QuTiP example: Physical implementation of Spin Chain Qubit model

Author: Anubhav Vardhan (anubhavvardhan@gmail.com)

For more information about QuTiP see http://qutip.org



In [1]:

    
%matplotlib inline



In [2]:

    
from qutip import *



In [3]:

    
from qutip.qip.models.circuitprocessor import *



In [4]:

    
from qutip.qip.models.spinchain import *

Hamiltonian:

$\displaystyle H = - \frac{1}{2}\sum_n^N h_n \sigma_z(n) - \frac{1}{2} \sum_n^{N-1} [ J_x^{(n)} \sigma_x(n) \sigma_x(n+1) + J_y^{(n)} \sigma_y(n) \sigma_y(n+1) +J_z^{(n)} \sigma_z(n) \sigma_z(n+1)]$

The linear and circular spin chain models employing the nearest neighbor interaction can be implemented using the SpinChain class.

Circuit Setup



In [5]:

    
N = 3
qc = QubitCircuit(N)

qc.add_gate("CNOT", targets=[0], controls=[2])

qc.png









    Out[5]:

The non-adjacent interactions are broken into a series of adjacent ones by the program automatically.



In [6]:

    
U_ideal = gate_sequence_product(qc.propagators())

U_ideal









    Out[6]:





Quantum object: dims = [[2, 2, 2], [2, 2, 2]], shape = [8, 8], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & 0.0 & 0.0\\0.0 & 0.0 & 1.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0\\0.0 & 0.0 & 0.0 & 0.0 & 1.0 & 0.0 & 0.0 & 0.0\\0.0 & 1.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & 0.0\\0.0 & 0.0 & 0.0 & 1.0 & 0.0 & 0.0 & 0.0 & 0.0\\\end{array}\right)\end{equation*}

Circular Spin Chain Model Implementation



In [7]:

    
p1 = CircularSpinChain(N, correct_global_phase=True)

U_list = p1.run(qc)

U_physical = gate_sequence_product(U_list)

U_physical.tidyup(atol=1e-5)









    Out[7]:





Quantum object: dims = [[2, 2, 2], [2, 2, 2]], shape = [8, 8], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}1.000 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.000 & 0.0 & 0.0\\0.0 & 0.0 & 1.000 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.000\\0.0 & 0.0 & 0.0 & 0.0 & 1.000 & 0.0 & 0.0 & 0.0\\0.0 & 1.000 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.000 & 0.0\\0.0 & 0.0 & 0.0 & 1.000 & 0.0 & 0.0 & 0.0 & 0.0\\\end{array}\right)\end{equation*}



In [8]:

    
(U_ideal - U_physical).norm()









    Out[8]:





0.0

The results obtained from the physical implementation agree with the ideal result.



In [9]:

    
p1.qc0.gates









    Out[9]:





[Gate(CNOT, targets=[0], controls=[2])]

The gates are first convert to gates with adjacent interactions moving in the direction with the least number of qubits in between.



In [10]:

    
p1.qc1.gates









    Out[10]:





[Gate(CNOT, targets=[0], controls=[2])]

They are then converted into the basis [ISWAP, RX, RZ]



In [11]:

    
p1.qc2.gates









    Out[11]:





[Gate(GLOBALPHASE, targets=None, controls=None),
 Gate(ISWAP, targets=[2, 0], controls=None),
 Gate(RZ, targets=[0], controls=None),
 Gate(RZ, targets=[2], controls=None),
 Gate(RX, targets=[2], controls=None),
 Gate(RZ, targets=[2], controls=None),
 Gate(RZ, targets=[2], controls=None),
 Gate(ISWAP, targets=[2, 0], controls=None),
 Gate(RZ, targets=[0], controls=None),
 Gate(RX, targets=[0], controls=None),
 Gate(RZ, targets=[0], controls=None),
 Gate(RZ, targets=[0], controls=None)]

The time for each applied gate:



In [12]:

    
p1.T_list









    Out[12]:





[1.25, 0.125, 0.125, 0.5, 0.125, 0.125, 1.25, 0.125, 0.5, 0.125, 0.125]

The pulse can be plotted as:



In [13]:

    
p1.plot_pulses();

Linear Spin Chain Model Implementation



In [14]:

    
p2 = LinearSpinChain(N, correct_global_phase=True)

U_list = p2.run(qc)

U_physical = gate_sequence_product(U_list)

U_physical.tidyup(atol=1e-5)









    Out[14]:





Quantum object: dims = [[2, 2, 2], [2, 2, 2]], shape = [8, 8], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}1.000 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.000 & 0.0 & 0.0\\0.0 & 0.0 & 1.000 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.000\\0.0 & 0.0 & 0.0 & 0.0 & 1.000 & 0.0 & 0.0 & 0.0\\0.0 & 1.000 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.000 & 0.0\\0.0 & 0.0 & 0.0 & 1.000 & 0.0 & 0.0 & 0.0 & 0.0\\\end{array}\right)\end{equation*}



In [15]:

    
(U_ideal - U_physical).norm()









    Out[15]:





0.0

The results obtained from the physical implementation agree with the ideal result.



In [16]:

    
p2.qc0.gates









    Out[16]:





[Gate(CNOT, targets=[0], controls=[2])]

The gates are first convert to gates with adjacent interactions moving in the direction with the least number of qubits in between.



In [17]:

    
p2.qc1.gates









    Out[17]:





[Gate(SWAP, targets=[0, 1], controls=None),
 Gate(CNOT, targets=[1], controls=[2]),
 Gate(SWAP, targets=[0, 1], controls=None)]

They are then converted into the basis [ISWAP, RX, RZ]



In [18]:

    
p2.qc2.gates









    Out[18]:





[Gate(GLOBALPHASE, targets=None, controls=None),
 Gate(ISWAP, targets=[0, 1], controls=None),
 Gate(RX, targets=[0], controls=None),
 Gate(ISWAP, targets=[0, 1], controls=None),
 Gate(RX, targets=[1], controls=None),
 Gate(ISWAP, targets=[1, 0], controls=None),
 Gate(RX, targets=[0], controls=None),
 Gate(GLOBALPHASE, targets=None, controls=None),
 Gate(ISWAP, targets=[2, 1], controls=None),
 Gate(RZ, targets=[1], controls=None),
 Gate(RZ, targets=[2], controls=None),
 Gate(RX, targets=[2], controls=None),
 Gate(RZ, targets=[2], controls=None),
 Gate(RZ, targets=[2], controls=None),
 Gate(ISWAP, targets=[2, 1], controls=None),
 Gate(RZ, targets=[1], controls=None),
 Gate(RX, targets=[1], controls=None),
 Gate(RZ, targets=[1], controls=None),
 Gate(RZ, targets=[1], controls=None),
 Gate(GLOBALPHASE, targets=None, controls=None),
 Gate(ISWAP, targets=[0, 1], controls=None),
 Gate(RX, targets=[0], controls=None),
 Gate(ISWAP, targets=[0, 1], controls=None),
 Gate(RX, targets=[1], controls=None),
 Gate(ISWAP, targets=[1, 0], controls=None),
 Gate(RX, targets=[0], controls=None)]

The time for each applied gate:



In [19]:

    
p2.T_list









    Out[19]:





[1.25,
 0.5,
 1.25,
 0.5,
 1.25,
 0.5,
 1.25,
 0.125,
 0.125,
 0.5,
 0.125,
 0.125,
 1.25,
 0.125,
 0.5,
 0.125,
 0.125,
 1.25,
 0.5,
 1.25,
 0.5,
 1.25,
 0.5]

The pulse can be plotted as:



In [20]:

    
p2.plot_pulses();

Software versions:



In [21]:

    
from qutip.ipynbtools import version_table
version_table()









    Out[21]:




Software Version
SciPy 0.14.1
Numpy 1.9.1
QuTiP 3.1.0
matplotlib 1.4.2
OS posix [linux]
Python 3.4.0 (default, Apr 11 2014, 13:05:11) 
[GCC 4.8.2]
Cython 0.21.2
IPython 2.3.1
Tue Jan 13 14:14:17 2015 JST

Software	Version
SciPy	0.14.1
Numpy	1.9.1
QuTiP	3.1.0
matplotlib	1.4.2
OS	posix [linux]
Python	3.4.0 (default, Apr 11 2014, 13:05:11) [GCC 4.8.2]
Cython	0.21.2
IPython	2.3.1
Tue Jan 13 14:14:17 2015 JST