def test_binary_decimal_to_float_conversion(self):
"""Tests converting binary decimals (e.g., 0.10 = 0.5 or 0.01 = 0.25) to floats, and vice versa."""
for num in np.linspace(0, 0.99, 25):
self.assertAlmostEqual(
QSVE.binary_decimal_to_float(QSVE.to_binary_decimal(num,
nbits=30),
big_endian=True), num)
class QuantumRecommendation:
"""Class for a quantum recommendation system."""
def __init__(self, preference_matrix, nprecision_bits=3):
"""Initializes a QuantumRecommendation.
Args:
preference_matrix
nprecision_bits : int
Number of precision qubits to use in QPE.
"""
self._matrix = deepcopy(preference_matrix)
self._precision = nprecision_bits
self._qsve = QSVE(preference_matrix)
@property
def matrix(self):
return self._matrix
@property
def num_users(self):
return len(self._matrix)
@property
def num_products(self):
return np.shape(self._matrix)[1]
def _threshold(self,
circuit,
register,
ctrl_string,
measure_flag_qubit=True):
"""Adds gates to the input circuit to perform a thresholding operation, which discards singular values
below some threshold, or minimum value.
Args:
circuit : qiskit.QuantumCircuit
Circuit to the thresholding operation (circuit) to.
register : qiskit.QuantumRegister
A register (in the input circuit) to threshold on. For recommendation systems, this will always be
the "singular value register."
minimum_value : float
A floating point value in the range [0, 1]. All singular values above minimum_value will be kept for
the recommendation, and values below will be discarded.
The thresholding circuit adds a register of three ancilla qubits to the circuit and has the following structure:
|0> --------O--------@-----------------------@--------O--------
| | | |
|0> --------O--------@-----------------------@--------O--------
| | | |
register |0> --------O--------@-----------------------@--------O--------
| | | |
|0> --------|--------|-----------------------|--------|--------
| | | |
|0> --------|--------|-----------------------|--------|--------
| | | |
| | | |
|0> -------[X]-------|---------@------------[X]-------|--------
| | |
threshold register |0> ----------------[X]--------|-----@---------------[X]-------
| |
|0> --------------------------[X]---[X]------------------------
The final qubit in the threshold_register is a "flag qubit" which determines whether or not to accept the
recommendation. That is, recommendation results are post-selected on this qubit.
Returns:
None
Modifies:
The input circuit.
"""
# Determine the number of controls
ncontrols = (ctrl_string + "1").index("1")
# Edge case in which no operations are added
if ncontrols == 0:
return
# Make sure there is at least one control
if ncontrols == len(ctrl_string):
raise ValueError(
"Argument ctrl_string should have at least one '1'.")
# Add the ancilla register of three qubits to the circuit
ancilla = QuantumRegister(3, name="threshold")
circuit.add_register(ancilla)
# ==============================
# Add the gates for thresholding
# ==============================
# Do the first "anti-controlled" Tofolli
for ii in range(ncontrols):
circuit.x(register[ii])
mct(circuit, register[:ncontrols], ancilla[0], None, mode="noancilla")
for ii in range(ncontrols):
circuit.x(register[ii])
# Do the second Tofolli
mct(circuit, register[:ncontrols], ancilla[1], None, mode="noancilla")
# Do the copy CNOTs in the ancilla register
circuit.cx(ancilla[0], ancilla[2])
circuit.cx(ancilla[1], ancilla[2])
# Uncompute the second Tofolli
mct(circuit, register[:ncontrols], ancilla[1], None, mode="noancilla")
# Uncompute the first "anti-controlled" Tofolli
for ii in range(ncontrols):
circuit.x(register[ii])
mct(circuit, register[:ncontrols], ancilla[0], None, mode="noancilla")
for ii in range(ncontrols):
circuit.x(register[ii])
# Add the "flag qubit" measurement, if desired
if measure_flag_qubit:
creg = ClassicalRegister(1, name="flag")
circuit.add_register(creg)
circuit.measure(ancilla[2], creg[0])
def create_circuit(self,
user,
threshold,
measurements=True,
return_registers=False,
logical_barriers=False,
swaps=True):
"""Returns the quantum circuit to recommend product(s) to a user.
Args:
user : numpy.ndarray
Mathematically, a vector whose length is equal to the number of columns in the preference matrix.
In the context of recommendation systems, this is a vector of "ratings" for "products,"
where each vector elements corresponds to a rating for product 0, 1, ..., N. Examples:
user = numpy.array([1, 0, 0, 0])
The user likes the first product, and we have no information about the other products.
user = numpy.array([0.5, 0.9, 0, 0])
The user is neutral about the first product, and rated the second product highly.
threshold : float in the interval [0, 1)
Only singular vectors with singular values greater than `threshold` are kept for making recommendations.
Note: Singular values are assumed to be normalized (via sigma / ||P||_F) to lie in the interval [0, 1).
This is related to the low rank approximation in the preference matrix P. The larger the threshold,
the lower the assumed rank of P. Examples:
threshold = 0.
All singular values are kept. This means we have "complete knowledge" about the user, and
recommendations are made solely based off their preferences. (No quantum circuit is needed.)
threshold = 0.5
All singular values greater than 0.5 are kept for recommendations.
threshold = 1 (invalid)
No singular values are kept for recommendations. This throws an error.
measurements : bool
Determines whether the product register is measured at the end of the circuit.
return_registers : bool
If True, registers in the circuit are returned in the order:
(1) Circuit.
(2) QPE register.
(3) User register.
(4) Product register.
(5) Classical register for product measurements (if measurements == True).
Note: Registers can always be accessed through the return circuit. This is provided for convenience.
logical_barriers : bool
Determines whether to place barriers in the circuit separating subroutines.
swaps : bool
If True, the product register qubits are swapped to put the measurement outcome in big endian.
Returns : qiskit.QuantumCircuit (and qiskit.QuantumRegisters, if desired)
A quantum circuit implementing the quantum recommendation systems algorithm.
"""
# Make sure the user is valid
self._validate_user(user)
# Convert the threshold value to the control string
ctrl_string = self._threshold_to_control_string(threshold)
# Create the QSVE circuit
circuit, qpe_register, user_register, product_register = self._qsve.create_circuit(
nprecision_bits=self._precision,
logical_barriers=logical_barriers,
load_row_norms=True,
init_state_col=user,
return_registers=True,
row_name="user",
col_name="product")
# Add the thresholding operation on the singular value (QPE) register
self._threshold(circuit,
qpe_register,
ctrl_string,
measure_flag_qubit=True)
# Add the inverse QSVE circuit
circuit += self._qsve.create_circuit(nprecision_bits=self._precision,
logical_barriers=logical_barriers,
load_row_norms=False,
init_state_col=None,
return_registers=False,
row_name="user",
col_name="product").inverse()
# Swap the qubits in the product register to put resulting bit string in big endian
if swaps:
for ii in range(len(product_register) // 2):
circuit.swap(product_register[ii], product_register[-ii - 1])
# Add measurements to the product register, if desired
if measurements:
creg = ClassicalRegister(len(product_register),
name="recommendation")
circuit.add_register(creg)
circuit.measure(product_register, creg)
if return_registers:
if measurements:
return circuit, qpe_register, user_register, product_register, creg
return circuit, qpe_register, user_register, product_register
return circuit
def run_and_return_counts(self, user, threshold, shots=10000):
"""Runs the quantum circuit recommending products for the given user and returns the raw counts."""
circuit = self.create_circuit(user,
threshold,
measurements=True,
logical_barriers=False)
job = execute(circuit,
BasicAer.get_backend("qasm_simulator"),
shots=shots)
results = job.result()
return results.get_counts()
def recommend(self,
user,
threshold,
shots=10000,
with_probabilities=True,
products_as_ints=True):
"""Returns a recommendation for a specified user."""
# Run the quantum recommendation and get the counts
counts = self.run_and_return_counts(user, threshold, shots)
# Remove all outcomes with flag qubit measured as zero (assuming the flag qubit is measured)
post_selected = []
for bits, count in counts.items():
# This checks if two different registers have been measured (i.e., checks if the flag qubit has been
# measured or not).
if " " in bits:
product, flag = bits.split()
if flag == "1":
post_selected.append((product, count))
else:
post_selected.append((bits, count))
# Get the number of post-selected measurements for normalization
new_shots = sum([x[1] for x in post_selected])
# Convert the bit string outcomes to ints
products = []
probs = []
for (product, count) in post_selected:
if products_as_ints:
product = self._binary_string_to_int(product, big_endian=True)
products.append(product)
if with_probabilities:
probs.append(count / new_shots)
if with_probabilities:
return products, probs
return products
def classical_recommendation(self, user, rank, quantum_format=True):
"""Returns a recommendation for a specified user via classical singular value decomposition.
Args:
user : numpy.ndarray
A vector whose length is equal to the number of columns in the preference matrix.
See help(QuantumRecommendation.create_circuit) for more context.
rank : int in the range (0, minimum dimension of preference matrix)
Specifies how many singular values (ordered in decreasing order) to keep in the recommendation.
quantum_format : bool
Specifies format of returned value(s).
If False, a vector of product recommendations is returned.
For example, vector = [1, 0, 0, 1] means the first and last products are recommended.
If True, two arguments are returned in the order:
(1) list<int>
List of integers specifying products to recommend.
(2) list<float>
List of floats specifying the probabilities to recommend products.
For example:
products = [0, 2], probabilities = [0.36, 0.64] means
product 0 is recommended with probability 0.36, product 2 is recommend with probability 0.64.
Returns : Union[numpy.ndarray, tuple(list<int>, list<float>)
See `quantum_format` above for more information.
"""
# Make sure the user and rank are ok
self._validate_user(user)
self._validate_rank(rank)
# Do the classical SVD
_, _, vmat = np.linalg.svd(self.matrix, full_matrices=True)
# Do the projection
recommendation = np.zeros_like(user, dtype=np.float64)
for ii in range(rank):
recommendation += np.dot(np.conj(vmat[ii]), user) * vmat[ii]
if np.allclose(recommendation, np.zeros_like(recommendation)):
raise RankError(
"Given rank is smaller than the rank of the preference matrix. Recommendations "
"cannot be made for all users.")
# Return the squared values for probabilities
probabilities = (recommendation /
np.linalg.norm(recommendation, ord=2))**2
# Return the vector if quantum_format is False
if not quantum_format:
return probabilities
# Format the same as the quantum recommendation
prods = []
probs = []
for (ii, p) in enumerate(probabilities):
if p > 0:
prods.append(ii)
probs.append(p)
return prods, probs
def _validate_user(self, user):
"""Validates a user (vector).
If an invalid user for the recommendation system is given, a UserError is thrown. Else, nothing happens.
Args:
user : numpy.ndarray
A vector representing a user in a recommendation system.
"""
# Make sure the user is of the correct type
if not isinstance(user, (list, tuple, np.ndarray)):
raise UserVectorError(
"Invalid type for user. Accepted types are list, tuple, and numpy.ndarray."
)
# Make sure the user vector has the correct length
if len(user) != self.num_products:
raise UserVectorError(
f"User vector should have length {self.num_products} but has length {len(user)}"
)
# Make sure at least one element of the user vector is nonzero
all_zeros = np.zeros_like(user)
if np.allclose(user, all_zeros):
raise UserVectorError(
"User vector is all zero and thus contains no information. "
"At least one element of the user vector must be nonzero.")
def _validate_rank(self, rank):
"""Throws a RankError if the rank is not valid, else nothing happens."""
if rank <= 0 or rank > self.num_users:
raise RankError(
"Rank must be in the range 0 < rank <= number of users.")
@staticmethod
def _to_binary_decimal(decimal, nbits=5):
"""Converts a decimal in base ten to a binary decimal string.
Args:
decimal : float
Floating point value in the interval [0, 1).
nbits : int
Number of bits to use in the binary decimal.
Return type:
str
"""
if 1 <= decimal < 0:
raise ValueError(
"Argument decimal should satisfy 0 <= decimal < 1.")
binary = ""
for ii in range(1, nbits + 1):
if decimal * 2**ii >= 1:
binary += "1"
decimal = (decimal * 10) % 1
else:
binary += "0"
return binary
@staticmethod
def _binary_string_to_int(bitstring, big_endian=True):
"""Returns the integer equivalent of the binary string.
Args:
bitstring : str
String of characters "1" and "0".
big_endian : bool
Dictates whether string is big endian (most significant bit first) or little endian.
"""
if not big_endian:
bitstring = str(reversed(bitstring))
val = 0
nbits = len(bitstring)
for (n, bit) in enumerate(bitstring):
if bit == "1":
val += 2**(nbits - n - 1)
return val
@staticmethod
def _rank_to_ctrl_string(rank, length):
"""Converts the rank to a ctrl_string for thresholding.
Args:
rank : int
Assumed rank of a system (cutoff for singular values).
length : int
Number of characters for the ctrl_string.
Returns : str
Binary string (base 2) representation of the rank with the given length.
"""
pass
def _threshold_to_control_string(self, threshold):
"""Returns a control string for the threshold circuit which keeps all values strictly above the threshold."""
# Make sure the threshold is ok
if threshold < 0 or threshold > 1:
raise ThresholdError(
"Argument threshold must satisfy 0 <= threshold <= 1.")
# Compute the angle 0 <= theta <= 1 for this threshold singular value
theta = 1 / np.pi * np.arccos(threshold)
# Return the binary decimal of theta
return self._qsve.to_binary_decimal(theta, nbits=self._precision)
def __str__(self):
return "Quantum Recommendation System with {} users and {} products.".format(
self.num_users, self.num_products)