# This code is part of Qiskit. # # (C) Copyright IBM 2020. # # This code is licensed under the Apache License, Version 2.0. You may # obtain a copy of this license in the LICENSE.txt file in the root directory # of this source tree or at http://www.apache.org/licenses/LICENSE-2.0. # # Any modifications or derivative works of this code must retain this # copyright notice, and modified files need to carry a notice indicating # that they have been altered from the originals. # pylint: disable=invalid-name """Symbolic waveform module. These are pulses which are described by symbolic equations for their envelopes and for their parameter constraints. """ from __future__ import annotations import functools import warnings from collections.abc import Mapping, Callable from copy import deepcopy from typing import Any import numpy as np import symengine as sym from qiskit.circuit.parameterexpression import ParameterExpression, ParameterValueType from qiskit.pulse.exceptions import PulseError from qiskit.pulse.library.pulse import Pulse from qiskit.pulse.library.waveform import Waveform from qiskit.utils.deprecate_pulse import deprecate_pulse_func def _lifted_gaussian( t: sym.Symbol, center: sym.Symbol | sym.Expr | complex, t_zero: sym.Symbol | sym.Expr | complex, sigma: sym.Symbol | sym.Expr | complex, ) -> sym.Expr: r"""Helper function that returns a lifted Gaussian symbolic equation. For :math:`\sigma=` ``sigma`` the symbolic equation will be .. math:: f(x) = \exp\left(-\frac12 \left(\frac{x - \mu}{\sigma}\right)^2 \right), with the center :math:`\mu=` ``duration/2``. Then, each output sample :math:`y` is modified according to: .. math:: y \mapsto \frac{y-y^*}{1.0-y^*}, where :math:`y^*` is the value of the un-normalized Gaussian at the endpoints of the pulse. This sets the endpoints to :math:`0` while preserving the amplitude at the center, i.e. :math:`y` is set to :math:`1.0`. Args: t: Symbol object representing time. center: Symbol or expression representing the middle point of the samples. t_zero: The value of t at which the pulse is lowered to 0. sigma: Symbol or expression representing Gaussian sigma. Returns: Symbolic equation. """ # Sympy automatically does expand. # This causes expression inconsistency after qpy round-trip serializing through sympy. # See issue for details: https://github.com/symengine/symengine.py/issues/409 t_shifted = (t - center).expand() t_offset = (t_zero - center).expand() gauss = sym.exp(-((t_shifted / sigma) ** 2) / 2) offset = sym.exp(-((t_offset / sigma) ** 2) / 2) return (gauss - offset) / (1 - offset) @functools.lru_cache(maxsize=None) def _is_amplitude_valid( envelope_lam: Callable, time: tuple[float, ...], *fargs: float ) -> bool | np.bool_: """A helper function to validate maximum amplitude limit. Result is cached for better performance. Args: envelope_lam: The SymbolicPulse's lambdified envelope_lam expression. time: The SymbolicPulse's time array, given as a tuple for hashability. fargs: The arguments for the lambdified envelope_lam, as given by `_get_expression_args`, except for the time array. Returns: Return True if no sample point exceeds 1.0 in absolute value. """ time = np.asarray(time, dtype=float) samples_norm = np.abs(envelope_lam(time, *fargs)) epsilon = 1e-7 # The value of epsilon mimics that of Waveform._clip() return np.all(samples_norm < 1.0 + epsilon) def _get_expression_args(expr: sym.Expr, params: dict[str, float]) -> list[np.ndarray | float]: """A helper function to get argument to evaluate expression. Args: expr: Symbolic expression to evaluate. params: Dictionary of parameter, which is a superset of expression arguments. Returns: Arguments passed to the lambdified expression. Raises: PulseError: When a free symbol value is not defined in the pulse instance parameters. """ args: list[np.ndarray | float] = [] for symbol in sorted(expr.free_symbols, key=lambda s: s.name): if symbol.name == "t": # 't' is a special parameter to represent time vector. # This should be place at first to broadcast other parameters # in symengine lambdify function. times = np.arange(0, params["duration"]) + 1 / 2 args.insert(0, times) continue try: args.append(params[symbol.name]) except KeyError as ex: raise PulseError( f"Pulse parameter '{symbol.name}' is not defined for this instance. " "Please check your waveform expression is correct." ) from ex return args class LambdifiedExpression: """Descriptor to lambdify symbolic expression with cache. When a new symbolic expression is assigned for the first time, :class:`.LambdifiedExpression` will internally lambdify the expressions and store the resulting callbacks in its cache. The next time it encounters the same expression it will return the cached callbacks thereby increasing the code's speed. Note that this class is a python `Descriptor`_, and thus not intended to be directly called by end-users. This class is designed to be attached to the :class:`.SymbolicPulse` as attributes for symbolic expressions. _`Descriptor`: https://docs.python.org/3/reference/datamodel.html#descriptors """ def __init__(self, attribute: str): """Create new descriptor. Args: attribute: Name of attribute of :class:`.SymbolicPulse` that returns the target expression to evaluate. """ self.attribute = attribute self.lambda_funcs: dict[int, Callable] = {} def __get__(self, instance, owner) -> Callable: expr = getattr(instance, self.attribute, None) if expr is None: raise PulseError(f"'{self.attribute}' of '{instance.pulse_type}' is not assigned.") key = hash(expr) if key not in self.lambda_funcs: self.__set__(instance, expr) return self.lambda_funcs[key] def __set__(self, instance, value): key = hash(value) if key not in self.lambda_funcs: params: list[Any] = [] for p in sorted(value.free_symbols, key=lambda s: s.name): if p.name == "t": # Argument "t" must be placed at first. This is a vector. params.insert(0, p) continue params.append(p) try: lamb = sym.lambdify(params, [value], real=False) def _wrapped_lamb(*args): if isinstance(args[0], np.ndarray): # When the args[0] is a vector ("t"), tile other arguments args[1:] # to prevent evaluation from looping over each element in t. t = args[0] args = np.hstack( ( t.reshape(t.size, 1), np.tile(args[1:], t.size).reshape(t.size, len(args) - 1), ) ) return lamb(args) func = _wrapped_lamb except RuntimeError: # Currently symengine doesn't support complex_double version for # several functions such as comparison operator and piecewise. # If expression contains these function, it fall back to sympy lambdify. # See https://github.com/symengine/symengine.py/issues/406 for details. import sympy func = sympy.lambdify(params, value) self.lambda_funcs[key] = func class SymbolicPulse(Pulse): r"""The pulse representation model with parameters and symbolic expressions. A symbolic pulse instance can be defined with an envelope and parameter constraints. Envelope and parameter constraints should be provided as symbolic expressions. Rather than creating a subclass, different pulse shapes can be distinguished by the instance attributes :attr:`SymbolicPulse.envelope` and :attr:`SymbolicPulse.pulse_type`. The symbolic expressions must be defined either with SymPy_ or Symengine_. Usually Symengine-based expression is much more performant for instantiation of the :class:`SymbolicPulse`, however, it doesn't support every functions available in SymPy. You may need to choose proper library depending on how you define your pulses. Symengine works in the most envelopes and constraints, and thus it is recommended to use this library especially when your program contains a lot of pulses. Also note that Symengine has the limited platform support and may not be available for your local system. Symengine is a required dependency for Qiskit on platforms that support it will always be installed along with Qiskit on macOS ``x86_64`` and ``arm64``, and Linux ``x86_64``, ``aarch64``, and ``ppc64le``. For 64-bit Windows users they will need to manual install it. For 32-bit platforms such as ``i686`` and ``armv7`` Linux, and on Linux ``s390x`` there are no pre-compiled packages available and to use symengine you'll need to compile it from source. If Symengine is not available in your environment SymPy will be used. .. _SymPy: https://www.sympy.org/en/index.html .. _Symengine: https://symengine.org .. _symbolic_pulse_envelope: .. rubric:: Envelope function The waveform at time :math:`t` is generated by the :meth:`get_waveform` according to .. math:: F(t, \Theta) = \times F(t, {\rm duration}, \overline{\rm params}) where :math:`\Theta` is the set of full pulse parameters in the :attr:`SymbolicPulse.parameters` dictionary which must include the :math:`\rm duration`. Note that the :math:`F` is an envelope of the waveform, and a programmer must provide this as a symbolic expression. :math:`\overline{\rm params}` can be arbitrary complex values as long as they pass :meth:`.validate_parameters` and your quantum backend can accept. The time :math:`t` and :math:`\rm duration` are in units of dt, i.e. sample time resolution, and this function is sampled with a discrete time vector in :math:`[0, {\rm duration}]` sampling the pulse envelope at every 0.5 dt (middle sampling strategy) when the :meth:`SymbolicPulse.get_waveform` method is called. The sample data is not generated until this method is called thus a symbolic pulse instance only stores parameter values and waveform shape, which greatly reduces memory footprint during the program generation. .. _symbolic_pulse_validation: .. rubric:: Pulse validation When a symbolic pulse is instantiated, the method :meth:`.validate_parameters` is called, and performs validation of the pulse. The validation process involves testing the constraint functions and the maximal amplitude of the pulse (see below). While the validation process will improve code stability, it will reduce performance and might create compatibility issues (particularly with JAX). Therefore, it is possible to disable the validation by setting the class attribute :attr:`.disable_validation` to ``True``. .. _symbolic_pulse_constraints: .. rubric:: Constraint functions Constraints on the parameters are defined with an instance attribute :attr:`SymbolicPulse.constraints` which can be provided through the constructor. The constraints value must be a symbolic expression, which is a function of parameters to be validated and must return a boolean value being ``True`` when parameters are valid. If there are multiple conditions to be evaluated, these conditions can be concatenated with logical expressions such as ``And`` and ``Or`` in SymPy or Symengine. The symbolic pulse instance can be played only when the constraint function returns ``True``. The constraint is evaluated when :meth:`.validate_parameters` is called. .. _symbolic_pulse_eval_condition: .. rubric:: Maximum amplitude validation When you play a pulse in a quantum backend, you might face the restriction on the power that your waveform generator can handle. Usually, the pulse amplitude is normalized by this maximum power, namely :math:`\max |F| \leq 1`. This condition is evaluated along with above constraints when you set ``limit_amplitude = True`` in the constructor. To evaluate maximum amplitude of the waveform, we need to call :meth:`get_waveform`. However, this introduces a significant overhead in the validation, and this cannot be ignored when you repeatedly instantiate symbolic pulse instances. :attr:`SymbolicPulse.valid_amp_conditions` provides a condition to skip this waveform validation, and the waveform is not generated as long as this condition returns ``True``, so that `healthy` symbolic pulses are created very quick. For example, for a simple pulse shape like ``amp * cos(f * t)``, we know that pulse amplitude is valid as long as ``amp`` remains less than magnitude 1.0. So ``abs(amp) <= 1`` could be passed as :attr:`SymbolicPulse.valid_amp_conditions` to skip doing a full waveform evaluation for amplitude validation. This expression is provided through the constructor. If this is not provided, the waveform is generated everytime when :meth:`.validate_parameters` is called. .. rubric:: Examples This is how a user can instantiate a symbolic pulse instance. In this example, we instantiate a custom `Sawtooth` envelope. .. code-block:: from qiskit.pulse.library import SymbolicPulse my_pulse = SymbolicPulse( pulse_type="Sawtooth", duration=100, parameters={"amp": 0.1, "freq": 0.05}, name="pulse1", ) Note that :class:`SymbolicPulse` can be instantiated without providing the envelope and constraints. However, this instance cannot generate waveforms without knowing the envelope definition. Now you need to provide the envelope. .. plot:: :alt: Output from the previous code. :include-source: import sympy from qiskit.pulse.library import SymbolicPulse t, amp, freq = sympy.symbols("t, amp, freq") envelope = 2 * amp * (freq * t - sympy.floor(1 / 2 + freq * t)) my_pulse = SymbolicPulse( pulse_type="Sawtooth", duration=100, parameters={"amp": 0.1, "freq": 0.05}, envelope=envelope, name="pulse1", ) my_pulse.draw() Likewise, you can define :attr:`SymbolicPulse.constraints` for ``my_pulse``. After providing the envelope definition, you can generate the waveform data. Note that it would be convenient to define a factory function that automatically accomplishes this procedure. .. code-block:: python def Sawtooth(duration, amp, freq, name): t, amp, freq = sympy.symbols("t, amp, freq") instance = SymbolicPulse( pulse_type="Sawtooth", duration=duration, parameters={"amp": amp, "freq": freq}, envelope=2 * amp * (freq * t - sympy.floor(1 / 2 + freq * t)), name=name, ) return instance You can also provide a :class:`Parameter` object in the ``parameters`` dictionary, or define ``duration`` with a :class:`Parameter` object when you instantiate the symbolic pulse instance. A waveform cannot be generated until you assign all unbounded parameters. Note that parameters will be assigned through the schedule playing the pulse. .. _symbolic_pulse_serialize: .. rubric:: Serialization The :class:`~SymbolicPulse` subclass can be serialized along with the symbolic expressions through :mod:`qiskit.qpy`. A user can therefore create a custom pulse subclass with a novel envelope and constraints, and then one can instantiate the class with certain parameters to run on a backend. This pulse instance can be saved in the QPY binary, which can be loaded afterwards even within the environment not having original class definition loaded. This mechanism also allows us to easily share a pulse program including custom pulse instructions with collaborators. """ __slots__ = ( "_pulse_type", "_params", "_envelope", "_constraints", "_valid_amp_conditions", ) disable_validation = False # Lambdify caches keyed on sympy expressions. Returns the corresponding callable. _envelope_lam = LambdifiedExpression("_envelope") _constraints_lam = LambdifiedExpression("_constraints") _valid_amp_conditions_lam = LambdifiedExpression("_valid_amp_conditions") @deprecate_pulse_func def __init__( self, pulse_type: str, duration: ParameterExpression | int, parameters: Mapping[str, ParameterExpression | complex] | None = None, name: str | None = None, limit_amplitude: bool | None = None, envelope: sym.Expr | None = None, constraints: sym.Expr | None = None, valid_amp_conditions: sym.Expr | None = None, ): """Create a parametric pulse. Args: pulse_type: Display name of this pulse shape. duration: Duration of pulse. parameters: Dictionary of pulse parameters that defines the pulse envelope. name: Display name for this particular pulse envelope. limit_amplitude: If ``True``, then limit the absolute value of the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. envelope: Pulse envelope expression. constraints: Pulse parameter constraint expression. valid_amp_conditions: Extra conditions to skip a full-waveform check for the amplitude limit. If this condition is not met, then the validation routine will investigate the full-waveform and raise an error when the amplitude norm of any data point exceeds 1.0. If not provided, the validation always creates a full-waveform. Raises: PulseError: When not all parameters are listed in the attribute :attr:`PARAM_DEF`. """ super().__init__( duration=duration, name=name, limit_amplitude=limit_amplitude, ) if parameters is None: parameters = {} self._pulse_type = pulse_type self._params = parameters self._envelope = envelope self._constraints = constraints self._valid_amp_conditions = valid_amp_conditions if not self.__class__.disable_validation: self.validate_parameters() def __getattr__(self, item): # Get pulse parameters with attribute-like access. params = object.__getattribute__(self, "_params") if item not in params: raise AttributeError(f"'{self.__class__.__name__}' object has no attribute '{item}'") return params[item] @property def pulse_type(self) -> str: """Return display name of the pulse shape.""" return self._pulse_type @property def envelope(self) -> sym.Expr: """Return symbolic expression for the pulse envelope.""" return self._envelope @property def constraints(self) -> sym.Expr: """Return symbolic expression for the pulse parameter constraints.""" return self._constraints @property def valid_amp_conditions(self) -> sym.Expr: """Return symbolic expression for the pulse amplitude constraints.""" return self._valid_amp_conditions def get_waveform(self) -> Waveform: r"""Return a Waveform with samples filled according to the formula that the pulse represents and the parameter values it contains. Since the returned array is a discretized time series of the continuous function, this method uses a midpoint sampler. For ``duration``, return: .. math:: \{f(t+0.5) \in \mathbb{C} | t \in \mathbb{Z} \wedge 0<=t<\texttt{duration}\} Returns: A waveform representation of this pulse. Raises: PulseError: When parameters are not assigned. PulseError: When expression for pulse envelope is not assigned. """ if self.is_parameterized(): raise PulseError("Unassigned parameter exists. All parameters must be assigned.") if self._envelope is None: raise PulseError("Pulse envelope expression is not assigned.") fargs = _get_expression_args(self._envelope, self.parameters) return Waveform(samples=self._envelope_lam(*fargs), name=self.name) def validate_parameters(self) -> None: """Validate parameters. Raises: PulseError: If the parameters passed are not valid. """ if self.is_parameterized(): return if self._constraints is not None: fargs = _get_expression_args(self._constraints, self.parameters) if not bool(self._constraints_lam(*fargs)): param_repr = ", ".join(f"{p}={v}" for p, v in self.parameters.items()) const_repr = str(self._constraints) raise PulseError( f"Assigned parameters {param_repr} violate following constraint: {const_repr}." ) if self._limit_amplitude: if self._valid_amp_conditions is not None: fargs = _get_expression_args(self._valid_amp_conditions, self.parameters) check_full_waveform = not bool(self._valid_amp_conditions_lam(*fargs)) else: check_full_waveform = True if check_full_waveform: # Check full waveform only when the condition is satisified or # evaluation condition is not provided. # This operation is slower due to overhead of 'get_waveform'. fargs = _get_expression_args(self._envelope, self.parameters) if not _is_amplitude_valid(self._envelope_lam, tuple(fargs.pop(0)), *fargs): param_repr = ", ".join(f"{p}={v}" for p, v in self.parameters.items()) raise PulseError( f"Maximum pulse amplitude norm exceeds 1.0 with parameters {param_repr}." "This can be overruled by setting Pulse.limit_amplitude." ) def is_parameterized(self) -> bool: """Return True iff the instruction is parameterized.""" return any(isinstance(val, ParameterExpression) for val in self.parameters.values()) @property def parameters(self) -> dict[str, Any]: params: dict[str, ParameterExpression | complex | int] = {"duration": self.duration} params.update(self._params) return params def __eq__(self, other: object) -> bool: if not isinstance(other, SymbolicPulse): return NotImplemented if self._pulse_type != other._pulse_type: return False if self._envelope != other._envelope: return False if self.parameters != other.parameters: return False return True def __repr__(self) -> str: param_repr = ", ".join(f"{p}={v}" for p, v in self.parameters.items()) name_repr = f", name='{self.name}'" if self.name is not None else "" return f"{self._pulse_type}({param_repr}{name_repr})" __hash__ = None class ScalableSymbolicPulse(SymbolicPulse): r"""Subclass of :class:`SymbolicPulse` for pulses with scalable envelope. Instance of :class:`ScalableSymbolicPulse` behaves the same as an instance of :class:`SymbolicPulse`, but its envelope is assumed to have a scalable form :math:`\text{amp}\times\exp\left(i\times\text{angle}\right)\times\text{F} \left(t,\text{parameters}\right)`, where :math:`\text{F}` is some function describing the rest of the envelope, and both `amp` and `angle` are real (float). Note that both `amp` and `angle` are stored in the :attr:`parameters` dictionary of the :class:`ScalableSymbolicPulse` instance. When two :class:`ScalableSymbolicPulse` objects are equated, instead of comparing `amp` and `angle` individually, only the complex amplitude :math:'\text{amp}\times\exp\left(i\times\text{angle}\right)' is compared. """ def __init__( self, pulse_type: str, duration: ParameterExpression | int, amp: ParameterValueType, angle: ParameterValueType, parameters: dict[str, ParameterExpression | complex] | None = None, name: str | None = None, limit_amplitude: bool | None = None, envelope: sym.Expr | None = None, constraints: sym.Expr | None = None, valid_amp_conditions: sym.Expr | None = None, ): """Create a scalable symbolic pulse. Args: pulse_type: Display name of this pulse shape. duration: Duration of pulse. amp: The magnitude of the complex amplitude of the pulse. angle: The phase of the complex amplitude of the pulse. parameters: Dictionary of pulse parameters that defines the pulse envelope. name: Display name for this particular pulse envelope. limit_amplitude: If ``True``, then limit the absolute value of the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. envelope: Pulse envelope expression. constraints: Pulse parameter constraint expression. valid_amp_conditions: Extra conditions to skip a full-waveform check for the amplitude limit. If this condition is not met, then the validation routine will investigate the full-waveform and raise an error when the amplitude norm of any data point exceeds 1.0. If not provided, the validation always creates a full-waveform. Raises: PulseError: If ``amp`` is complex. """ if isinstance(amp, complex): raise PulseError( "amp represents the magnitude of the complex amplitude and can't be complex" ) if not isinstance(parameters, dict): parameters = {"amp": amp, "angle": angle} else: parameters = deepcopy(parameters) parameters["amp"] = amp parameters["angle"] = angle super().__init__( pulse_type=pulse_type, duration=duration, parameters=parameters, name=name, limit_amplitude=limit_amplitude, envelope=envelope, constraints=constraints, valid_amp_conditions=valid_amp_conditions, ) # pylint: disable=too-many-return-statements def __eq__(self, other: object) -> bool: if not isinstance(other, ScalableSymbolicPulse): return NotImplemented if self._pulse_type != other._pulse_type: return False if self._envelope != other._envelope: return False complex_amp1 = self.amp * np.exp(1j * self.angle) complex_amp2 = other.amp * np.exp(1j * other.angle) if isinstance(complex_amp1, ParameterExpression) or isinstance( complex_amp2, ParameterExpression ): if complex_amp1 != complex_amp2: return False else: if not np.isclose(complex_amp1, complex_amp2): return False for key, value in self.parameters.items(): if key not in ["amp", "angle"] and value != other.parameters[key]: return False return True class _PulseType(type): """Metaclass to warn at isinstance check.""" def __instancecheck__(cls, instance): cls_alias = getattr(cls, "alias", None) # TODO promote this to Deprecation warning in future. # Once type information usage is removed from user code, # we will convert pulse classes into functions. warnings.warn( "Typechecking with the symbolic pulse subclass will be deprecated. " f"'{cls_alias}' subclass instance is turned into SymbolicPulse instance. " f"Use self.pulse_type == '{cls_alias}' instead.", PendingDeprecationWarning, ) if not isinstance(instance, SymbolicPulse): return False return instance.pulse_type == cls_alias def __getattr__(cls, item): # For pylint. A SymbolicPulse subclass must implement several methods # such as .get_waveform and .validate_parameters. # In addition, they conventionally offer attribute-like access to the pulse parameters, # for example, instance.amp returns instance._params["amp"]. # If pulse classes are directly instantiated, pylint yells no-member # since the pulse class itself implements nothing. These classes just # behave like a factory by internally instantiating the SymbolicPulse and return it. # It is not realistic to write disable=no-member across qiskit packages. return NotImplemented class Gaussian(metaclass=_PulseType): r"""A lifted and truncated pulse envelope shaped according to the Gaussian function whose mean is centered at the center of the pulse (duration / 2): .. math:: \begin{aligned} f'(x) &= \exp\Bigl( -\frac12 \frac{{(x - \text{duration}/2)}^2}{\text{sigma}^2} \Bigr)\\ f(x) &= \text{A} \times \frac{f'(x) - f'(-1)}{1-f'(-1)}, \quad 0 \le x < \text{duration} \end{aligned} where :math:`f'(x)` is the gaussian waveform without lifting or amplitude scaling, and :math:`\text{A} = \text{amp} \times \exp\left(i\times\text{angle}\right)`. """ alias = "Gaussian" def __new__( cls, duration: int | ParameterValueType, amp: ParameterValueType, sigma: ParameterValueType, angle: ParameterValueType = 0.0, name: str | None = None, limit_amplitude: bool | None = None, ) -> ScalableSymbolicPulse: """Create new pulse instance. Args: duration: Pulse length in terms of the sampling period `dt`. amp: The magnitude of the amplitude of the Gaussian envelope. sigma: A measure of how wide or narrow the Gaussian peak is; described mathematically in the class docstring. angle: The angle of the complex amplitude of the Gaussian envelope. Default value 0. name: Display name for this pulse envelope. limit_amplitude: If ``True``, then limit the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. Returns: ScalableSymbolicPulse instance. """ parameters = {"sigma": sigma} # Prepare symbolic expressions _t, _duration, _amp, _sigma, _angle = sym.symbols("t, duration, amp, sigma, angle") _center = _duration / 2 envelope_expr = ( _amp * sym.exp(sym.I * _angle) * _lifted_gaussian(_t, _center, _duration + 1, _sigma) ) consts_expr = _sigma > 0 valid_amp_conditions_expr = sym.Abs(_amp) <= 1.0 return ScalableSymbolicPulse( pulse_type=cls.alias, duration=duration, amp=amp, angle=angle, parameters=parameters, name=name, limit_amplitude=limit_amplitude, envelope=envelope_expr, constraints=consts_expr, valid_amp_conditions=valid_amp_conditions_expr, ) @deprecate_pulse_func def __init__(self): pass class GaussianSquare(metaclass=_PulseType): """A square pulse with a Gaussian shaped risefall on both sides lifted such that its first sample is zero. Exactly one of the ``risefall_sigma_ratio`` and ``width`` parameters has to be specified. If ``risefall_sigma_ratio`` is not None and ``width`` is None: .. math:: \\begin{aligned} \\text{risefall} &= \\text{risefall\\_sigma\\_ratio} \\times \\text{sigma}\\\\ \\text{width} &= \\text{duration} - 2 \\times \\text{risefall} \\end{aligned} If ``width`` is not None and ``risefall_sigma_ratio`` is None: .. math:: \\text{risefall} = \\frac{\\text{duration} - \\text{width}}{2} In both cases, the lifted gaussian square pulse :math:`f'(x)` is defined as: .. math:: \\begin{aligned} f'(x) &= \\begin{cases}\ \\exp\\biggl(-\\frac12 \\frac{(x - \\text{risefall})^2}{\\text{sigma}^2}\\biggr)\ & x < \\text{risefall}\\\\ 1\ & \\text{risefall} \\le x < \\text{risefall} + \\text{width}\\\\ \\exp\\biggl(-\\frac12\ \\frac{{\\bigl(x - (\\text{risefall} + \\text{width})\\bigr)}^2}\ {\\text{sigma}^2}\ \\biggr)\ & \\text{risefall} + \\text{width} \\le x\ \\end{cases}\\\\ f(x) &= \\text{A} \\times \\frac{f'(x) - f'(-1)}{1-f'(-1)},\ \\quad 0 \\le x < \\text{duration} \\end{aligned} where :math:`f'(x)` is the gaussian square waveform without lifting or amplitude scaling, and :math:`\\text{A} = \\text{amp} \\times \\exp\\left(i\\times\\text{angle}\\right)`. """ alias = "GaussianSquare" def __new__( cls, duration: int | ParameterValueType, amp: ParameterValueType, sigma: ParameterValueType, width: ParameterValueType | None = None, angle: ParameterValueType = 0.0, risefall_sigma_ratio: ParameterValueType | None = None, name: str | None = None, limit_amplitude: bool | None = None, ) -> ScalableSymbolicPulse: """Create new pulse instance. Args: duration: Pulse length in terms of the sampling period `dt`. amp: The magnitude of the amplitude of the Gaussian and square pulse. sigma: A measure of how wide or narrow the Gaussian risefall is; see the class docstring for more details. width: The duration of the embedded square pulse. angle: The angle of the complex amplitude of the pulse. Default value 0. risefall_sigma_ratio: The ratio of each risefall duration to sigma. name: Display name for this pulse envelope. limit_amplitude: If ``True``, then limit the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. Returns: ScalableSymbolicPulse instance. Raises: PulseError: When width and risefall_sigma_ratio are both empty or both non-empty. """ # Convert risefall_sigma_ratio into width which is defined in OpenPulse spec if width is None and risefall_sigma_ratio is None: raise PulseError( "Either the pulse width or the risefall_sigma_ratio parameter must be specified." ) if width is not None and risefall_sigma_ratio is not None: raise PulseError( "Either the pulse width or the risefall_sigma_ratio parameter can be specified" " but not both." ) if width is None and risefall_sigma_ratio is not None: width = duration - 2.0 * risefall_sigma_ratio * sigma parameters = {"sigma": sigma, "width": width} # Prepare symbolic expressions _t, _duration, _amp, _sigma, _width, _angle = sym.symbols( "t, duration, amp, sigma, width, angle" ) _center = _duration / 2 _sq_t0 = _center - _width / 2 _sq_t1 = _center + _width / 2 _gaussian_ledge = _lifted_gaussian(_t, _sq_t0, -1, _sigma) _gaussian_redge = _lifted_gaussian(_t, _sq_t1, _duration + 1, _sigma) envelope_expr = ( _amp * sym.exp(sym.I * _angle) * sym.Piecewise( (_gaussian_ledge, _t <= _sq_t0), (_gaussian_redge, _t >= _sq_t1), (1, True) ) ) consts_expr = sym.And(_sigma > 0, _width >= 0, _duration >= _width) valid_amp_conditions_expr = sym.Abs(_amp) <= 1.0 return ScalableSymbolicPulse( pulse_type=cls.alias, duration=duration, amp=amp, angle=angle, parameters=parameters, name=name, limit_amplitude=limit_amplitude, envelope=envelope_expr, constraints=consts_expr, valid_amp_conditions=valid_amp_conditions_expr, ) @deprecate_pulse_func def __init__(self): pass @deprecate_pulse_func def GaussianSquareDrag( duration: int | ParameterExpression, amp: float | ParameterExpression, sigma: float | ParameterExpression, beta: float | ParameterExpression, width: float | ParameterExpression | None = None, angle: float | ParameterExpression | None = 0.0, risefall_sigma_ratio: float | ParameterExpression | None = None, name: str | None = None, limit_amplitude: bool | None = None, ) -> ScalableSymbolicPulse: """A square pulse with a Drag shaped rise and fall This pulse shape is similar to :class:`~.GaussianSquare` but uses :class:`~.Drag` for its rise and fall instead of :class:`~.Gaussian`. The addition of the DRAG component of the rise and fall is sometimes helpful in suppressing the spectral content of the pulse at frequencies near to, but slightly offset from, the fundamental frequency of the drive. When there is a spectator qubit close in frequency to the fundamental frequency, suppressing the drive at the spectator's frequency can help avoid unwanted excitation of the spectator. Exactly one of the ``risefall_sigma_ratio`` and ``width`` parameters has to be specified. If ``risefall_sigma_ratio`` is not ``None`` and ``width`` is ``None``: .. math:: \\begin{aligned} \\text{risefall} &= \\text{risefall\\_sigma\\_ratio} \\times \\text{sigma}\\\\ \\text{width} &= \\text{duration} - 2 \\times \\text{risefall} \\end{aligned} If ``width`` is not None and ``risefall_sigma_ratio`` is None: .. math:: \\text{risefall} = \\frac{\\text{duration} - \\text{width}}{2} Gaussian :math:`g(x, c, σ)` and lifted gaussian :math:`g'(x, c, σ)` curves can be written as: .. math:: \\begin{aligned} g(x, c, σ) &= \\exp\\Bigl(-\\frac12 \\frac{(x - c)^2}{σ^2}\\Bigr)\\\\ g'(x, c, σ) &= \\frac{g(x, c, σ)-g(-1, c, σ)}{1-g(-1, c, σ)} \\end{aligned} From these, the lifted DRAG curve :math:`d'(x, c, σ, β)` can be written as .. math:: d'(x, c, σ, β) = g'(x, c, σ) \\times \\Bigl(1 + 1j \\times β \\times\ \\Bigl(-\\frac{x - c}{σ^2}\\Bigr)\\Bigr) The lifted gaussian square drag pulse :math:`f'(x)` is defined as: .. math:: \\begin{aligned} f'(x) &= \\begin{cases}\ \\text{A} \\times d'(x, \\text{risefall}, \\text{sigma}, \\text{beta})\ & x < \\text{risefall}\\\\ \\text{A}\ & \\text{risefall} \\le x < \\text{risefall} + \\text{width}\\\\ \\text{A} \\times \\times d'(\ x - (\\text{risefall} + \\text{width}),\ \\text{risefall},\ \\text{sigma},\ \\text{beta}\ )\ & \\text{risefall} + \\text{width} \\le x\ \\end{cases}\\\\ \\end{aligned} where :math:`\\text{A} = \\text{amp} \\times \\exp\\left(i\\times\\text{angle}\\right)`. Args: duration: Pulse length in terms of the sampling period `dt`. amp: The amplitude of the DRAG rise and fall and of the square pulse. sigma: A measure of how wide or narrow the DRAG risefall is; see the class docstring for more details. beta: The DRAG correction amplitude. width: The duration of the embedded square pulse. angle: The angle in radians of the complex phase factor uniformly scaling the pulse. Default value 0. risefall_sigma_ratio: The ratio of each risefall duration to sigma. name: Display name for this pulse envelope. limit_amplitude: If ``True``, then limit the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. Returns: ScalableSymbolicPulse instance. Raises: PulseError: When width and risefall_sigma_ratio are both empty or both non-empty. """ # Convert risefall_sigma_ratio into width which is defined in OpenPulse spec if width is None and risefall_sigma_ratio is None: raise PulseError( "Either the pulse width or the risefall_sigma_ratio parameter must be specified." ) if width is not None and risefall_sigma_ratio is not None: raise PulseError( "Either the pulse width or the risefall_sigma_ratio parameter can be specified" " but not both." ) if width is None and risefall_sigma_ratio is not None: width = duration - 2.0 * risefall_sigma_ratio * sigma parameters = {"sigma": sigma, "width": width, "beta": beta} # Prepare symbolic expressions _t, _duration, _amp, _sigma, _beta, _width, _angle = sym.symbols( "t, duration, amp, sigma, beta, width, angle" ) _center = _duration / 2 _sq_t0 = _center - _width / 2 _sq_t1 = _center + _width / 2 _gaussian_ledge = _lifted_gaussian(_t, _sq_t0, -1, _sigma) _gaussian_redge = _lifted_gaussian(_t, _sq_t1, _duration + 1, _sigma) _deriv_ledge = -(_t - _sq_t0) / (_sigma**2) * _gaussian_ledge _deriv_redge = -(_t - _sq_t1) / (_sigma**2) * _gaussian_redge envelope_expr = ( _amp * sym.exp(sym.I * _angle) * sym.Piecewise( (_gaussian_ledge + sym.I * _beta * _deriv_ledge, _t <= _sq_t0), (_gaussian_redge + sym.I * _beta * _deriv_redge, _t >= _sq_t1), (1, True), ) ) consts_expr = sym.And(_sigma > 0, _width >= 0, _duration >= _width) valid_amp_conditions_expr = sym.And(sym.Abs(_amp) <= 1.0, sym.Abs(_beta) < _sigma) return ScalableSymbolicPulse( pulse_type="GaussianSquareDrag", duration=duration, amp=amp, angle=angle, parameters=parameters, name=name, limit_amplitude=limit_amplitude, envelope=envelope_expr, constraints=consts_expr, valid_amp_conditions=valid_amp_conditions_expr, ) @deprecate_pulse_func def gaussian_square_echo( duration: int | ParameterValueType, amp: float | ParameterExpression, sigma: float | ParameterExpression, width: float | ParameterExpression | None = None, angle: float | ParameterExpression | None = 0.0, active_amp: float | ParameterExpression | None = 0.0, active_angle: float | ParameterExpression | None = 0.0, risefall_sigma_ratio: float | ParameterExpression | None = None, name: str | None = None, limit_amplitude: bool | None = None, ) -> SymbolicPulse: """An echoed Gaussian square pulse with an active tone overlaid on it. The Gaussian Square Echo pulse is composed of three pulses. First, a Gaussian Square pulse :math:`f_{echo}(x)` with amplitude ``amp`` and phase ``angle`` playing for half duration, followed by a second Gaussian Square pulse :math:`-f_{echo}(x)` with opposite amplitude and same phase playing for the rest of the duration. Third a Gaussian Square pulse :math:`f_{active}(x)` with amplitude ``active_amp`` and phase ``active_angle`` playing for the entire duration. The Gaussian Square Echo pulse :math:`g_e()` can be written as: .. math:: \\begin{aligned} g_e(x) &= \\begin{cases}\ f_{\\text{active}} + f_{\\text{echo}}(x)\ & x < \\frac{\\text{duration}}{2}\\\\ f_{\\text{active}} - f_{\\text{echo}}(x)\ & \\frac{\\text{duration}}{2} < x\ \\end{cases}\\\\ \\end{aligned} One case where this pulse can be used is when implementing a direct CNOT gate with a cross-resonance superconducting qubit architecture. When applying this pulse to the target qubit, the active portion can be used to cancel IX terms from the cross-resonance drive while the echo portion can reduce the impact of a static ZZ coupling. Exactly one of the ``risefall_sigma_ratio`` and ``width`` parameters has to be specified. If ``risefall_sigma_ratio`` is not ``None`` and ``width`` is ``None``: .. math:: \\begin{aligned} \\text{risefall} &= \\text{risefall\\_sigma\\_ratio} \\times \\text{sigma}\\\\ \\text{width} &= \\text{duration} - 2 \\times \\text{risefall} \\end{aligned} If ``width`` is not None and ``risefall_sigma_ratio`` is None: .. math:: \\text{risefall} = \\frac{\\text{duration} - \\text{width}}{2} References: 1. |citation1|_ .. _citation1: https://iopscience.iop.org/article/10.1088/2058-9565/abe519 .. |citation1| replace:: *Jurcevic, P., Javadi-Abhari, A., Bishop, L. S., Lauer, I., Bogorin, D. F., Brink, M., Capelluto, L., G{\"u}nl{\"u}k, O., Itoko, T., Kanazawa, N. & others Demonstration of quantum volume 64 on a superconducting quantum computing system. (Section V)* Args: duration: Pulse length in terms of the sampling period `dt`. amp: The amplitude of the rise and fall and of the echoed pulse. sigma: A measure of how wide or narrow the risefall is; see the class docstring for more details. width: The duration of the embedded square pulse. angle: The angle in radians of the complex phase factor uniformly scaling the echoed pulse. Default value 0. active_amp: The amplitude of the active pulse. active_angle: The angle in radian of the complex phase factor uniformly scaling the active pulse. Default value 0. risefall_sigma_ratio: The ratio of each risefall duration to sigma. name: Display name for this pulse envelope. limit_amplitude: If ``True``, then limit the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. Returns: ScalableSymbolicPulse instance. Raises: PulseError: When width and risefall_sigma_ratio are both empty or both non-empty. """ # Convert risefall_sigma_ratio into width which is defined in OpenPulse spec if width is None and risefall_sigma_ratio is None: raise PulseError( "Either the pulse width or the risefall_sigma_ratio parameter must be specified." ) if width is not None and risefall_sigma_ratio is not None: raise PulseError( "Either the pulse width or the risefall_sigma_ratio parameter can be specified" " but not both." ) if width is None and risefall_sigma_ratio is not None: width = duration - 2.0 * risefall_sigma_ratio * sigma parameters = { "amp": amp, "angle": angle, "sigma": sigma, "width": width, "active_amp": active_amp, "active_angle": active_angle, } # Prepare symbolic expressions ( _t, _duration, _amp, _sigma, _active_amp, _width, _angle, _active_angle, ) = sym.symbols("t, duration, amp, sigma, active_amp, width, angle, active_angle") # gaussian square echo for rotary tone _center = _duration / 4 _width_echo = (_duration - 2 * (_duration - _width)) / 2 _sq_t0 = _center - _width_echo / 2 _sq_t1 = _center + _width_echo / 2 _gaussian_ledge = _lifted_gaussian(_t, _sq_t0, -1, _sigma) _gaussian_redge = _lifted_gaussian(_t, _sq_t1, _duration / 2 + 1, _sigma) envelope_expr_p = ( _amp * sym.exp(sym.I * _angle) * sym.Piecewise( (_gaussian_ledge, _t <= _sq_t0), (_gaussian_redge, _t >= _sq_t1), (1, True), ) ) _center_echo = _duration / 2 + _duration / 4 _sq_t0_echo = _center_echo - _width_echo / 2 _sq_t1_echo = _center_echo + _width_echo / 2 _gaussian_ledge_echo = _lifted_gaussian(_t, _sq_t0_echo, _duration / 2 - 1, _sigma) _gaussian_redge_echo = _lifted_gaussian(_t, _sq_t1_echo, _duration + 1, _sigma) envelope_expr_echo = ( -1 * _amp * sym.exp(sym.I * _angle) * sym.Piecewise( (_gaussian_ledge_echo, _t <= _sq_t0_echo), (_gaussian_redge_echo, _t >= _sq_t1_echo), (1, True), ) ) envelope_expr = sym.Piecewise( (envelope_expr_p, _t <= _duration / 2), (envelope_expr_echo, _t >= _duration / 2), (0, True) ) # gaussian square for active cancellation tone _center_active = _duration / 2 _sq_t0_active = _center_active - _width / 2 _sq_t1_active = _center_active + _width / 2 _gaussian_ledge_active = _lifted_gaussian(_t, _sq_t0_active, -1, _sigma) _gaussian_redge_active = _lifted_gaussian(_t, _sq_t1_active, _duration + 1, _sigma) envelope_expr_active = ( _active_amp * sym.exp(sym.I * _active_angle) * sym.Piecewise( (_gaussian_ledge_active, _t <= _sq_t0_active), (_gaussian_redge_active, _t >= _sq_t1_active), (1, True), ) ) envelop_expr_total = envelope_expr + envelope_expr_active consts_expr = sym.And( _sigma > 0, _width >= 0, _duration >= _width, _duration / 2 >= _width_echo ) # Check validity of amplitudes valid_amp_conditions_expr = sym.And(sym.Abs(_amp) + sym.Abs(_active_amp) <= 1.0) return SymbolicPulse( pulse_type="gaussian_square_echo", duration=duration, parameters=parameters, name=name, limit_amplitude=limit_amplitude, envelope=envelop_expr_total, constraints=consts_expr, valid_amp_conditions=valid_amp_conditions_expr, ) @deprecate_pulse_func def GaussianDeriv( duration: int | ParameterValueType, amp: float | ParameterExpression, sigma: float | ParameterExpression, angle: float | ParameterExpression | None = 0.0, name: str | None = None, limit_amplitude: bool | None = None, ) -> ScalableSymbolicPulse: """An unnormalized Gaussian derivative pulse. The Gaussian function is centered around the halfway point of the pulse, and the envelope of the pulse is given by: .. math:: f(x) = -\\text{A}\\frac{x-\\mu}{\\text{sigma}^{2}}\\exp \\left[-\\left(\\frac{x-\\mu}{2\\text{sigma}}\\right)^{2}\\right] , 0 <= x < duration where :math:`\\text{A} = \\text{amp} \\times\\exp\\left(i\\times\\text{angle}\\right)`, and :math:`\\mu=\\text{duration}/2`. Args: duration: Pulse length in terms of the sampling period `dt`. amp: The magnitude of the amplitude of the pulse (the value of the corresponding Gaussian at the midpoint `duration`/2). sigma: A measure of how wide or narrow the corresponding Gaussian peak is in terms of `dt`; described mathematically in the class docstring. angle: The angle in radians of the complex phase factor uniformly scaling the pulse. Default value 0. name: Display name for this pulse envelope. limit_amplitude: If ``True``, then limit the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. Returns: ScalableSymbolicPulse instance. """ parameters = {"sigma": sigma} # Prepare symbolic expressions _t, _duration, _amp, _angle, _sigma = sym.symbols("t, duration, amp, angle, sigma") envelope_expr = ( -_amp * sym.exp(sym.I * _angle) * ((_t - (_duration / 2)) / _sigma**2) * sym.exp(-(1 / 2) * ((_t - (_duration / 2)) / _sigma) ** 2) ) consts_expr = _sigma > 0 valid_amp_conditions_expr = sym.Abs(_amp / _sigma) <= sym.exp(1 / 2) return ScalableSymbolicPulse( pulse_type="GaussianDeriv", duration=duration, amp=amp, angle=angle, parameters=parameters, name=name, limit_amplitude=limit_amplitude, envelope=envelope_expr, constraints=consts_expr, valid_amp_conditions=valid_amp_conditions_expr, ) class Drag(metaclass=_PulseType): """The Derivative Removal by Adiabatic Gate (DRAG) pulse is a standard Gaussian pulse with an additional Gaussian derivative component and lifting applied. It can be calibrated either to reduce the phase error due to virtual population of the :math:`|2\\rangle` state during the pulse or to reduce the frequency spectrum of a standard Gaussian pulse near the :math:`|1\\rangle\\leftrightarrow|2\\rangle` transition, reducing the chance of leakage to the :math:`|2\\rangle` state. .. math:: \\begin{aligned} g(x) &= \\exp\\Bigl(-\\frac12 \\frac{(x - \\text{duration}/2)^2}{\\text{sigma}^2}\\Bigr)\\\\ g'(x) &= \\text{A}\\times\\frac{g(x)-g(-1)}{1-g(-1)}\\\\ f(x) &= g'(x) \\times \\Bigl(1 + 1j \\times \\text{beta} \\times\ \\Bigl(-\\frac{x - \\text{duration}/2}{\\text{sigma}^2}\\Bigr) \\Bigr), \\quad 0 \\le x < \\text{duration} \\end{aligned} where :math:`g(x)` is a standard unlifted Gaussian waveform, :math:`g'(x)` is the lifted :class:`~qiskit.pulse.library.Gaussian` waveform, and :math:`\\text{A} = \\text{amp} \\times \\exp\\left(i\\times\\text{angle}\\right)`. References: 1. |citation1|_ .. _citation1: https://link.aps.org/doi/10.1103/PhysRevA.83.012308 .. |citation1| replace:: *Gambetta, J. M., Motzoi, F., Merkel, S. T. & Wilhelm, F. K. Analytic control methods for high-fidelity unitary operations in a weakly nonlinear oscillator. Phys. Rev. A 83, 012308 (2011).* 2. |citation2|_ .. _citation2: https://link.aps.org/doi/10.1103/PhysRevLett.103.110501 .. |citation2| replace:: *F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K. Wilhelm Phys. Rev. Lett. 103, 110501 – Published 8 September 2009.* """ alias = "Drag" def __new__( cls, duration: int | ParameterValueType, amp: ParameterValueType, sigma: ParameterValueType, beta: ParameterValueType, angle: ParameterValueType = 0.0, name: str | None = None, limit_amplitude: bool | None = None, ) -> ScalableSymbolicPulse: """Create new pulse instance. Args: duration: Pulse length in terms of the sampling period `dt`. amp: The magnitude of the amplitude of the DRAG envelope. sigma: A measure of how wide or narrow the Gaussian peak is; described mathematically in the class docstring. beta: The correction amplitude. angle: The angle of the complex amplitude of the DRAG envelope. Default value 0. name: Display name for this pulse envelope. limit_amplitude: If ``True``, then limit the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. Returns: ScalableSymbolicPulse instance. """ parameters = {"sigma": sigma, "beta": beta} # Prepare symbolic expressions _t, _duration, _amp, _sigma, _beta, _angle = sym.symbols( "t, duration, amp, sigma, beta, angle" ) _center = _duration / 2 _gauss = _lifted_gaussian(_t, _center, _duration + 1, _sigma) _deriv = -(_t - _center) / (_sigma**2) * _gauss envelope_expr = _amp * sym.exp(sym.I * _angle) * (_gauss + sym.I * _beta * _deriv) consts_expr = _sigma > 0 valid_amp_conditions_expr = sym.And(sym.Abs(_amp) <= 1.0, sym.Abs(_beta) < _sigma) return ScalableSymbolicPulse( pulse_type="Drag", duration=duration, amp=amp, angle=angle, parameters=parameters, name=name, limit_amplitude=limit_amplitude, envelope=envelope_expr, constraints=consts_expr, valid_amp_conditions=valid_amp_conditions_expr, ) @deprecate_pulse_func def __init__(self): pass class Constant(metaclass=_PulseType): """A simple constant pulse, with an amplitude value and a duration: .. math:: f(x) = \\text{amp}\\times\\exp\\left(i\\text{angle}\\right) , 0 <= x < duration f(x) = 0 , elsewhere """ alias = "Constant" def __new__( cls, duration: int | ParameterValueType, amp: ParameterValueType, angle: ParameterValueType = 0.0, name: str | None = None, limit_amplitude: bool | None = None, ) -> ScalableSymbolicPulse: """Create new pulse instance. Args: duration: Pulse length in terms of the sampling period `dt`. amp: The magnitude of the amplitude of the square envelope. angle: The angle of the complex amplitude of the square envelope. Default value 0. name: Display name for this pulse envelope. limit_amplitude: If ``True``, then limit the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. Returns: ScalableSymbolicPulse instance. """ # Prepare symbolic expressions _t, _amp, _duration, _angle = sym.symbols("t, amp, duration, angle") # Note this is implemented using Piecewise instead of just returning amp # directly because otherwise the expression has no t dependence and sympy's # lambdify will produce a function f that for an array t returns amp # instead of amp * np.ones(t.shape). # # See: https://github.com/sympy/sympy/issues/5642 envelope_expr = ( _amp * sym.exp(sym.I * _angle) * sym.Piecewise((1, sym.And(_t >= 0, _t <= _duration)), (0, True)) ) valid_amp_conditions_expr = sym.Abs(_amp) <= 1.0 return ScalableSymbolicPulse( pulse_type="Constant", duration=duration, amp=amp, angle=angle, name=name, limit_amplitude=limit_amplitude, envelope=envelope_expr, valid_amp_conditions=valid_amp_conditions_expr, ) @deprecate_pulse_func def __init__(self): pass @deprecate_pulse_func def Sin( duration: int | ParameterExpression, amp: float | ParameterExpression, phase: float | ParameterExpression, freq: float | ParameterExpression | None = None, angle: float | ParameterExpression | None = 0.0, name: str | None = None, limit_amplitude: bool | None = None, ) -> ScalableSymbolicPulse: """A sinusoidal pulse. The envelope of the pulse is given by: .. math:: f(x) = \\text{A}\\sin\\left(2\\pi\\text{freq}x+\\text{phase}\\right) , 0 <= x < duration where :math:`\\text{A} = \\text{amp} \\times\\exp\\left(i\\times\\text{angle}\\right)`. Args: duration: Pulse length in terms of the sampling period `dt`. amp: The magnitude of the amplitude of the sinusoidal wave. Wave range is [-`amp`,`amp`]. phase: The phase of the sinusoidal wave (note that this is not equivalent to the angle of the complex amplitude) freq: The frequency of the sinusoidal wave, in terms of 1 over sampling period. If not provided defaults to a single cycle (i.e :math:'\\frac{1}{\\text{duration}}'). The frequency is limited to the range :math:`\\left(0,0.5\\right]` (the Nyquist frequency). angle: The angle in radians of the complex phase factor uniformly scaling the pulse. Default value 0. name: Display name for this pulse envelope. limit_amplitude: If ``True``, then limit the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. Returns: ScalableSymbolicPulse instance. """ if freq is None: freq = 1 / duration parameters = {"freq": freq, "phase": phase} # Prepare symbolic expressions _t, _duration, _amp, _angle, _freq, _phase = sym.symbols("t, duration, amp, angle, freq, phase") envelope_expr = _amp * sym.exp(sym.I * _angle) * sym.sin(2 * sym.pi * _freq * _t + _phase) consts_expr = sym.And(_freq > 0, _freq < 0.5) # This might fail for waves shorter than a single cycle valid_amp_conditions_expr = sym.Abs(_amp) <= 1.0 return ScalableSymbolicPulse( pulse_type="Sin", duration=duration, amp=amp, angle=angle, parameters=parameters, name=name, limit_amplitude=limit_amplitude, envelope=envelope_expr, constraints=consts_expr, valid_amp_conditions=valid_amp_conditions_expr, ) @deprecate_pulse_func def Cos( duration: int | ParameterExpression, amp: float | ParameterExpression, phase: float | ParameterExpression, freq: float | ParameterExpression | None = None, angle: float | ParameterExpression | None = 0.0, name: str | None = None, limit_amplitude: bool | None = None, ) -> ScalableSymbolicPulse: """A cosine pulse. The envelope of the pulse is given by: .. math:: f(x) = \\text{A}\\cos\\left(2\\pi\\text{freq}x+\\text{phase}\\right) , 0 <= x < duration where :math:`\\text{A} = \\text{amp} \\times\\exp\\left(i\\times\\text{angle}\\right)`. Args: duration: Pulse length in terms of the sampling period `dt`. amp: The magnitude of the amplitude of the cosine wave. Wave range is [-`amp`,`amp`]. phase: The phase of the cosine wave (note that this is not equivalent to the angle of the complex amplitude). freq: The frequency of the cosine wave, in terms of 1 over sampling period. If not provided defaults to a single cycle (i.e :math:'\\frac{1}{\\text{duration}}'). The frequency is limited to the range :math:`\\left(0,0.5\\right]` (the Nyquist frequency). angle: The angle in radians of the complex phase factor uniformly scaling the pulse. Default value 0. name: Display name for this pulse envelope. limit_amplitude: If ``True``, then limit the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. Returns: ScalableSymbolicPulse instance. """ if freq is None: freq = 1 / duration parameters = {"freq": freq, "phase": phase} # Prepare symbolic expressions _t, _duration, _amp, _angle, _freq, _phase = sym.symbols("t, duration, amp, angle, freq, phase") envelope_expr = _amp * sym.exp(sym.I * _angle) * sym.cos(2 * sym.pi * _freq * _t + _phase) consts_expr = sym.And(_freq > 0, _freq < 0.5) # This might fail for waves shorter than a single cycle valid_amp_conditions_expr = sym.Abs(_amp) <= 1.0 return ScalableSymbolicPulse( pulse_type="Cos", duration=duration, amp=amp, angle=angle, parameters=parameters, name=name, limit_amplitude=limit_amplitude, envelope=envelope_expr, constraints=consts_expr, valid_amp_conditions=valid_amp_conditions_expr, ) @deprecate_pulse_func def Sawtooth( duration: int | ParameterExpression, amp: float | ParameterExpression, phase: float | ParameterExpression, freq: float | ParameterExpression | None = None, angle: float | ParameterExpression | None = 0.0, name: str | None = None, limit_amplitude: bool | None = None, ) -> ScalableSymbolicPulse: """A sawtooth pulse. The envelope of the pulse is given by: .. math:: f(x) = 2\\text{A}\\left[g\\left(x\\right)- \\lfloor g\\left(x\\right)+\\frac{1}{2}\\rfloor\\right] where :math:`\\text{A} = \\text{amp} \\times\\exp\\left(i\\times\\text{angle}\\right)`, :math:`g\\left(x\\right)=x\\times\\text{freq}+\\frac{\\text{phase}}{2\\pi}`, and :math:`\\lfloor ...\\rfloor` is the floor operation. Args: duration: Pulse length in terms of the sampling period `dt`. amp: The magnitude of the amplitude of the sawtooth wave. Wave range is [-`amp`,`amp`]. phase: The phase of the sawtooth wave (note that this is not equivalent to the angle of the complex amplitude) freq: The frequency of the sawtooth wave, in terms of 1 over sampling period. If not provided defaults to a single cycle (i.e :math:'\\frac{1}{\\text{duration}}'). The frequency is limited to the range :math:`\\left(0,0.5\\right]` (the Nyquist frequency). angle: The angle in radians of the complex phase factor uniformly scaling the pulse. Default value 0. name: Display name for this pulse envelope. limit_amplitude: If ``True``, then limit the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. Returns: ScalableSymbolicPulse instance. """ if freq is None: freq = 1 / duration parameters = {"freq": freq, "phase": phase} # Prepare symbolic expressions _t, _duration, _amp, _angle, _freq, _phase = sym.symbols("t, duration, amp, angle, freq, phase") lin_expr = _t * _freq + _phase / (2 * sym.pi) envelope_expr = 2 * _amp * sym.exp(sym.I * _angle) * (lin_expr - sym.floor(lin_expr + 1 / 2)) consts_expr = sym.And(_freq > 0, _freq < 0.5) # This might fail for waves shorter than a single cycle valid_amp_conditions_expr = sym.Abs(_amp) <= 1.0 return ScalableSymbolicPulse( pulse_type="Sawtooth", duration=duration, amp=amp, angle=angle, parameters=parameters, name=name, limit_amplitude=limit_amplitude, envelope=envelope_expr, constraints=consts_expr, valid_amp_conditions=valid_amp_conditions_expr, ) @deprecate_pulse_func def Triangle( duration: int | ParameterExpression, amp: float | ParameterExpression, phase: float | ParameterExpression, freq: float | ParameterExpression | None = None, angle: float | ParameterExpression | None = 0.0, name: str | None = None, limit_amplitude: bool | None = None, ) -> ScalableSymbolicPulse: """A triangle wave pulse. The envelope of the pulse is given by: .. math:: f(x) = \\text{A}\\left[\\text{sawtooth}\\left(x\\right)\\right] , 0 <= x < duration where :math:`\\text{A} = \\text{amp} \\times\\exp\\left(i\\times\\text{angle}\\right)`, and :math:`\\text{sawtooth}\\left(x\\right)` is a sawtooth wave with the same frequency as the triangle wave, but a phase shifted by :math:`\\frac{\\pi}{2}`. Args: duration: Pulse length in terms of the sampling period `dt`. amp: The magnitude of the amplitude of the triangle wave. Wave range is [-`amp`,`amp`]. phase: The phase of the triangle wave (note that this is not equivalent to the angle of the complex amplitude) freq: The frequency of the triangle wave, in terms of 1 over sampling period. If not provided defaults to a single cycle (i.e :math:'\\frac{1}{\\text{duration}}'). The frequency is limited to the range :math:`\\left(0,0.5\\right]` (the Nyquist frequency). angle: The angle in radians of the complex phase factor uniformly scaling the pulse. Default value 0. name: Display name for this pulse envelope. limit_amplitude: If ``True``, then limit the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. Returns: ScalableSymbolicPulse instance. """ if freq is None: freq = 1 / duration parameters = {"freq": freq, "phase": phase} # Prepare symbolic expressions _t, _duration, _amp, _angle, _freq, _phase = sym.symbols("t, duration, amp, angle, freq, phase") lin_expr = _t * _freq + _phase / (2 * sym.pi) - 0.25 sawtooth_expr = 2 * (lin_expr - sym.floor(lin_expr + 1 / 2)) envelope_expr = _amp * sym.exp(sym.I * _angle) * (-2 * sym.Abs(sawtooth_expr) + 1) consts_expr = sym.And(_freq > 0, _freq < 0.5) # This might fail for waves shorter than a single cycle valid_amp_conditions_expr = sym.Abs(_amp) <= 1.0 return ScalableSymbolicPulse( pulse_type="Triangle", duration=duration, amp=amp, angle=angle, parameters=parameters, name=name, limit_amplitude=limit_amplitude, envelope=envelope_expr, constraints=consts_expr, valid_amp_conditions=valid_amp_conditions_expr, ) @deprecate_pulse_func def Square( duration: int | ParameterValueType, amp: float | ParameterExpression, phase: float | ParameterExpression, freq: float | ParameterExpression | None = None, angle: float | ParameterExpression | None = 0.0, name: str | None = None, limit_amplitude: bool | None = None, ) -> ScalableSymbolicPulse: """A square wave pulse. The envelope of the pulse is given by: .. math:: f(x) = \\text{A}\\text{sign}\\left[\\sin \\left(2\\pi x\\times\\text{freq}+\\text{phase}\\right)\\right] , 0 <= x < duration where :math:`\\text{A} = \\text{amp} \\times\\exp\\left(i\\times\\text{angle}\\right)`, and :math:`\\text{sign}` is the sign function with the convention :math:`\\text{sign}\\left(0\\right)=1`. Args: duration: Pulse length in terms of the sampling period ``dt``. amp: The magnitude of the amplitude of the square wave. Wave range is :math:`\\left[-\\texttt{amp},\\texttt{amp}\\right]`. phase: The phase of the square wave (note that this is not equivalent to the angle of the complex amplitude). freq: The frequency of the square wave, in terms of 1 over sampling period. If not provided defaults to a single cycle (i.e :math:`\\frac{1}{\\text{duration}}`). The frequency is limited to the range :math:`\\left(0,0.5\\right]` (the Nyquist frequency). angle: The angle in radians of the complex phase factor uniformly scaling the pulse. Default value 0. name: Display name for this pulse envelope. limit_amplitude: If ``True``, then limit the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. Returns: ScalableSymbolicPulse instance. """ if freq is None: freq = 1 / duration parameters = {"freq": freq, "phase": phase} # Prepare symbolic expressions _t, _duration, _amp, _angle, _freq, _phase = sym.symbols("t, duration, amp, angle, freq, phase") _x = _freq * _t + _phase / (2 * sym.pi) envelope_expr = ( _amp * sym.exp(sym.I * _angle) * (2 * (2 * sym.floor(_x) - sym.floor(2 * _x)) + 1) ) consts_expr = sym.And(_freq > 0, _freq < 0.5) # This might fail for waves shorter than a single cycle valid_amp_conditions_expr = sym.Abs(_amp) <= 1.0 return ScalableSymbolicPulse( pulse_type="Square", duration=duration, amp=amp, angle=angle, parameters=parameters, name=name, limit_amplitude=limit_amplitude, envelope=envelope_expr, constraints=consts_expr, valid_amp_conditions=valid_amp_conditions_expr, ) @deprecate_pulse_func def Sech( duration: int | ParameterValueType, amp: float | ParameterExpression, sigma: float | ParameterExpression, angle: float | ParameterExpression | None = 0.0, name: str | None = None, zero_ends: bool | None = True, limit_amplitude: bool | None = None, ) -> ScalableSymbolicPulse: """An unnormalized sech pulse. The sech function is centered around the halfway point of the pulse, and the envelope of the pulse is given by: .. math:: f(x) = \\text{A}\\text{sech}\\left( \\frac{x-\\mu}{\\text{sigma}}\\right) , 0 <= x < duration where :math:`\\text{A} = \\text{amp} \\times\\exp\\left(i\\times\\text{angle}\\right)`, and :math:`\\mu=\\text{duration}/2`. If `zero_ends` is set to `True`, the output `y` is modified: .. math:: y\\left(x\\right) \\mapsto \\text{A}\\frac{y-y^{*}}{\\text{A}-y^{*}}, where :math:`y^{*}` is the value of :math:`y` at the endpoints (at :math:`x=-1 and :math:`x=\\text{duration}+1`). This shifts the endpoints value to zero, while also rescaling to preserve the amplitude at `:math:`\\text{duration}/2``. Args: duration: Pulse length in terms of the sampling period `dt`. amp: The magnitude of the amplitude of the pulse (the value at the midpoint `duration`/2). sigma: A measure of how wide or narrow the sech peak is in terms of `dt`; described mathematically in the class docstring. angle: The angle in radians of the complex phase factor uniformly scaling the pulse. Default value 0. name: Display name for this pulse envelope. zero_ends: If True, zeros the ends at x = -1, x = `duration` + 1, but rescales to preserve `amp`. Default value True. limit_amplitude: If ``True``, then limit the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. Returns: ScalableSymbolicPulse instance. """ parameters = {"sigma": sigma} # Prepare symbolic expressions _t, _duration, _amp, _angle, _sigma = sym.symbols("t, duration, amp, angle, sigma") complex_amp = _amp * sym.exp(sym.I * _angle) envelope_expr = complex_amp * sym.sech((_t - (_duration / 2)) / _sigma) if zero_ends: shift_val = complex_amp * sym.sech((-1 - (_duration / 2)) / _sigma) envelope_expr = complex_amp * (envelope_expr - shift_val) / (complex_amp - shift_val) consts_expr = _sigma > 0 valid_amp_conditions_expr = sym.Abs(_amp) <= 1.0 return ScalableSymbolicPulse( pulse_type="Sech", duration=duration, amp=amp, angle=angle, parameters=parameters, name=name, limit_amplitude=limit_amplitude, envelope=envelope_expr, constraints=consts_expr, valid_amp_conditions=valid_amp_conditions_expr, ) @deprecate_pulse_func def SechDeriv( duration: int | ParameterValueType, amp: float | ParameterExpression, sigma: float | ParameterExpression, angle: float | ParameterExpression | None = 0.0, name: str | None = None, limit_amplitude: bool | None = None, ) -> ScalableSymbolicPulse: """An unnormalized sech derivative pulse. The sech function is centered around the halfway point of the pulse, and the envelope of the pulse is given by: .. math:: f(x) = \\text{A}\\frac{d}{dx}\\left[\\text{sech} \\left(\\frac{x-\\mu}{\\text{sigma}}\\right)\\right] , 0 <= x < duration where :math:`\\text{A} = \\text{amp} \\times\\exp\\left(i\\times\\text{angle}\\right)`, :math:`\\mu=\\text{duration}/2`, and :math:`d/dx` is a derivative with respect to `x`. Args: duration: Pulse length in terms of the sampling period `dt`. amp: The magnitude of the amplitude of the pulse (the value of the corresponding sech function at the midpoint `duration`/2). sigma: A measure of how wide or narrow the corresponding sech peak is, in terms of `dt`; described mathematically in the class docstring. angle: The angle in radians of the complex phase factor uniformly scaling the pulse. Default value 0. name: Display name for this pulse envelope. limit_amplitude: If ``True``, then limit the amplitude of the waveform to 1. The default is ``True`` and the amplitude is constrained to 1. Returns: ScalableSymbolicPulse instance. """ parameters = {"sigma": sigma} # Prepare symbolic expressions _t, _duration, _amp, _angle, _sigma = sym.symbols("t, duration, amp, angle, sigma") time_argument = (_t - (_duration / 2)) / _sigma sech_deriv = -sym.tanh(time_argument) * sym.sech(time_argument) / _sigma envelope_expr = _amp * sym.exp(sym.I * _angle) * sech_deriv consts_expr = _sigma > 0 valid_amp_conditions_expr = sym.Abs(_amp) / _sigma <= 2.0 return ScalableSymbolicPulse( pulse_type="SechDeriv", duration=duration, amp=amp, angle=angle, parameters=parameters, name=name, limit_amplitude=limit_amplitude, envelope=envelope_expr, constraints=consts_expr, valid_amp_conditions=valid_amp_conditions_expr, )