values of the quadratic form are parameterized.

## 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 `CircuitInstruction`

s 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

`= 378`

### 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 `SwapGate`

s 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** (opens in a new tab) – 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.

## Methods

### required_result_qubits

`static required_result_qubits(quadratic, linear, offset)`

Get the number of required result qubits.

**Parameters**

**quadratic**(*ndarray*(opens in a new tab)*|**List*(opens in a new tab)*[**List*(opens in a new tab)*[**float*(opens in a new tab)*]]*) – A matrix containing the quadratic coefficients.**linear**(*ndarray*(opens in a new tab)*|**List*(opens in a new tab)*[**float*(opens in a new tab)*]*) – An array containing the linear coefficients.**offset**(*float*(opens in a new tab)) – A constant offset.

**Returns**

The number of qubits needed to represent the value of the quadratic form in twos complement.

**Return type**