Decomposition of the Toffoli gate in terms of CNOT and single-qubit rotations

Copyright (C) 2011 and later, Paul D. Nation & Robert J. Johansson

This notebooks demonstrates how a toffoli gate can be rewritten in terms of CNOT gates and single qubit gates, and verifies the equivalence of the two gate sequences by comparing their matrix representations. For more information about the toffoli decomposition, see Nielsen & Chuang, Sec. 4.3, p178.

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]:
from qutip import *
In [2]:
q = QubitCircuit(3, reverse_states=False)
q.add_gate("TOFFOLI", controls=[0, 2], targets=[1])
In [3]:
q.png
Out[3]:
In [4]:
U = gate_sequence_product(q.propagators())

U.tidyup()
Out[4]:
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 & 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 & 1.0 & 0.0 & 0.0\\\end{array}\right)\end{equation*}
In [5]:
q2 = q.resolve_gates()
In [6]:
q2.png
Out[6]:
In [7]:
U2 = gate_sequence_product(q2.propagators())

U2.tidyup()
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 & 1.000 & 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 & 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 & 1.000 & 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 & 0.0 & 0.0 & 1.000 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.000 & 0.0 & 0.0\\\end{array}\right)\end{equation*}
In [8]:
U == U2
Out[8]:
True

Versions

In [9]:
%reload_ext version_information

%version_information numpy, cython, scipy, matplotlib, qutip
Out[9]:
SoftwareVersion
Python3.4.0 64bit [GCC 4.8.2]
IPython2.3.1
OSLinux 3.16.0 29 generic x86_64 with Ubuntu 14.10 utopic
numpy1.9.1
cython0.21.2
scipy0.14.1
matplotlib1.4.2
qutip3.1.0
Tue Jan 13 13:21:05 2015 JST