quantum/lib_q_computer_math.py

778 lines
25 KiB
Python

import random
from collections import defaultdict
from functools import reduce
from typing import Union
import numpy as np
from numbers import Complex
import sympy
from load_test import sizeof_fmt
ListOrNdarray = Union[list, np.ndarray]
REPR_TENSOR_OP = ""
REPR_GREEK_PSI = "ψ"
REPR_GREEK_PHI = "φ"
REPR_MATH_KET = ""
REPR_MATH_BRA = ""
REPR_MATH_SQRT = ""
REPR_MATH_SUBSCRIPT_NUMBERS = "₀₁₂₃₄₅₆₇₈₉"
# Keep a reference to already used states for naming
UNIVERSE_STATES = []
class Matrix(object):
"""Wraps a Matrix... it's for my understanding, this could easily probably be np.array"""
def __init__(self, m: ListOrNdarray = None, *args, **kwargs):
"""
Can be initialized with a matrix, e.g. for |0> this is [[0],[1]]
:param m: a matrix representing the quantum state
:param name: the name of the state (optional)
"""
if m is None:
self.m = np.array([])
elif type(m) is list:
self.m = np.array(m)
elif type(m) is np.ndarray:
self.m = m
else:
raise TypeError("m needs to be a list or ndarray type")
def __add__(self, other):
return Matrix(self.m + other.m)
def __eq__(self, other):
if isinstance(other, Complex):
return bool(np.allclose(self.m, other))
elif isinstance(other, TwoQubitPartial):
return False
return np.allclose(self.m, other.m)
def __mul_or_kron__(self, other):
"""
Multiplication with a number is Linear op;
with another state is a composition via the Kronecker/Tensor product
"""
if isinstance(other, Complex):
return Matrix(self.m * other)
elif isinstance(other, Matrix):
return self.__class__(np.kron(self.m, other.m))
raise NotImplementedError
def __rmul__(self, other):
return self.__mul_or_kron__(other)
def __mul__(self, other):
return self.__mul_or_kron__(other)
def __or__(self, other):
"""Define inner product: <self|other>
"""
m = np.dot(self._conjugate_transpose(), other.m)
try:
return self.__class__(m)
except:
try:
return other.__class__(m)
except:
...
return Matrix(m)
def __len__(self):
return len(self.m)
def outer(self, other):
"""Define outer product |0><0|"""
return Matrix(np.outer(self.m, other.m))
def x(self, other):
"""Define outer product |0><0| looks like |0x0| which is 0.x(0)"""
return self.outer(other)
def _conjugate_transpose(self):
return self.m.transpose().conjugate()
def _complex_conjugate(self):
return self.m.conjugate()
def conjugate_transpose(self):
return Matrix(self._conjugate_transpose())
def complex_conjugate(self):
return Matrix(self._complex_conjugate())
class Vector(Matrix):
def __init__(self, m: ListOrNdarray = None, *args, **kwargs):
super().__init__(m, *args, **kwargs)
if not self._is_vector():
raise TypeError("Not a vector")
def _is_vector(self):
return self.m.shape[1] == 1
class State(Vector):
def __init__(self, m: ListOrNdarray = None, name: str = '', *args, **kwargs):
"""State vector representing quantum state"""
super().__init__(m, *args, **kwargs)
self.name = name
if not self._is_normalized():
raise TypeError("Not a normalized state vector")
def _is_normalized(self):
return np.isclose(np.sum(np.abs(self.m ** 2)), 1.0)
@staticmethod
def _normalize_angles(theta, phi):
# theta is between [0 and pi]
theta = theta.real % np.pi if theta.real != np.pi else theta.real
# phi is between (0 and 2*pi]
phi = phi.real % (2 * np.pi)
return theta, phi
@classmethod
def from_angles(cls, theta, phi):
theta, phi = cls._normalize_angles(theta, phi)
m0 = np.cos(theta / 2)
m1 = np.sin(theta / 2) * np.power(np.e, (1j * phi))
m = m0 * _0 + m1 * _1
return cls(m.m)
def to_angles(self):
if not self.m.shape == (2, 1):
raise Exception("State needs to be 2x1 matrix")
m0, m1 = self.m[0][0], self.m[1][0]
# theta is between 0 and pi
theta = 2 * np.arccos(m0)
if theta < 0:
theta += np.pi
assert 0 <= theta <= np.pi
div = np.sin(theta / 2)
if div == 0:
# here is doesn't matter what phi is as phase at the poles is arbitrary
phi = 0
else:
exp = m1 / div
phi = np.log(complex(exp)) / 1j
if phi < 0:
phi += 2 * np.pi
assert 0 <= phi <= 2 * np.pi
return theta, phi
def __repr__(self):
if not self.name:
for well_known_state in well_known_states:
if self == well_known_state:
return repr(well_known_state)
for used_state in UNIVERSE_STATES:
if self == used_state:
return repr(used_state)
next_state_sub = ''.join([REPR_MATH_SUBSCRIPT_NUMBERS[int(d)] for d in str(len(UNIVERSE_STATES))])
self.name = '{}{}'.format(REPR_GREEK_PSI, next_state_sub)
UNIVERSE_STATES.append(self)
matrix_rep = "{}".format(self.m).replace('[', '').replace(']', '').replace('\n', '|').strip()
state_name = '|{}> = {}'.format(self.name, matrix_rep)
return state_name
def norm(self):
"""Norm/Length of the vector = sqrt(<self|self>)"""
return self.length()
def length(self):
"""Norm/Length of the vector = sqrt(<self|self>)"""
return np.sqrt((self | self).m).item(0)
def is_orthogonal(self, other):
"""If the inner (dot) product is zero, this vector is orthogonal to other"""
return self | other == 0
def get_prob(self, j):
"""pr(j) = |<e_j|self>|^2"""
# j_th basis vector
e_j = State([[1] if i == int(j) else [0] for i in range(len(self.m))])
return np.absolute((e_j | self).m.item(0)) ** 2
def measure(self):
weights = [self.get_prob(j) for j in range(len(self))]
format_str = "{:0" + str(int(np.ceil(np.log2(len(weights))))) + "b}"
choices = [format_str.format(i) for i in range(len(weights))]
return random.choices(choices, weights)[0]
class Operator(object):
def __init__(self, func=None, *args, **kwargs):
"""An Operator turns one function into another"""
self.func = func
def on(self, *args, **kwargs):
return self(*args, **kwargs)
def __call__(self, *args, **kwargs):
return self.func(*args, **kwargs)
class LinearOperator(Operator):
def __init__(self, func=None, *args, **kwargs):
"""Linear operators satisfy f(x+y) = f(x) + f(y) and a*f(x) = f(a*x)"""
super().__init__(func, *args, **kwargs)
if not self._is_linear():
raise TypeError("Not a linear operator")
def _is_linear(self):
# TODO: How to verify if the func is linear?
# in case of Unitary Operator, self.func is a lambda that takes a Matrix (assumes has .m component)
return True
# a, b = sympy.symbols('a, b')
# expr, vars_ = a+b, [a, b]
# for x in vars_:
# for y in vars_:
# try:
# if not sympy.Eq(sympy.diff(expr, x, y), 0):
# return False
# except TypeError:
# return False
# return True
class SquareMatrix(Matrix):
def __init__(self, m: ListOrNdarray, *args, **kwargs):
super().__init__(m, *args, **kwargs)
if not self._is_square():
raise TypeError("Not a Square matrix")
def _is_square(self):
return self.m.shape[0] == self.m.shape[1]
class UnitaryMatrix(SquareMatrix):
def __init__(self, m: ListOrNdarray, *args, **kwargs):
"""Represents a Unitary matrix that satisfies UU+ = I"""
super().__init__(m, *args, **kwargs)
if not self._is_unitary():
raise TypeError("Not a Unitary matrix")
def _is_unitary(self):
"""Checks if the matrix product of itself with conjugate transpose is Identity UU+ = I"""
UU_ = np.dot(self._conjugate_transpose(), self.m)
I = np.eye(self.m.shape[0])
return np.isclose(UU_, I).all()
class UnitaryOperator(LinearOperator, UnitaryMatrix):
def __init__(self, m: ListOrNdarray, name: str = '', *args, **kwargs):
"""UnitaryOperator inherits from both LinearOperator and a Unitary matrix
It is used to act on a State vector by defining the operator to be the dot product"""
self.name = name
UnitaryMatrix.__init__(self, m=m, *args, **kwargs)
LinearOperator.__init__(self, func=self.operator_func, *args, **kwargs)
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)
# TODO - 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_ = TwoQubitPartial("C")
x_ = 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
def verify_step(self, step):
if not (step.count(self.A) == 1 and step.count(self.B) == 1):
raise RuntimeError("Both CONTROL and TARGET need to be defined in the same step exactly once")
def compose(self, step, state):
# TODO: Hacky way to do CNOT
# Should generalize for a 2-Qubit gate
# _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')
_p = State([[1 / np.sqrt(2)],
[1 / np.sqrt(2)]],
name='+')
_m = State([[1 / np.sqrt(2)],
[- 1 / np.sqrt(2)]],
name='-')
_00 = State([[1],
[0],
[0],
[0]],
name='00')
_11 = State([[0],
[0],
[0],
[1]],
name='11')
well_known_states = [_0, _1, _p, _m]
_ = 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")
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], ],
TwoQubitPartial("C"),
TwoQubitPartial("x"),
I,
X)
C, x = CNOT.A, CNOT.B
# TODO: End Hacky way to define 2-qbit gate
###########################################################
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
# II: Dynamics | The evolution of a closed system is described by a unitary transformation
#
# Operators turn one vector into another
# the times 2 operator should return the times two multiplication
_times_2 = Operator(lambda x: 2 * x)
assert _times_2.on(5) == 10
assert _times_2(5) == 10
# 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]])
# III: Measurement | A quantum measurement is described by an orthonormal basis |e_j>
# for state space. If the initial state of the system is |psi>
# then we get outcome j with probability pr(j) = |<e_j|psi>|^2
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
# IV: Compositing | tensor/kronecker product when composing
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...
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_angles()
s = State.from_angles(*angles)
assert q == s
assert State.from_angles(0, 0) == _0
assert State.from_angles(np.pi, 0) == _1
assert State.from_angles(np.pi / 2, 0) == _p
assert State.from_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_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):
if C in step or x in step:
if not (step.count(C) == 1 and step.count(x) == 1):
raise RuntimeError("Both CONTROL and TARGET need to be defined in the same step exactly once")
# TODO: Hacky way to do CNOT
# Should generalize for a 2-Qubit gate
# _0.x(_0) * Matrix(I.m) + _1.x(_1) * Matrix(X.m)
outer_0, outer_1 = [], []
for s in step:
if s == C:
outer_0.append(_0.x(_0))
outer_1.append(_1.x(_1))
elif s == x:
outer_0.append(Matrix(I.m))
outer_1.append(Matrix(X.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)
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])
qc.add_row([_, _])
qc.add_row([_, x])
qc.print()
qp = QuantumProcessor(qc)
qp.get_sample(100)
if __name__ == "__main__":
test()
test_to_from_angles()
test_quantum_processor()
# print(_1 | _0)
# qubit = _00 + _11
# print(qubit)
# bell = CNOT.on(qubit)
# HI = I*H
# print(bell)