def grover_invert(f, y, n):
"""Grover's algorithm for inverting a general function f that maps a
sequence of n bits (represented as an int whose binary representation is the
bit sequence) to another sequence of bits.
Args:
f: Function to invert
y: Value of the function at which to evaluate the inverse.
Returns:
The input x such that f(x) = y. If more than one input suffices, it
returns one at random. If no input suffices, returns -1.
"""
if n <= 0:
raise ValueError('n must be positive')
Hn = hadamard_gate(n)
Ui = _function_oracle(f, y, n)
Ud = grover_diffusion_operator(n)
MAX_ITER = 50
count = 0
x = None
# Repeat until a solution is found or the iteration limit is reached
while count < MAX_ITER and (x is None or f(x) != y):
q = QubitSystem(n) # system of n bits in state |0>
# apply Hadamard gate to create uniform superposition of basis states
Hn * q
for _ in range(_r(2**n)):
Ui * q # apply operator that flips the sign of the matching index
Ud * q # apply Grover's diffusion operator
x = q.measure()
count += 1
return x if f(x) == y else -1
def shor_factor(n):
"""Shor's factorization algorithm.
Args:
n: Integer >=2 to factor.
Returns:
If n is composite, a non-trivial factor of n. If n is prime, returns 1.
Raises:
ValueError if n is <= 1.
"""
if n <= 1:
raise ValueError('n must be at least 2')
if is_prime(n):
return 1
if n % 2 == 0:
return 2 # even numbers > 2 are trivial
# Need to check that n is not a power of an integer for algorithm to work
root = int_root(n)
if root != -1:
return root
# choose m s.t. n^2 <= 2^m < 2*n^2
# log2(n^2) <= m <= log2(2 * n^2) = 1 + log2(n^2)
m = ceil(log(n**2, 2))
ny = ceil(log(n - 1, 2)) # number of qubits in output of function f
I = QuantumGate(eye(2**ny))
H = kron(hadamard_gate(m), I)
MAX_ITER = 10
niter = 0
while True:
a = n - 1 # arbitrary integer coprime to n
# Initialize a system of qubits long enough to represent the integers 0
# to 2^m - 1 alongside an integer up to n - 1, then apply Hadamard gate
# to the first m qubits to create a uniform superposition.
q = QubitSystem(m + ny)
H * q
# Apply the function f(x) = a^x (mod n) to the system
f = lambda x: pow(a, x, n)
Uf = function_gate(f, m, ny)
Uf * q
# Find the period of f via quantum Fourier transform
qft(q)
r = 1. / q.measure() # period = 1 / frequency
niter += 1
if niter >= MAX_ITER or (r % 2 == 0 and (a**(r / 2)) % n != -1):
break
return gcd(a**(r / 2) + 1, n)
def test_controlled_gate(self):
# gate that permutes the coefficients on a two-qubit system
P = QuantumGate(array([[0., 0., 0., 1.],
[1., 0., 0., 0.],
[0., 1., 0., 0.],
[0., 0., 1., 0.]]))
q1 = QubitSystem(3, 0b011)
q2 = QubitSystem(3, 0b101)
q3 = QubitSystem(3, 0b101)
# controlled gate permutes the second two bits iff the first bit is one
CG = controlled_gate(P)
CG * q1
self.assertEqual(q1.measure(), 0b011)
CG * q2
self.assertEqual(q2.measure(), 0b110)
# test Toffoli gate, a.k.a. controlled-controlled-not gate
T = toffoli_gate()
T * q1
self.assertEqual(q1.measure(), 0b011)
T * q2
self.assertEqual(q2.measure(), 0b111)
# Test Fredkin gate, a.k.a. controlled swap gate
F = fredkin_gate()
F * q1
self.assertEqual(q1.measure(), 0b011)
F * q3
self.assertEqual(q3.measure(), 0b110)
def test_function_gate(self):
f = lambda x: x >> 2
m, k = 4, 2
Uf = function_gate(f, m, k)
# Initial state - M-bit binary string for 6 plus random K-bit register
q = QubitSystem(m + k, 0b011010)
# Expected output state: first M bits unchanged, followed by initial
# string added to f(x) mod 2 (where x = first M bits)
Uf * q
self.assertEqual(q.measure(), 0b011011)
def test_x_gate(self):
X = x_gate()
self.assertTrue((X.matrix() == array([[0., 1.], [1., 0.]])).all())
# Test that applying the X-gate to a qubit in state |0> converts it to
# state |1>, and vice versa
q = QubitSystem()
X * q
self.assertEqual(q.measure(), 1)
X * q
self.assertEqual(q.measure(), 0)
def test_tensor_product(self):
# Test that |1> (x) |0> = |10>
q0 = QubitSystem(1, 0)
q1 = QubitSystem(1, 1)
q1 * q0
self.assertEqual(q1.measure(), 0b10)
# Test that |0100> (x) |101> = |0100101>
q4 = QubitSystem(4, 0b0100)
q5 = QubitSystem(3, 0b101)
q4 * q5
self.assertEqual(q4.measure(), 0b0100101)
def test_measure(self):
# First, test measurement of a system in a pure state
STATE = '01010'
N = len(STATE)
q = QubitSystem(N, int('0b' + STATE, 2))
self.assertEqual(q.measure(), int('0b' + STATE, 2))
self.assertEqual(q.smeasure(), STATE)
for i in range(N):
self.assertEqual(q.measure(i + 1), int(STATE[i]))
# Test probabilistic measurement: repeatedly measure superpositions of
# basis states and test that the outcome varies randomly.
N = 2
NTRIALS = 100
# number of 1's in the first and second bit, respectively
nsuccess1 = 0
nsuccess2 = 0
for i in range(NTRIALS):
q = QubitSystem(N) # set the qubits to |00>
hadamard_gate(N) * q # equal superposition of four basis states
nsuccess1 += q.measure(1)
nsuccess2 += q.measure(2)
# Test that repeated measurement gives the same result
state = q.measure()
self.assertEqual(q.measure(), state)
self.assertEqual(q.measure(), state)
# Test that the number of 1's that appeared in the first and second
# qubits is approximately half the total (within approx. 99% confidence
# interval)
CUTOFF = NTRIALS * 0.15
self.assertTrue(abs(nsuccess1 - NTRIALS / 2.) < CUTOFF)
self.assertTrue(abs(nsuccess2 - NTRIALS / 2.) < CUTOFF)
def grover_search(match_text, lst):
"""Grover's quantum algorithm for searching.
Args:
match_text: Text to find in the list lst.
lst: List of strings to search to find a string matching match_text.
Returns:
The index i of the item such that lst[i] is the same string as
match_text. The lines must match exactly; it is not enough for the text
to be contained in the line. If two or more lines match, it will only
return one of the line numbers. Returns -1 if no matching line is found,
i.e. the algorithm fails to find a solution.
"""
if len(lst) <= 0:
raise ValueError('List must be of positive length')
n = len(lst)
N = int(ceil(log(n, 2))) # number of qubits needed
Hn = hadamard_gate(N)
Ui = _search_oracle(match_text, lst)
Ud = grover_diffusion_operator(N)
MAX_ITER = 50
count = 0
index = n
# Repeat until a solution is found or the iteration limit is reached
while count < MAX_ITER and (index >= n or lst[index] != match_text):
q = QubitSystem(N) # system of log2(n) bits in state |0>
# apply Hadamard gate to create uniform superposition of basis states
Hn * q
for _ in range(_r(2**N)):
Ui * q # apply operator that flips the sign of the matching index
Ud * q # apply Grover's diffusion operator
index = q.measure()
count += 1
return index if index < n and lst[index] == match_text else -1
def test_act(self):
q1 = QubitSystem(2)
q2 = QubitSystem(2, 0b10)
SWAP = swap_gate()
CNOT = cnot_gate()
# SWAP * |00> = |00>
SWAP * q1
self.assertEqual(q1.measure(), 0b00)
# SWAP * |10> = |01>, and vice versa
SWAP * q2
self.assertEqual(q2.measure(), 0b01)
SWAP * q2
self.assertEqual(q2.measure(), 0b10)
# CNOT * |00> = |00>
CNOT * q1
self.assertEqual(q1.measure(), 0b00)
# CNOT * |10> = |11> and vice versa
CNOT * q2
self.assertEqual(q2.measure(), 0b11)
CNOT * q2
self.assertEqual(q2.measure(), 0b10)
# Test applying a gate to a subset of the qubits in the system
q = QubitSystem(4) # state 0000
X = x_gate()
X.act(q, 3)
self.assertEqual(q.measure(), 0b0010)
X.act(q, 4)
self.assertEqual(q.measure(), 0b0011)
SWAP.act(q, 2)
self.assertEqual(q.measure(), 0b0101)
SWAP * q
self.assertEqual(q.measure(), 0b1001)