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

GitHub(opens in a new tab)

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

$Q(x) = x^T A x + x^T b + c$

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.

$|x\rangle_n |0\rangle_m \mapsto |x\rangle_n |(Q(x) + 2^m) \mod 2^m \rangle_m$

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

Raises

## Attributes

### ancillas

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

### calibrations

Return calibration dictionary.

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

### clbits

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

### data

The circuit data (instructions and context).

Returns

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

Return type

QuantumCircuitData

### global_phase

The global phase of the current circuit scope in radians.

### instances

Default value: 328

### 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 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.

### num_ancillas

Return the number of ancilla qubits.

### num_captured_vars

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.

### num_clbits

Return number of classical bits.

### num_declared_vars

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.

### num_input_vars

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.

### num_parameters

The number of parameter objects in the circuit.

### num_qubits

Return number of qubits.

### num_vars

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

This is the length of the iter_vars() iterable.

### 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

Default value: 'circuit'

### qubits

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

### name

Type: str

A human-readable name for the circuit.

### qregs

Type: list[QuantumRegister]

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

### cregs

Type: list[ClassicalRegister]

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

### duration

Type: int | float | None

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

### unit

The unit that duration is specified in.

## Methods

### required_result_qubits

static required_result_qubits(quadratic, linear, offset)

GitHub(opens in a new tab)

Get the number of required result qubits.

Parameters

Returns

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

Return type

int(opens in a new tab)