# Source code for qutip.core.energy_restricted

    from .dimensions import Space
    from .states import state_number_enumerate
    from . import data as _data
    from . import Qobj, qdiags
    import numpy as np
    import scipy.sparse
    from .. import settings
    import math
    import numbers
    import itertools
    import warnings

    __all__ = ['enr_state_dictionaries', 'enr_nstates',
               'enr_fock', 'enr_thermal_dm', 'enr_destroy', 'enr_identity',
               'enr_ptrace', 'enr_tensor']

    def enr_state_dictionaries(dims, excitations):
        """
        Return the number of states, and lookup-dictionaries for translating
        a state tuple to a state index, and vice versa, for a system with a given
        number of components and maximum number of excitations.

        Parameters
        ----------
        dims: list
            A list with the number of states in each sub-system.

        excitations : integer
            The maximum numbers of dimension

        Returns
        -------
        nstates, state2idx, idx2state: integer, dict, dict
            The number of states `nstates`, a dictionary for looking up state
            indices from a state tuple, and a dictionary for looking up state
            state tuples from state indices. state2idx and idx2state are reverses
            of each other, i.e., ``state2idx[idx2state[idx]] = idx`` and
            ``idx2state[state2idx[state]] = state``.
        """
        nstates = 0
        state2idx = {}
        idx2state = {}

        for state in state_number_enumerate(dims, excitations):
            state2idx[state] = nstates
            idx2state[nstates] = state
            nstates += 1

        return nstates, state2idx, idx2state

    def enr_nstates(dims, excitations):
        """
        Directly compute the number of states for a system with a given number of
        components and maximum number of excitations, using the inclusion-exclusion
        principle. Much faster than enumerating all states.

        Parameters
        ----------
        dims: list of integers
            A list with the number of states in each sub-system.

        excitations : integer
            The maximum number excitations across all sub-systems.

        Returns
        -------
        nstates: integer
            The number of states in the excitation-number restricted state space
        """
        if len(dims) == 0:
            return 1
        m = len(dims)
        kmax = min((excitations, sum(dims)))//min(dims)
        if all(d == dims[0] for d in dims):  # this situation can be solved faster
            return sum(
                (-1)**k * math.comb(m, k) * math.comb(excitations-k*dims[0]+m, m)
                for k in range(kmax+1))
        else:  # in general, need to iterate over all subsets
            return sum((-1)**k * math.comb(excitations + m - sum(subset), m)
                       for k in range(kmax+1)
                       for subset in itertools.combinations(dims, k)
                       if sum(subset) < excitations+m)

    class EnrSpace(Space):
        _stored_dims = {}

        def __init__(self, dims, excitations):
            self.dims = tuple(dims)
            self.n_excitations = excitations
            enr_dicts = enr_state_dictionaries(dims, excitations)
            self.size, self.state2idx, self.idx2state = enr_dicts
            self.issuper = False
            self.superrep = None
            self._pure_dims = False

        def __eq__(self, other):
            return (
                self is other
                or (
                    type(other) is type(self)
                    and self.dims == other.dims
                    and self.n_excitations == other.n_excitations
                )
            )

        def __hash__(self):
            return hash((self.dims, self.n_excitations))

        def __repr__(self):
            return f"EnrSpace({self.dims}, {self.n_excitations})"

        def as_list(self):
            return list(self.dims)

        def dims2idx(self, dims):
            return self.state2idx[tuple(dims)]

        def idx2dims(self, idx):
            return list(self.idx2state[idx])

        def step(self):
            raise TypeError("ENR Space does not have homogenous array steps")

        def drop_scalar_dims(self):
            return EnrSpace(
                [d for d in self.dims if d != 1],
                self.n_excitations
            )

    def enr_fock(dims, excitations, state, *, dtype=None):
        """
        Generate the Fock state representation in a excitation-number restricted
        state space. The `dims` argument is a list of integers that define the
        number of quantums states of each component of a composite quantum system,
        and the `excitations` specifies the maximum number of excitations for
        the basis states that are to be included in the state space. The `state`
        argument is a tuple of integers that specifies the state (in the number
        basis representation) for which to generate the Fock state representation.

        Parameters
        ----------
        dims : list
            A list of the dimensions of each subsystem of a composite quantum
            system.

        excitations : integer
            The maximum number of excitations that are to be included in the
            state space.

        state : list of integers
            The state in the number basis representation.

        dtype : type or str, optional
            Storage representation. Any data-layer known to `qutip.data.to` is
            accepted.

        Returns
        -------
        ket : Qobj
            A Qobj instance that represent a Fock state in the exication-number-
            restricted state space defined by `dims` and `exciations`.

        """
        dtype = _data._parse_default_dtype(dtype, "dense")
        nstates, state2idx, _ = enr_state_dictionaries(dims, excitations)
        try:
            data = _data.one_element[dtype](
                (nstates, 1),
                (state2idx[tuple(state)], 0),
                1
            )
        except KeyError:
            msg = (
                "state tuple " + str(tuple(state))
                + " is not in the restricted state space."
            )
            raise ValueError(msg) from None
        return Qobj(data, dims=[EnrSpace(dims, excitations), [1]*len(dims)],
                    copy=False, dtype=dtype)

    def enr_thermal_dm(dims, excitations, n, *, dtype=None):
        """
        Generate the density operator for a thermal state in the excitation-number-
        restricted state space defined by the `dims` and `exciations` arguments.
        See the documentation for enr_fock for a more detailed description of
        these arguments. The temperature of each mode in dims is specified by
        the average number of excitatons `n`.

        Parameters
        ----------
        dims : list
            A list of the dimensions of each subsystem of a composite quantum
            system.

        excitations : integer
            The maximum number of excitations that are to be included in the
            state space.

        n : integer
            The average number of exciations in the thermal state. `n` can be
            a float (which then applies to each mode), or a list/array of the same
            length as dims, in which each element corresponds specifies the
            temperature of the corresponding mode.

        dtype : type or str, optional
            Storage representation. Any data-layer known to `qutip.data.to` is
            accepted.

        Returns
        -------
        dm : Qobj
            Thermal state density matrix.
        """
        dtype = _data._parse_default_dtype(dtype, "sparse")
        nstates, _, idx2state = enr_state_dictionaries(dims, excitations)
        enr_dims = [EnrSpace(dims, excitations)] * 2
        if not isinstance(n, (list, np.ndarray)):
            n = np.ones(len(dims)) * n
        else:
            n = np.asarray(n)

        diags = [np.prod((n / (n + 1)) ** np.array(state))
                 for idx, state in idx2state.items()]
        diags /= np.sum(diags)
        out = qdiags(diags, 0, dims=enr_dims,
                     shape=(nstates, nstates), dtype=dtype)
        out._isherm = True
        return out

    def enr_destroy(dims, excitations, *, dtype=None):
        """
        Generate annilation operators for modes in a excitation-number-restricted
        state space. For example, consider a system consisting of 4 modes, each
        with 5 states. The total hilbert space size is 5**4 = 625. If we are
        only interested in states that contain up to 2 excitations, we only need
        to include states such as

            (0, 0, 0, 0)
            (0, 0, 0, 1)
            (0, 0, 0, 2)
            (0, 0, 1, 0)
            (0, 0, 1, 1)
            (0, 0, 2, 0)
            ...

        This function creates annihilation operators for the 4 modes that act
        within this state space:

            a1, a2, a3, a4 = enr_destroy([5, 5, 5, 5], excitations=2)

        From this point onwards, the annihiltion operators a1, ..., a4 can be
        used to setup a Hamiltonian, collapse operators and expectation-value
        operators, etc., following the usual pattern.

        Parameters
        ----------
        dims : list
            A list of the dimensions of each subsystem of a composite quantum
            system.

        excitations : integer
            The maximum number of excitations that are to be included in the
            state space.

        dtype : type or str, optional
            Storage representation. Any data-layer known to `qutip.data.to` is
            accepted.

        Returns
        -------
        a_ops : list of qobj
            A list of annihilation operators for each mode in the composite
            quantum system described by dims.
        """
        dtype = _data._parse_default_dtype(dtype, "sparse")
        nstates, state2idx, idx2state = enr_state_dictionaries(dims, excitations)
        enr_dims = [EnrSpace(dims, excitations)] * 2

        a_ops = [scipy.sparse.lil_matrix((nstates, nstates), dtype=np.complex128)
                 for _ in dims]

        for n1, state1 in idx2state.items():
            for idx, s in enumerate(state1):
                # if s > 0, the annihilation operator of mode idx has a non-zero
                # entry with one less excitation in mode idx in the final state
                if s > 0:
                    state2 = state1[:idx] + (s-1,) + state1[idx+1:]
                    n2 = state2idx[state2]
                    a_ops[idx][n2, n1] = np.sqrt(s)

        return [
            Qobj(a, dims=enr_dims, isunitary=False, isherm=False, dtype=dtype)
            for a in a_ops
        ]

    def enr_identity(dims, excitations, *, dtype=None):
        """
        Generate the identity operator for the excitation-number restricted
        state space defined by the `dims` and `exciations` arguments. See the
        docstring for enr_fock for a more detailed description of these arguments.

        Parameters
        ----------
        dims : list
            A list of the dimensions of each subsystem of a composite quantum
            system.

        excitations : integer
            The maximum number of excitations that are to be included in the
            state space.

        dtype : type or str, optional
            Storage representation. Any data-layer known to `qutip.data.to` is
            accepted.

        Returns
        -------
        op : Qobj
            A Qobj instance that represent the identity operator in the
            exication-number-restricted state space defined by `dims` and
            `exciations`.
        """
        dtype = _data._parse_default_dtype(dtype, "diagonal")
        dims = EnrSpace(dims, excitations)
        return Qobj(_data.identity[dtype](dims.size),
                    dims=[dims, dims],
                    isherm=True,
                    isunitary=True,
                    copy=False,
                    dtype=dtype)

    def enr_ptrace(rho, sel):
        """
        Trace out the modes not in `sel`, with input and output in the excitation-
        number restricted state space. Only for density matrices with CSR dtype.
        Be aware that the partial trace behaves weirdly in ENR space, since the
        size of the subsystems' spaces depend on other subsystems' states. e.g.,
        it does not invert the tensor multiplication with the identity operator.

        Parameters
        ----------
        rho : Qobj
            The input density matrix.

        sel : list of integers
            The indices of the modes to keep.

        Returns
        -------
        out : Qobj
            A Qobj instance that represents the partially-traced `rho` in the
            excitation-number restricted state. The maximum number of
            excitations is the same as for the input.
        """
        if rho.shape[0] != rho.shape[1]:
            raise ValueError(
                "enr_ptrace is only defined for square density matrices")
        if not isinstance(rho._dims[0], EnrSpace):
            raise TypeError("Not an ENR operator")
        try:
            sel = sorted(sel)
        except TypeError:
            if not isinstance(sel, numbers.Integral):
                raise TypeError(
                    "selection must be an integer or list of integers"
                ) from None
            sel = [sel]

        dims = rho.dims[0]
        if (len(sel) == len(dims)):
            # no need to trace out anything
            return rho
        if len(sel) == 0:
            # trace out all modes
            return Qobj(rho.tr(), dtype="csr")

        dtype = rho.dtype
        if dtype is _data.Dense:
            warnings.warn("enr_ptrace may be slow for dense matrices.")

        excitations = rho._dims[0].n_excitations
        mat = rho.to(_data.CSR).data.as_scipy().tocoo()

        # get the new dimensions
        dims_new = [dims[i] for i in sel]
        toremove = set(range(len(dims))) - set(sel)
        # initialize the new matrix
        space_new = EnrSpace(dims_new, excitations)
        out = scipy.sparse.dok_matrix((space_new.size, space_new.size),
                                      dtype="complex128")

        # loop over nonzero elements of the input, (it's sparse)
        for row_idx_old, col_idx_old, val in zip(mat.row, mat.col, mat.data):
            # find the states corresponding to the old indices
            row_state_old = rho._dims[0].idx2dims(row_idx_old)
            col_state_old = rho._dims[0].idx2dims(col_idx_old)

            # only keep if on diagonal or if diagonal wrt systems to be removed
            if (row_idx_old == col_idx_old or all(
                    row_state_old[ii] == col_state_old[ii] for ii in toremove)):
                # find the new indices for the kept subsystems
                row_state_new = tuple(row_state_old[ii] for ii in sel)
                col_state_new = tuple(col_state_old[ii] for ii in sel)

                row_idx_new = space_new.dims2idx(row_state_new)
                col_idx_new = space_new.dims2idx(col_state_new)
                out[row_idx_new, col_idx_new] += val

        # convert to Qobj
        return Qobj(out, dims=[space_new]*2, dtype=dtype)

    def enr_tensor(*args: Qobj, newexcitations: int = None,
                   truncate=False, verbose=True):
        """
        Calculates the tensor product of input operators in the excitation-
        number restricted state space.
        Be aware that tensor multiplication behaves weirdly in ENR space, since the
        size of the subsystems' spaces depend on other subsystems' states.
        Taking a tensor product with the identity is not inverted by partial trace!

        Parameters
        ----------
        args : Qobj's
            The input operators to be tensor-multiplied.

        newexcitations : integer, optional
            The maximum number of excitations for the output operator. If not
            given, the minimum of the input excitations is used.

        truncate : bool, optional
            If True, the function will ignore entries that are not in the new
            restricted state space. If False, such entries will raise an error.

        verbose : bool, optional
            If True, the function will print a message if any entries are
            truncated.

        Returns
        -------
        obj : qobj
            A composite quantum object.
        """
        if not args:
            raise TypeError("Requires at least one input argument")
        if len(args) == 1 and isinstance(args[0], Qobj):
            return args[0].copy()
        if len(args) == 1:
            try:
                args = tuple(args[0])
            except TypeError:
                raise TypeError("requires Qobj operands") from None
        if not all(isinstance(q, Qobj) for q in args):
            raise TypeError("requires Qobj operands")

        excitations = [(getattr(q._dims[0], 'n_excitations', None)
                        or q._dims[1].n_excitations) for q in args]

        # get the new max excitations
        newexcitations = newexcitations or min(excitations)
        newdims = [tuple(itertools.chain(*(q.dims[i] for q in args)))
                   for i in range(2)]

        trunccount = 0
        out = {((), ()): 1.0}  # initialize a dict to tensor the first Qobj with

        # loop over the the operators
        for q in args:
            out, trunccount = _enr_tensor_qobj_with_dict(
                out, q.to(_data.CSR), newexcitations, truncate, trunccount)

        if trunccount > 0 and verbose:
            print(f"Truncated {trunccount} entries.")

        dtype = args[0].dtype if all(q.dtype == args[0].dtype for q in args) \
            else None
        dtype = _data._parse_default_dtype(dtype, "sparse")
        return _enr_qobj_from_dict(out, newdims, newexcitations,
                                   isherm=all(q.isherm for q in args), dtype=dtype)

    def _enr_tensor_qobj_with_dict(d: dict, q: Qobj, newexcitations: int,
                                   truncate, trunccount=0):
        """
        Helper function to do the tensor product between a dictionary (which
        represents a Qobj) and a Qobj.
        """
        out = {}
        mat = q.data.as_scipy().tocoo()

        # loop over nonzero elements of the qobj
        for row_idx, col_idx, val in zip(mat.row, mat.col, mat.data):
            # convert the indices to the state tuples
            row_state = tuple(q._dims[0].idx2dims(row_idx))
            col_state = tuple(q._dims[1].idx2dims(col_idx))

            # loop over the nonzero elements of the dictionary
            for (row_state_old, col_state_old), oldval in d.items():
                # concatenate the states
                row_state_new = row_state_old + row_state
                col_state_new = col_state_old + col_state

                if (sum(row_state_new) <= newexcitations
                        and sum(col_state_new) <= newexcitations):
                    out[row_state_new, col_state_new] = oldval * val
                elif truncate:
                    trunccount += 1
                else:
                    missingstates = (str(state) for state
                                     in [row_state_new, col_state_new]
                                     if sum(state) > newexcitations)
                    msg = (
                        "state " + ", ".join(missingstates) +
                        " is not in the new restricted state space. " +
                        "Set `truncate=True` to truncate these entries."
                    )
                    raise ValueError(msg) from None

        return (out, trunccount)

    def _enr_qobj_from_dict(op_dict, dims, excitations, **kwargs):
        rowspace, row_idxlist = _get_space_and_indices(
            op_dict, dims, 0, excitations)
        colspace, col_idxlist = _get_space_and_indices(
            op_dict, dims, 1, excitations)

        op_coo = scipy.sparse.coo_matrix(
            (list(op_dict.values()), (row_idxlist, col_idxlist)),
            shape=(rowspace.size, colspace.size), dtype="complex128")

        return Qobj(op_coo, dims=[rowspace, colspace], **kwargs)

    def _get_space_and_indices(op_dict, dims, dim_idx, excitations):
        if np.prod(dims[dim_idx]) == 1:  # 1-dimensional (e.g. ket)
            space = Space(list(dims[dim_idx]))
            indices = [0]*len(op_dict)
        else:
            space = EnrSpace(dims[dim_idx], excitations)
            indices = [space.dims2idx(coord_state[dim_idx])
                       for coord_state in op_dict.keys()]
        return space, indices