PauliList Class

[dsvandet]

These notes are related to the PauliList class representing a list of Pauli Operators

Introduction

These notes are in regards to satisfying the following requirement:

• Requirement: An class that represents a list of Pauli operators

This requirement is can be located at:

Core QEC representations // Requirement: How to represent a list of Pauli Operators?

There are two related topics for discussion here: Pauli and PauliList classes from quantum_info

Comments on Current Pauli and PauliList classes

A list of Pauli operators can be used as (for example)

• a generating set for a Stabilizer or Gauge group
• a general list of Pauli operators for computations
• a Pauli Frame
• or a whole host of others uses

It should be noted here that I am using the term operator in the more general abstract mathematical object rather than just a matrix representation as often used in Qiskit.

The quantum_info module within qiskit terra provides some classes that could be used for this purpose. Of interest for this requirement is the set of capabilities in the operators directory. The symplectic folder holds the following classes

• BasePauli class: Symplectic representation of a list of N-qubit Paulis. Base class for Pauli and PauliTable
• Pauli class: This class represents an operator $P$ from the full $n$-qubit Pauli group
• PauliTable class: Symplectic representation of a list Pauli operators.
• StabilizerTable class: Symplectic representation of a list Stabilizer operators

The PauliTable class cannot store any phases and the StabilizerTable can only store a restricted class of phases meaning that you cannot have a list of general Pauli operators with any (allowable Pauli phase $-1, 1, -i, i$). An issue Terra-Issue #5993 was raised and a new PauliList class is being developed. The PR for this is Terra-PR-pauli-list. This PR has now be completed.

As it stands currently PauliList will replace both StabilizerTable and PauliTable. PauliList is a subclass of the modified BasePauli which in the PR above now can store a list of Pauli Operators with all allowed phases.

PauliList should be suitable as a base for the needs in qiskit-qec.

Examples:

from qiskit.quantum_info.operators import Pauli
from qiskit.quantum_info.operators import PauliList

a = Pauli('-iXYZ')
b = PauliList(['XYY','iYYY','-iXXX'])

a.phase
array([1])

a._phase
array([2])

b._x
array([[ True,  True, False],
       [ True,  True,  True],
       [False, False, False]])

Location of Code

Currently these changes are staged at dsvandet/qiskit-terra/tree/paulilist_representations.

Modifications to PauliList Class

• Allow a PauliList to be created from a lists of Pauli's with a different number of qubits. So now you can say PauliList(['X','ZY']) and get the equivalent form PauliList(['IX', 'ZY']). This is essential when using index notation for Pauli strings. For example my_list = PauliList(["X1X2X3X4", "Y9Y6X3X1"]) will give you PauliList(['IIIIIXXXXI', 'YIIYIIXIXI']). You can specify the number of quits in index form using something like PauliList("X1X2X3", "Z3X7I23") if needed.