First things first
To load the qutip modules, we must first call the import statement:
In [1]: from qutip import *
that will load all of the user available functions. Often, we also need to import the NumPy and Matplotlib libraries with:
In [2]: import numpy as np In [3]: import matplotlib.pyplot as plt
The quantum object class
The key difference between classical and quantum mechanics lies in the use of operators instead of numbers as variables. Moreover, we need to specify state vectors and their properties. Therefore, in computing the dynamics of quantum systems we need a data structure that is capable of encapsulating the properties of a quantum operator and ket/bra vectors. The quantum object class, qutip.Qobj, accomplishes this using matrix representation.
Note
If you are running QuTiP from a python script you must use the print function to view the Qobj attributes.
States and operators
Manually specifying the data for each quantum object is inefficient. Even more so when most objects correspond to commonly used types such as the ladder operators of a harmonic oscillator, the Pauli spin operators for a two-level system, or state vectors such as Fock states. Therefore, QuTiP includes predefined objects for a variety of states in states module:
| States | Command | Inputs |
|---|---|---|
| Fock ket | states.fock(N,m) | N = number of levels in Hilbert space, m = level containing excitation |
| Coherent ket | states.coherent(N,alpha) | alpha = complex number (eigenvalue) for requested coherent state |
| Fock density matrix | states.fock_dm(N,m) | same as Fock state ket |
| Coherent density matrix | states.coherent_dm(N,alpha) | same as Coherent state ket |
| Thermal density matrix | states.thermal_dm(N,n) | n = particle number expectation value |
and operators in the operators module:
| Operators | Command | Inputs |
|---|---|---|
| Commutator | operators.commutator(A, B, kind) | Kind = ‘normal’ or ‘anti’. |
| Identity | operators.qeye(N) | N = number of levels in Hilbert space. |
| Annihilation | operators.destroy(N) | same as above |
| Creation | operators.create(N) | same as above |
Qobj attributes
We have seen that a quantum object has several internal attributes, such as data, dims, and shape. These can be accessed in the following way:
In [4]: a = destroy(4)
In [5]: a.dims Out[5]: [[4], [4]]
In [6]: a.shape Out[6]: (4, 4)
In general, the attributes (properties) of a Qobj object (or any Python class) can be retrieved using the Q.attribute notation. In addition to the attributes shown with the print function, the Qobj class also has the following:
| Property | Attribute | Description |
|---|---|---|
| Data | Q.data | Matrix representing state or operator |
| Dimensions | Q.dims | List keeping track of shapes for individual components of a multipartite system (for tensor products and partial traces). |
| Shape | Q.shape | Dimensions of underlying data matrix. |
| is Hermitian? | Q.isherm | Is the operator Hermitian or not? |
| Type | Q.type | Is object of type ‘ket, ‘bra’, ‘oper’, or ‘super’? |
The data attribute returns a message stating that the data is a sparse matrix. All Qobj instances store their data as a sparse matrix to save memory. To access the underlying dense matrix one needs to use the qutip.Qobj.full function as described below.
Qobj Math
The rules for mathematical operations on Qobj instances are similar to standard matrix arithmetic. In addition, the logic operators is equal == and is not equal != are also supported.
In [7]: a + 5
Adds the matrix a with an identity matrix of the same size multiplied by 5. In other words an equivalent to the above math is:
In [8]: a + 5*qeye(4)
The matrix as well as scalar products are performed by '*'.
Functions operating on Qobj class
Like attributes, the quantum object class has defined functions (methods) that operate on Qobj class instances. For a general quantum object Q:
| Function | Command | Description |
|---|---|---|
| Conjugate | Q.conj() | Conjugate of quantum object. |
| Dagger | Q.dag() | Returns adjoint (dagger) of object. |
| Diagonal | Q.diag() | Returns the diagonal elements. |
| Eigenenergies | Q.eigenenergies() | Eigenenergies (values) of operator. |
| Eigenstates | Q.eigenstates() | Returns eigenvalues and eigenvectors. |
| Groundstate | Q.groundstate() | Eigenval & eigket of Qobj groundstate. |
| Overlap | Q.overlap(state) | Overlap between current Qobj and a given state. |
| Projector | Q.proj() | Form projector operator from given ket or bra vector. |
| Trace | Q.tr() | Returns trace of quantum object. |
| Transpose | Q.trans() | Transpose of quantum object. |