Source code for sympy.physics.quantum.qft
"""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
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)
