"""An implementation of qubits and gates acting on them.
Todo:
* Update docstrings.
* Update tests.
* Implement apply using decompose.
* Implement represent using decompose or something smarter. For this to
work we first have to implement represent for SWAP.
* Decide if we want upper index to be inclusive in the constructor.
* Fix the printing of Rk gates in plotting.
"""
from __future__ import print_function, division
from sympy import Expr, Matrix, exp, I, pi, Integer, Symbol
from sympy.core.compatibility import u, range
from sympy.functions import sqrt
from sympy.physics.quantum.qapply import qapply
from sympy.physics.quantum.qexpr import QuantumError, QExpr
from sympy.matrices import eye
from sympy.physics.quantum.tensorproduct import matrix_tensor_product
from sympy.physics.quantum.gate import (
Gate, HadamardGate, SwapGate, OneQubitGate, CGate, PhaseGate, TGate, ZGate
)
__all__ = [
'QFT',
'IQFT',
'RkGate',
'Rk'
]
#-----------------------------------------------------------------------------
# Fourier stuff
#-----------------------------------------------------------------------------
[docs]class RkGate(OneQubitGate):
"""This is the R_k gate of the QTF."""
gate_name = u'Rk'
gate_name_latex = u'R'
def __new__(cls, *args):
if len(args) != 2:
raise QuantumError(
'Rk gates only take two arguments, got: %r' % args
)
# For small k, Rk gates simplify to other gates, using these
# substitutions give us familiar results for the QFT for small numbers
# of qubits.
target = args[0]
k = args[1]
if k == 1:
return ZGate(target)
elif k == 2:
return PhaseGate(target)
elif k == 3:
return TGate(target)
args = cls._eval_args(args)
inst = Expr.__new__(cls, *args)
inst.hilbert_space = cls._eval_hilbert_space(args)
return inst
@classmethod
def _eval_args(cls, args):
# Fall back to this, because Gate._eval_args assumes that args is
# all targets and can't contain duplicates.
return QExpr._eval_args(args)
@property
def k(self):
return self.label[1]
@property
def targets(self):
return self.label[:1]
@property
def gate_name_plot(self):
return r'$%s_%s$' % (self.gate_name_latex, str(self.k))
def get_target_matrix(self, format='sympy'):
if format == 'sympy':
return Matrix([[1, 0], [0, exp(Integer(2)*pi*I/(Integer(2)**self.k))]])
raise NotImplementedError(
'Invalid format for the R_k gate: %r' % format)
Rk = RkGate
class Fourier(Gate):
"""Superclass of Quantum Fourier and Inverse Quantum Fourier Gates."""
@classmethod
def _eval_args(self, args):
if len(args) != 2:
raise QuantumError(
'QFT/IQFT only takes two arguments, got: %r' % args
)
if args[0] >= args[1]:
raise QuantumError("Start must be smaller than finish")
return Gate._eval_args(args)
def _represent_default_basis(self, **options):
return self._represent_ZGate(None, **options)
def _represent_ZGate(self, basis, **options):
"""
Represents the (I)QFT In the Z Basis
"""
nqubits = options.get('nqubits', 0)
if nqubits == 0:
raise QuantumError(
'The number of qubits must be given as nqubits.')
if nqubits < self.min_qubits:
raise QuantumError(
'The number of qubits %r is too small for the gate.' % nqubits
)
size = self.size
omega = self.omega
#Make a matrix that has the basic Fourier Transform Matrix
arrayFT = [[omega**(
i*j % size)/sqrt(size) for i in range(size)] for j in range(size)]
matrixFT = Matrix(arrayFT)
#Embed the FT Matrix in a higher space, if necessary
if self.label[0] != 0:
matrixFT = matrix_tensor_product(eye(2**self.label[0]), matrixFT)
if self.min_qubits < nqubits:
matrixFT = matrix_tensor_product(
matrixFT, eye(2**(nqubits - self.min_qubits)))
return matrixFT
@property
def targets(self):
return range(self.label[0], self.label[1])
@property
def min_qubits(self):
return self.label[1]
@property
def size(self):
"""Size is the size of the QFT matrix"""
return 2**(self.label[1] - self.label[0])
@property
def omega(self):
return Symbol('omega')
[docs]class QFT(Fourier):
"""The forward quantum Fourier transform."""
gate_name = u'QFT'
gate_name_latex = u'QFT'
[docs] def decompose(self):
"""Decomposes QFT into elementary gates."""
start = self.label[0]
finish = self.label[1]
circuit = 1
for level in reversed(range(start, finish)):
circuit = HadamardGate(level)*circuit
for i in range(level - start):
circuit = CGate(level - i - 1, RkGate(level, i + 2))*circuit
for i in range((finish - start)//2):
circuit = SwapGate(i + start, finish - i - 1)*circuit
return circuit
def _apply_operator_Qubit(self, qubits, **options):
return qapply(self.decompose()*qubits)
def _eval_inverse(self):
return IQFT(*self.args)
@property
def omega(self):
return exp(2*pi*I/self.size)
[docs]class IQFT(Fourier):
"""The inverse quantum Fourier transform."""
gate_name = u'IQFT'
gate_name_latex = u'{QFT^{-1}}'
[docs] def decompose(self):
"""Decomposes IQFT into elementary gates."""
start = self.args[0]
finish = self.args[1]
circuit = 1
for i in range((finish - start)//2):
circuit = SwapGate(i + start, finish - i - 1)*circuit
for level in range(start, finish):
for i in reversed(range(level - start)):
circuit = CGate(level - i - 1, RkGate(level, -i - 2))*circuit
circuit = HadamardGate(level)*circuit
return circuit
def _eval_inverse(self):
return QFT(*self.args)
@property
def omega(self):
return exp(-2*pi*I/self.size)
