QuTiP example: Quantum Gates and their usage

Author: Anubhav Vardhan ([email protected])

User-defined gate added by: Boxi Li ([email protected])

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

Note: The circuit image visualizations require ImageMagick for display.

ImageMagick can be easily installed with the command conda install imagemagick if you have conda installed. Otherwise, please follow the installation instructions on the ImageMagick documentation.

In [1]:
%matplotlib inline
In [2]:
from IPython.display import Image
In [3]:
from numpy import pi
import numpy as np
In [4]:
from qutip import *
from qutip.qip.operations import *
from qutip.qip.circuit import QubitCircuit, Gate

Introduction

Gates in QuTiP and their representation

Controlled-PHASE

In [5]:
cphase(pi/2)
Out[5]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}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 & 1.0j\\\end{array}\right)\end{equation*}
In [6]:
q = QubitCircuit(2, reverse_states=False)
q.add_gate("CSIGN", controls=[0], targets=[1])
q.png
Out[6]:

Rotation about X-axis

In [7]:
rx(pi/2)
Out[7]:
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}0.707 & -0.707j\\-0.707j & 0.707\\\end{array}\right)\end{equation*}
In [8]:
q = QubitCircuit(1, reverse_states=False)
q.add_gate("RX", targets=[0], arg_value=pi/2, arg_label=r'\frac{\pi}{2}')
q.png
Out[8]:

Rotation about Y-axis

In [9]:
ry(pi/2)
Out[9]:
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}0.707 & -0.707\\0.707 & 0.707\\\end{array}\right)\end{equation*}
In [10]:
q = QubitCircuit(1, reverse_states=False)
q.add_gate("RY", targets=[0], arg_value=pi/2, arg_label=r'\frac{\pi}{2}')
q.png
Out[10]:

Rotation about Z-axis

In [11]:
rz(pi/2)
Out[11]:
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}(0.707-0.707j) & 0.0\\0.0 & (0.707+0.707j)\\\end{array}\right)\end{equation*}
In [12]:
q = QubitCircuit(1, reverse_states=False)
q.add_gate("RZ", targets=[0], arg_value=pi/2, arg_label=r'\frac{\pi}{2}')
q.png
Out[12]:

CNOT

In [13]:
cnot()
Out[13]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\0.0 & 1.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 1.0\\0.0 & 0.0 & 1.0 & 0.0\\\end{array}\right)\end{equation*}
In [14]:
q = QubitCircuit(2, reverse_states=False)
q.add_gate("CNOT", controls=[0], targets=[1])
q.png
Out[14]:

CSIGN

In [15]:
csign()
Out[15]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}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 & -1.0\\\end{array}\right)\end{equation*}
In [16]:
q = QubitCircuit(2, reverse_states=False)
q.add_gate("CSIGN", controls=[0], targets=[1])
q.png
Out[16]:

Berkeley

In [17]:
berkeley()
Out[17]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}0.924 & 0.0 & 0.0 & 0.383j\\0.0 & 0.383 & 0.924j & 0.0\\0.0 & 0.924j & 0.383 & 0.0\\0.383j & 0.0 & 0.0 & 0.924\\\end{array}\right)\end{equation*}
In [18]:
q = QubitCircuit(2, reverse_states=False)
q.add_gate("BERKELEY", targets=[0, 1])
q.png
Out[18]:

SWAPalpha

In [19]:
swapalpha(pi/2)
Out[19]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\0.0 & (0.610-0.488j) & (0.390+0.488j) & 0.0\\0.0 & (0.390+0.488j) & (0.610-0.488j) & 0.0\\0.0 & 0.0 & 0.0 & 1.0\\\end{array}\right)\end{equation*}

FREDKIN

In [20]:
fredkin()
Out[20]:
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 & 1.0 & 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 & 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 & 1.0 & 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 & 0.0 & 0.0 & 1.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0\\\end{array}\right)\end{equation*}

TOFFOLI

In [21]:
toffoli()
Out[21]:
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 & 1.0 & 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 & 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 & 1.0 & 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 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & 0.0\\\end{array}\right)\end{equation*}

SWAP

In [22]:
swap()
Out[22]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 1.0 & 0.0\\0.0 & 1.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 1.0\\\end{array}\right)\end{equation*}

ISWAP

In [23]:
iswap()
Out[23]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 1.0j & 0.0\\0.0 & 1.0j & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 1.0\\\end{array}\right)\end{equation*}

SQRTiSWAP

In [24]:
sqrtiswap()
Out[24]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.707 & 0.707j & 0.0\\0.0 & 0.707j & 0.707 & 0.0\\0.0 & 0.0 & 0.0 & 1.0\\\end{array}\right)\end{equation*}

SQRTSWAP

In [25]:
sqrtswap()
Out[25]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\0.0 & (0.500+0.500j) & (0.500-0.500j) & 0.0\\0.0 & (0.500-0.500j) & (0.500+0.500j) & 0.0\\0.0 & 0.0 & 0.0 & 1.0\\\end{array}\right)\end{equation*}

SQRTNOT

In [26]:
sqrtnot()
Out[26]:
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}(0.500+0.500j) & (0.500-0.500j)\\(0.500-0.500j) & (0.500+0.500j)\\\end{array}\right)\end{equation*}

HADAMARD

In [27]:
snot()
Out[27]:
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.707 & 0.707\\0.707 & -0.707\\\end{array}\right)\end{equation*}

PHASEGATE

In [28]:
phasegate(pi/2)
Out[28]:
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0\\0.0 & 1.0j\\\end{array}\right)\end{equation*}

GLOBALPHASE

In [29]:
globalphase(pi/2)
Out[29]:
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}1.0j & 0.0\\0.0 & 1.0j\\\end{array}\right)\end{equation*}

Mølmer–Sørensen gate

In [30]:
molmer_sorensen(pi/2)
Out[30]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}0.707 & 0.0 & 0.0 & -0.707j\\0.0 & 0.707 & -0.707j & 0.0\\0.0 & -0.707j & 0.707 & 0.0\\-0.707j & 0.0 & 0.0 & 0.707\\\end{array}\right)\end{equation*}

Qubit rotation gate

In [31]:
qrot(pi/2, pi/4)
Out[31]:
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}0.707 & (-0.500-0.500j)\\(0.500-0.500j) & 0.707\\\end{array}\right)\end{equation*}

Expanding gates to larger qubit registers

The example above show how to generate matrice representations of the gates implemented in QuTiP, in their minimal qubit requirements. If the same gates is to be represented in a qubit register of size $N$, the optional keywork argument N can be specified when calling the gate function. For example, to generate the matrix for the CNOT gate for a $N=3$ bit register:

In [32]:
cnot(N=3)
Out[32]:
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 & 1.0 & 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 & 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 & 1.0 & 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 & 1.0 & 0.0 & 0.0\\\end{array}\right)\end{equation*}
In [33]:
q = QubitCircuit(3, reverse_states=False)
q.add_gate("CNOT", controls=[1], targets=[2])
q.png
Out[33]:

Furthermore, the control and target qubits (when applicable) can also be similarly specified using keyword arguments control and target (or in some cases controls or targets):

In [34]:
cnot(N=3, control=2, target=0)
Out[34]:
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*}
In [35]:
q = QubitCircuit(3, reverse_states=False)
q.add_gate("CNOT", controls=[0], targets=[2])
q.png
Out[35]:

Setup of a Qubit Circuit

The gates implemented in QuTiP can be used to build any qubit circuit using the class QubitCircuit. The output can be obtained in the form of a unitary matrix or a latex representation.

In the following example, we take a SWAP gate. It is known that a swap gate is equivalent to three CNOT gates applied in the given format.

In [36]:
N = 2
qc0 = QubitCircuit(N)
qc0.add_gate("SWAP", [0, 1], None)
qc0.png
Out[36]:
In [37]:
U_list0 = qc0.propagators()
U0 = gate_sequence_product(U_list0)
U0
Out[37]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 1.0 & 0.0\\0.0 & 1.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 1.0\\\end{array}\right)\end{equation*}
In [38]:
qc1 = QubitCircuit(N)
qc1.add_gate("CNOT", 0, 1)
qc1.add_gate("CNOT", 1, 0)
qc1.add_gate("CNOT", 0, 1)
qc1.png
Out[38]:
In [39]:
U_list1 = qc1.propagators()
U1 = gate_sequence_product(U_list1)
U1
Out[39]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 1.0 & 0.0\\0.0 & 1.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 1.0\\\end{array}\right)\end{equation*}

In place of manually converting the SWAP gate to CNOTs, it can be automatically converted using an inbuilt function in QubitCircuit

In [40]:
qc2 = qc0.resolve_gates("CNOT")
qc2.png
Out[40]:
In [41]:
U_list2 = qc2.propagators()
U2 = gate_sequence_product(U_list2)
U2
Out[41]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 1.0 & 0.0\\0.0 & 1.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 1.0\\\end{array}\right)\end{equation*}

From QuTiP 4.4, we can also add gate at arbitrary position in a circuit.

In [ ]:
qc1.add_gate("CSIGN", index=1)
qc1.png

Example of basis transformation

In [42]:
qc3 = QubitCircuit(3)
qc3.add_gate("CNOT", 1, 0)
qc3.add_gate("RX", 0, None, pi/2, r"\pi/2")
qc3.add_gate("RY", 1, None, pi/2, r"\pi/2")
qc3.add_gate("RZ", 2, None, pi/2, r"\pi/2")
qc3.add_gate("ISWAP", [1, 2])
qc3.png
Out[42]:
In [43]:
U3 = gate_sequence_product(qc3.propagators())
U3
Out[43]:
Quantum object: dims = [[2, 2, 2], [2, 2, 2]], shape = (8, 8), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}(0.354-0.354j) & 0.0 & (-0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (-0.354-0.354j) & 0.0\\(0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0\\0.0 & (-0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (-0.354-0.354j) & 0.0 & (0.354+0.354j)\\0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j)\\(-0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (-0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0\\(0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0\\0.0 & (0.354+0.354j) & 0.0 & (-0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (-0.354+0.354j)\\0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j)\\\end{array}\right)\end{equation*}

The transformation can either be only in terms of 2-qubit gates:

In [44]:
qc4 = qc3.resolve_gates("CNOT")
qc4.png
Out[44]:
In [45]:
U4 = gate_sequence_product(qc4.propagators())
U4
Out[45]:
Quantum object: dims = [[2, 2, 2], [2, 2, 2]], shape = (8, 8), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}(0.354-0.354j) & 0.0 & (-0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (-0.354-0.354j) & 0.0\\(0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0\\0.0 & (-0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (-0.354-0.354j) & 0.0 & (0.354+0.354j)\\0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j)\\(-0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (-0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0\\(0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0\\0.0 & (0.354+0.354j) & 0.0 & (-0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (-0.354+0.354j)\\0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j)\\\end{array}\right)\end{equation*}
In [46]:
qc5 = qc3.resolve_gates("ISWAP")
qc5.png
Out[46]:
In [47]:
U5 = gate_sequence_product(qc5.propagators())
U5
Out[47]:
Quantum object: dims = [[2, 2, 2], [2, 2, 2]], shape = (8, 8), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}(0.354-0.354j) & 0.0 & (-0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (-0.354-0.354j) & 0.0\\(0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0\\0.0 & (-0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (-0.354-0.354j) & 0.0 & (0.354+0.354j)\\0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j)\\(-0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (-0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0\\(0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0\\0.0 & (0.354+0.354j) & 0.0 & (-0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (-0.354+0.354j)\\0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j)\\\end{array}\right)\end{equation*}

Or the transformation can be in terms of any 2 single qubit rotation gates along with the 2-qubit gate.

In [48]:
qc6 = qc3.resolve_gates(["ISWAP", "RX", "RY"])
qc6.png
Out[48]:
In [49]:
U6 = gate_sequence_product(qc6.propagators())
U6
Out[49]:
Quantum object: dims = [[2, 2, 2], [2, 2, 2]], shape = (8, 8), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}(0.354-0.354j) & 0.0 & (-0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (-0.354-0.354j) & 0.0\\(0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0\\0.0 & (-0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (-0.354-0.354j) & 0.0 & (0.354+0.354j)\\0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j)\\(-0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (-0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0\\(0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0\\0.0 & (0.354+0.354j) & 0.0 & (-0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (-0.354+0.354j)\\0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j)\\\end{array}\right)\end{equation*}
In [50]:
qc7 = qc3.resolve_gates(["CNOT", "RZ", "RX"])
qc7.png
Out[50]:
In [51]:
U7 = gate_sequence_product(qc7.propagators())
U7
Out[51]:
Quantum object: dims = [[2, 2, 2], [2, 2, 2]], shape = (8, 8), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}(0.354-0.354j) & 0.0 & (-0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (-0.354-0.354j) & 0.0\\(0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0\\0.0 & (-0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (-0.354-0.354j) & 0.0 & (0.354+0.354j)\\0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j)\\(-0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (-0.354+0.354j) & 0.0 & (0.354-0.354j) & 0.0\\(0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j) & 0.0\\0.0 & (0.354+0.354j) & 0.0 & (-0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (-0.354+0.354j)\\0.0 & (0.354-0.354j) & 0.0 & (0.354-0.354j) & 0.0 & (0.354+0.354j) & 0.0 & (0.354+0.354j)\\\end{array}\right)\end{equation*}

Resolving non-adjacent interactions

Interactions between non-adjacent qubits can be resolved by QubitCircuit to a series of adjacent interactions, which is useful for systems such as spin chain models.

In [52]:
qc8 = QubitCircuit(3)
qc8.add_gate("CNOT", 2, 0)
qc8.png
Out[52]:
In [53]:
U8 = gate_sequence_product(qc8.propagators())
U8
Out[53]:
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 & 1.0 & 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 & 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 & 1.0 & 0.0 & 0.0\\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 & 1.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & 0.0\\\end{array}\right)\end{equation*}
In [54]:
qc9 = qc8.adjacent_gates()
qc9.png
Out[54]:
In [55]:
U9 = gate_sequence_product(qc9.propagators())
U9
Out[55]:
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 & 1.0 & 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 & 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 & 1.0 & 0.0 & 0.0\\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 & 1.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & 0.0\\\end{array}\right)\end{equation*}
In [56]:
qc10 = qc9.resolve_gates("CNOT")
qc10.png
Out[56]:
In [57]:
U10 = gate_sequence_product(qc10.propagators())
U10
Out[57]:
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 & 1.0 & 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 & 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 & 1.0 & 0.0 & 0.0\\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 & 1.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & 0.0\\\end{array}\right)\end{equation*}

Adding gate in the middle of a circuit

From QuTiP 4.4 one can add a gate at an arbitrary position of a circuit. All one needs to do is to specify the parameter index. With this, we can also add the same gate at multiple positions at the same time.

In [58]:
qc = QubitCircuit(1)
qc.add_gate("RX", targets=1)
qc.add_gate("RX", targets=1)
qc.add_gate("RY", targets=1, index=[1,0])
qc.gates
Out[58]:
[Gate(RY, targets=[1], controls=None),
 Gate(RX, targets=[1], controls=None),
 Gate(RY, targets=[1], controls=None),
 Gate(RX, targets=[1], controls=None)]

User defined gates

From QuTiP 4.4 on, user defined gates can be defined by a python function that takes at most one parameter and return a Qobj, the dimension of the Qobj has to match the qubit system.

In [59]:
def user_gate1(arg_value):
    # controlled rotation X
    mat = np.zeros((4, 4), dtype=np.complex)
    mat[0, 0] = mat[1, 1] = 1.
    mat[2:4, 2:4] = rx(arg_value)
    return Qobj(mat, dims=[[2, 2], [2, 2]])

def user_gate2():
    # S gate
    mat = np.array([[1.,   0],
                    [0., 1.j]])
    return Qobj(mat, dims=[[2], [2]])

To let the QubitCircuit process those gates, we need to modify its attribute QubitCircuit.user_gates, which is a python dictionary in the form {name: gate_function}.

In [60]:
qc = QubitCircuit(2)
qc.user_gates = {"CTRLRX": user_gate1, 
                 "S"     : user_gate2}

When calling the add_gate method, the target qubits and the argument need to be given.

In [64]:
# qubit 0 controls qubit 1
qc.add_gate("CTRLRX", targets=[0,1], arg_value=pi/2)
# qubit 1 controls qubit 0
qc.add_gate("CTRLRX", targets=[1,0], arg_value=pi/2)
# a gate can also be added using the Gate class
g_T = Gate("S", targets=[1])
qc.add_gate("S", targets=[1])
props = qc.propagators()
In [65]:
props[0] # qubit 0 controls qubit 1
Out[65]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\0.0 & 1.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.707 & -0.707j\\0.0 & 0.0 & -0.707j & 0.707\\\end{array}\right)\end{equation*}
In [66]:
props[1] # qubit 1 controls qubit 0
Out[66]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.707 & 0.0 & -0.707j\\0.0 & 0.0 & 1.0 & 0.0\\0.0 & -0.707j & 0.0 & 0.707\\\end{array}\right)\end{equation*}
In [67]:
props[2] # S  gate acts on qubit 1
Out[67]:
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\0.0 & 1.0j & 0.0 & 0.0\\0.0 & 0.0 & 1.0 & 0.0\\0.0 & 0.0 & 0.0 & 1.0j\\\end{array}\right)\end{equation*}

Software versions

In [68]:
from qutip.ipynbtools import version_table
version_table()
Out[68]:
SoftwareVersion
QuTiP4.5.0.dev0+0f412cd4
Numpy1.18.1
SciPy1.4.1
matplotlib3.1.3
Cython0.29.15
Number of CPUs2
BLAS InfoINTEL MKL
IPython7.12.0
Python3.7.6 (default, Jan 8 2020, 19:59:22) [GCC 7.3.0]
OSposix [linux]
Mon Mar 02 11:35:36 2020 CET
In [ ]: