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elegans.py
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elegans.py
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# file to implement the q-elegans model as a circuit which should be able to arbitrarily approximate any circuit
import numpy as np
from datetime import datetime
import matplotlib.pyplot as plt
from functools import partial
from oscars_toolbox.trabbit import trabbit
# ------ define gates ------ #
# jones matrices for single qubit rotation; see Simon and Mukunda that show HQQ = SU(2)
def R(alpha): return np.matrix([[np.cos(alpha), np.sin(alpha)], [-np.sin(alpha), np.cos(alpha)]])
def Rp(alpha): return np.matrix([[np.sin(alpha), np.cos(alpha)], [np.cos(alpha), np.sin(alpha)]])
def H(theta): return np.matrix([[np.cos(2*theta), np.sin(2*theta)], [np.sin(2*theta), -np.cos(2*theta)]])
def Q(alpha): return R(alpha) @ np.matrix(np.diag([np.exp(np.pi / 4 * 1j), np.exp(-np.pi / 4 * 1j)])) @ R(-alpha)
def Rz(theta): return np.matrix(np.diag([np.exp(-1j * theta / 2), np.exp(1j * theta / 2)]))
# pauli x, identity, and hadamard
X = np.matrix([[0, 1], [1, 0]])
I2 = np.eye(2)
Had = np.matrix([[1, 1], [1, -1]]) / np.sqrt(2)
# function for CNOT
def CNOT(N, i, j, theta):
'''Returns the CNOT gate on N qubits with control qubit i and target qubit j.
Params:
N: number of qubits
i: control qubit
j: target qubit
theta1: angle of rotation for H gate
theta2: angle of rotation for Q gate
Note: for validation purposes, np.kron(np.kron(P0, I2), I2) + np.kron(np.kron(P1, I2), X) is solution for CNOT(3, 0, 2, 0)
'''
assert i < j, 'This construction requires i < j'
# define projection operators: |0><0|, |1><1|
P0 = np.array([[1, 0], [0, 0]])
P1 = np.array([[0, 0], [0, 1]])
# initialize gate
gate = 1
# loop over all qubits: |0>
for k in range(N):
if k == i:
# control
gate = np.kron(gate, P0)
else:
# add identity for non-control and non-target qubits
gate = np.kron(gate, I2)
# second part of the gate for when control qubit is |1>
gate1 = 1
for k in range(N):
if k == i:
gate1 = np.kron(gate1, P1)
elif k == j: # target part
# gate1 = np.kron(gate1, (H(theta1/2) @ Q(theta2/2) @ Q(theta3/2)) @ X @ (H(-theta1/2) @ Q(-theta2/2) @ Q(-theta3/2))) # if control is |0>, then target is identity
gate1 = np.kron(gate1, Rp(theta))
else:
gate1 = np.kron(gate1, I2)
# combine the two parts
gate = gate + gate1
return gate
# ------ define circuit ------ #
def get_circuit(n, params, config=0):
'''Returns a parametrized circuit that can approximate any unitary on n qubits.
Params:
n: number of qubits
params: parameters for the circuit
config: configuration of the circuit. 0 is only HQQ, 1 is CNOT + HQQ, 2 is HQQ + CNOT + HQQ
'''
def apply_HQQ(HQQ_params, circuit):
# loop through all qubits
for i in range(n):
# apply HQQ at ith qubit, I2 everywhere else
# build up that term and then multiply to the circuit
term = 1
for j in range(n):
if j == i:
term = np.kron(term, H(HQQ_params[i, 0]) @ Q(HQQ_params[i, 1]) @ Q(HQQ_params[i, 2]))
else:
term = np.kron(term, I2)
circuit = circuit @ term
return circuit
def apply_CNOT(CNOT_params, circuit):
# loop through all qubits as control
for i in range(n):
# loop through all qubits as target
for j in range(n):
# apply CNOT, which in general is defined: CNOT = |0><0| x I + |1><1| x X. we extend it
if i < j:
circuit = circuit @ CNOT(n, i, j, CNOT_params[i, j])
elif i > j:
circuit = circuit @ CNOT(n, j, i, CNOT_params[i, j])
return circuit
assert config in [0, 1, 2], f'config must be 0, 1, or 2. you have = {config}'
# first apply Had to all qubits
circuit = Had
for _ in range(n-1):
circuit = np.kron(circuit, Had)
if config == 0:
# apply HQQ gates on all qubits
HQQ_params = params.reshape((n, 3))
circuit = apply_HQQ(HQQ_params, circuit)
elif config == 1:
CNOT_params = params[:n**2].reshape((n, n))
HQQ_params = params[n**2:].reshape((n, 3))
circuit = apply_CNOT(CNOT_params, circuit)
circuit = apply_HQQ(HQQ_params, circuit)
elif config == 2:
HQQ_params = params[:3*n].reshape((n, 3))
CNOT_params = params[3*n:3*n+n**2].reshape((n, n))
HQQ_params2 = params[3*n+n**2:].reshape((n, 3))
circuit = apply_HQQ(HQQ_params, circuit)
circuit = apply_CNOT(CNOT_params, circuit)
circuit = apply_HQQ(HQQ_params2, circuit)
return circuit
# ------ define cost function ------ #
def loss_func(params, target, config):
'''Returns the loss function for a given target unitary.'''
n = int(np.log2(target.shape[0]))
circuit = get_circuit(n, params, config)
return np.linalg.norm(circuit - target)
# print(np.sqrt(np.abs(np.trace(circuit @ target.conj().T))**2))
# return 1-np.sqrt(np.abs(np.trace(circuit @ target.conj().T))**2)
def random_func(size=1):
rand = np.random.uniform(0, 2*np.pi, size=size)
if size == 1:
return rand[0]
else:
return rand
def find_params(target, config=0):
'''Returns the parameters that minimize the loss function for a given target unitary.'''
# initialize parameters
n = int(np.log2(target.shape[0]))
# minimize loss function
loss_func_param = lambda params: loss_func(params, target, config)
if config == 0:
random_gen = partial(random_func, size=3*n)
bounds = [(0, 2*np.pi)] * 3*n
elif config == 1:
random_gen = partial(random_func, size=n**2+3*n)
bounds = [(0, 2*np.pi)] * (n**2 + 3*n)
elif config == 2:
random_gen = partial(random_func, size=3*n+n**2+3*n)
bounds = [(0, 2*np.pi)] * (3*n + n**2 + 3*n)
return trabbit(loss_func=loss_func_param, random_gen=random_gen, bounds=bounds, alpha=1, temperature=.01, verbose=True)
# ------ test ------ #
def random_circuit(n, d, I2_prob = 0.2, H_prob = 0.2, Q_prob = 0.2, HQQ_prob = 0.2, CNOT_prob = 0.2):
'''Returns a random quantum cictui on n qubits for a depth d.'''
# initialize random unitary
unitary = Had
for _ in range(n-1):
unitary = np.kron(unitary, Had)
p = np.array([I2_prob, H_prob, Q_prob, HQQ_prob, CNOT_prob])
p /= np.sum(p)
# loop through all layers
for _ in range(d):
# choose which gates to apply; if CNOT, chose random control and target qubits
gates = np.random.choice(['I2', 'H', 'Q', 'HQQ', 'CNOT'], size=n, p=p)
term = np.array([1])
# initialize list of CNOT gates to apply at the end
CNOT_gates = []
for j, gate in enumerate(gates):
if gate == 'I2':
term = np.kron(term, I2)
elif gate == 'H':
term = np.kron(term, H(random_func()))
elif gate == 'Q':
term = np.kron(term, Q(random_func()))
elif gate == 'HQQ':
term = np.kron(term, H(random_func()) @ Q(random_func()) @ Q(random_func()))
elif gate == 'CNOT':
i = np.random.choice(n, size=1, replace=False)
if j < i:
i_temp = i
i = j
j = i_temp
elif j == i:
# choose another qubit
while j == i:
j = np.random.choice(n, size=1, replace=False)
if j < i:
i_temp = i
i = j
j = i_temp
CNOT_gates.append(CNOT(n, i, j, random_func()))
# figure out how many I2 to add to make dimensions match
if len(CNOT_gates) > 0:
for _ in range(len(CNOT_gates)):
term = np.kron(term, I2)
for CNOT_gate in CNOT_gates:
term = term @ CNOT_gate
# apply term to unitary
unitary = unitary @ term
return unitary
# ----- measure circuits ------ #
def get_entropy(density_matrix):
'''Returns the entropy of a given density matrix.'''
# get eigenvalues
eigvals = np.linalg.eigvals(density_matrix)
# get probabilities
probs = np.abs(eigvals)**2
# return entropy. fully entangled circuit
return -np.sum(probs * np.log(probs))
# ------ plot matrix ------ #
def print_matrix(matrix, title=None, savefig=False):
'''Prints a matrix in a nice way.'''
# get magnnitude and phase
mag = np.abs(matrix)
phase = np.angle(matrix)
# where magnitude is 0, phase is 0
phase[mag == 0] = 0
# plot with colorbar
fig, ax = plt.subplots(2, 1, figsize=(5, 8))
im0 = ax[0].imshow(mag)
im1 = ax[1].imshow(phase)
ax[0].set_title('Magnitude')
ax[1].set_title('Phase')
fig.colorbar(im0, ax=ax[0])
fig.colorbar(im1, ax=ax[1])
if title is not None:
fig.suptitle(title)
plt.tight_layout()
date = str(datetime.now()).split('.')
if savefig and title is not None:
plt.savefig(f'elegans_figs/{title}_{date}.pdf')
elif savefig:
plt.savefig(f'elegans_figs/{date}.pdf')
plt.show()
def compare_matrices(A, B, title=None, savefig=False):
'''Prints two matrices side by side.'''
# get magnnitude and phase
mag_A = np.abs(A)
phase_A = np.angle(A)
# where magnitude is 0, phase is 0
phase_A[mag_A == 0] = 0
mag_B = np.abs(B)
phase_B = np.angle(B)
# where magnitude is 0, phase is 0
phase_B[mag_B == 0] = 0
# plot with colorbar
fig, ax = plt.subplots(2, 2, figsize=(10, 8))
im0 = ax[0, 0].imshow(mag_A)
im1 = ax[1, 0].imshow(phase_A)
im2 = ax[0, 1].imshow(mag_B)
im3 = ax[1, 1].imshow(phase_B)
ax[0, 0].set_title('Magnitude')
ax[1, 0].set_title('Phase')
ax[0, 1].set_title('Magnitude')
ax[1, 1].set_title('Phase')
fig.colorbar(im0, ax=ax[0, 0])
fig.colorbar(im1, ax=ax[1, 0])
fig.colorbar(im2, ax=ax[0, 1])
fig.colorbar(im3, ax=ax[1, 1])
if title is not None:
fig.suptitle(title)
plt.tight_layout()
date = str(datetime.now()).split('.')
if savefig and title is not None:
plt.savefig(f'elegans_figs/{title}_{date}.pdf')
elif savefig:
plt.savefig(f'elegans_figs/{date}.pdf')
if __name__ == '__main__':
N = 5
d = 5
circuit = random_circuit(N, d, CNOT_prob=0)
x_best, loss_best = find_params(circuit, config=2)
print(x_best)
print(loss_best)
approx = get_circuit(N, x_best)
compare_matrices(circuit, approx, title=f'N={N}, d={d}, loss={loss_best}', savefig=True)