Source code for mitiq.zne.inference

# Copyright (C) 2020 Unitary Fund
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"""Classes corresponding to different zero-noise extrapolation methods."""
from abc import ABC, abstractmethod
from copy import deepcopy
from typing import (
    Any,
    Callable,
    cast,
    Dict,
    List,
    Optional,
    Sequence,
    Tuple,
    Union,
)
import warnings

from matplotlib.figure import Figure
import matplotlib.pyplot as plt
import numpy as np
from numpy.lib.polynomial import RankWarning
from scipy.optimize import curve_fit, OptimizeWarning
from cirq import Circuit

from mitiq._typing import QPROGRAM, QuantumResult
from mitiq.observable import Observable
from mitiq.executor import Executor
from mitiq.zne.scaling import fold_gates_at_random
from mitiq.interface import accept_any_qprogram_as_input


ExtrapolationResult = Union[
    float,  # The zero-noise value.
    Tuple[
        float,  # The zero-noise value.
        Optional[float],  # The (estimated) error on the zero-noise value.
        List[float],  # Optimal parameters found during fitting.
        Optional[np.ndarray],  # Covariance of fitting parameters.
        Callable[[float], float],  # Function that was fit.
    ],
]


[docs]class ExtrapolationError(Exception): """Error raised by :class:`.Factory` objects when the extrapolation fit fails. """ pass
_EXTR_ERR = ( "The extrapolation fit failed to converge." " The problem may be solved by switching to a more stable" " extrapolation model such as `LinearFactory`." )
[docs]class ExtrapolationWarning(Warning): """Warning raised by :class:`.Factory` objects when the extrapolation fit is ill-conditioned. """ pass
_EXTR_WARN = ( "The extrapolation fit may be ill-conditioned." " Likely, more data points are necessary to fit the parameters" " of the model." ) DATA_MISSING_ERR = ( "Data is either ill-defined or not enough to evaluate the required" " information. Please make sure that the 'run' and 'reduce' methods" " have been called and that enough expectation values have been measured." )
[docs]class ConvergenceWarning(Warning): """Warning raised by :class:`.Factory` objects when their `run_classical` method fails to converge. """ pass
@accept_any_qprogram_as_input def _check_circuit_length(circuit: Circuit) -> None: """Raises a warning if the circuit is too short.""" if len(list(circuit.all_operations())) < 5: warnings.warn( "The input circuit is very short. " "This may reduce the accuracy of noise scaling." )
[docs]def mitiq_curve_fit( ansatz: Callable[..., float], scale_factors: Sequence[float], exp_values: Sequence[float], init_params: Optional[List[float]] = None, ) -> Tuple[List[float], np.ndarray]: """Fits the ansatz to the (scale factor, expectation value) data using ``scipy.optimize.curve_fit``, returning the optimal parameters and covariance matrix of the parameters. Args: ansatz: The model function used for zero-noise extrapolation. The first argument is the noise scale variable, the remaining arguments are the parameters to fit. scale_factors: The array of noise scale factors. exp_values: The array of expectation values. init_params: Initial guess for the parameters. If None, the initial values are set to 1. Returns: The array of optimal parameters and the covariance matrix of the parameters. If the fit is ill-conditioned, the covariance matrix may contain np.inf elements. Raises: ExtrapolationError: If the extrapolation fit fails. ExtrapolationWarning: If the extrapolation fit is ill-conditioned. """ try: with warnings.catch_warnings(record=True) as warn_list: opt_params, params_cov = curve_fit( ansatz, scale_factors, exp_values, p0=init_params ) for warn in warn_list: # replace OptimizeWarning with ExtrapolationWarning if warn.category is OptimizeWarning: warn.category = ExtrapolationWarning warn.message = _EXTR_WARN # type: ignore # re-raise all warnings warnings.warn_explicit( warn.message, warn.category, warn.filename, warn.lineno ) except RuntimeError: raise ExtrapolationError(_EXTR_ERR) from None return list(opt_params), params_cov
[docs]def mitiq_polyfit( scale_factors: Sequence[float], exp_values: Sequence[float], deg: int, weights: Optional[Sequence[float]] = None, ) -> Tuple[List[float], Optional[np.ndarray]]: """Fits the ansatz to the (scale factor, expectation value) data using ``numpy.polyfit``, returning the optimal parameters and covariance matrix of the parameters. Args: scale_factors: The array of noise scale factors. exp_values: The array of expectation values. deg: The degree of the polynomial fit. weights: Optional array of weights for each sampled point. This is used to make a weighted least squares fit. Returns: The optimal parameters and covariance matrix of the parameters. If there is not enough data to estimate the covariance matrix, it is returned as None. Raises: ExtrapolationWarning: If the extrapolation fit is ill-conditioned. """ with warnings.catch_warnings(record=True) as warn_list: try: opt_params, params_cov = np.polyfit( scale_factors, exp_values, deg, w=weights, cov=True ) except (ValueError, np.linalg.LinAlgError): opt_params = np.polyfit(scale_factors, exp_values, deg, w=weights) params_cov = None for warn in warn_list: # replace RankWarning with ExtrapolationWarning if warn.category is RankWarning: warn.category = ExtrapolationWarning warn.message = _EXTR_WARN # type: ignore # re-raise all warnings warnings.warn_explicit( warn.message, warn.category, warn.filename, warn.lineno ) return list(opt_params), params_cov
[docs]class Factory(ABC): """Abstract base class which performs the classical parts of zero-noise extrapolation. This minimally includes: * scaling circuits, * sending jobs to execute, * collecting the results, * fitting the collected data, * Extrapolating to the zero-noise limit. If all scale factors are set a priori, the jobs can be batched. This is handled by a BatchedFactory. If the next scale factor depends on the previous history of results, jobs are run sequentially. This is handled by an AdaptiveFactory. """ def __init__(self) -> None: self._instack: List[Dict[str, float]] = [] self._outstack: List[float] = [] self._opt_params: Optional[List[float]] = None self._params_cov: Optional[np.ndarray] = None self._zne_limit: Optional[float] = None self._zne_error: Optional[float] = None self._zne_curve: Optional[Callable[[float], float]] = None self._already_reduced = False self._options: Dict[str, Optional[float]] = {}
[docs] def get_scale_factors(self) -> np.ndarray: """Returns the scale factors at which the factory has computed expectation values. """ return np.array( [params.get("scale_factor") for params in self._instack] )
[docs] def get_expectation_values(self) -> np.ndarray: """Returns the expectation values computed by the factory.""" return np.array(self._outstack)
[docs] def get_optimal_parameters(self) -> np.ndarray: """Returns the optimal model parameters produced by the extrapolation fit. """ if self._opt_params is None: raise ValueError(DATA_MISSING_ERR) return np.array(self._opt_params)
[docs] def get_parameters_covariance(self) -> np.ndarray: """Returns the covariance matrix of the model parameters produced by the extrapolation fit. """ if self._params_cov is None: raise ValueError(DATA_MISSING_ERR) return np.array(self._params_cov)
[docs] def get_zero_noise_limit(self) -> float: """Returns the last evaluation of the zero-noise limit computed by the factory. To re-evaluate its value, the method 'reduce' should be called first. """ if self._zne_limit is None: raise ValueError(DATA_MISSING_ERR) return self._zne_limit
[docs] def get_zero_noise_limit_error(self) -> float: """Returns the extrapolation error representing the uncertainty affecting the zero-noise limit. It is deduced by error propagation from the covariance matrix associated to the fit parameters. Note: this quantity is only related to the ability of the model to fit the measured data. Therefore, it may underestimate the actual error existing between the zero-noise limit and the true ideal expectation value. """ if self._zne_error is None: raise ValueError(DATA_MISSING_ERR) return self._zne_error
[docs] def get_extrapolation_curve(self) -> Callable[[float], float]: """Returns the extrapolation curve, i.e., a function which inputs a noise scale factor and outputs the associated expectation value. This function is the solution of the regression problem used to evaluate the zero-noise extrapolation. """ if self._zne_curve is None: raise ValueError(DATA_MISSING_ERR) return self._zne_curve
[docs] @abstractmethod def run( self, qp: QPROGRAM, executor: Union[Executor, Callable[..., QuantumResult]], observable: Optional[Observable] = None, scale_noise: Callable[ [QPROGRAM, float], QPROGRAM ] = fold_gates_at_random, num_to_average: int = 1, ) -> "Factory": """Calls the executor function on noise-scaled quantum circuit and stores the results. Args: qp: Quantum circuit to scale noise in. executor: A ``mitiq.Executor`` or a function which inputs a (list of) quantum circuits and outputs a (list of) ``mitiq.QuantumResult`` s. observable: Observable to compute the expectation value of. If None, the `executor` must return an expectation value. Otherwise, the `QuantumResult` returned by `executor` is used to compute the expectation of the observable. scale_noise: Function which inputs a quantum circuit and outputs a noise-scaled quantum circuit. num_to_average: Number of times the executor function is called on each noise-scaled quantum circuit. """ raise NotImplementedError
[docs] @abstractmethod def run_classical( self, scale_factor_to_expectation_value: Callable[..., float], ) -> "Factory": """Calls the function scale_factor_to_expectation_value at each scale factor of the factory, and stores the results. Args: scale_factor_to_expectation_value: A function which inputs a scale factor and outputs an expectation value. This does not have to involve a quantum processor making this a "classical analogue" of the run method. """ raise NotImplementedError
@abstractmethod def reduce(self) -> float: raise NotImplementedError
[docs] def push( self, instack_val: Dict[str, float], outstack_val: float ) -> "Factory": """Appends "instack_val" to "self._instack" and "outstack_val" to "self._outstack". Each time a new expectation value is computed this method should be used to update the internal state of the Factory. """ if self._already_reduced: warnings.warn( "You are pushing new data into a factory object despite its " ".reduce() method has already been called. Please make " "sure your intention is to append new data to the stack of " "previous data. Otherwise, the method .reset() can be used " "to clean the internal state of the factory.", ExtrapolationWarning, ) self._instack.append(instack_val) self._outstack.append(outstack_val) return self
[docs] def plot_data(self) -> Figure: """Returns a figure which is a scatter plot of (x, y) data where x are scale factors at which expectation values have been computed, and y are the associated expectation values. Returns: fig: A 2D scatter plot described above. """ fig = plt.figure(figsize=(7, 5)) ax = plt.gca() plt.plot( self.get_scale_factors(), self.get_expectation_values(), "o", markersize=10, markeredgecolor="black", alpha=0.8, label="Data", ) ax.grid(True) plt.xlabel("Noise scale factor") plt.ylabel("Expectation value") return fig
[docs] def plot_fit(self) -> Figure: """Returns a figure which plots the experimental data as well as the best fit curve. Returns: fig: A figure which plots the best fit curve as well as the data. """ fig = self.plot_data() smooth_scale_factors = np.linspace(0, self.get_scale_factors()[-1], 20) smooth_expectations = [ self.get_extrapolation_curve()(scale_factor) for scale_factor in smooth_scale_factors ] plt.xlim(left=0) fig.axes[0].plot( smooth_scale_factors, smooth_expectations, "--", lw=2, color="black", label="Best fit", ) return fig
[docs] def reset(self) -> "Factory": """Resets the internal state of the Factory.""" self._instack = [] self._outstack = [] self._opt_params = None self._params_cov = None self._zne_limit = None self._zne_error = None self._already_reduced = False return self
[docs]class BatchedFactory(Factory, ABC): """Abstract class of a non-adaptive Factory initialized with a pre-determined set of scale factors. Specific (non-adaptive) extrapolation algorithms are derived from this class by defining the `reduce` method. """ def __init__( self, scale_factors: Sequence[float], shot_list: Optional[List[int]] = None, ) -> None: """Constructs a BatchedFactory. Args: scale_factors: Sequence of noise scale factors at which expectation values should be measured. shot_list: Optional sequence of integers corresponding to the number of samples taken for each expectation value. If this argument is explicitly passed to the factory, it must have the same length of scale_factors and the executor function must accept "shots" as a valid keyword argument. Raises: ValueError: If the number of scale factors is less than 2. TypeError: If shot_list is provided and has any non-integer values. """ if len(scale_factors) < 2: raise ValueError("At least 2 scale factors are necessary.") if shot_list and ( not isinstance(shot_list, Sequence) or not all([isinstance(shots, int) for shots in shot_list]) ): raise TypeError( "The optional argument shot_list must be None " "or a valid iterator of integers." ) if shot_list and (len(scale_factors) != len(shot_list)): raise IndexError( "The arguments scale_factors and shot_list" " must have the same length." f" But len(scale_factors) is {len(scale_factors)}" f" and len(shot_list) is {len(shot_list)}." ) self._scale_factors = scale_factors self._shot_list = shot_list super(BatchedFactory, self).__init__()
[docs] @staticmethod @abstractmethod def extrapolate(*args: Any, **kwargs: Any) -> ExtrapolationResult: """Returns the extrapolation to the zero-noise limit.""" raise NotImplementedError
[docs] def reduce(self) -> float: """Evaluates the zero-noise limit found by fitting according to the factory's extrapolation method. Returns: The zero-noise limit. """ ( self._zne_limit, self._zne_error, self._opt_params, self._params_cov, self._zne_curve, ) = self.extrapolate( # type: ignore self.get_scale_factors(), self.get_expectation_values(), full_output=True, **self._options, ) self._already_reduced = True return self._zne_limit
[docs] def run( self, qp: QPROGRAM, executor: Union[Executor, Callable[..., QuantumResult]], observable: Optional[Observable] = None, scale_noise: Callable[ [QPROGRAM, float], QPROGRAM ] = fold_gates_at_random, num_to_average: int = 1, ) -> "BatchedFactory": """Computes the expectation values at each scale factor and stores them in the factory. If the executor returns a single expectation value, the circuits are run sequentially. If the executor is batched and returns a list of expectation values (one for each circuit), then the circuits are sent to the backend as a single job. To detect if an executor is batched, it must be annotated with a return type that is one of the following: * Iterable[float] * List[float] * Sequence[float] * Tuple[float] * numpy.ndarray Args: qp: Quantum circuit to run. executor: A ``mitiq.Executor`` or a function which inputs a (list of) quantum circuits and outputs a (list of) ``mitiq.QuantumResult`` s. observable: Observable to compute the expectation value of. If None, the `executor` must return an expectation value. Otherwise, the `QuantumResult` returned by `executor` is used to compute the expectation of the observable. scale_noise: Noise scaling function. num_to_average: The number of circuits executed for each noise scale factor. This parameter can be used to increase the precision of the "executor" or to average the effect of a non-deterministic "scale_noise" function. """ self.reset() self._batch_populate_instack() _check_circuit_length(qp) # Get all noise-scaled circuits to run. to_run = self._generate_circuits(qp, scale_noise, num_to_average) # Run all circuits. if not isinstance(executor, Executor): executor = Executor(executor) # Get the list of keywords associated to each circuit in "to_run". kwargs_list = self._get_keyword_args(num_to_average) # If there are different keyword args, run each circuit individually. # https://stackoverflow.com/questions/1151658/python-hashable-dicts. class HashableDict(Dict[Any, Any]): def __hash__(self) -> int: # type: ignore[override] return hash(tuple(sorted(self.items()))) if len(set(HashableDict(kwargs) for kwargs in kwargs_list)) != 1: res = [] for circuit, kwargs in zip(to_run, kwargs_list): res.extend( executor.evaluate( circuit, observable, force_run_all=True, **kwargs ) ) else: # Else, run all circuits. res = executor.evaluate( to_run, observable, force_run_all=True, **kwargs_list[0] ) # Reshape "res" to have "num_to_average" columns reshaped = np.array(res).reshape((-1, num_to_average)) # Average the "num_to_average" columns self._outstack = np.average(reshaped, axis=1) return self
[docs] def run_classical( self, scale_factor_to_expectation_value: Callable[..., float] ) -> "BatchedFactory": """Computes expectation values by calling the input function at each scale factor. Args: scale_factor_to_expectation_value: Function mapping a noise scale factor to an expectation value. If shot_list is not None, "shots" must be an argument of this function. """ self.reset() self._batch_populate_instack() kwargs_list = self._get_keyword_args(num_to_average=1) self._outstack = [ scale_factor_to_expectation_value(scale_factor, **kwargs) for scale_factor, kwargs in zip(self._scale_factors, kwargs_list) ] return self
def _generate_circuits( self, circuit: QPROGRAM, scale_noise: Callable[[QPROGRAM, float], QPROGRAM], num_to_average: int = 1, ) -> List[QPROGRAM]: """Returns all noise-scaled circuits to run. Args: circuit: Base circuit to scale noise in. scale_noise: Noise scaling function. num_to_average: Number of times to call scale_noise at each scale factor. """ to_run = [] for scale_factor in self.get_scale_factors(): for _ in range(num_to_average): to_run.append(scale_noise(circuit, scale_factor)) return to_run def _batch_populate_instack(self) -> None: """Populates the instack with all computed values.""" if self._shot_list: self._instack = [ {"scale_factor": scale, "shots": shots} for scale, shots in zip(self._scale_factors, self._shot_list) ] else: self._instack = [ {"scale_factor": scale} for scale in self._scale_factors ] def _get_keyword_args(self, num_to_average: int) -> List[Dict[str, Any]]: """Returns a list of keyword dictionaries to be used for executing the circuits generated by the method "_generate_circuits". Args: num_to_average: The number of times the same keywords are used for each scale factor. This should correspond to the number of circuits executed for each scale factor. Returns: The output list of keyword dictionaries. """ params = deepcopy(self._instack) for d in params: _ = d.pop("scale_factor") # Repeat each keyword num_to_average times return [k for k in params for _ in range(num_to_average)]
[docs]class AdaptiveFactory(Factory, ABC): """Abstract class designed to adaptively produce a new noise scaling parameter based on a historical stack of previous noise scale parameters ("self._instack") and previously estimated expectation values ("self._outstack"). Specific zero-noise extrapolation algorithms which are adaptive are derived from this class. """
[docs] @abstractmethod def next(self) -> Dict[str, float]: """Returns a dictionary of parameters to execute a circuit at.""" raise NotImplementedError
[docs] @abstractmethod def is_converged(self) -> bool: """Returns True if all needed expectation values have been computed, else False. """ raise NotImplementedError
[docs] @abstractmethod def reduce(self) -> float: """Returns the extrapolation to the zero-noise limit.""" raise NotImplementedError
[docs] def run_classical( self, scale_factor_to_expectation_value: Callable[..., float], max_iterations: int = 100, ) -> "AdaptiveFactory": """Evaluates a sequence of expectation values until enough data is collected (or iterations reach "max_iterations"). Args: scale_factor_to_expectation_value: Function mapping a noise scale factor to an expectation value. If shot_list is not None, "shots" must be an argument of this function. max_iterations: Maximum number of iterations (optional). Default: 100. Raises: ConvergenceWarning: If iteration loop stops before convergence. """ # Reset the instack, outstack, and optimal parameters self.reset() counter = 0 while not self.is_converged() and counter < max_iterations: next_in_params = self.next() next_exec_params = deepcopy(next_in_params) # Get next scale factor and remove it from next_exec_params scale_factor = next_exec_params.pop("scale_factor") next_expval = scale_factor_to_expectation_value( scale_factor, **next_exec_params ) self.push(next_in_params, next_expval) counter += 1 if counter == max_iterations: warnings.warn( "Factory iteration loop stopped before convergence. " f"Maximum number of iterations ({max_iterations}) " "was reached.", ConvergenceWarning, ) return self
[docs] def run( self, qp: QPROGRAM, executor: Union[Executor, Callable[..., QuantumResult]], observable: Optional[Observable] = None, scale_noise: Callable[ [QPROGRAM, float], QPROGRAM ] = fold_gates_at_random, num_to_average: int = 1, max_iterations: int = 100, ) -> "AdaptiveFactory": """Evaluates a sequence of expectation values by executing quantum circuits until enough data is collected (or iterations reach "max_iterations"). Args: qp: Circuit to mitigate. executor: A ``mitiq.Executor`` or a function which inputs a (list of) quantum circuits and outputs a (list of) ``mitiq.QuantumResult`` s. observable: Observable to compute the expectation value of. If None, the `executor` must return an expectation value. Otherwise, the `QuantumResult` returned by `executor` is used to compute the expectation of the observable. scale_noise: Function that scales the noise level of a quantum circuit. num_to_average: Number of times expectation values are computed by the executor after each call to scale_noise, then averaged. max_iterations: Maximum number of iterations (optional). """ _check_circuit_length(qp) if not isinstance(executor, Executor): executor = Executor(executor) def scale_factor_to_expectation_value( scale_factor: float, **exec_params: Any ) -> float: """Evaluates the quantum expectation value for a given scale_factor and other executor parameters.""" to_run = [ scale_noise(qp, scale_factor) for _ in range(num_to_average) ] expectation_values = executor.evaluate( # type: ignore[union-attr] to_run, observable, force_run_all=True, **exec_params ) return np.average(expectation_values) return self.run_classical( scale_factor_to_expectation_value, max_iterations )
[docs]class PolyFactory(BatchedFactory): """Factory object implementing a zero-noise extrapolation algorithm based on a polynomial fit. Args: scale_factors: Sequence of noise scale factors at which expectation values should be measured. order: Extrapolation order (degree of the polynomial fit). It cannot exceed len(scale_factors) - 1. shot_list: Optional sequence of integers corresponding to the number of samples taken for each expectation value. If this argument is explicitly passed to the factory, it must have the same length of scale_factors and the executor function must accept "shots" as a valid keyword argument. Raises: ValueError: If data is not consistent with the extrapolation model. ExtrapolationWarning: If the extrapolation fit is ill-conditioned. Note: RichardsonFactory and LinearFactory are special cases of PolyFactory. """ def __init__( self, scale_factors: Sequence[float], order: int, shot_list: Optional[List[int]] = None, ) -> None: """Instantiates a new object of this Factory class.""" if order > len(scale_factors) - 1: raise ValueError( "The extrapolation order cannot exceed len(scale_factors) - 1." ) super(PolyFactory, self).__init__(scale_factors, shot_list) self._options = {"order": order}
[docs] @staticmethod def extrapolate( # type: ignore scale_factors: Sequence[float], exp_values: Sequence[float], order: int, full_output: bool = False, ) -> ExtrapolationResult: """Static method which evaluates a polynomial extrapolation to the zero-noise limit. Args: scale_factors: The array of noise scale factors. exp_values: The array of expectation values. order: The extrapolation order (degree of the polynomial fit). full_output: If False (default), only the zero-noise limit is returned. If True, additional information about the extrapolated limit is returned too. Returns: The extrapolated zero-noise limit. If full_output is True, also returns * standard deviation of the extrapolated zero-noise limit, * optimal parameters of the best-fit model, * parameter covariance matrix of best-fit model, * best-fit model as a Callable[[float], float] function. Raises: ExtrapolationWarning: If the extrapolation fit is ill-conditioned. Note: This static method computes the zero-noise limit from input parameters. To compute the zero-noise limit from the Factory parameters, use the ``reduce`` method. """ opt_params, params_cov = mitiq_polyfit( scale_factors, exp_values, order ) zne_limit = opt_params[-1] if not full_output: return zne_limit zne_error = None if params_cov is not None: if params_cov.shape == (order + 1, order + 1): zne_error = np.sqrt(params_cov[order, order]) def zne_curve(scale_factor: float) -> float: return np.polyval(opt_params, scale_factor) return zne_limit, zne_error, opt_params, params_cov, zne_curve
[docs]class RichardsonFactory(BatchedFactory): """Factory object implementing Richardson extrapolation. Args: scale_factors: Sequence of noise scale factors at which expectation values should be measured. shot_list: Optional sequence of integers corresponding to the number of samples taken for each expectation value. If this argument is explicitly passed to the factory, it must have the same length of scale_factors and the executor function must accept "shots" as a valid keyword argument. Raises: ValueError: If data is not consistent with the extrapolation model. ExtrapolationWarning: If the extrapolation fit is ill-conditioned. """
[docs] @staticmethod def extrapolate( # type: ignore scale_factors: Sequence[float], exp_values: Sequence[float], full_output: bool = False, ) -> ExtrapolationResult: """Static method which evaluates the Richardson extrapolation to the zero-noise limit. Args: scale_factors: The array of noise scale factors. exp_values: The array of expectation values. full_output: If False (default), only the zero-noise limit is returned. If True, additional results are returned too. Returns: The extrapolated zero-noise limit. If full_output is True, also returns * standard deviation of the extrapolated zero-noise limit, * optimal parameters of the best-fit model, * parameter covariance matrix of best-fit model, * best-fit model as a Callable[[float], float] function. Raises: ExtrapolationWarning: If the extrapolation fit is ill-conditioned. Note: This static method computes the zero-noise limit from input parameters. To compute the zero-noise limit from the Factory parameters, use the ``reduce`` method. """ # Richardson extrapolation is a particular case of a polynomial fit # with order equal to the number of data points minus 1. order = len(scale_factors) - 1 return PolyFactory.extrapolate( scale_factors, exp_values, order, full_output )
[docs]class FakeNodesFactory(BatchedFactory): """Factory object implementing a modified version [De2020polynomial]_ of Richardson extrapolation. In this version the original set of scale factors is mapped to a new set of fake nodes, known as Chebyshev-Lobatto points. This method may give a better interpolation for particular types of curves and if the number of scale factors is large (> 10). One should be aware that, in many other cases, the fake nodes extrapolation method is usually not superior to standard Richardson extrapolation. Args: scale_factors: Sequence of noise scale factors at which expectation values should be measured. shot_list: Optional sequence of integers corresponding to the number of samples taken for each expectation value. If this argument is explicitly passed to the factory, it must have the same length of scale_factors and the executor function must accept "shots" as a valid keyword argument. Raises: ValueError: If data is not consistent with the extrapolation model. ExtrapolationWarning: If the extrapolation fit is ill-conditioned. .. [De2020polynomial] : S.De Marchia. F. Marchetti, E.Perracchionea and D.Poggialia, "Polynomial interpolation via mapped bases without resampling," *Journ of Comp. and App. Math.* **364**, 112347 (2020), (https://www.sciencedirect.com/science/article/abs/pii/S0377042719303449). """
[docs] @staticmethod def extrapolate( # type: ignore scale_factors: Sequence[float], exp_values: Sequence[float], full_output: bool = False, ) -> ExtrapolationResult: if not FakeNodesFactory._is_equally_spaced(scale_factors): raise ValueError("The scale factors must be equally spaced.") # Define interval [a, b] for which the scale_factors are mapped to a = 0.0 b = min(scale_factors) + max(scale_factors) # Mapping to the fake nodes fake_nodes = FakeNodesFactory._map_to_fake_nodes(scale_factors, a, b) if not full_output: return RichardsonFactory.extrapolate(fake_nodes, exp_values) ( zne_limit, zne_error, opt_params, params_cov, zne_curve, ) = RichardsonFactory.extrapolate( # type: ignore[misc] fake_nodes, exp_values, True ) # Convert zne_curve from the "fake node space" to the real space. # Note: since a=0.0, this conversion is not necessary for zne_limit. def new_curve(scale_factor: float) -> float: """Get real zne_curve from the curve based on fake nodes.""" return zne_curve( FakeNodesFactory._map_to_fake_nodes([scale_factor], a, b)[0] ) return zne_limit, zne_error, opt_params, params_cov, new_curve
@staticmethod def _map_to_fake_nodes( x: Union[Sequence[float]], a: float, b: float ) -> Sequence[float]: """ A function that maps inputs to Chebyshev-Lobatto points. Based on the function [De2020polynomial]_: S(x) = (a - b)/2 * cos(pi * (x - a)/(b - a)) + (a + b)/2. Where a and b are the endpoints of the interval [a, b] of CL points we are mapping to. Args: x: Sequence[float]: Set of values to be mapped to CL points. a: A float representing the interval starting at a. b: A float representing the interval ending at b. Returns: A new sequence of fake nodes (Chebyshev-Lobatto points). .. [De2020polynomial]: S.De Marchia. F. Marchetti, E.Perracchionea and D.Poggialia, "Polynomial interpolation via mapped bases without resampling," *Journ of Comp. and App. Math.* **364**, 112347 (2020), (https://www.sciencedirect.com/science/article/abs/pii/S0377042719303449). """ # The mapping function def mapping(_x: float) -> float: return (a - b) / 2 * np.cos(np.pi * (_x - a) / (b - a)) + ( a + b ) / 2 return [mapping(y) for y in x] @staticmethod def _is_equally_spaced(arr: Sequence[float]) -> bool: """Checks if the sequence is equally spaced.""" diff_arr = np.diff(np.sort(arr)) return np.allclose(diff_arr, diff_arr[0])
[docs]class LinearFactory(BatchedFactory): """ Factory object implementing zero-noise extrapolation based on a linear fit. Args: scale_factors: Sequence of noise scale factors at which expectation values should be measured. shot_list: Optional sequence of integers corresponding to the number of samples taken for each expectation value. If this argument is explicitly passed to the factory, it must have the same length of scale_factors and the executor function must accept "shots" as a valid keyword argument. Raises: ValueError: If data is not consistent with the extrapolation model. ExtrapolationWarning: If the extrapolation fit is ill-conditioned. """
[docs] @staticmethod def extrapolate( # type: ignore scale_factors: Sequence[float], exp_values: Sequence[float], full_output: bool = False, ) -> ExtrapolationResult: """Static method which evaluates the linear extrapolation to the zero-noise limit. Args: scale_factors: The array of noise scale factors. exp_values: The array of expectation values. full_output: If False (default), only the zero-noise limit is returned. If True, additional results are returned too. Returns: The extrapolated zero-noise limit. If full_output is True, also returns * standard deviation of the extrapolated zero-noise limit, * optimal parameters of the best-fit model, * parameter covariance matrix of best-fit model, * best-fit model as a Callable[[float], float] function. Raises: ExtrapolationWarning: If the extrapolation fit is ill-conditioned. Note: This static method computes the zero-noise limit from input parameters. To compute the zero-noise limit from the Factory parameters, use the ``reduce`` method. """ # Linear extrapolation is equivalent to a polynomial fit with order=1 return PolyFactory.extrapolate( scale_factors, exp_values, 1, full_output )
[docs]class ExpFactory(BatchedFactory): """ Factory object implementing a zero-noise extrapolation algorithm assuming an exponential ansatz y(x) = a + b * exp(-c * x), with c > 0. If y(x->inf) is unknown, the ansatz y(x) is fitted with a non-linear optimization. If y(x->inf) is given and avoid_log=False, the exponential model is mapped into a linear model by a logarithmic transformation. Args: scale_factors: Sequence of noise scale factors at which expectation values should be measured. asymptote: Infinite-noise limit (optional argument). avoid_log: If set to True, the exponential model is not linearized with a logarithm and a non-linear fit is applied even if asymptote is not None. The default value is False. shot_list: Optional sequence of integers corresponding to the number of samples taken for each expectation value. If this argument is explicitly passed to the factory, it must have the same length of scale_factors and the executor function must accept "shots" as a valid keyword argument. Raises: ValueError: If data is not consistent with the extrapolation model. ExtrapolationError: If the extrapolation fit fails. ExtrapolationWarning: If the extrapolation fit is ill-conditioned. """ def __init__( self, scale_factors: Sequence[float], asymptote: Optional[float] = None, avoid_log: bool = False, shot_list: Optional[List[int]] = None, ) -> None: """Instantiate an new object of this Factory class.""" super(ExpFactory, self).__init__(scale_factors, shot_list) if not (asymptote is None or isinstance(asymptote, float)): raise ValueError( "The argument 'asymptote' must be either a float or None" ) self._options = { "asymptote": asymptote, "avoid_log": avoid_log, }
[docs] @staticmethod def extrapolate( # type: ignore scale_factors: Sequence[float], exp_values: Sequence[float], asymptote: Optional[float] = None, avoid_log: bool = False, eps: float = 1.0e-6, full_output: bool = False, ) -> ExtrapolationResult: """Static method which evaluates the extrapolation to the zero-noise limit assuming an exponential ansatz y(x) = a + b * exp(-c * x), with c > 0. If y(x->inf) is unknown, the ansatz y(x) is fitted with a non-linear optimization. If y(x->inf) is given and avoid_log=False, the exponential model is mapped into a linear model by a logarithmic transformation. Args: scale_factors: The array of noise scale factors. exp_values: The array of expectation values. asymptote: The infinite-noise limit y(x->inf) (optional argument). avoid_log: If set to True, the exponential model is not linearized with a logarithm and a non-linear fit is applied even if asymptote is not None. The default value is False. eps: Epsilon to regularize log(sign(scale_factors - asymptote)) when the argument is to close to zero or negative. full_output: If False (default), only the zero-noise limit is returned. If True, additional information about the extrapolated limit is returned too. Returns: The extrapolated zero-noise limit. If full_output is True, also returns * standard deviation of the extrapolated zero-noise limit, * optimal parameters of the best-fit model, * parameter covariance matrix of best-fit model, * best-fit model as a Callable[[float], float] function. Raises: ValueError: If the arguments are not consistent with the extrapolation model. ExtrapolationError: If the extrapolation fit fails. ExtrapolationWarning: If the extrapolation fit is ill-conditioned. Note: This static method computes the zero-noise limit from input parameters. To compute the zero-noise limit from the Factory parameters, use the ``reduce`` method. """ return PolyExpFactory.extrapolate( scale_factors, exp_values, order=1, asymptote=asymptote, avoid_log=avoid_log, eps=eps, full_output=full_output, )
[docs]class PolyExpFactory(BatchedFactory): """ Factory object implementing a zero-noise extrapolation algorithm assuming an (almost) exponential ansatz with a non linear exponent y(x) = a + sign * exp(z(x)), where z(x) is a polynomial of a given order. The parameter "sign" is a sign variable which can be either 1 or -1, corresponding to decreasing and increasing exponentials, respectively. The parameter "sign" is automatically deduced from the data. If y(x->inf) is unknown, the ansatz y(x) is fitted with a non-linear optimization. If y(x->inf) is given and avoid_log=False, the exponential model is mapped into a polynomial model by logarithmic transformation. Args: scale_factors: Sequence of noise scale factors at which expectation values should be measured. order: Extrapolation order (degree of the polynomial z(x)). It cannot exceed len(scale_factors) - 1. If asymptote is None, order cannot exceed len(scale_factors) - 2. asymptote: The infinite-noise limit y(x->inf) (optional argument). avoid_log: If set to True, the exponential model is not linearized with a logarithm and a non-linear fit is applied even if asymptote is not None. The default value is False. shot_list: Optional sequence of integers corresponding to the number of samples taken for each expectation value. If this argument is explicitly passed to the factory, it must have the same length of scale_factors and the executor function must accept "shots" as a valid keyword argument. Raises: ValueError: If data is not consistent with the extrapolation model. ExtrapolationError: If the extrapolation fit fails. ExtrapolationWarning: If the extrapolation fit is ill-conditioned. """ def __init__( self, scale_factors: Sequence[float], order: int, asymptote: Optional[float] = None, avoid_log: bool = False, shot_list: Optional[List[int]] = None, ) -> None: """Instantiates a new object of this Factory class.""" super(PolyExpFactory, self).__init__(scale_factors, shot_list) if not (asymptote is None or isinstance(asymptote, float)): raise ValueError( "The argument 'asymptote' must be either a float or None" ) self._options = { "order": order, "asymptote": asymptote, "avoid_log": avoid_log, }
[docs] @staticmethod def extrapolate( # type: ignore scale_factors: Sequence[float], exp_values: Sequence[float], order: int, asymptote: Optional[float] = None, avoid_log: bool = False, eps: float = 1.0e-6, full_output: bool = False, ) -> ExtrapolationResult: """Static method which evaluates the extrapolation to the zero-noise limit with an exponential ansatz (whose exponent is a polynomial of degree "order"). The exponential ansatz is y(x) = a + sign * exp(z(x)) where z(x) is a polynomial and "sign" is either +1 or -1 corresponding to decreasing and increasing exponentials, respectively. The parameter "sign" is automatically deduced from the data. It is also assumed that z(x-->inf) = -inf, such that y(x-->inf) --> a. If asymptote is None, the ansatz y(x) is fitted with a non-linear optimization. If asymptote is given and avoid_log=False, a linear fit with respect to z(x) := log[sign * (y(x) - asymptote)] is performed. Args: scale_factors: The array of noise scale factors. exp_values: The array of expectation values. asymptote: The infinite-noise limit y(x->inf) (optional argument). order: The degree of the polynomial z(x). avoid_log: If set to True, the exponential model is not linearized with a logarithm and a non-linear fit is applied even if asymptote is not None. The default value is False. eps: Epsilon to regularize log(sign(scale_factors - asymptote)) when the argument is to close to zero or negative. full_output: If False (default), only the zero-noise limit is returned. If True, additional information about the extrapolated limit is returned too. Returns: The extrapolated zero-noise limit. If full_output is True, also returns * standard deviation of the extrapolated zero-noise limit, * optimal parameters of the best-fit model, * parameter covariance matrix of best-fit model, * best-fit model as a Callable[[float], float] function. Raises: ValueError: If the arguments are not consistent with the extrapolation model. ExtrapolationError: If the extrapolation fit fails. ExtrapolationWarning: If the extrapolation fit is ill-conditioned. Note: This static method computes the zero-noise limit from input parameters. To compute the zero-noise limit from the Factory parameters, use the ``reduce`` method. """ # Shift is 0 if asymptote is given, 1 if asymptote is not given shift = int(asymptote is None) # Check arguments error_str = ( "Data is not enough: at least two data points are necessary." ) if scale_factors is None or exp_values is None: raise ValueError(error_str) if len(scale_factors) != len(exp_values) or len(scale_factors) < 2: raise ValueError(error_str) if order > len(scale_factors) - (1 + shift): raise ValueError( "Extrapolation order is too high. " "The order cannot exceed the number" f" of data points minus {1 + shift}." ) if not np.allclose(np.real(exp_values), exp_values): raise ValueError( f"Cannot extrapolate: Some expectation values in {exp_values} " f"have non-zero imaginary part." ) exp_values = np.real(exp_values) # Initialize default errors zne_error = None params_cov = None # Deduce "sign" parameter of the exponential ansatz linear_params, _ = mitiq_polyfit(scale_factors, exp_values, deg=1) sign = np.sign(-linear_params[0]) def _ansatz_unknown(x: float, *coeffs: float) -> float: """Ansatz of generic order with unknown asymptote.""" # Coefficients of the polynomial to be exponentiated z_coeffs = coeffs[2:][::-1] return coeffs[0] + coeffs[1] * np.exp(x * np.polyval(z_coeffs, x)) def _ansatz_known(x: float, *coeffs: float) -> float: """Ansatz of generic order with known asymptote.""" # Coefficients of the polynomial to be exponentiated z_coeffs = coeffs[1:][::-1] return asymptote + coeffs[0] * np.exp(x * np.polyval(z_coeffs, x)) # CASE 1: asymptote is None. if asymptote is None: # First guess for the parameters p_zero = [0.0, sign, -1.0] + [0.0 for _ in range(order - 1)] opt_params, params_cov = mitiq_curve_fit( _ansatz_unknown, scale_factors, exp_values, p_zero ) # The zero noise limit is ansatz(0)= asympt + b zne_limit = opt_params[0] + opt_params[1] def zne_curve(scale_factor: float) -> float: return _ansatz_unknown(scale_factor, *opt_params) # Use propagation of errors to calculate zne_error if params_cov is not None: if params_cov.shape == (order + 2, order + 2): zne_error = np.sqrt( params_cov[0, 0] + 2 * params_cov[0, 1] + params_cov[1, 1] ) if full_output: return ( zne_limit, zne_error, opt_params, params_cov, zne_curve, ) return zne_limit # CASE 2: asymptote is given and "avoid_log" is True if avoid_log: # First guess for the parameters p_zero = [sign, -1.0] + [0.0 for _ in range(order - 1)] opt_params, params_cov = mitiq_curve_fit( _ansatz_known, scale_factors, exp_values, p_zero ) # The zero noise limit is ansatz(0)= asymptote + b zne_limit = asymptote + opt_params[0] def zne_curve(scale_factor: float) -> float: return _ansatz_known(scale_factor, *opt_params) # Use propagation of errors to calculate zne_error if params_cov is not None: if params_cov.shape == (order + 1, order + 1): zne_error = np.sqrt(params_cov[0, 0]) opt_params = [asymptote] + list(opt_params) if full_output: return ( zne_limit, zne_error, opt_params, params_cov, zne_curve, ) return zne_limit # CASE 3: asymptote is given and "avoid_log" is False # Polynomial fit of z(x). shifted_y = [max(sign * (y - asymptote), eps) for y in exp_values] zstack = np.log(shifted_y) # Get coefficients {z_j} of z(x)= z_0 + z_1*x + z_2*x**2... # Note: coefficients are ordered from high powers to powers of x # Weights "w" are used to compensate for error propagation # after the log transformation y --> z z_coefficients, param_cov = mitiq_polyfit( scale_factors, zstack, deg=order, weights=np.sqrt(np.abs(shifted_y)), ) # The zero noise limit is ansatz(0) zne_limit = asymptote + sign * np.exp(z_coefficients[-1]) def _zne_curve(scale_factor: float) -> float: return asymptote + sign * np.exp( np.polyval(z_coefficients, scale_factor) ) # Use propagation of errors to calculate zne_error if params_cov is not None: if params_cov.shape == (order + 1, order + 1): zne_error = np.exp(z_coefficients[-1]) * np.sqrt( params_cov[order + 1, order + 1] ) # Parameters from low order to high order opt_params = [asymptote] + list(z_coefficients[::-1]) if full_output: return zne_limit, zne_error, opt_params, params_cov, _zne_curve return zne_limit
# Keep a log of the optimization process storing: # noise value(s), expectation value(s), parameters, and zero limit OptimizationHistory = List[ Tuple[List[Dict[str, float]], List[float], List[float], float] ]
[docs]class AdaExpFactory(AdaptiveFactory): """Factory object implementing an adaptive zero-noise extrapolation algorithm assuming an exponential ansatz y(x) = a + b * exp(-c * x), with c > 0. The noise scale factors are are chosen adaptively at each step, depending on the history of collected results. If y(x->inf) is unknown, the ansatz y(x) is fitted with a non-linear optimization. If y(x->inf) is given and avoid_log=False, the exponential model is mapped into a linear model by logarithmic transformation. Args: steps: The number of optimization steps. At least 3 are necessary. scale_factor: The second noise scale factor (the first is always 1.0). Further scale factors are adaptively determined. asymptote: The infinite-noise limit y(x->inf) (optional argument). avoid_log: If set to True, the exponential model is not linearized with a logarithm and a non-linear fit is applied even if asymptote is not None. The default value is False. max_scale_factor: Maximum noise scale factor. Default is 6.0. Raises: ValueError: If data is not consistent with the extrapolation model. ExtrapolationError: If the extrapolation fit fails. ExtrapolationWarning: If the extrapolation fit is ill-conditioned. """ _SHIFT_FACTOR = 1.27846 _EPSILON = 1.0e-9 def __init__( self, steps: int, scale_factor: float = 2.0, asymptote: Optional[float] = None, avoid_log: bool = False, max_scale_factor: float = 6.0, ) -> None: """Instantiate a new object of this Factory class.""" super(AdaExpFactory, self).__init__() if not (asymptote is None or isinstance(asymptote, float)): raise ValueError( "The argument 'asymptote' must be either a float or None" ) if scale_factor <= 1: raise ValueError( "The argument 'scale_factor' must be strictly larger than one." ) if steps < 3 + int(asymptote is None): raise ValueError( "The argument 'steps' must be an integer" " greater or equal to 3. " "If 'asymptote' is None, 'steps' must be" " greater or equal to 4." ) if max_scale_factor <= 1: raise ValueError( "The argument 'max_scale_factor' must be" " strictly larger than one." ) self._steps = steps self._scale_factor = scale_factor self.asymptote = asymptote self.avoid_log = avoid_log self.max_scale_factor = max_scale_factor self.history: OptimizationHistory = []
[docs] def next(self) -> Dict[str, float]: """Returns a dictionary of parameters to execute a circuit at.""" # The 1st scale factor is always 1 if len(self._instack) == 0: return {"scale_factor": 1.0} # The 2nd scale factor is self._scale_factor if len(self._instack) == 1: return {"scale_factor": self._scale_factor} # If asymptote is None we use 2 * scale_factor as third noise parameter if (len(self._instack) == 2) and (self.asymptote is None): return {"scale_factor": 2 * self._scale_factor} with warnings.catch_warnings(): # This is an intermediate fit, so we suppress its warning messages warnings.simplefilter("ignore", ExtrapolationWarning) # Call reduce() to fit the exponent and save it in self.history self.reduce() # The next line avoids warnings after intermediate extrapolations self._already_reduced = False # Get the most recent fitted parameters from self.history _, _, params, _ = self.history[-1] # The exponent parameter is the 3rd element of params exponent = params[2] # Further noise scale factors are determined with # an adaptive rule which depends on self.exponent next_scale_factor = min( 1.0 + self._SHIFT_FACTOR / np.abs(exponent + self._EPSILON), self.max_scale_factor, ) return {"scale_factor": next_scale_factor}
[docs] def is_converged(self) -> bool: """Returns True if all the needed expectation values have been computed, else False. """ if len(self._outstack) != len(self._instack): raise IndexError( f"The length of 'self._instack' ({len(self._instack)}) " f"and 'self._outstack' ({len(self._outstack)}) must be equal." ) return len(self._outstack) == self._steps
[docs] @staticmethod def extrapolate( # type: ignore scale_factors: Sequence[float], exp_values: Sequence[float], asymptote: Optional[float] = None, avoid_log: bool = False, eps: float = 1.0e-6, full_output: bool = False, ) -> ExtrapolationResult: """Static method which evaluates the extrapolation to the zero-noise limit assuming an exponential ansatz y(x) = a + b * exp(-c * x), with c > 0. If y(x->inf) is unknown, the ansatz y(x) is fitted with a non-linear optimization. If y(x->inf) is given and avoid_log=False, the exponential model is mapped into a linear model by a logarithmic transformation. Args: scale_factors: The array of noise scale factors. exp_values: The array of expectation values. asymptote: The infinite-noise limit y(x->inf) (optional argument). avoid_log: If set to True, the exponential model is not linearized with a logarithm and a non-linear fit is applied even if asymptote is not None. The default value is False. eps: Epsilon to regularize log(sign(scale_factors - asymptote)) when the argument is to close to zero or negative. full_output: If False (default), only the zero-noise limit is returned. If True, additional results are returned too. Returns: The extrapolated zero-noise limit. If full_output is True, also returns * standard deviation of the extrapolated zero-noise limit, * optimal parameters of the best-fit model, * parameter covariance matrix of best-fit model, * best-fit model as a Callable[[float], float] function. Raises: ValueError: If the arguments are not consistent with the extrapolation model. ExtrapolationError: If the extrapolation fit fails. ExtrapolationWarning: If the extrapolation fit is ill-conditioned. Note: This static method computes the zero-noise limit from input parameters. To compute the zero-noise limit from the Factory parameters, use the ``reduce`` method. """ return ExpFactory.extrapolate( scale_factors, exp_values, asymptote=asymptote, avoid_log=avoid_log, eps=eps, full_output=full_output, )
[docs] def reduce(self) -> float: """Returns the zero-noise limit found by fitting an exponential model to the internal data stored in the factory. Returns: The zero-noise limit. """ ( self._zne_limit, self._zne_error, self._opt_params, self._params_cov, self._zne_curve, ) = self.extrapolate( # type: ignore cast(Sequence[float], self.get_scale_factors()), cast(Sequence[float], self.get_expectation_values()), asymptote=self.asymptote, avoid_log=self.avoid_log, full_output=True, ) # Update optimization history self.history.append( (self._instack, self._outstack, self._opt_params, self._zne_limit) ) self._already_reduced = True return self._zne_limit