# Overview of operator classes

In Qiskit, quantum operators are represented using classes from the quantum_info module. The most important operator class is SparsePauliOp, which represents a general quantum operator as a linear combination of Pauli strings. SparsePauliOp is the class most commonly used to represent quantum observables. The rest of this page explains how to use SparsePauliOp and other operator classes.

[1] :
import numpy as np
from qiskit.quantum_info.operators import Operator, Pauli, SparsePauliOp

## SparsePauliOp

The SparsePauliOp class represents a linear combination of Pauli strings. There are several ways to initialize a SparsePauliOp, but the most flexible way is to use the from_sparse_list method, as demonstrated in the following code cell. The from_sparse_list accepts a list of (pauli_string, qubit_indices, coefficient) triplets.

[2] :
op1 = SparsePauliOp.from_sparse_list(
[("ZX", [1, 4], 1.0), ("YY", [0, 3], -1 + 1j)], num_qubits=5
)
op1

Output:

SparsePauliOp(['XIIZI', 'IYIIY'],
coeffs=[ 1.+0.j, -1.+1.j])


SparsePauliOp supports arithmetic operations, as demonstrated in the following code cell.

[3] :
op2 = SparsePauliOp.from_sparse_list(
[("XXZ", [0, 1, 4], 1 + 2j), ("ZZ", [1, 2], -1 + 1j)], num_qubits=5
)

print("op1 + op2:")
print(op1 + op2)
print()
# Multiplication by a scalar
print("2 * op1:")
print(2 * op1)
print()
# Operator multiplication (composition)
print("op1 @ op2:")
print(op1 @ op2)
print()
# Tensor product
print("op1.tensor(op2):")
print(op1.tensor(op2))

Output:

op1 + op2:
SparsePauliOp(['XIIZI', 'IYIIY', 'ZIIXX', 'IIZZI'],
coeffs=[ 1.+0.j, -1.+1.j,  1.+2.j, -1.+1.j])

2 * op1:
SparsePauliOp(['XIIZI', 'IYIIY'],
coeffs=[ 2.+0.j, -2.+2.j])

op1 @ op2:
SparsePauliOp(['YIIYX', 'XIZII', 'ZYIXZ', 'IYZZY'],
coeffs=[ 1.+2.j, -1.+1.j, -1.+3.j,  0.-2.j])

op1.tensor(op2):
SparsePauliOp(['XIIZIZIIXX', 'XIIZIIIZZI', 'IYIIYZIIXX', 'IYIIYIIZZI'],
coeffs=[ 1.+2.j, -1.+1.j, -3.-1.j,  0.-2.j])


## Pauli

The Pauli class represents a single Pauli string with an optional phase coefficient from the set $\set{+1, i, -1, -i}$. A Pauli can be initialized by passing a string of characters from the set {"I", "X", "Y", "Z"}, optionally prefixed by one of {"", "i", "-", "-i"} to represent the phase coefficient.

[4] :
op1 = Pauli("iXX")
op1

Output:

Pauli('iXX')


The following code cell demonstrates the use of some attributes and methods.

[5] :
print(f"Dimension of {op1}: {op1.dim}")
print(f"Phase of {op1}: {op1.phase}")
print(f"Matrix representation of {op1}: \n {op1.to_matrix()}")

Output:

Dimension of iXX: (4, 4)
Phase of iXX: 3
Matrix representation of iXX:
[[0.+0.j 0.+0.j 0.+0.j 0.+1.j]
[0.+0.j 0.+0.j 0.+1.j 0.+0.j]
[0.+0.j 0.+1.j 0.+0.j 0.+0.j]
[0.+1.j 0.+0.j 0.+0.j 0.+0.j]]


Pauli objects possess a number of other methods to manipulate the operators such as determining its adjoint, whether it (anti)commutes with another Pauli, and computing the dot product with another Pauli. Refer to the API documentation for more info.

## Operator

The Operator class represents a general linear operator. Unlike SparsePauliOp, Operator stores the linear operator as a dense matrix. Because the memory required to store a dense matrix scales exponentially with the number of qubits, the Operator class is only suitable for use with a small number of qubits.

You can initialize an Operator by directly passing a Numpy array storing the matrix of the operator. For example, the following code cell creates a two-qubit Pauli XX operator:

[6] :
XX = Operator(
np.array(
[
[0, 0, 0, 1],
[0, 0, 1, 0],
[0, 1, 0, 0],
[1, 0, 0, 0],
]
)
)
XX

Output:

Operator([[0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],
[0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],
[0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],
[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]],
input_dims=(2, 2), output_dims=(2, 2))


The operator object stores the underlying matrix, and the input and output dimension of subsystems.

• data: To access the underlying Numpy array, you can use the Operator.data property.
• dims: To return the total input and output dimension of the operator, you can use the Operator.dim property. Note: the output is returned as a tuple (input_dim, output_dim), which is the reverse of the shape of the underlying matrix.
[7] :
XX.data

Output:

array([[0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],
[0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],
[0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],
[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]])

[8] :
input_dim, output_dim = XX.dim
input_dim, output_dim

Output:

(4, 4)


The operator class also keeps track of subsystem dimensions, which can be used for composing operators together. These can be accessed using the input_dims and output_dims functions.

For $2^N$ by $2^M$ operators, the input and output dimensions are automatically assumed to be M-qubit and N-qubit:

[9] :
op = Operator(np.random.rand(2**1, 2**2))
print("Input dimensions:", op.input_dims())
print("Output dimensions:", op.output_dims())

Output:

Input dimensions: (2, 2)
Output dimensions: (2,)


If the input matrix is not divisible into qubit subsystems, then it will be stored as a single-qubit operator. For example, for a $6\times6$ matrix:

[10] :
op = Operator(np.random.rand(6, 6))
print("Input dimensions:", op.input_dims())
print("Output dimensions:", op.output_dims())

Output:

Input dimensions: (6,)
Output dimensions: (6,)


The input and output dimension can also be manually specified when initializing a new operator:

[11] :
# Force input dimension to be (4,) rather than (2, 2)
op = Operator(np.random.rand(2**1, 2**2), input_dims=[4])
print("Input dimensions:", op.input_dims())
print("Output dimensions:", op.output_dims())

Output:

Input dimensions: (4,)
Output dimensions: (2,)

[12] :
# Specify system is a qubit and qutrit
op = Operator(np.random.rand(6, 6), input_dims=[2, 3], output_dims=[2, 3])
print("Input dimensions:", op.input_dims())
print("Output dimensions:", op.output_dims())

Output:

Input dimensions: (2, 3)
Output dimensions: (2, 3)


You can also extract just the input or output dimensions of a subset of subsystems using the input_dims and output_dims functions:

[13] :
print("Dimension of input system 0:", op.input_dims([0]))
print("Dimension of input system 1:", op.input_dims([1]))

Output:

Dimension of input system 0: (2,)
Dimension of input system 1: (3,)


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