# QobjEvo: time-dependent quantum objects¶

Made by Eric Giguere, updated by Jake Lishman

In [1]:
# Basic setup
import numpy as np
import qutip

size = 4
t = 1.0
a = qutip.destroy(size)
n = qutip.num(size)
Id = qutip.qeye(size)


## Motivation¶

A Qobj is a generic quantum object, but it only represents constant objects. When we need to represent time-dependent objects, the relevant functions take a list of (Qobj, <time-dependence>) tuples, where the time dependence is a function, a string or an array. Inside the function, this is then converted into a new, time-dependent class QobjEvo. We then use this for several compilation and optimisation steps to ensure that the solvers run quickly.

You do not need to use QobjEvo, but if you are planning to reuse some large operators for a long-running integration, you can instantiate one yourself and manually call the C compilation methods. This will have a one-off start-up cost of a couple of seconds, but will make all uses of it faster afterwards. This trade-off is often worthwhile if the integration will take over around 15 minutes, even if you do not reuse the object.

## Instantiation¶

QobjEvo is instantiated with the same time-dependent list that is passed to, say, mesolve. This doesn't even have to be time dependent, but constant ones will not have significant benefits over varying ones. The three types of scalar time dependence are:

• function
• string
• array

This can represent objects of the form $$A(t) = \sum_k f_k(t) A_k$$ where the $f_k(t)$ are time-dependent scalars, and the $A_k$ are constant Qobj objects. The list then looks like

[A0, [A1, f1], [A2, f2], ...]

where all the Ak are constant Qobjs, and the fk are time dependences in one of the available forms.

It is not recommended to mix formats within one QobjEvo, as the available optimisations will typically be reduced.

In [2]:
constant_form = qutip.QobjEvo([n])


### Function time dependence¶

This should be a valid Python function with the signature

(t: float, args: dict) -> complex


where t is the time, args is a dictionary containing arguments which you can change without needing a new QobjEvo, and the return value is the complex value of $f_k$. We'll look more at args later.

In [3]:
def cos_t(t, args):
return np.cos(t)

function_form = qutip.QobjEvo([n, [a + ad, cos_t]])


If you need something more complex, such as a state with memory or to build a parametrised set of functions where the arguments will not change once set, you can use a class which implements __call__.

In [4]:
class callable_time_dependence:

def __call__(self, t, args):

callable_form = qutip.QobjEvo([n, [a + ad, callable_time_dependence(2)]])


### String time dependence¶

This should be a valid single Python expression that evaluates to a complex. Roughly, if you could do eval(x) and get a valid complex, you will be fine. In addition to t being the time, the following symbols are also defined in scope with their usual definitions:

sin  cos  tan   asin  acos  atan
sinh cosh tanh  asinh acosh atanh
exp  log  log10 erf   zerf  sqrt
real imag conj  abs   norm  arg
proj pi

In addition, np refers to numpy and spe to scipy.special.

String dependence can be compiled down to C code if Cython is available (or to pure Python if it isn't). Compiling to C takes a comparatively long time, but if you plan to reuse the same time dependence a lot, this will produce the fastest code.

In [5]:
string_form = qutip.QobjEvo([n, [a + ad, "cos(t)"]])


### Array dependence¶

If the time dependence is particularly costly to compute, you may pass an array containing the value of the functions evaluated at different times, and separately pass the corresponding list of times to the tlist keyword argument of the QobjEvo constructor. All times inbetween will be interpolated with cubic splines.

The times in tlist must be sorted, but they don't need to be evenly distributed. If you use this for more than one entry, all entries must have the same values for tlist (you can only pass one).

In [6]:
tlist = np.linspace(0, 10, 101)
values = np.cos(tlist)
array_form = qutip.QobjEvo([n, [a + ad, values]], tlist=tlist)


## Evaluation¶

No matter what type of time dependence was used, or even if only a constant was created, you can always call your QobjEvo instance like a function to get the value at that time. This will return a Qobj.

In [7]:
constant_form(2)

Out[7]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & 0.0 & 0.0 & 0.0\\0.0 & 1.0 & 0.0 & 0.0\\0.0 & 0.0 & 2.0 & 0.0\\0.0 & 0.0 & 0.0 & 3.0\\\end{array}\right)\end{equation*}
In [8]:
function_form(2)

Out[8]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & -0.416 & 0.0 & 0.0\\-0.416 & 1.0 & -0.589 & 0.0\\0.0 & -0.589 & 2.0 & -0.721\\0.0 & 0.0 & -0.721 & 3.0\\\end{array}\right)\end{equation*}
In [9]:
callable_form(2)

Out[9]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & 1.584 & 0.0 & 0.0\\1.584 & 1.0 & 2.240 & 0.0\\0.0 & 2.240 & 2.0 & 2.743\\0.0 & 0.0 & 2.743 & 3.0\\\end{array}\right)\end{equation*}
In [10]:
string_form(2)

Out[10]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & -0.416 & 0.0 & 0.0\\-0.416 & 1.0 & -0.589 & 0.0\\0.0 & -0.589 & 2.0 & -0.721\\0.0 & 0.0 & -0.721 & 3.0\\\end{array}\right)\end{equation*}
In [11]:
array_form(2)

Out[11]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & -0.416 & 0.0 & 0.0\\-0.416 & 1.0 & -0.589 & 0.0\\0.0 & -0.589 & 2.0 & -0.721\\0.0 & 0.0 & -0.721 & 3.0\\\end{array}\right)\end{equation*}

## Compilation¶

If you have Cython available, QobjEvo can compile itself down to C code for speed. This will be most successful with the string format of time dependence. The compilation process is likely to be slow, but afterwards the QobjEvo should return values significantly faster.

Compilation is done with the QobjEvo.compile() method, and modifies the object in-place. Calling the same method again will not force a recompilation. This is important for re-using compiled QobjEvo objects - the solvers will not need to recompile each time the same object is passed.

In [12]:
string_form.compile()
string_form(4)

Out[12]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & -0.654 & 0.0 & 0.0\\-0.654 & 1.0 & -0.924 & 0.0\\0.0 & -0.924 & 2.0 & -1.132\\0.0 & 0.0 & -1.132 & 3.0\\\end{array}\right)\end{equation*}

## Arguments¶

Data can be passed to the coefficient functions or strings using an args dictionary. In the functional form, this is passed as an explicit parameter. In the string form, args acts like additional variable definitions for the scope of the function. In other words, 'x + 2' will work and return a value if, for example, args is {'x': 1}.
The keys of args should always be strings representing valid Python identifiers (variable names), and they should not begin with an underscore (_).

Values can be any type, but if Cython compilation is used with string dependence, there will be a large performance penalty for using anything other than valid C numeric types (including complex), numpy arrays, or any object that Cython cannot call natively.

You can pass args directly at instantiation, and these will be used in every call, unless you specifically override them using the args keyword argument in the call. The overriding is temporary, and all subsequent calls will use the values given at initialisation.

### Function form¶

In [13]:
def coeff_with_args(t, args):
return t + args["delta"]

td_args = qutip.QobjEvo([Id, coeff_with_args], args={"delta": 1.0})
td_args(2)

Out[13]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}3.0 & 0.0 & 0.0 & 0.0\\0.0 & 3.0 & 0.0 & 0.0\\0.0 & 0.0 & 3.0 & 0.0\\0.0 & 0.0 & 0.0 & 3.0\\\end{array}\right)\end{equation*}
In [14]:
# Temporarily overriding the arguments.
td_args(2, args={"delta": 10})

Out[14]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}12.0 & 0.0 & 0.0 & 0.0\\0.0 & 12.0 & 0.0 & 0.0\\0.0 & 0.0 & 12.0 & 0.0\\0.0 & 0.0 & 0.0 & 12.0\\\end{array}\right)\end{equation*}
In [15]:
# A subsequent regular call will still use the args given at initialisation.
td_args(2)

Out[15]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}3.0 & 0.0 & 0.0 & 0.0\\0.0 & 3.0 & 0.0 & 0.0\\0.0 & 0.0 & 3.0 & 0.0\\0.0 & 0.0 & 0.0 & 3.0\\\end{array}\right)\end{equation*}

### String form¶

In [16]:
td_args_str = qutip.QobjEvo([Id, "t + delta"], args={"delta": 1.0})
td_args_str(2)

Out[16]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}3.0 & 0.0 & 0.0 & 0.0\\0.0 & 3.0 & 0.0 & 0.0\\0.0 & 0.0 & 3.0 & 0.0\\0.0 & 0.0 & 0.0 & 3.0\\\end{array}\right)\end{equation*}
In [17]:
td_args_str(2, args={"delta": 10})

Out[17]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}12.0 & 0.0 & 0.0 & 0.0\\0.0 & 12.0 & 0.0 & 0.0\\0.0 & 0.0 & 12.0 & 0.0\\0.0 & 0.0 & 0.0 & 12.0\\\end{array}\right)\end{equation*}

### Using objects¶

The argument value need not just be a number. Even Cython-compiled strings can accept functions which Cython can call natively, such as the core numpy functions.

In [18]:
td_args_str = qutip.QobjEvo([Id, "f(t)"], args={"f": np.cos})
td_args_str.compile()
td_args_str(0.0)

Out[18]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\0.0 & 1.0 & 0.0 & 0.0\\0.0 & 0.0 & 1.0 & 0.0\\0.0 & 0.0 & 0.0 & 1.0\\\end{array}\right)\end{equation*}
In [19]:
td_args_str(np.pi)

Out[19]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}-1.0 & 0.0 & 0.0 & 0.0\\0.0 & -1.0 & 0.0 & 0.0\\0.0 & 0.0 & -1.0 & 0.0\\0.0 & 0.0 & 0.0 & -1.0\\\end{array}\right)\end{equation*}

### Dynamic arguments¶

When QobjEvo is used in the solvers, certain dynamic arguments will be populated at each iteration, if and only if their names are present in the args dictionary used at QobjEvo intialisation. The initial values of all of these dynamic arguments will be a representation of 0 in the corresponding type, for example "state" will be qzero() of the correct dimensions.

There several of these "magic" variables, mostly revolving around the state currently being evolved:

• "state" or "state_qobj": a Qobj of the current state.
• "state_mat": a dense 2D np.ndarray of the state as a matrix, similar to state.full()
• "state_vec": a dense 1D np.ndarray of the state as a vector. This only generally makes sense for kets.
• "expect_op_<n>": complex, where <n> is an index into e_ops, the current expectation value of e_ops[n] (the <> should not appear, e.g. "expect_op_0").
• "collapse": a list of (t: float, n: int) indicating the time t a collapse occurred, and which of c_ops caused it. Only present when using mcsolve.
In [20]:
args = {"state": None}

def print_args(t, args):
print("\n".join(['"' + key + '":\n' +
repr(value) for key, value in args.items()]))
return t

td_args = qutip.QobjEvo([Id, print_args], args=args)
td_args(0)

"state":
Quantum object: dims = [[4], [1]], shape = (4, 1), type = ket
Qobj data =
[[0.]
[0.]
[0.]
[0.]]

Out[20]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0\\\end{array}\right)\end{equation*}
In [21]:
# The state keyword argument is typically unused.
# Here it just simulates being inside a solver at a particular state.
td_args(0, state=qutip.basis(4, 2))

"state":
Quantum object: dims = [[4], [1]], shape = (4, 1), type = ket
Qobj data =
[[0.]
[0.]
[1.]
[0.]]

Out[21]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0\\\end{array}\right)\end{equation*}

## Mathematics¶

QobjEvo supports the basic mathematical operations which make sense for time-dependent quantum objects:

• addition of QobjEvo and Qobj
• subtraction of QobjEvo and Qobj
• product with QobjEvo, Qobj or scalars
• division by a scalar
• negation: -x
• conjugation: QobjEvo.conj()
• adjoint (dagger): QobjEvo.dag()
• transpose: QobjEvo.trans
In [22]:
(array_form * 2)(0)

Out[22]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & 2.0 & 0.0 & 0.0\\2.0 & 2.0 & 2.828 & 0.0\\0.0 & 2.828 & 4.0 & 3.464\\0.0 & 0.0 & 3.464 & 6.0\\\end{array}\right)\end{equation*}
In [23]:
(array_form + 1 + a)(0)

Out[23]:
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 2.0 & 0.0 & 0.0\\1.0 & 2.0 & 2.828 & 0.0\\0.0 & 1.414 & 3.0 & 3.464\\0.0 & 0.0 & 1.732 & 4.0\\\end{array}\right)\end{equation*}

## Superoperators¶

The functions in qutip.superoperator can also be used for QobjEvo. Of particular importance is liouvillian, as mesolve can take the Liouvillian as its first argument (and will be faster this way), in place of separated Hamiltonians and collapse operators.

In [24]:
liouv = qutip.liouvillian(array_form, c_ops=[constant_form])
liouv(0)

Out[24]:
Quantum object: dims = [[[4], [4]], [[4], [4]]], shape = (16, 16), type = super, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}0.0 & -1.0j & 0.0 & 0.0 & 1.0j & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\-1.0j & (-0.500-1.0j) & -1.414j & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & -1.414j & (-2.0-2.0j) & -1.732j & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & -1.732j & (-4.500-3.0j) & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\1.0j & 0.0 & 0.0 & 0.0 & (-0.500+1.0j) & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \vdots\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & (-0.500-1.0j) & 0.0 & 0.0 & 0.0 & 1.732j\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & (-4.500+3.0j) & -1.0j & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & -1.0j & (-2.0+2.0j) & -1.414j & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & -1.414j & (-0.500+1.0j) & -1.732j\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 1.732j & 0.0 & 0.0 & -1.732j & 0.0\\\end{array}\right)\end{equation*}

## Epilogue¶

In [25]:
qutip.about()

QuTiP: Quantum Toolbox in Python
================================
Copyright (c) QuTiP team 2011 and later.
Current admin team: Alexander Pitchford, Nathan Shammah, Shahnawaz Ahmed, Neill Lambert, Eric Giguère, Boxi Li, Jake Lishman, Simon Cross and Asier Galicia.
Board members: Daniel Burgarth, Robert Johansson, Anton F. Kockum, Franco Nori and Will Zeng.
Original developers: R. J. Johansson & P. D. Nation.
Currently developed through wide collaboration. See https://github.com/qutip for details.

QuTiP Version:      4.7.1.dev0+9098716
Numpy Version:      1.22.4
Scipy Version:      1.8.1
Cython Version:     0.29.32
Matplotlib Version: 3.5.2
Python Version:     3.10.4
Number of CPUs:     2
BLAS Info:          Generic
OPENMP Installed:   False
INTEL MKL Ext:      False
Platform Info:      Linux (x86_64)
Installation path:  /home/runner/work/qutip-tutorials/qutip-tutorials/qutip/qutip
================================================================================
For your convenience a bibtex reference can be easily generated using qutip.cite()