Enable piece-wise-constant functions (#3645)

* Initial draft of time dependent hamiltonian

* Allow creation of TDHamiltonian by multiplication of fn and Observable

* Import TDHamiltonian as qml.ops.TDHamiltonian

* Remove top-level import due to circular imports

* Reorganize H_drift and H_ts

* Add docstrings

* Fix addition bug for Tensor and Observable

* Update docstring

* Rename TDHamiltonian to ParametrizedHamiltonian

* Rename file parametrized_hamiltonian.py

* Calling H(params, t) returns Operator instead of matrix

* Remove inheritance from Observable

* Change variable names and docstring comments to reflect switch from Time-Dependent to Parametrized

* Docstring example

* Move from qubit module to math_op module

* Fix bug when calling ParametrizedHamiltonian if H_fixed is None

* Update __add__ method

* Add tests

* Update tests_passing_pylint

* update tests for pylint

* Switch from isfunction to callable in Observable.__mul__

* Return 0 instead of None if _get_terms is empty

* Apply docstring suggestions from code review

Co-authored-by: Albert Mitjans <a.mitjanscoma@gmail.com>
Co-authored-by: Korbinian Kottmann <43949391+Qottmann@users.noreply.github.com>

* Clean up based on code review suggestions

* Remove assumption that ops are Observables

* Switch from pH.H_fixed to pH.H_fixed()

* Support addition as Operator+ParametrizedHamiltonian

* Support creating ParametrizedHamiltonian via qml.ops.dot

* Test for qutrit ParametrizedHamiltonian

* Add wires argument to __call__

* Incorporate code review suggestions

* Add pwc_from_array

* Add pwc_from_function

* Examples in docstrings

* Change call signature on pwc_from_array to (dt, index)

* Change call signature on pwc_from_smooth to (dt, num_bins)

* Deal with jax.numpy import

* Update pennylane/operation.py

* Remove unintentional edits

* Remove unintended changes

* Switch arg from dt to t

* Move from math to pulse module

* Return 0 outside of time interval

* Fix bug where first bin is twice as large as subsequent bins and extends to before t1

* Add tests

* Clarifying change in docstring

* Reorganize tests

* Update tests

* changelog

* Update tests

* Rename t in pwc to timespan to differentiate from t in (params, t) from the resulting callable

* Rename t in pwc to timespan to differentiate from t in (params, t) from the resulting callable

* Reinstate import tests

* Update docstring

* Mark as xfail for now

* Update tests

* test codecov

* clean up

* Apply suggestions from code review

* Apply suggestions from code review

* Move files pt1

* Update example

* Fix import statement

* Use jnp.concatenate instead of appending to list

Co-authored-by: Albert Mitjans <a.mitjanscoma@gmail.com>
Co-authored-by: Korbinian Kottmann <43949391+Qottmann@users.noreply.github.com>

doc/releases/changelog-dev.md

@@ -241,6 +241,55 @@
   4: ─────────────────╰RY(0.81)─╰SWAP─────────────────╰RY(0.06)─╰SWAP─────────────────┤  State
   ```
 
+
+* Added `pwc` as a convenience function for defining a `ParametrizedHamiltonian`.
+  This function can be used to create a callable coefficient by setting 
+  the timespan over which the function should be non-zero. The resulting callable 
+  can be passed an array of parameters and a time.
+  [(#3645)](https://github.com/PennyLaneAI/pennylane/pull/3645)
+
+  ```pycon
+  >>> timespan = (2, 4)
+  >>> f = pwc(timespan)
+  >>> f * qml.PauliX(0)
+  ParametrizedHamiltonian: terms=1
+  ``` 
+  The `params` array will be used as bin values evenly distributed over the timespan, 
+  and the parameter `t` will determine which of the bins is returned.
+
+  ```pycon
+  >>> f(params=[1.2, 2.3, 3.4, 4.5], t=3.9)
+  DeviceArray(4.5, dtype=float32) 
+  >>> f(params=[1.2, 2.3, 3.4, 4.5], t=6)  # zero outside the range (2, 4) 
+  DeviceArray(0., dtype=float32)
+  ```
+
+* Added `pwc_from_function` as a decorator for defining a `ParametrizedHamiltonian`.
+  This function can be used to decorate a function and create a piecewise constant
+  approximation of it. 
+  [(#3645)](https://github.com/PennyLaneAI/pennylane/pull/3645)
+
+  ```pycon
+  >>> @pwc_from_function(t=(2, 4), num_bins=10)
+  ... def f1(p, t):
+  ...     return p * t
+  ```
+  The resulting function approximates the same of `p**2 * t` on the interval `t=(2, 4)` 
+  in 10 bins, and returns zero outside the interval.
+
+  ```pycon
+  # t=2 and t=2.1 are within the same bin
+  >>> f1(3, 2), f1(3, 2.1)
+  (DeviceArray(6., dtype=float32), DeviceArray(6., dtype=float32))
+  # next bin
+  >>> f1(3, 2.2)
+  DeviceArray(6.6666665, dtype=float32)
+  # outside the interval t=(2, 4)
+  >>> f1(3, 5)
+  DeviceArray(0., dtype=float32)
+  ```
+
+
 <h3>Improvements</h3>
 
 * `qml.purity` is added as a measurement process for purity

pennylane/pulse/__init__.py

@@ -15,6 +15,6 @@
 This module contains classes and functions used in pulse programming.
 """
 
-from .convenience_functions import constant, rect
+from .convenience_functions import constant, rect, pwc, pwc_from_function
 from .parametrized_evolution import ParametrizedEvolution
 from .parametrized_hamiltonian import ParametrizedHamiltonian

pennylane/pulse/convenience_functions.py

@@ -13,6 +13,7 @@
 # limitations under the License.
 """This file contains convenience functions for pulse programming."""
 from typing import Callable, List, Tuple, Union
+import numpy as np
 
 has_jax = True
 try:
@@ -115,3 +116,132 @@ def f(p, t):
         return _f(p, t)
 
     return f
+
+
+def pwc(timespan):
+    """Creates a function that is piecewise-constant in time.
+
+    Args:
+            timespan(Union[float, tuple(float, float)]: The timespan defining the region where the function is non-zero.
+              If an integer is provided, the timespan is defined as ``(0, timespan)``.
+
+    Returns:
+            func: a function that takes two arguments, an array of trainable parameters and a `float` defining the
+            time at which the function is evaluated. When called, the function uses the array of parameters to
+            create evenly sized bins within the ``timespan``, with each bin value set by an element of the array.
+            It then selects the value of the parameter array corresponding to the specified time, based on the
+            assigned binning.
+
+    **Example**
+
+    >>> timespan = (1, 3)
+    >>> f1 = pwc(timespan)
+
+    The resulting function ``f1`` has the call signature ``f1(params, t)``. If passed an array of parameters and
+    a time, it will assign the array as the constants in the piecewise function, and select the constant corresponding
+    to the specified time, based on the time interval defined by ``timespan``.
+
+    >>> params = [10, 11, 12, 13, 14]
+    >>> f1(params, 2)
+    Array(12, dtype=int32)
+
+    >>> f1(params, 2.1)  # same bin
+    Array(12, dtype=int32)
+
+    >>> f1(params, 2.5)  # next bin
+    Array(13, dtype=int32)
+    """
+    if not has_jax:
+        raise ImportError(
+            "Module jax is required for any pulse-related convenience function. "
+            "You can install jax via: pip install jax"
+        )
+
+    if isinstance(timespan, tuple):
+        t1, t2 = timespan
+    else:
+        t1 = 0
+        t2 = timespan
+
+    def func(params, t):
+        num_bins = len(params)
+        params = jnp.concatenate([jnp.array(params), jnp.zeros(1)])
+        # get idx from timestamp, then set idx=0 if idx is out of bounds for the array
+        idx = num_bins / (t2 - t1) * (t - t1)
+        idx = jnp.where((idx >= 0) & (idx <= num_bins), jnp.array(idx, dtype=int), -1)
+
+        return params[idx]
+
+    return func
+
+
+def pwc_from_function(timespan, num_bins):
+    """
+    Decorator to turn a smooth function into a piecewise constant function.
+
+    Args:
+            timespan(Union[float, tuple(float)]): The timespan defining the region where the function is non-zero.
+              If an integer is provided, the timespan is defined as ``(0, timespan)``.
+            num_bins(int): number of bins for time-binning the function
+
+    Returns:
+            a function that takes some smooth function ``f(params, t)`` and converts it to a
+            piecewise constant function spanning time ``t`` in `num_bins` bins.
+
+    **Example**
+
+    .. code-block:: python3
+
+        def smooth_function(params, t):
+            return params[0] * t + params[1]
+
+        timespan = 10
+        num_bins = 10
+
+        binned_function = pwc_from_function(timespan, num_bins)(f0)
+
+    >>> binned_function([2, 4], 3), smooth_function([2, 4], 3)  # t = 3
+    (DeviceArray(10.666666, dtype=float32), DeviceArray(10, dtype=int32))
+
+    >>> binned_function([2, 4], 3.2), smooth_function([2, 4], 3.2)  # t = 3.2
+    (DeviceArray(10.666666, dtype=float32), DeviceArray(10.4, dtype=float32))
+
+    >>> binned_function([2, 4], 4.5), smooth_function([2, 4], 4.5)  # t = 4.5
+    (DeviceArray(12.888889, dtype=float32), DeviceArray(13., dtype=float32))
+
+    The same effect can be achieved by decorating the smooth function:
+
+    >>> @pwc_from_function(timespan, num_bins)
+    ... def fn(params, t):
+    ...      return params[0] * t + params[1]
+    >>> fn([2, 4], 3)
+    DeviceArray(10.666666, dtype=float32)
+
+    """
+    if not has_jax:
+        raise ImportError(
+            "Module jax is required for any pulse-related convenience function. "
+            "You can install jax via: pip install jax"
+        )
+
+    if isinstance(timespan, tuple):
+        t1, t2 = timespan
+    else:
+        t1 = 0
+        t2 = timespan
+
+    def inner(fn):
+        time_bins = np.linspace(t1, t2, num_bins)
+
+        def wrapper(params, t):
+            constants = jnp.array(list(fn(params, time_bins)) + [0])
+
+            idx = num_bins / (t2 - t1) * (t - t1)
+            # check interval is within 0 to num_bins, then cast to int, to avoid casting outcomes between -1 and 0 as 0
+            idx = jnp.where((idx >= 0) & (idx <= num_bins), jnp.array(idx, dtype=int), -1)
+
+            return constants[idx]
+
+        return wrapper
+
+    return inner