Christopher Granade
University of Sydney
$\newcommand{\ket}[1]{\left|#1\right\rangle}$
$\newcommand{\bra}[1]{\left\langle#1\right|}$
$\newcommand{\cnot}{{\scriptstyle \rm CNOT}}$
$\newcommand{\Tr}{\operatorname{Tr}}$
In this notebook, we will demonstrate the qubit_clifford_group generator, which yields each element of the Clifford group on a single qubit.
We enable a few features such that this notebook runs in both Python 2 and 3.
In [1]:
from __future__ import division, print_function
In [2]:
import numpy as np
import qutip as qt
from qutip.ipynbtools import version_table
from IPython.display import display
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%matplotlib inline
In [4]:
qt.settings.colorblind_safe = True
The Clifford group $\mathcal{C}_n$ on $n$ qubits is the group of automorphisms of the Pauli group $\mathcal{P}_n$, $\mathcal{C}_n = \{U | \forall P \in \mathcal{P}_n: U P U^\dagger \in \mathcal{P}_n\}$, and is generated by the CNOT, Hadamard and phase gates. The Clifford group is very useful in a number of different contexts in quantum information such as error correction and randomized benchmarking. In particular, the Clifford group is often of interest because the Gottesman-Knill theorem gives that a circuit made up only of Clifford gates, Pauli preparations and Pauli measurements can be efficiently simulated with classical resources. This simulation proceeds by representing Clifford group elements by their action on a generating set for the Pauli group--- this representation requires only polynomially many classical bits with the number of qubits.
At times, however, we will need representations of Clifford group elements as unitary operators on the full Hilbert space. This is useful, for instance, in reasoning about how imperfect Clifford group elements behave. QuTiP provides a generator, qubit_clifford_group, that iterates over the elements of the Clifford group on a single qubit.
In [5]:
cliffords = list(qt.qubit_clifford_group())
This generator works by using the decomposition exploited by Ross and Selinger in their circuit synthesis package. In particular, they use that the single-qubit Clifford group can be characterized as
\begin{equation}
\mathcal{C}_1 = \left\{
\omega^i E^j X^k S^\ell |
i \in \operatorname{range}(8),\
j \in \operatorname{range}(3),\
k \in \operatorname{range}(2),\
\ell \in \operatorname{range}(4)
\right\},
\end{equation}
where $\omega = \mathrm{e}^{2\pi \mathrm{i} / 8}$, $S$ is the $\pi / 2$ phase gate, $E = \omega^3 H S^3$, $H$ is the Hadamard gate and $X$ is the familiar Pauli matrix. The function qubit_clifford_group yields this characterization, save for the iteration over powers of \omega.
In [6]:
display("S", cliffords[1])
display("X", cliffords[4])
display("E", cliffords[8])
These three gates can be understood in terms of their actions on the Pauli group. To demonstrate, we plot the superoperator representations for each of $S$, $X$ and $E$ in the Pauli basis as Hinton diagrams.
In [7]:
qt.visualization.hinton(qt.to_super(cliffords[1]));
qt.visualization.hinton(qt.to_super(cliffords[4]));
qt.visualization.hinton(qt.to_super(cliffords[8]));
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version_table()
Out[8]:
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