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# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2020.
#
# obtain a copy of this license in the LICENSE.txt file in the root directory
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.

"""A circuit implementing a quadratic form on binary variables."""

from typing import Union, Optional, List

import numpy as np

from qiskit.circuit import QuantumCircuit, QuantumRegister, ParameterExpression
from ..basis_change import QFT

[docs]class QuadraticForm(QuantumCircuit): r"""Implements a quadratic form on binary variables encoded in qubit registers. A quadratic form on binary variables is a quadratic function :math:Q acting on a binary variable of :math:n bits, :math:x = x_0 ... x_{n-1}. For an integer matrix :math:A, an integer vector :math:b and an integer :math:c the function can be written as .. math:: Q(x) = x^T A x + x^T b + c If :math:A, :math:b or :math:c contain scalar values, this circuit computes only an approximation of the quadratic form. Provided with :math:m qubits to encode the value, this circuit computes :math:Q(x) \mod 2^m in [two's complement](https://stackoverflow.com/questions/1049722/what-is-2s-complement) representation. .. math:: |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 :math: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, :math: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 :math:Q(x) is too large to be represented with m qubits, the resulting bitstring is :math:(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 <https://arxiv.org/pdf/1912.04088.pdf>_ """ def __init__( self, num_result_qubits: Optional[int] = None, quadratic: Optional[ Union[np.ndarray, List[List[Union[float, ParameterExpression]]]] ] = None, linear: Optional[Union[np.ndarray, List[Union[float, ParameterExpression]]]] = None, offset: Optional[Union[float, ParameterExpression]] = None, little_endian: bool = True, ) -> None: r""" Args: num_result_qubits: The number of qubits to encode the result. Called :math:m in the class documentation. quadratic: A matrix containing the quadratic coefficients, :math:A. linear: An array containing the linear coefficients, :math:b. offset: A constant offset, :math:c. little_endian: Encode the result in little endianness. 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. """ # check inputs match if quadratic is not None and linear is not None: if len(quadratic) != len(linear): raise ValueError("Mismatching sizes of quadratic and linear.") # temporarily set quadratic and linear to [] instead of None so we can iterate over them if quadratic is None: quadratic = [] if linear is None: linear = [] if offset is None: offset = 0 num_input_qubits = np.max([1, len(linear), len(quadratic)]) # deduce number of result bits if not added if num_result_qubits is None: # check no value is parameterized if ( any(any(isinstance(q_ij, ParameterExpression) for q_ij in q_i) for q_i in quadratic) or any(isinstance(l_i, ParameterExpression) for l_i in linear) or isinstance(offset, ParameterExpression) ): raise ValueError( "If the number of result qubits is not specified, the quadratic " "form matrices/vectors/offset may not be parameterized." ) num_result_qubits = self.required_result_qubits(quadratic, linear, offset) qr_input = QuantumRegister(num_input_qubits) qr_result = QuantumRegister(num_result_qubits) circuit = QuantumCircuit(qr_input, qr_result, name="Q(x)") # set quadratic and linear again to None if they were None if len(quadratic) == 0: quadratic = None if len(linear) == 0: linear = None scaling = np.pi * 2 ** (1 - num_result_qubits) # initial QFT (just hadamards) circuit.h(qr_result) if little_endian: qr_result = qr_result[::-1] # constant coefficient if offset != 0: for i, q_i in enumerate(qr_result): circuit.p(scaling * 2**i * offset, q_i) # the linear part consists of the vector and the diagonal of the # matrix, since x_i * x_i = x_i, as x_i is a binary variable for j in range(num_input_qubits): value = linear[j] if linear is not None else 0 value += quadratic[j][j] if quadratic is not None else 0 if value != 0: for i, q_i in enumerate(qr_result): circuit.cp(scaling * 2**i * value, qr_input[j], q_i) # the quadratic part adds A_ij and A_ji as x_i x_j == x_j x_i if quadratic is not None: for j in range(num_input_qubits): for k in range(j + 1, num_input_qubits): value = quadratic[j][k] + quadratic[k][j] if value != 0: for i, q_i in enumerate(qr_result): circuit.mcp(scaling * 2**i * value, [qr_input[j], qr_input[k]], q_i) # add the inverse QFT iqft = QFT(num_result_qubits, do_swaps=False).inverse().reverse_bits() circuit.compose(iqft, qubits=qr_result[:], inplace=True) super().__init__(*circuit.qregs, name="Q(x)") self.compose(circuit.to_gate(), qubits=self.qubits, inplace=True)
[docs] @staticmethod def required_result_qubits( quadratic: Union[np.ndarray, List[List[float]]], linear: Union[np.ndarray, List[float]], offset: float, ) -> int: """Get the number of required result qubits. Args: quadratic: A matrix containing the quadratic coefficients. linear: An array containing the linear coefficients. offset: A constant offset. Returns: The number of qubits needed to represent the value of the quadratic form in twos complement. """ bounds = [] # bounds = [minimum value, maximum value] for condition in [lambda x: x < 0, lambda x: x > 0]: bound = 0.0 bound += sum(sum(q_ij for q_ij in q_i if condition(q_ij)) for q_i in quadratic) bound += sum(l_i for l_i in linear if condition(l_i)) bound += offset if condition(offset) else 0 bounds.append(bound) # the minimum number of qubits is the number of qubits needed to represent # the minimum/maximum value plus one sign qubit num_qubits_for_min = int(np.ceil(np.log2(max(-bounds[0], 1)))) num_qubits_for_max = int(np.ceil(np.log2(bounds[1] + 1))) num_result_qubits = 1 + max(num_qubits_for_min, num_qubits_for_max) return num_result_qubits