quadratic_form

quadratic_form#

Functionality#

This program implements a quadratic form on binary variables encoded in qubit registers.

A quadratic form on binary variables is a quadratic function \(Q(x)\) acting on a binary variable of \(n\) bits, \(x=x_{n-1}\cdots x_n\). For an integer matrix \(A\), an integer vector \(b\) and an integer \(c\), the function can be written as

\[ Q(x) = x^{\top}Ax + x^{\top}b + c \]

If \(A\in\mathbb{Z}^{n \times n}, b\in\mathbb{Z}^{n \times 1}\) or \(c\in\mathbb{Z}\) contain scalar values, this circuit computes only an approximation of the quadratic form.

quadratic_form.py#

Callee: QuadraticForm

Provided with \(m\) qubits to encode the value, this circuit computes \(Q(x) \mod 2^m\) in two’s complement representation:

\[ U_{\text{QuadraticForm}}: \ket{0}_m\ket{x}_n \mapsto \ket{(Q(x) + 2^m) \mod 2^m}_m\ket{x}_n \]

We use two’s complement bit pattern to interpret the \(m\)-bit output as a signed integer. If the most significant bit is 0, the positive value is stored as \(Q(x)\mod 2^m\). Otherwise, the negative value is stored as \((Q(x) + 2^m) \mod 2^m\). Equivalently, the rule is:

For example, the value of \(Q(x)=3\) is ‘011’, where the first 0 indicates a positive value and 11 is the value 3. \(Q(x)=-3\) would be transformed as \(-3+2^3=5\), i.e., ‘101’.

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 [32], Fig. 6.

Test expectations:

The probability of the basis state which corresponds with bitstring \((Q(x) + 2^m) \mod 2^m\) will be 1 upon measurement. When \(Q(x)\) is negative, the corresponding state is \(2^m+Q(x)\).

Doc reference: [33] [34]

API#

qolumbina.programs.quadratic_form.quadratic_form#

A circuit implementing a quadratic form on binary variables.

class QuadraticForm(num_result_qubits=None, quadratic=None, linear=None, offset=None, little_endian=True)[source]#

Bases: QuantumCircuit

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

Parameters:

  • num_result_qubits (Optional[int]) – The number of qubits used to encode the result of the quadratic form, i.e., \(m\).

  • quadratic (Optional[Union[np.ndarray, List[List[Union[float, ParameterExpression]]]]]) – The quadratic coefficients matrix \(A\). The default is None. If None, it is assumed to be a zero matrix.

  • linear (Optional[Union[np.ndarray, List[Union[float, ParameterExpression]]]]) – The linear coefficients vector \(b\). The default is None. If None, it is assumed to be a zero vector.

  • offset (Optional[Union[float, ParameterExpression]]) – the offset scalar \(c\). The default is None. If None, it is assumed to be zero.

  • little_endian (bool) – Whether the input qubits are in little-endian order. Default is True.

Raises:

  • ValueError – If linear and quadratic have mismatching sizes.

  • ValueError – If num_result_qubits is unspecified but cannot be determined because some values of the quadratic form are parameterized.

static required_result_qubits(quadratic, linear, offset)[source]#

Get the number of required result qubits.

Parameters:

  • quadratic (ndarray | List[List[float]]) – A matrix containing the quadratic coefficients.

  • linear (ndarray | List[float]) – An array containing the linear coefficients.

  • offset (float) – A constant offset.

Returns:

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

Return type:

int

class QuadraticFormGate(num_result_qubits=None, quadratic=None, linear=None, offset=None, label='Q(x)')[source]#

Bases: Gate

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

Parameters:

  • num_result_qubits (int | None)

  • quadratic (Sequence[Sequence[float]] | None)

  • linear (Sequence[Sequence[float]] | None)

  • offset (float | None)

  • label (str)

static required_result_qubits(quadratic, linear, offset)[source]#

Get the number of required result qubits.

Parameters:

  • quadratic (Sequence[Sequence[float]]) – A matrix containing the quadratic coefficients.

  • linear (Sequence[float]) – An array containing the linear coefficients.

  • offset (float) – A constant offset.

Returns:

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

Return type:

int

validate_parameter(parameter)[source]#
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