# QuTiP example: Physical implementation of Spin Chain Qubit model¶

Author: Anubhav Vardhan ([email protected])

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.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]:
SoftwareVersion
SciPy0.14.1
Numpy1.9.1
QuTiP3.1.0
matplotlib1.4.2
OSposix [linux]
Python3.4.0 (default, Apr 11 2014, 13:05:11) [GCC 4.8.2]
Cython0.21.2
IPython2.3.1
Tue Jan 13 14:14:17 2015 JST