Project: s41467-024-46879-4 | Function: sample (python)

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**DESCRIPTION:** Implements simulated annealing sampling for Equilibrium Propagation phases on Ising machines, supporting both free phase annealing and reverse annealing for nudge phase with custom beta schedules

**KEY TERMS:** {
  "beta_schedule": "annealing temperature schedule for exploration control",
  "num_reads": "number of samples per annealing run (10 in experiments)",
  "seed": "random seed for reproducible sampling",
  "h": "bias fields encoding input data and nudging signals",
  "J": "coupling matrix representing synaptic weights",
  "initial_states": "starting configuration for reverse annealing"
}

**CONTEXT CODE:** 
    Examples:
        This example looks at a sampler's parameters and some of their values.

        >>> import neal
        >>> sampler = neal.SimulatedAnnealingSampler()
        >>> for kwarg in sorted(sampler.parameters):
        ...     print(kwarg)
        beta_range
        beta_schedule_type
        initial_states
        initial_states_generator
        interrupt_function
        num_reads
        num_sweeps
        seed
        >>> sampler.parameters['beta_range']
        []
        >>> sampler.parameters['beta_schedule_type']
        ['beta_schedule_options']

    """

    properties = None
    """dict: A dict containing any additional information about the sampler.

    Examples:
        This example looks at the values set for a sampler property.

        >>> import neal
        >>> sampler = neal.SimulatedAnnealingSampler()
        >>> sampler.properties['beta_schedule_options']
        ('linear', 'geometric')

    """

    def __init__(self):
        # create a local copy in case folks for some reason want to modify them
        self.parameters = {'beta_range': [],
                           'num_reads': [],
                           'num_sweeps': [],
                           'beta_schedule_type': ['beta_schedule_options'],
                           'seed': [],
                           'interrupt_function': [],
                           'initial_states': [],
                           'initial_states_generator': [],
                           }
        self.properties = {'beta_schedule_options': ('linear', 'geometric')
                           }

    def sample(self, bqm, beta_range=None, num_reads=None, num_sweeps=1000,
               beta_schedule_type="geometric", seed=None, interrupt_function=None,
               initial_states=None, initial_states_generator="random", reverse = False, fraction_annealed = 0, **kwargs):
        """Sample from a binary quadratic model using an implemented sample method.

        Args:
            bqm (:class:`dimod.BinaryQuadraticModel`):
                The binary quadratic model to be sampled.

            beta_range (tuple, optional):
                A 2-tuple defining the beginning and end of the beta schedule, where beta is the
                inverse temperature. The schedule is applied linearly in beta. Default range is set
                based on the total bias associated with each node.

            num_reads (int, optional, default=len(initial_states) or 1):
                Number of reads. Each read is generated by one run of the simulated
                annealing algorithm. If `num_reads` is not explicitly given, it is
                selected to match the number of initial states given. If initial states
                are not provided, only one read is performed.

            num_sweeps (int, optional, default=1000):
                Number of sweeps or steps.

            beta_schedule_type (string, optional, default='geometric'):
                Beta schedule type, or how the beta values are interpolated between
                the given 'beta_range'. Supported values are:

                * linear
                * geometric

            seed (int, optional):
                Seed to use for the PRNG. Specifying a particular seed with a constant
                set of parameters produces identical results. If not provided, a random seed
                is chosen.

            initial_states (samples-like, optional, default=None):
                One or more samples, each defining an initial state for all the
                problem variables. Initial states are given one per read, but
                if fewer than `num_reads` initial states are defined,
                additional values are generated as specified by
                `initial_states_generator`. See func:`.as_samples` for a
                description of "samples-like".

            initial_states_generator (str, 'none'/'tile'/'random', optional, default='random'):
                Defines the expansion of `initial_states` if fewer than
                `num_reads` are specified:

                * "none":
                    If the number of initial states specified is smaller than
                    `num_reads`, raises ValueError.

                * "tile":
                    Reuses the specified initial states if fewer than `num_reads`
                    or truncates if greater.

                * "random":
                    Expands the specified initial states with randomly generated
                    states if fewer than `num_reads` or truncates if greater.

            interrupt_function (function, optional):
                If provided, interrupt_function is called with no parameters
                between each sample of simulated annealing. If the function
                returns True, then simulated annealing will terminate and return
                with all of the samples and energies found so far.

        Returns:
            :obj:`dimod.Response`: A `dimod` :obj:`~dimod.Response` object.

        Examples:
            This example runs simulated annealing on a binary quadratic model with some
            different input parameters.

            >>> import dimod
            >>> import neal
            ...
            >>> sampler = neal.SimulatedAnnealingSampler()
            >>> bqm = dimod.BinaryQuadraticModel({'a': .5, 'b': -.5}, {('a', 'b'): -1}, 0.0, dimod.SPIN)
            >>> # Run with default parameters
            >>> sampleset = sampler.sample(bqm)
            >>> # Run with specified parameters
            >>> sampleset = sampler.sample(bqm, seed=1234, beta_range=[0.1, 4.2],
            ...                                num_reads=1, num_sweeps=20,
            ...                                beta_schedule_type='geometric')
            >>> # Reuse a seed
            >>> a1 = next((sampler.sample(bqm, seed=88)).samples())['a']
            >>> a2 = next((sampler.sample(bqm, seed=88)).samples())['a']
            >>> a1 == a2
            True

        """
        # MASKED: Implements simulated annealing sampling for Equilibrium Propagation phases on Ising machines, supporting both free phase annealing and reverse annealing for nudge phase with custom beta schedules implementation
        raise NotImplementedError("This function is masked for testing purposes.")


Neal = SimulatedAnnealingSampler


def _default_ising_beta_range(h, J):
    """Determine the starting and ending beta from h J



Complete the function implementation replacing "raise NotImplementedError".