Revisions to Grover search for basis states with complex coefficents and multiple marked states

Improved typesetting.

Source Link

edited Jan 6, 2020 at 22:13

14.9k
4
31
73

This is the circuit I used: first I initialized the initial state as 1 / sqrt(8),1 / sqrt(8),1 / sqrt(8),1 / sqrt(8),1 / sqrt(8),1 /sqrt(8),1 /sqrt(8),1 / sqrt(8),0,0,0,0,0,0,0,0 (a four-qubit system) as

$$ |x\rangle=\frac{1}{\sqrt{8}}(1,1,1,1,1,1,1 ,1,0,0,0,0,0,0,0,0)^T $$

then I applied this inverse QFT circuit. The resultant state vector is 0. 707+0.j, 0.088-0.444j, 0, 0.088-0.059j, 0, 0.088-0.132j, 0, 0.088-0.018j, 0, 0.088+0.018j, 0, 0.088+0.132j, 0, 0.088+0.059j, 0, 0.088+0.444j (rounded values)

$$ (0.707+0i, 0.088-0.444i, 0, 0.088-0.059i, 0, 0.088-0.132i, 0, 0.088-0.018i, 0, 0.088+0.018i, 0, 0.088+0.132i, 0, 0.088+0.059i, 0, 0.088+0.444i)^T $$

Note: values are rounded

Then I negated the states of interest 0000$|0000\rangle$, 0001$|0001\rangle$, 1111$1111\rangle$ by using this circuit (I believe you guys can tell which part of it is for which states if now please let me know:

Then I applied the diffusion operator. Notice that I negated the phase of 0000$|0000\rangle$ instead of the rest. This resulted in a 180 degree$180^o$ global phase difference. In the Grover search case where all coefficients are real the result for measurement is identical to marking all the rest states but I'm not sure for complex-valued coefficient will there be any difference.

uncomment #circ.draw() to plot the circuitNote: uncomment #circ.draw() to plot the circuit

This is the circuit I used: first I initialized the initial state as 1 / sqrt(8),1 / sqrt(8),1 / sqrt(8),1 / sqrt(8),1 / sqrt(8),1 /sqrt(8),1 /sqrt(8),1 / sqrt(8),0,0,0,0,0,0,0,0 (a four-qubit system)

then I applied this inverse QFT circuit. The resultant state vector is 0. 707+0.j, 0.088-0.444j, 0, 0.088-0.059j, 0, 0.088-0.132j, 0, 0.088-0.018j, 0, 0.088+0.018j, 0, 0.088+0.132j, 0, 0.088+0.059j, 0, 0.088+0.444j (rounded values)

Then I negated the states of interest 0000, 0001, 1111 by using this circuit (I believe you guys can tell which part of it is for which states if now please let me know:

Then I applied the diffusion operator. Notice that I negated the phase of 0000 instead of the rest. This resulted in a 180 degree global phase difference. In the Grover search case where all coefficients are real the result for measurement is identical to marking all the rest states but I'm not sure for complex-valued coefficient will there be any difference.

uncomment #circ.draw() to plot the circuit

This is the circuit I used: first I initialized the initial state (a four-qubit system) as

$$ |x\rangle=\frac{1}{\sqrt{8}}(1,1,1,1,1,1,1 ,1,0,0,0,0,0,0,0,0)^T $$

then I applied this inverse QFT circuit. The resultant state vector is

$$ (0.707+0i, 0.088-0.444i, 0, 0.088-0.059i, 0, 0.088-0.132i, 0, 0.088-0.018i, 0, 0.088+0.018i, 0, 0.088+0.132i, 0, 0.088+0.059i, 0, 0.088+0.444i)^T $$

Note: values are rounded

Then I negated the states of interest $|0000\rangle$, $|0001\rangle$, $1111\rangle$ by using this circuit (I believe you guys can tell which part of it is for which states if now please let me know:

Then I applied the diffusion operator. Notice that I negated the phase of $|0000\rangle$ instead of the rest. This resulted in a $180^o$ global phase difference. In the Grover search case where all coefficients are real the result for measurement is identical to marking all the rest states but I'm not sure for complex-valued coefficient will there be any difference.

Note: uncomment #circ.draw() to plot the circuit

added 435 characters in body

Source Link

edited Jun 13, 2019 at 2:53

Sam

437
2
8

import numpy as np
import math
from qiskit import *
%matplotlib inline

import math
from qiskit import QuantumRegister, ClassicalRegister, QuantumCircuit, execute, BasicAer


#initialization
q = QuantumRegister(5, "q")
c = ClassicalRegister(4, 'c')
circ = QuantumCircuit(q, c, name="initializer_circ")

desired_vector = [
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   0,
   0,
   0,
   0,
   0,
   0,
   0,
   0]


circ.initialize(desired_vector, [q[0], q[1], q[2], q[3]])

#circ.draw()


#IQFT
sub_q = QuantumRegister(4)
sub_circ = QuantumCircuit(sub_q, name='IQFT')
sub_circ.swap(sub_q[0], sub_q[3])
for j in range(4):
   for k in range(j):
       sub_circ.cu1(math.pi/float(2**(j-k)), sub_q[j], sub_q[k])
   sub_circ.h(sub_q[j])
sub_circ=sub_circ.inverse()
sub_inst = sub_circ.to_instruction()
circ.append(sub_inst, [q[0],q[1],q[2],q[3]])
#circ.draw()


#phase inverstion of |0000>
circ.x(q[0])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])
circ.h(q[3])
circ.ccx(q[0], q[1], q[4])
circ.ccx(q[2], q[4], q[3])
circ.ccx(q[0], q[1], q[4])
circ.h(q[3])
circ.x(q[0])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])

#circ.draw()



#phase inverstion of |0001>
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])
circ.h(q[3])
circ.ccx(q[0], q[1], q[4])
circ.ccx(q[2], q[4], q[3])
circ.ccx(q[0], q[1], q[4])
circ.h(q[3])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])

#circ.draw()


#phase inverstion of |1111>
circ.h(q[3])
circ.ccx(q[0], q[1], q[4])
circ.ccx(q[2], q[4], q[3])
circ.ccx(q[0], q[1], q[4])
circ.h(q[3])

#circ.draw()



#Diffusion Operator
circ.h(q[0])
circ.h(q[1])
circ.h(q[2])
circ.h(q[3])
circ.x(q[0])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])
circ.h(q[3])
circ.ccx(q[0], q[1], q[4])
circ.ccx(q[2], q[4], q[3])
circ.ccx(q[0], q[1], q[4])
circ.h(q[3])
circ.x(q[0])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])
circ.h(q[0])
circ.h(q[1])
circ.h(q[2])
circ.h(q[3])
#circ.draw() 


#check the state vector at any stage run the following code:

# Import Aer
from qiskit import BasicAer

# Run the quantum circuit on a statevector simulator backend
backend = BasicAer.get_backend('statevector_simulator')
# Create a Quantum Program for execution 
job = execute(circ, backend)
result = job.result()
outputstate = result.get_statevector(circ, decimals=3)
print(outputstate)
print(np.absolute(outputstate))

import numpy as np
import math
from qiskit import *
%matplotlib inline

import math
from qiskit import QuantumRegister, ClassicalRegister, QuantumCircuit, execute, BasicAer


#initialization
q = QuantumRegister(5, "q")
c = ClassicalRegister(4, 'c')
circ = QuantumCircuit(q, c, name="initializer_circ")

desired_vector = [
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   0,
   0,
   0,
   0,
   0,
   0,
   0,
   0]


circ.initialize(desired_vector, [q[0], q[1], q[2], q[3]])

#circ.draw()


#IQFT
sub_q = QuantumRegister(4)
sub_circ = QuantumCircuit(sub_q, name='IQFT')
sub_circ.swap(sub_q[0], sub_q[3])
for j in range(4):
   for k in range(j):
       sub_circ.cu1(math.pi/float(2**(j-k)), sub_q[j], sub_q[k])
   sub_circ.h(sub_q[j])
sub_circ=sub_circ.inverse()
sub_inst = sub_circ.to_instruction()
circ.append(sub_inst, [q[0],q[1],q[2],q[3]])
#circ.draw()


#phase inverstion of |0000>
circ.x(q[0])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])
circ.h(q[3])
circ.ccx(q[0], q[1], q[4])
circ.ccx(q[2], q[4], q[3])
circ.ccx(q[0], q[1], q[4])
circ.h(q[3])
circ.x(q[0])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])

#circ.draw()



#phase inverstion of |0001>
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])
circ.h(q[3])
circ.ccx(q[0], q[1], q[4])
circ.ccx(q[2], q[4], q[3])
circ.ccx(q[0], q[1], q[4])
circ.h(q[3])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])

#circ.draw()


#phase inverstion of |1111>
circ.h(q[3])
circ.ccx(q[0], q[1], q[4])
circ.ccx(q[2], q[4], q[3])
circ.ccx(q[0], q[1], q[4])
circ.h(q[3])

#circ.draw()



#Diffusion Operator
circ.h(q[0])
circ.h(q[1])
circ.h(q[2])
circ.h(q[3])
circ.x(q[0])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])
circ.h(q[3])
circ.ccx(q[0], q[1], q[4])
circ.ccx(q[2], q[4], q[3])
circ.ccx(q[0], q[1], q[4])
circ.h(q[3])
circ.x(q[0])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])
circ.h(q[0])
circ.h(q[1])
circ.h(q[2])
circ.h(q[3])
#circ.draw()

import numpy as np
import math
from qiskit import *
%matplotlib inline

import math
from qiskit import QuantumRegister, ClassicalRegister, QuantumCircuit, execute, BasicAer


#initialization
q = QuantumRegister(5, "q")
c = ClassicalRegister(4, 'c')
circ = QuantumCircuit(q, c, name="initializer_circ")

desired_vector = [
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   1 / math.sqrt(8),
   0,
   0,
   0,
   0,
   0,
   0,
   0,
   0]


circ.initialize(desired_vector, [q[0], q[1], q[2], q[3]])

#circ.draw()


#IQFT
sub_q = QuantumRegister(4)
sub_circ = QuantumCircuit(sub_q, name='IQFT')
sub_circ.swap(sub_q[0], sub_q[3])
for j in range(4):
   for k in range(j):
       sub_circ.cu1(math.pi/float(2**(j-k)), sub_q[j], sub_q[k])
   sub_circ.h(sub_q[j])
sub_circ=sub_circ.inverse()
sub_inst = sub_circ.to_instruction()
circ.append(sub_inst, [q[0],q[1],q[2],q[3]])
#circ.draw()


#phase inverstion of |0000>
circ.x(q[0])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])
circ.h(q[3])
circ.ccx(q[0], q[1], q[4])
circ.ccx(q[2], q[4], q[3])
circ.ccx(q[0], q[1], q[4])
circ.h(q[3])
circ.x(q[0])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])

#circ.draw()



#phase inverstion of |0001>
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])
circ.h(q[3])
circ.ccx(q[0], q[1], q[4])
circ.ccx(q[2], q[4], q[3])
circ.ccx(q[0], q[1], q[4])
circ.h(q[3])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])

#circ.draw()


#phase inverstion of |1111>
circ.h(q[3])
circ.ccx(q[0], q[1], q[4])
circ.ccx(q[2], q[4], q[3])
circ.ccx(q[0], q[1], q[4])
circ.h(q[3])

#circ.draw()



#Diffusion Operator
circ.h(q[0])
circ.h(q[1])
circ.h(q[2])
circ.h(q[3])
circ.x(q[0])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])
circ.h(q[3])
circ.ccx(q[0], q[1], q[4])
circ.ccx(q[2], q[4], q[3])
circ.ccx(q[0], q[1], q[4])
circ.h(q[3])
circ.x(q[0])
circ.x(q[1])
circ.x(q[2])
circ.x(q[3])
circ.h(q[0])
circ.h(q[1])
circ.h(q[2])
circ.h(q[3])
#circ.draw() 


#check the state vector at any stage run the following code:

# Import Aer
from qiskit import BasicAer

# Run the quantum circuit on a statevector simulator backend
backend = BasicAer.get_backend('statevector_simulator')
# Create a Quantum Program for execution 
job = execute(circ, backend)
result = job.result()
outputstate = result.get_statevector(circ, decimals=3)
print(outputstate)
print(np.absolute(outputstate))