1200 lines
41 KiB
Python
1200 lines
41 KiB
Python
import random
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from collections import defaultdict
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from functools import reduce
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from numbers import Complex
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from typing import Union
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import numpy as np
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import sympy as sp
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from load_test import sizeof_fmt
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ListOrNdarray = Union[list, np.ndarray]
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REPR_TENSOR_OP = "⊗"
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REPR_GREEK_BETA = "β"
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REPR_GREEK_PSI = "ψ"
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REPR_GREEK_PHI = "φ"
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REPR_MATH_KET = "⟩"
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REPR_MATH_BRA = "⟨"
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REPR_MATH_SQRT = "√"
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REPR_MATH_SUBSCRIPT_NUMBERS = "₀₁₂₃₄₅₆₇₈₉"
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# Keep a reference to already used states for naming
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UNIVERSE_STATES = []
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class Matrix(object):
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"""Wraps a Matrix... it's for my understanding, this could easily probably be np.array"""
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def __init__(self, m: ListOrNdarray = None, *args, **kwargs):
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"""
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Can be initialized with a matrix, e.g. for |0> this is [[0],[1]]
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:param m: a matrix representing the quantum state
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:param name: the name of the state (optional)
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"""
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if m is None:
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self.m = np.array([])
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elif type(m) is list:
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self.m = np.array(m)
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elif type(m) is np.ndarray:
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self.m = m
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else:
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raise TypeError("m needs to be a list or ndarray type")
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def __add__(self, other):
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return Matrix(self.m + other.m)
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def __eq__(self, other):
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if isinstance(other, Complex):
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return bool(np.allclose(self.m, other))
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elif isinstance(other, TwoQubitPartial):
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return False
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return np.allclose(self.m, other.m)
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def __mul_or_kron__(self, other):
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"""
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Multiplication with a number is Linear op;
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with another state is a composition via the Kronecker/Tensor product
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"""
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if isinstance(other, Complex):
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return Matrix(self.m * other)
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elif isinstance(other, Matrix):
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return self.__class__(np.kron(self.m, other.m))
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raise NotImplementedError
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def __rmul__(self, other):
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return self.__mul_or_kron__(other)
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def __mul__(self, other):
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return self.__mul_or_kron__(other)
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def __or__(self, other):
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"""Define inner product: <self|other>
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"""
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m = np.dot(self._conjugate_transpose(), other.m)
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try:
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return self.__class__(m)
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except:
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try:
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return other.__class__(m)
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except:
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...
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return Matrix(m)
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def __len__(self):
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return len(self.m)
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def outer(self, other):
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"""Define outer product |0><0|"""
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return Matrix(np.outer(self.m, other.m))
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def x(self, other):
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"""Define outer product |0><0| looks like |0x0| which is 0.x(0)"""
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return self.outer(other)
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def _conjugate_transpose(self):
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return self.m.transpose().conjugate()
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def _complex_conjugate(self):
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return self.m.conjugate()
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def conjugate_transpose(self):
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return Matrix(self._conjugate_transpose())
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def complex_conjugate(self):
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return Matrix(self._complex_conjugate())
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@property
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def human_m(self):
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return humanize(self.m)
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def __repr__(self):
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return str(self.m)
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class HorizontalVector(Matrix):
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"""Horizontal vector is basically a list"""
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def __init__(self, m: ListOrNdarray = None, *args, **kwargs):
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super().__init__(m, *args, **kwargs)
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if not self._is_vector():
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raise TypeError("Not a vector")
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def _is_vector(self):
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return len(self.m.shape) == 1
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class Vector(Matrix):
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def __init__(self, m: ListOrNdarray = None, *args, **kwargs):
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super().__init__(m, *args, **kwargs)
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if not self._is_vector():
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raise TypeError("Not a vector")
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def _is_vector(self):
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return self.m.shape[1] == 1
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class State(Vector):
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def __init__(self, m: ListOrNdarray = None, name: str = '', *args, **kwargs):
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"""State vector representing quantum state"""
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super().__init__(m, *args, **kwargs)
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self.name = name
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self.measurement_result = None
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if not self._is_normalized():
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raise TypeError("Not a normalized state vector")
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def _is_normalized(self):
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return np.isclose(np.sum(np.abs(self.m ** 2)), 1.0)
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@staticmethod
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def _normalize_angles(theta, phi):
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# theta is between [0 and pi]
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theta = theta.real % np.pi if theta.real != np.pi else theta.real
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# phi is between (0 and 2*pi]
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phi = phi.real % (2 * np.pi)
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return theta, phi
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@classmethod
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def from_bloch_angles(cls, theta, phi):
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"""Creates a state from angles in Bloch sphere"""
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theta, phi = cls._normalize_angles(theta, phi)
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m0 = np.cos(theta / 2)
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m1 = np.sin(theta / 2) * np.power(np.e, (1j * phi))
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m = m0 * _0 + m1 * _1
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return cls(m.m)
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def to_bloch_angles(self):
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"""Returns the angles of this state on the Bloch sphere"""
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if not self.m.shape == (2, 1):
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raise Exception("State needs to describe only 1 qubit on the bloch sphere (2x1 matrix)")
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m0, m1 = self.m[0][0], self.m[1][0]
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# theta is between 0 and pi
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theta = 2 * np.arccos(m0)
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if theta < 0:
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theta += np.pi
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assert 0 <= theta <= np.pi
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div = np.sin(theta / 2)
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if div == 0:
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# here is doesn't matter what phi is as phase at the poles is arbitrary
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phi = 0
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else:
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exp = m1 / div
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phi = np.log(complex(exp)) / 1j
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if phi < 0:
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phi += 2 * np.pi
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assert 0 <= phi <= 2 * np.pi
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return theta, phi
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def rotate_x(self, theta):
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return Rx(theta).on(self)
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def rotate_y(self, theta):
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return Ry(theta).on(self)
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def rotate_z(self, theta):
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return Rz(theta).on(self)
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def __repr__(self):
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if not self.name:
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if np.count_nonzero(self.m == 1):
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fmt_str = self.get_fmt_of_element()
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index = np.where(self.m == [1])[0][0]
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self.name = fmt_str.format(index)
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else:
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for well_known_state in well_known_states:
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try:
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if self == well_known_state:
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return repr(well_known_state)
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except:
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...
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for used_state in UNIVERSE_STATES:
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try:
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if self == used_state:
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return repr(used_state)
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except:
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...
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next_state_sub = ''.join([REPR_MATH_SUBSCRIPT_NUMBERS[int(d)] for d in str(len(UNIVERSE_STATES))])
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self.name = '{}{}'.format(REPR_GREEK_PSI, next_state_sub)
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UNIVERSE_STATES.append(self)
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# matrix_rep = "{}".format(self.m).replace('[', '').replace(']', '').replace('\n', '|').strip()
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# state_name = '|{}> = {}'.format(self.name, matrix_rep)
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state_name = '|{}>'.format(self.name)
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return state_name
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def norm(self):
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"""Norm/Length of the vector = sqrt(<self|self>)"""
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return self.length()
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def length(self):
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"""Norm/Length of the vector = sqrt(<self|self>)"""
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return np.sqrt((self | self).m).item(0)
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def is_orthogonal(self, other):
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"""If the inner (dot) product is zero, this vector is orthogonal to other"""
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return self | other == 0
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def get_prob(self, j):
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"""pr(j) = |<e_j|self>|^2"""
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# j_th basis vector
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e_j = State([[1] if i == int(j) else [0] for i in range(len(self.m))])
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return np.absolute((e_j | self).m.item(0)) ** 2
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def get_fmt_of_element(self):
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return "{:0" + str(int(np.ceil(np.log2(len(self))))) + "b}"
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def measure(self):
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"""
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Measures in the computational basis.
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Irreversable operation. Measuring again will result in the same result
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TODO: Generalize the method so it takes a basis
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TODO: Should we memoize per basis?
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If it's measured twice, should it return the same state?
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What about if measured twice but in different bases?
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E.g. measure1 -> computation -> A
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measure2 -> +/- basis -> B
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measure3 -> computation -> should it return A again or random weighted?
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measure4 -> +/- basis -> should it return B again or random weighted?
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:return: binary representation of the measured qubit (e.g. "011")
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"""
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if self.measurement_result:
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return self.measurement_result
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weights = [self.get_prob(j) for j in range(len(self))]
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format_str = self.get_fmt_of_element()
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choices = [format_str.format(i) for i in range(len(weights))]
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self.measurement_result = random.choices(choices, weights)[0]
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return self.measurement_result
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def measure_with_op(self, mo):
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"""
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Measures with a measurement operator mo
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TODO: Can't define `mo: MeasurementOperator` because in python you can't declare classes before defining them
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"""
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m = mo.on(self) / np.sqrt(mo.get_prob(self))
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return State(m)
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def measure_partial(self, qubit_n):
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"""Partial measurement of state
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Measures the n-th qubit with probability sum(a_n),
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adjusting the probability of the rest of the state
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"""
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max_qubits = int(np.log2(len(self)))
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if not (0 < qubit_n <= max_qubits):
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raise Exception("Partial measurement of qubit_n must be between 1 and {}".format(max_qubits))
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format_str = self.get_fmt_of_element()
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# e.g. for state |000>:
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# ['000', '001', '010', '011', '100', '101', '110', '111']
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bin_repr = [format_str.format(i) for i in range(len(self))]
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# e.g. for qbit_n = 2
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# [0, 0, 1, 1, 0, 0, 1, 1]
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partial_measurement_of_qbit = [int(b[qubit_n - 1]) for b in bin_repr]
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# [0, 1, 4, 5]
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indexes_for_p_0 = [i for i, index in enumerate(partial_measurement_of_qbit) if index==0]
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weights_0 = [self.get_prob(j) for j in indexes_for_p_0]
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choices_0 = [format_str.format(i) for i in indexes_for_p_0]
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measurement_result = random.choices(choices_0, weights_0)[0]
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# TODO: Verify if this is the correct collapse to lower dimension after partial measurement
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# https://www.youtube.com/watch?v=MG_9JWsrKtM&list=PL1826E60FD05B44E4&index=16
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normalization_factor = np.sqrt(np.sum([self.m[i][0] for i in indexes_for_p_0]))
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self.m = [
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[(self.m[i][0]**2)/normalization_factor] for i in indexes_for_p_0
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]
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return measurement_result
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def pretty_print(self):
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format_str = self.get_fmt_of_element() + " | {}"
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for i, element in enumerate(self.m):
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print(format_str.format(i, humanize_num(element[0])))
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@classmethod
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def from_string(cls, q):
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try:
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bin_repr = q[1:-1]
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dec_bin = int(bin_repr, 2)
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bin_length = 2 ** len(bin_repr)
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arr = [[1] if i == dec_bin else [0] for i in range(bin_length)]
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except:
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raise Exception("State from string should be of the form |00..01> with all numbers either 0 or 1")
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return cls(arr)
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@staticmethod
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def get_random_state_angles():
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phi = np.random.uniform(0, 2 * np.pi)
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t = np.random.uniform(0, 1)
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theta = np.arccos(1 - 2 * t)
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return theta, phi
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def get_bloch_coordinates(self):
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theta, phi = self.to_bloch_angles()
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x = np.sin(theta) * np.cos(phi)
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y = np.sin(theta) * np.sin(phi)
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z = np.cos(theta)
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return [x, y, z]
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def test_measure_partial():
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state = s("|000>")
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state.measure_partial(2)
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def normalize_state(state_vector: ListOrNdarray):
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"""Normalize a state by dividing by the square root of sum of the squares of states"""
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norm_coef = np.sqrt(np.sum(np.array(state_vector) ** 2))
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if norm_coef == 0:
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raise TypeError("zero-sum vector")
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return state_vector / norm_coef
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def s(q, name=None):
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"""Helper method for creating state easily"""
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if type(q) == str:
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# e.g. |000>
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if q[0] == '|' and q[-1] == '>':
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state = State.from_string(q)
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state.name = name
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return state
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elif type(q) == list:
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# e.g. s([1,0]) => |0>
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return State(np.reshape(q, (len(q), 1)), name=name)
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return State(q, name=name)
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def humanize_num(fl, tolerance=1e-3):
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if np.abs(fl.imag) < tolerance:
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fl = fl.real
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if np.abs(fl.real) < tolerance:
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fl = 0 + 1j * fl.imag
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try:
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return sp.nsimplify(fl, [sp.pi], tolerance, full=True)
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except:
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return sp.nsimplify(fl, [sp.pi], tolerance)
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def pp(m, tolerance=1e-3):
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for element in m:
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print(humanize_num(element, tolerance=tolerance))
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def humanize(m):
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rv = []
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for element in m:
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if type(element) in [np.ndarray, list]:
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rv.append(humanize(element))
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else:
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rv.append(humanize_num(element))
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return rv
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class Operator(object):
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def __init__(self, func: sp.Lambda, *args, **kwargs):
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"""An Operator turns one function into another"""
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self.func = func
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def on(self, *args):
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return self(*args)
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def __call__(self, *args):
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return self.func(*args)
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class LinearOperator(Operator):
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def __init__(self, func: sp.Lambda, *args, **kwargs):
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"""Linear operators satisfy f(x+y) = f(x) + f(y) and a*f(x) = f(a*x)"""
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super().__init__(func, *args, **kwargs)
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if not self._is_linear():
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raise TypeError("Not a linear operator")
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def _is_linear(self):
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# A function is (jointly) linear in a given set of variables
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# if all second-order derivatives are identically zero (including mixed ones).
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# https://stackoverflow.com/questions/36283548/check-if-an-equation-is-linear-for-a-specific-set-of-variables
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expr, vars_ = self.func.expr, self.func.variables
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for x in vars_:
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for y in vars_:
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try:
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if not sp.Eq(sp.diff(expr, x, y), 0):
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return False
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except TypeError:
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return False
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return True
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def test_linear_operator():
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# Operators turn one vector into another
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# the times 2 operator should return the times two multiplication
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_x = sp.Symbol('x')
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_times_2 = LinearOperator(sp.Lambda(_x, 2 * _x))
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assert _times_2.on(5) == 10
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assert _times_2(5) == 10
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assert_raises(TypeError, "Not a linear operator", LinearOperator, sp.Lambda(_x, _x ** 2))
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class SquareMatrix(Matrix):
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def __init__(self, m: ListOrNdarray, *args, **kwargs):
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super().__init__(m, *args, **kwargs)
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if not self._is_square():
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raise TypeError("Not a Square matrix")
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def _is_square(self):
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return self.m.shape[0] == self.m.shape[1]
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class LinearTransformation(LinearOperator, Matrix):
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def __init__(self, m: ListOrNdarray, func=None, name: str = '', *args, **kwargs):
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"""LinearTransformation (or linear map) inherits from both LinearOperator and a Matrix
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It is used to act on a State vector by defining the operator to be the dot product"""
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self.name = name
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Matrix.__init__(self, m=m, *args, **kwargs)
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self.func = func or self.operator_func
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LinearOperator.__init__(self, func=func, *args, **kwargs)
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def _is_linear(self):
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# Every matrix transformation is a linear transformation
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# https://www.mathbootcamps.com/proof-every-matrix-transformation-is-a-linear-transformation/
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return True
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def operator_func(self, other):
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return Vector(np.dot(self.m, other.m))
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def get_eigens(self):
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""" Returns (eigenvalue, eigenvector)
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M|v> = λ|v> ->
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M :is the operator (self)
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|v> :is eigenstate of M
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λ :is the corresponding eigenvalue
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"""
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eigenvalues, eigenvectors = np.linalg.eig(self.m)
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rv = []
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for i in range(0, len(eigenvectors)):
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eigenvalue = eigenvalues[i]
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eigenvector = HorizontalVector(eigenvectors[:, i])
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rv.append((eigenvalue, eigenvector))
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return rv
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class UnitaryMatrix(SquareMatrix):
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def __init__(self, m: ListOrNdarray, *args, **kwargs):
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"""Represents a Unitary matrix that satisfies UU+ = I"""
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super().__init__(m, *args, **kwargs)
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if not self._is_unitary():
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raise TypeError("Not a Unitary matrix")
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def _is_unitary(self):
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"""Checks if the matrix product of itself with conjugate transpose is Identity UU+ = I"""
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UU_ = np.dot(self._conjugate_transpose(), self.m)
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||
I = np.eye(self.m.shape[0])
|
||
return np.isclose(UU_, I).all()
|
||
|
||
|
||
class HermitianMatrix(SquareMatrix):
|
||
def __init__(self, m: ListOrNdarray, *args, **kwargs):
|
||
"""Represents a Hermitian matrix that satisfies U=U+"""
|
||
super().__init__(m, *args, **kwargs)
|
||
if not self._is_hermitian():
|
||
raise TypeError("Not a Hermitian matrix")
|
||
|
||
def _is_hermitian(self):
|
||
"""Checks if its equal to the conjugate transpose U = U+"""
|
||
return np.isclose(self.m, self._conjugate_transpose()).all()
|
||
|
||
|
||
class UnitaryOperator(LinearTransformation, UnitaryMatrix):
|
||
def __init__(self, m: ListOrNdarray, name: str = '', *args, **kwargs):
|
||
"""UnitaryOperator inherits from both LinearTransformation and a Unitary matrix
|
||
It is used to act on a State vector by defining the operator to be the dot product"""
|
||
UnitaryMatrix.__init__(self, m=m, *args, **kwargs)
|
||
LinearTransformation.__init__(self, m=m, func=self.operator_func, *args, **kwargs)
|
||
self.name = name
|
||
|
||
def operator_func(self, other):
|
||
return State(np.dot(self.m, other.m))
|
||
|
||
def __repr__(self):
|
||
if self.name:
|
||
return '-{}-'.format(self.name)
|
||
return str(self.m)
|
||
|
||
|
||
class HermitianOperator(LinearTransformation, HermitianMatrix):
|
||
def __init__(self, m: ListOrNdarray, name: str = '', *args, **kwargs):
|
||
"""HermitianMatrix inherits from both LinearTransformation and a Hermitian matrix
|
||
It is used to act on a State vector by defining the operator to be the dot product"""
|
||
self.name = name
|
||
HermitianMatrix.__init__(self, m=m, *args, **kwargs)
|
||
LinearOperator.__init__(self, func=self.operator_func, *args, **kwargs)
|
||
|
||
def operator_func(self, other):
|
||
"""This might not return a normalized state vector so don't wrap it in State"""
|
||
return np.dot(self.m, other.m)
|
||
|
||
def __repr__(self):
|
||
if self.name:
|
||
return '-{}-'.format(self.name)
|
||
return str(self.m)
|
||
|
||
|
||
# How to add a CNOT gate to the Quantum Processor?
|
||
# Imagine if I have to act on a 3-qubit computer and CNOT(q1, q3)
|
||
#
|
||
# Decomposed CNOT :
|
||
# reverse engineered from
|
||
# https://quantumcomputing.stackexchange.com/questions/4252/how-to-derive-the-cnot-matrix-for-a-3-qbit-system-where-the-control-target-qbi
|
||
#
|
||
# CNOT(q1, I, q2):
|
||
# |0><0| x I_2 x I_2 + |1><1| x I_2 x X
|
||
# np.kron(np.kron(np.outer(_0.m, _0.m), np.eye(2)), np.eye(2)) + np.kron(np.kron(np.outer(_1.m, _1.m), np.eye(2)), X.m)
|
||
#
|
||
#
|
||
# CNOT(q1, q2):
|
||
# |0><0| x I + |1><1| x X
|
||
# np.kron(np.outer(_0.m, _0.m), np.eye(2)) + np.kron(np.outer(_1.m, _1.m), X.m)
|
||
# _0.x(_0) * Matrix(I.m) + _1.x(_1) * Matrix(X.m)
|
||
|
||
class TwoQubitPartial(object):
|
||
def __init__(self, rpr):
|
||
self.rpr = rpr
|
||
self.operator = None
|
||
|
||
def __repr__(self):
|
||
return str("-{}-".format(self.rpr))
|
||
|
||
|
||
C_partial = TwoQubitPartial("C")
|
||
x_partial = TwoQubitPartial("x")
|
||
|
||
|
||
class TwoQubitOperator(UnitaryOperator):
|
||
def __init__(self, m: ListOrNdarray, A: TwoQubitPartial, B: TwoQubitPartial,
|
||
A_p: UnitaryOperator, B_p: UnitaryOperator, *args, **kwargs):
|
||
super().__init__(m, *args, **kwargs)
|
||
A.operator, B.operator = self, self
|
||
self.A = A
|
||
self.B = B
|
||
self.A_p = A_p
|
||
self.B_p = B_p
|
||
|
||
A.operator = self
|
||
B.operator = self
|
||
|
||
def verify_step(self, step):
|
||
if not (step.count(self.A) == 1 and step.count(self.B) == 1):
|
||
raise RuntimeError("Both {} and {} need to be defined in the same step exactly once".format(
|
||
self.A, self.B
|
||
))
|
||
|
||
def compose(self, step, state):
|
||
# _0.x(_0) * Matrix(I.m) + _1.x(_1) * Matrix(X.m)
|
||
outer_0, outer_1 = [], []
|
||
for s in step:
|
||
if s == self.A:
|
||
outer_0.append(_0.x(_0))
|
||
outer_1.append(_1.x(_1))
|
||
elif s == self.B:
|
||
outer_0.append(Matrix(self.A_p.m))
|
||
outer_1.append(Matrix(self.B_p.m))
|
||
else:
|
||
outer_0.append(Matrix(s.m))
|
||
outer_1.append(Matrix(s.m))
|
||
return reduce((lambda x, y: x * y), outer_0) + reduce((lambda x, y: x * y), outer_1)
|
||
|
||
|
||
"""
|
||
Define States and Operators
|
||
"""
|
||
|
||
_0 = State([[1],
|
||
[0]],
|
||
name='0')
|
||
|
||
_1 = State([[0],
|
||
[1]],
|
||
name='1')
|
||
|
||
# |+> and |-> states
|
||
_p = State([[1 / np.sqrt(2)],
|
||
[1 / np.sqrt(2)]],
|
||
name='+')
|
||
|
||
_m = State([[1 / np.sqrt(2)],
|
||
[- 1 / np.sqrt(2)]],
|
||
name='-')
|
||
|
||
# 4 Bell states
|
||
b_00 = State([[1 / np.sqrt(2)],
|
||
[0],
|
||
[0],
|
||
[1 / np.sqrt(2)]],
|
||
name='B00')
|
||
|
||
b_01 = State([[0],
|
||
[1 / np.sqrt(2)],
|
||
[1 / np.sqrt(2)],
|
||
[0]],
|
||
name='B01')
|
||
|
||
b_10 = State([[1 / np.sqrt(2)],
|
||
[0],
|
||
[0],
|
||
[-1 / np.sqrt(2)]],
|
||
name='B10')
|
||
|
||
b_11 = State([[0],
|
||
[1 / np.sqrt(2)],
|
||
[-1 / np.sqrt(2)],
|
||
[0]],
|
||
name='B11')
|
||
|
||
well_known_states = [_p, _m, b_00, b_01, b_10, b_11]
|
||
|
||
_ = I = UnitaryOperator([[1, 0],
|
||
[0, 1]],
|
||
name="-")
|
||
|
||
X = UnitaryOperator([[0, 1],
|
||
[1, 0]],
|
||
name="X")
|
||
|
||
Y = UnitaryOperator([[0, -1j],
|
||
[1j, 0]],
|
||
name="Y")
|
||
|
||
Z = UnitaryOperator([[1, 0],
|
||
[0, -1]],
|
||
name="Z")
|
||
|
||
|
||
# These are rotations that are specified commonly e.g. in
|
||
# https://www.quantum-inspire.com/kbase/rotation-operators/
|
||
# http://www.vcpc.univie.ac.at/~ian/hotlist/qc/talks/bloch-sphere-rotations.pdf
|
||
# and elsewhere DO NOT equate X, Y and Z for theta=np.pi
|
||
# However - they are correct up to a global phase which is all that matters for measurement purposes
|
||
#
|
||
|
||
def Rx(theta):
|
||
return UnitaryOperator([[np.cos(theta / 2), -1j * np.sin(theta / 2)],
|
||
[-1j * np.sin(theta / 2), np.cos(theta / 2)]],
|
||
name="Rx")
|
||
|
||
|
||
def Ry(theta):
|
||
return UnitaryOperator([[np.cos(theta / 2), -np.sin(theta / 2)],
|
||
[np.sin(theta / 2), np.cos(theta / 2)]],
|
||
name="Ry")
|
||
|
||
|
||
def Rz(theta):
|
||
return UnitaryOperator([[np.power(np.e, -1j * theta / 2), 0],
|
||
[0, np.power(np.e, 1j * theta / 2)]],
|
||
name="Rz")
|
||
|
||
|
||
H = UnitaryOperator([[1 / np.sqrt(2), 1 / np.sqrt(2)],
|
||
[1 / np.sqrt(2), -1 / np.sqrt(2)], ],
|
||
name="H")
|
||
|
||
CNOT = TwoQubitOperator([[1, 0, 0, 0],
|
||
[0, 1, 0, 0],
|
||
[0, 0, 0, 1],
|
||
[0, 0, 1, 0], ],
|
||
C_partial, x_partial, I, X)
|
||
|
||
|
||
def assert_raises(exception, msg, callable, *args, **kwargs):
|
||
try:
|
||
callable(*args, **kwargs)
|
||
except exception as e:
|
||
assert str(e) == msg
|
||
return
|
||
assert False
|
||
|
||
|
||
def assert_not_raises(exception, msg, callable, *args, **kwargs):
|
||
try:
|
||
callable(*args, **kwargs)
|
||
except exception as e:
|
||
if str(e) == msg:
|
||
assert False
|
||
|
||
|
||
def test_unitary_hermitian():
|
||
# Unitary is UU+ = I; Hermitian is U = U+
|
||
# Matrixes could be either, neither or both
|
||
# Quantum operators (gates) are described *only* by unitary transformations
|
||
# Hermitian operators are used for measurement operators - https://towardsdatascience.com/understanding-basics-of-measurements-in-quantum-computation-4c885879eba0
|
||
h_not_u = [
|
||
[1, 0],
|
||
[0, 2],
|
||
]
|
||
assert_not_raises(TypeError, "Not a Hermitian matrix", HermitianMatrix, h_not_u)
|
||
assert_raises(TypeError, "Not a Unitary matrix", UnitaryMatrix, h_not_u)
|
||
|
||
u_not_h = [
|
||
[1, 0],
|
||
[0, 1j],
|
||
]
|
||
assert_raises(TypeError, "Not a Hermitian matrix", HermitianMatrix, u_not_h)
|
||
assert_not_raises(TypeError, "Not a Unitary matrix", UnitaryMatrix, u_not_h)
|
||
|
||
u_and_h = [
|
||
[0, 1],
|
||
[1, 0],
|
||
]
|
||
assert_not_raises(TypeError, "Not a Hermitian matrix", HermitianMatrix, u_and_h)
|
||
assert_not_raises(TypeError, "Not a Unitary matrix", UnitaryMatrix, u_and_h)
|
||
|
||
not_u_not_h = [
|
||
[1, 2],
|
||
[0, 1],
|
||
]
|
||
assert_raises(TypeError, "Not a Hermitian matrix", HermitianMatrix, not_u_not_h)
|
||
assert_raises(TypeError, "Not a Unitary matrix", UnitaryMatrix, not_u_not_h)
|
||
|
||
|
||
class MeasurementOperator(HermitianOperator):
|
||
"""Measurement operators are Hermitians: <ψ|M†_m M_m|ψ>"""
|
||
|
||
@classmethod
|
||
def create_from_prob(cls, matrix: Matrix, *args, **kwargs):
|
||
"""returns |M†_m><M_m|"""
|
||
return cls(matrix.conjugate_transpose().x(matrix).m, *args, **kwargs)
|
||
|
||
def get_prob(self, state: State):
|
||
"""Returns result of <ψ|M†_m M_m|ψ>
|
||
"""
|
||
# <ψ|
|
||
state_ct = state.conjugate_transpose()
|
||
# This is: <ψ| . M†_m M_m . |ψ>
|
||
return state_ct.m.dot(self.m.dot(state.m)).item()
|
||
|
||
|
||
def test_measurement_ops():
|
||
m0 = MeasurementOperator.create_from_prob(Matrix([1, 0]))
|
||
m1 = MeasurementOperator.create_from_prob(Matrix([0, 1]))
|
||
assert m0 == Matrix([[1, 0],
|
||
[0, 0]])
|
||
assert m1 == Matrix([[0, 0],
|
||
[0, 1]])
|
||
|
||
# p(0) -> probability of measurement to yield a 0
|
||
assert m0.get_prob(_0) == 1.0
|
||
assert m1.get_prob(_0) == 0.0
|
||
assert m0.get_prob(_1) == 0.0
|
||
assert m1.get_prob(_1) == 1.0
|
||
|
||
# Post-state measurement of qubit with operator
|
||
assert _p.measure_with_op(m0) == _0
|
||
assert _p.measure_with_op(m1) == _1
|
||
assert _m.measure_with_op(m0) == _0
|
||
assert _m.measure_with_op(m1) == s([0, -1])
|
||
|
||
|
||
def abs_squared(x):
|
||
return np.abs(x) ** 2
|
||
|
||
|
||
def testRotPauli():
|
||
# From http://www.vcpc.univie.ac.at/~ian/hotlist/qc/talks/bloch-sphere.pdf
|
||
# / slide 11:
|
||
# However, the only measurable quantities are the probabilities
|
||
# |α|**2 and |β|**2, so multiplying the state by an arbitrary factor
|
||
# exp(iγ) (a global phase) has no observable consequences
|
||
assert np.allclose(abs_squared(Rx(np.pi).m), abs_squared(X.m))
|
||
assert np.allclose(abs_squared(Ry(np.pi).m), abs_squared(Y.m))
|
||
assert np.allclose(abs_squared(Rz(np.pi).m), abs_squared(Z.m))
|
||
|
||
|
||
def test_eigenstuff():
|
||
assert LinearTransformation(m=[[1, 0], [0, 0]]).get_eigens() == \
|
||
[(1.0, HorizontalVector([1., 0.])), (0., HorizontalVector([0., 1.]))]
|
||
|
||
|
||
def test():
|
||
# Test properties of Hilbert vector space
|
||
# The four postulates of Quantum Mechanics
|
||
# I: States | Associated to any physical system is a complex vector space
|
||
# known as the state space of the system. If the system is closed
|
||
# then the system is described completely by its state vector
|
||
# which is a unit vector in the space.
|
||
# Mathematically, this vector space is also a function space
|
||
assert _0 + _1 == _1 + _0 # commutativity of vector addition
|
||
assert _0 + (_1 + _p) == (_0 + _1) + _p # associativity of vector addition
|
||
assert 8 * (_0 + _1) == 8 * _0 + 8 * _1 # Linear when multiplying by constants
|
||
assert _0 | _0 == 1 # parallel have 1 product
|
||
assert _0 | _1 == 0 # orthogonal have 0 product
|
||
assert _0.is_orthogonal(_1)
|
||
assert _1 | (8 * _0) == 8 * (_1 | _0) # Inner product is linear multiplied by constants
|
||
assert _p | (_1 + _0) == (_p | _1) + (_p | _0) # Inner product is linear in superpos of vectors
|
||
|
||
assert np.isclose(_1.length(), 1.0) # all of the vector lengths are normalized
|
||
assert np.isclose(_0.length(), 1.0)
|
||
assert np.isclose(_p.length(), 1.0)
|
||
|
||
assert _0 | _1 == (_1 | _0).complex_conjugate() # non-commutative inner product
|
||
|
||
test_to_from_angles()
|
||
|
||
# II: Dynamics | The evolution of a closed system is described by a unitary transformation
|
||
#
|
||
test_linear_operator()
|
||
|
||
test_eigenstuff()
|
||
|
||
# Understanding the difference between unitary and hermitians
|
||
test_unitary_hermitian()
|
||
|
||
# Pauli X gate flips the |0> to |1> and the |1> to |0>
|
||
assert X | _1 == _0
|
||
assert X | _0 == _1
|
||
|
||
# Test the Y Pauli operator with complex number literal notation
|
||
assert Y | _0 == State([[0],
|
||
[1j]])
|
||
assert Y | _1 == State([[-1j],
|
||
[0]])
|
||
|
||
# Test Pauli rotation gates
|
||
testRotPauli()
|
||
|
||
# III: Measurement | A quantum measurement is described by an orthonormal basis |e_j>
|
||
# for state space. If the initial state of the system is |ψ>
|
||
# then we get outcome j with probability pr(j) = |<e_j|ψ>|^2
|
||
# Note: The postulates are applicable on closed, isolated systems.
|
||
# Systems that are closed and are described by unitary time evolution by a Hamiltonian
|
||
# can be measured by projective measurements. Systems are not closed in reality and hence
|
||
# are immeasurable using projective measurements.
|
||
# POVM (Positive Operator-Valued Measure) is a restriction on the projective measurements,
|
||
# such that it encompasses everything except the environment.
|
||
|
||
assert _0.get_prob(0) == 1 # Probability for |0> in 0 is 1
|
||
assert _0.get_prob(1) == 0 # Probability for |0> in 1 is 0
|
||
|
||
assert _1.get_prob(0) == 0 # Probability for |1> in 0 is 0
|
||
assert _1.get_prob(1) == 1 # Probability for |1> in 1 is 1
|
||
|
||
assert np.isclose(_p.get_prob(0), 0.5) # Probability for |+> in 0 is 0.5
|
||
assert np.isclose(_p.get_prob(1), 0.5) # Probability for |+> in 1 is 0.5
|
||
|
||
# Test measurement operators
|
||
test_measurement_ops()
|
||
|
||
# IV: Compositing | The state space of a composite physical system
|
||
# is the tensor product of the state spaces
|
||
# of the component physical systems.
|
||
# i.e. if we have systems numbered 1 through n, ψ_i,
|
||
# then the joint state is |ψ_1> ⊗ |ψ_2> ⊗ ... ⊗ |ψ_n>
|
||
assert _0 * _0 == State([[1], [0], [0], [0]])
|
||
assert _0 * _1 == State([[0], [1], [0], [0]])
|
||
assert _1 * _0 == State([[0], [0], [1], [0]])
|
||
assert _1 * _1 == State([[0], [0], [0], [1]])
|
||
|
||
# CNOT applies a control qubit on a target.
|
||
# If the control is a |0>, target remains unchanged.
|
||
# If the control is a |1>, target is flipped.
|
||
assert CNOT.on(_0 * _0) == _0 * _0
|
||
assert CNOT.on(_0 * _1) == _0 * _1
|
||
assert CNOT.on(_1 * _0) == _1 * _1
|
||
assert CNOT.on(_1 * _1) == _1 * _0
|
||
|
||
# ALL FOUR NOW - Create a Bell state (I) that has H|0> (II), measures (III) and composition in CNOT (IV)
|
||
# Bell state
|
||
# First - create a superposition
|
||
H = UnitaryOperator([[1 / np.sqrt(2), 1 / np.sqrt(2)],
|
||
[1 / np.sqrt(2), -1 / np.sqrt(2)], ])
|
||
superpos = H | _0
|
||
assert superpos == _p
|
||
|
||
# Then CNOT the superposition with a |0> qubit
|
||
bell = CNOT | (superpos * _0)
|
||
assert bell == State([[1 / np.sqrt(2)],
|
||
[0.],
|
||
[0.],
|
||
[1 / np.sqrt(2)], ])
|
||
|
||
assert np.isclose(bell.get_prob(0b00), 0.5) # Probability for bell in 00 is 0.5
|
||
assert np.isclose(bell.get_prob(0b01), 0) # Probability for bell in 01 is 0.
|
||
assert np.isclose(bell.get_prob(0b10), 0) # Probability for bell in 10 is 0.
|
||
assert np.isclose(bell.get_prob(0b11), 0.5) # Probability for bell in 11 is 0.5
|
||
|
||
################################
|
||
# TODO: Don't know where outer product fits - something about density operator?
|
||
assert _0.x(_0) == Matrix([[1, 0],
|
||
[0, 0]])
|
||
assert _0.x(_1) == Matrix([[0, 1],
|
||
[0, 0]])
|
||
assert _1.x(_0) == Matrix([[0, 0],
|
||
[1, 0]])
|
||
assert _1.x(_1) == Matrix([[0, 0],
|
||
[0, 1]])
|
||
|
||
|
||
def test_to_from_angles():
|
||
for q in [_0, _1, _p, _m]:
|
||
angles = q.to_bloch_angles()
|
||
s = State.from_bloch_angles(*angles)
|
||
assert q == s
|
||
|
||
assert State.from_bloch_angles(0, 0) == _0
|
||
assert State.from_bloch_angles(np.pi, 0) == _1
|
||
assert State.from_bloch_angles(np.pi / 2, 0) == _p
|
||
assert State.from_bloch_angles(np.pi / 2, np.pi) == _m
|
||
for theta, phi, qbit in [
|
||
(0, 0, _0),
|
||
(np.pi, 0, _1),
|
||
(np.pi / 2, 0, _p),
|
||
(np.pi / 2, np.pi, _m),
|
||
]:
|
||
s = State.from_bloch_angles(theta, phi)
|
||
assert s == qbit
|
||
|
||
|
||
def naive_load_test(N):
|
||
import os
|
||
import psutil
|
||
import gc
|
||
from time import time
|
||
from sys import getsizeof
|
||
|
||
print("{:>10} {:>10} {:>10} {:>10} {:>10} {:>10} {:>10} {:>10} {:>10}".format(
|
||
"qbits",
|
||
"kron_len",
|
||
"mem_used",
|
||
"mem_per_q",
|
||
"getsizeof",
|
||
"getsiz/len",
|
||
"nbytes",
|
||
"nbytes/len",
|
||
"time"))
|
||
|
||
_0 = State([[1], [0]], name='0')
|
||
|
||
process = psutil.Process(os.getpid())
|
||
mem_init = process.memory_info().rss
|
||
for i in range(2, N + 1):
|
||
start = time()
|
||
m = _0
|
||
for _ in range(i):
|
||
m = m * _0
|
||
len_m = len(m)
|
||
elapsed = time() - start
|
||
mem_b = process.memory_info().rss - mem_init
|
||
print("{:>10} {:>10} {:>10} {:>10} {:>10} {:>10} {:>10} {:>10} {:>10}".format(
|
||
i,
|
||
len(m),
|
||
sizeof_fmt(mem_b),
|
||
sizeof_fmt(mem_b / len_m),
|
||
sizeof_fmt(getsizeof(m)),
|
||
sizeof_fmt(getsizeof(m) / len_m),
|
||
sizeof_fmt(m.m.nbytes),
|
||
sizeof_fmt(m.m.nbytes / len_m),
|
||
np.round(elapsed, 2)))
|
||
|
||
gc.collect()
|
||
|
||
|
||
class QuantumCircuit(object):
|
||
def __init__(self, n_qubits: int, initial_steps=None):
|
||
self.n_qubits = n_qubits
|
||
if not initial_steps:
|
||
self.steps = [[_0 for _ in range(n_qubits)], ]
|
||
else:
|
||
self.steps = initial_steps
|
||
self._called_add_row = 0
|
||
|
||
@property
|
||
def rows(self):
|
||
return self._called_add_row
|
||
|
||
def add_row_step(self, row: int, step: int, qbit_state):
|
||
if len(self.steps) <= step:
|
||
self.steps += [[I for _ in range(self.n_qubits)] for _ in range(len(self.steps) - step + 1)]
|
||
self.steps[step][row] = qbit_state
|
||
|
||
def add_step(self, step_data: list):
|
||
if len(step_data) != self.n_qubits:
|
||
raise RuntimeError("Length of step is: {}, should be: {}".format(len(step_data), self.n_qubits))
|
||
step_i = len(step_data)
|
||
for row, qubit_state in enumerate(step_data):
|
||
self.add_row_step(row, step_i, qubit_state)
|
||
|
||
def add_steps(self, steps_data: list):
|
||
for step_data in steps_data:
|
||
self.add_step(step_data)
|
||
|
||
def add_row(self, row_data: list):
|
||
if self._called_add_row >= self.n_qubits:
|
||
raise RuntimeError("Adding more rows than qubits")
|
||
for step_i, qubit_state in enumerate(row_data):
|
||
self.add_row_step(self._called_add_row, step_i + 1, qubit_state)
|
||
self._called_add_row += 1
|
||
|
||
def add_rows(self, rows_data: list):
|
||
for row_data in rows_data:
|
||
self.add_row(row_data)
|
||
|
||
def compose_quantum_state(self, step):
|
||
partials = [op for op in step if isinstance(op, TwoQubitPartial)]
|
||
# TODO: No more than 1 TwoQubitGate **OR** UnitaryOperator can be used in a step
|
||
for partial in partials:
|
||
two_qubit_op = partial.operator
|
||
two_qubit_op.verify_step()
|
||
return two_qubit_op.compose(step)
|
||
return reduce((lambda x, y: x * y), step)
|
||
|
||
def step(self):
|
||
if self.current_step == 0 and self.current_state == self.HALT_STATE:
|
||
self.current_state = self.RUNNING_STATE
|
||
if self.current_step >= len(self.steps):
|
||
self.current_state = self.HALT_STATE
|
||
raise RuntimeWarning("Halted")
|
||
running_step = self.steps[self.current_step]
|
||
step_quantum_state = self.compose_quantum_state(running_step)
|
||
if not self.current_quantum_state:
|
||
self.current_quantum_state = step_quantum_state
|
||
else:
|
||
self.current_quantum_state = step_quantum_state | self.current_quantum_state
|
||
self.current_step += 1
|
||
|
||
def run(self):
|
||
for _ in self.steps:
|
||
self.step()
|
||
self.current_state = self.HALT_STATE
|
||
|
||
def reset(self):
|
||
self.current_step = 0
|
||
self.current_quantum_state = None
|
||
self.current_state = self.HALT_STATE
|
||
|
||
def print(self):
|
||
print("=" * 3 * len(self.steps))
|
||
for line_no in range(self.n_qubits):
|
||
line = ''
|
||
for step in self.steps:
|
||
state_repr = repr(step[line_no])
|
||
line += state_repr
|
||
print(line)
|
||
print("=" * 3 * len(self.steps))
|
||
|
||
|
||
class QuantumProcessor(object):
|
||
HALT_STATE = "HALT"
|
||
RUNNING_STATE = "RUNNING"
|
||
|
||
def __init__(self, circuit: QuantumCircuit):
|
||
if circuit.rows != circuit.n_qubits:
|
||
raise Exception("Declared circuit with n_qubits: {} but called add_row: {}".format(
|
||
circuit.n_qubits, circuit.rows
|
||
))
|
||
self.circuit = circuit
|
||
self.c_step = 0
|
||
self.c_q_state = None
|
||
self.c_state = self.HALT_STATE
|
||
self.reset()
|
||
|
||
def compose_quantum_state(self, step):
|
||
if any([type(s) is TwoQubitPartial for s in step]):
|
||
two_qubit_gates = filter(lambda s: type(s) is TwoQubitPartial, step)
|
||
state = []
|
||
for two_qubit_gate in two_qubit_gates:
|
||
two_qubit_gate.operator.verify_step(step)
|
||
state = two_qubit_gate.operator.compose(step, state)
|
||
return state
|
||
return reduce((lambda x, y: x * y), step)
|
||
|
||
def step(self):
|
||
if self.c_step == 0 and self.c_state == self.HALT_STATE:
|
||
self.c_state = self.RUNNING_STATE
|
||
if self.c_step >= len(self.circuit.steps):
|
||
self.c_state = self.HALT_STATE
|
||
raise RuntimeWarning("Halted")
|
||
step_quantum_state = self.get_next_step()
|
||
if not self.c_q_state:
|
||
self.c_q_state = step_quantum_state
|
||
else:
|
||
self.c_q_state = State((step_quantum_state | self.c_q_state).m)
|
||
self.c_step += 1
|
||
|
||
def get_next_step(self):
|
||
running_step = self.circuit.steps[self.c_step]
|
||
step_quantum_state = self.compose_quantum_state(running_step)
|
||
return step_quantum_state
|
||
|
||
def run(self):
|
||
for _ in self.circuit.steps:
|
||
self.step()
|
||
self.c_state = self.HALT_STATE
|
||
|
||
def reset(self):
|
||
self.c_step = 0
|
||
self.c_q_state = None
|
||
self.c_state = self.HALT_STATE
|
||
|
||
def measure(self):
|
||
if self.c_state != self.HALT_STATE:
|
||
raise RuntimeError("Processor is still running")
|
||
return self.c_q_state.measure()
|
||
|
||
def run_n(self, n: int):
|
||
for i in range(n):
|
||
self.run()
|
||
result = self.measure()
|
||
print("Run {}: {}".format(i, result))
|
||
self.reset()
|
||
|
||
def get_sample(self, n: int):
|
||
rv = defaultdict(int)
|
||
for i in range(n):
|
||
self.run()
|
||
result = self.measure()
|
||
rv[result] += 1
|
||
self.reset()
|
||
return rv
|
||
|
||
@staticmethod
|
||
def print_sample(rv):
|
||
for k, v in sorted(rv.items(), key=lambda x: x[0]):
|
||
print("{}: {}".format(k, v))
|
||
|
||
|
||
def test_quantum_processor():
|
||
# Produce Bell state between 0 and 2 qubit
|
||
qc = QuantumCircuit(3)
|
||
qc.add_row([H, C_partial])
|
||
qc.add_row([_, _])
|
||
qc.add_row([_, x_partial])
|
||
qc.print()
|
||
qp = QuantumProcessor(qc)
|
||
qp.print_sample(qp.get_sample(100))
|
||
|
||
|
||
def test_light():
|
||
# TODO: Are these measurement operators the correct way to represent hor/ver/diag filter?
|
||
# No, becuase they are not unitaries
|
||
hor_filter = MeasurementOperator.create_from_prob(Matrix(_0.m), name='h')
|
||
diag_filter = MeasurementOperator.create_from_prob(Matrix(_p.m), name='d')
|
||
ver_filter = MeasurementOperator.create_from_prob(Matrix(_1.m), name='v')
|
||
|
||
# create random light/photon
|
||
random_pol = [[np.random.uniform(0, 1)], [np.random.uniform(0, 1)]]
|
||
random_light = State(normalize_state(random_pol))
|
||
qc = QuantumCircuit(1, initial_steps=[[random_light]])
|
||
|
||
# add three filters as operators
|
||
# qc.add_row([hor_filter, ver_filter])
|
||
qc.add_row([hor_filter, diag_filter, ver_filter])
|
||
qc.print()
|
||
qp = QuantumProcessor(qc)
|
||
# TODO: This doesn't work - dealing with non-normalized state vectors
|
||
# - When measured in horizontal filter, the state vector has a positive |0> and 0 for |1>
|
||
qp.print_sample(qp.get_sample(100))
|
||
|
||
|
||
if __name__ == "__main__":
|
||
# test()
|
||
# test_quantum_processor()
|
||
# test_light()
|
||
test_measure_partial()
|