Source code for qibo.hamiltonians.hamiltonians

"""Module defining Hamiltonian classes."""

from itertools import chain
from typing import Optional

import numpy as np
import sympy

from qibo.backends import PyTorchBackend, _check_backend
from qibo.config import EINSUM_CHARS, log, raise_error
from qibo.hamiltonians.abstract import AbstractHamiltonian
from qibo.symbols import Z


[docs]class Hamiltonian(AbstractHamiltonian): """Hamiltonian based on a dense or sparse matrix representation. Args: nqubits (int): number of quantum bits. matrix (ndarray): Matrix representation of the Hamiltonian in the computational basis as an array of shape :math:`2^{n} \\times 2^{n}`. Sparse matrices based on ``scipy.sparse`` for ``numpy`` / ``qibojit`` backends (or on ``tf.sparse`` for the ``tensorflow`` backend) are also supported. backend (:class:`qibo.backends.abstract.Backend`, optional): backend to be used in the execution. If ``None``, it uses :class:`qibo.backends.GlobalBackend`. Defaults to ``None``. """ def __init__(self, nqubits, matrix, backend=None): from qibo.backends import _check_backend self.backend = _check_backend(backend) if not ( isinstance(matrix, self.backend.tensor_types) or self.backend.is_sparse(matrix) ): raise_error( TypeError, f"Matrix of invalid type {type(matrix)} given during Hamiltonian initialization", ) matrix = self.backend.cast(matrix) super().__init__() self.nqubits = nqubits self.matrix = matrix self._eigenvalues = None self._eigenvectors = None self._exp = {"a": None, "result": None} @property def matrix(self): """Returns the full matrix representation. For :math:`n` qubits, can be a dense :math:`2^{n} \\times 2^{n}` array or a sparse matrix, depending on how the Hamiltonian was created. """ return self._matrix @matrix.setter def matrix(self, mat): shape = tuple(mat.shape) if shape != 2 * (2**self.nqubits,): raise_error( ValueError, f"The Hamiltonian is defined for {self.nqubits} qubits " + f"while the given matrix has shape {shape}.", ) self._matrix = mat
[docs] @classmethod def from_symbolic(cls, symbolic_hamiltonian, symbol_map, backend=None): """Creates a :class:`qibo.hamiltonian.Hamiltonian` from a symbolic Hamiltonian. We refer to :ref:`How to define custom Hamiltonians using symbols? <symbolicham-example>` for more details. Args: symbolic_hamiltonian (sympy.Expr): full Hamiltonian written with ``sympy`` symbols. symbol_map (dict): Dictionary that maps each symbol that appears in the Hamiltonian to a pair ``(target, matrix)``. backend (:class:`qibo.backends.abstract.Backend`, optional): backend to be used in the execution. If ``None``, it uses :class:`qibo.backends.GlobalBackend`. Defaults to ``None``. Returns: :class:`qibo.hamiltonians.SymbolicHamiltonian`: object that implements the Hamiltonian represented by the given symbolic expression. """ log.warning( "`Hamiltonian.from_symbolic` and the use of symbol maps is " "deprecated. Please use `SymbolicHamiltonian` and Qibo symbols " "to construct Hamiltonians using symbols." ) return SymbolicHamiltonian( symbolic_hamiltonian, symbol_map=symbol_map, backend=backend )
[docs] def eigenvalues(self, k=6): if self._eigenvalues is None: self._eigenvalues = self.backend.calculate_eigenvalues(self.matrix, k) return self._eigenvalues
[docs] def eigenvectors(self, k=6): if self._eigenvectors is None: self._eigenvalues, self._eigenvectors = self.backend.calculate_eigenvectors( self.matrix, k ) return self._eigenvectors
[docs] def exp(self, a): from qibo.quantum_info.linalg_operations import ( # pylint: disable=C0415 matrix_exponentiation, ) if self._exp.get("a") != a: self._exp["a"] = a self._exp["result"] = matrix_exponentiation( a, self.matrix, self._eigenvectors, self._eigenvalues, self.backend ) return self._exp.get("result")
[docs] def expectation(self, state, normalize=False): if isinstance(state, self.backend.tensor_types): state = self.backend.cast(state) shape = tuple(state.shape) if len(shape) == 1: # state vector return self.backend.calculate_expectation_state(self, state, normalize) if len(shape) == 2: # density matrix return self.backend.calculate_expectation_density_matrix( self, state, normalize ) raise_error( ValueError, "Cannot calculate Hamiltonian expectation value " + f"for state of shape {shape}", ) raise_error( TypeError, "Cannot calculate Hamiltonian expectation " + f"value for state of type {type(state)}", )
[docs] def expectation_from_samples(self, freq, qubit_map=None): obs = self.matrix if ( self.backend.np.count_nonzero( obs - self.backend.np.diag(self.backend.np.diagonal(obs)) ) != 0 ): raise_error(NotImplementedError, "Observable is not diagonal.") keys = list(freq.keys()) if qubit_map is None: qubit_map = list(range(int(np.log2(len(obs))))) counts = np.array(list(freq.values())) / sum(freq.values()) expval = 0 size = len(qubit_map) for j, k in enumerate(keys): index = 0 for i in qubit_map: index += int(k[qubit_map.index(i)]) * 2 ** (size - 1 - i) expval += obs[index, index] * counts[j] return self.backend.np.real(expval)
[docs] def eye(self, dim: Optional[int] = None): """Generate Identity matrix with dimension ``dim``""" if dim is None: dim = int(self.matrix.shape[0]) return self.backend.cast(self.backend.matrices.I(dim), dtype=self.matrix.dtype)
[docs] def energy_fluctuation(self, state): """ Evaluate energy fluctuation: .. math:: \\Xi_{k}(\\mu) = \\sqrt{\\bra{\\mu} \\, H^{2} \\, \\ket{\\mu} - \\bra{\\mu} \\, H \\, \\ket{\\mu}^2} \\, . for a given state :math:`\\ket{\\mu}`. Args: state (ndarray): quantum state to be used to compute the energy fluctuation. Returns: float: Energy fluctuation value. """ state = self.backend.cast(state) energy = self.expectation(state) h = self.matrix h2 = Hamiltonian(nqubits=self.nqubits, matrix=h @ h, backend=self.backend) average_h2 = self.backend.calculate_expectation_state(h2, state, normalize=True) return self.backend.np.sqrt(self.backend.np.abs(average_h2 - energy**2))
def __add__(self, o): if isinstance(o, self.__class__): if self.nqubits != o.nqubits: raise_error( RuntimeError, "Only hamiltonians with the same number of qubits can be added.", ) new_matrix = self.matrix + o.matrix elif isinstance(o, self.backend.numeric_types): new_matrix = self.matrix + o * self.eye() else: raise_error( NotImplementedError, f"Hamiltonian addition to {type(o)} not implemented.", ) return self.__class__(self.nqubits, new_matrix, backend=self.backend) def __sub__(self, o): if isinstance(o, self.__class__): if self.nqubits != o.nqubits: raise_error( RuntimeError, "Only hamiltonians with the same number of qubits can be subtracted.", ) new_matrix = self.matrix - o.matrix elif isinstance(o, self.backend.numeric_types): new_matrix = self.matrix - o * self.eye() else: raise_error( NotImplementedError, f"Hamiltonian subtraction to {type(o)} not implemented.", ) return self.__class__(self.nqubits, new_matrix, backend=self.backend) def __rsub__(self, o): if isinstance(o, self.__class__): # pragma: no cover # impractical case because it will be handled by `__sub__` if self.nqubits != o.nqubits: raise_error( RuntimeError, "Only hamiltonians with the same number of qubits can be added.", ) new_matrix = o.matrix - self.matrix elif isinstance(o, self.backend.numeric_types): new_matrix = o * self.eye() - self.matrix else: raise_error( NotImplementedError, f"Hamiltonian subtraction to {type(o)} not implemented.", ) return self.__class__(self.nqubits, new_matrix, backend=self.backend) def __mul__(self, o): if isinstance(o, self.backend.tensor_types): o = complex(o) elif not isinstance(o, self.backend.numeric_types): raise_error( NotImplementedError, f"Hamiltonian multiplication to {type(o)} not implemented.", ) new_matrix = self.matrix * o r = self.__class__(self.nqubits, new_matrix, backend=self.backend) o = self.backend.cast(o) if self._eigenvalues is not None: if self.backend.np.real(o) >= 0: # TODO: check for side effects K.qnp r._eigenvalues = o * self._eigenvalues elif not self.backend.is_sparse(self.matrix): axis = (0,) if isinstance(self.backend, PyTorchBackend) else 0 r._eigenvalues = o * self.backend.np.flip(self._eigenvalues, axis) if self._eigenvectors is not None: if self.backend.np.real(o) > 0: # TODO: see above r._eigenvectors = self._eigenvectors elif o == 0: r._eigenvectors = self.eye(int(self._eigenvectors.shape[0])) return r def __matmul__(self, o): if isinstance(o, self.__class__): matrix = self.backend.calculate_hamiltonian_matrix_product( self.matrix, o.matrix ) return self.__class__(self.nqubits, matrix, backend=self.backend) if isinstance(o, self.backend.tensor_types): return self.backend.calculate_hamiltonian_state_product(self.matrix, o) raise_error( NotImplementedError, f"Hamiltonian matmul to {type(o)} not implemented.", )
[docs]class SymbolicHamiltonian(AbstractHamiltonian): """Hamiltonian based on a symbolic representation. Calculations using symbolic Hamiltonians are either done directly using the given ``sympy`` expression as it is (``form``) or by parsing the corresponding ``terms`` (which are :class:`qibo.core.terms.SymbolicTerm` objects). The latter approach is more computationally costly as it uses a ``sympy.expand`` call on the given form before parsing the terms. For this reason the ``terms`` are calculated only when needed, for example during Trotterization. The dense matrix of the symbolic Hamiltonian can be calculated directly from ``form`` without requiring ``terms`` calculation (see :meth:`qibo.core.hamiltonians.SymbolicHamiltonian.calculate_dense` for details). Args: form (sympy.Expr): Hamiltonian form as a ``sympy.Expr``. Ideally the Hamiltonian should be written using Qibo symbols. See :ref:`How to define custom Hamiltonians using symbols? <symbolicham-example>` example for more details. symbol_map (dict): Dictionary that maps each ``sympy.Symbol`` to a tuple of (target qubit, matrix representation). This feature is kept for compatibility with older versions where Qibo symbols were not available and may be deprecated in the future. It is not required if the Hamiltonian is constructed using Qibo symbols. The symbol_map can also be used to pass non-quantum operator arguments to the symbolic Hamiltonian, such as the parameters in the :meth:`qibo.hamiltonians.models.MaxCut` Hamiltonian. backend (:class:`qibo.backends.abstract.Backend`, optional): backend to be used in the execution. If ``None``, it uses :class:`qibo.backends.GlobalBackend`. Defaults to ``None``. """ def __init__(self, form=None, nqubits=None, symbol_map={}, backend=None): super().__init__() self._form = None self._terms = None self.constant = 0 # used only when we perform calculations using ``_terms`` self._dense = None self.symbol_map = symbol_map # if a symbol in the given form is not a Qibo symbol it must be # included in the ``symbol_map`` from qibo.symbols import Symbol # pylint: disable=import-outside-toplevel self._qiboSymbol = Symbol # also used in ``self._get_symbol_matrix`` from qibo.backends import _check_backend self.backend = _check_backend(backend) if form is not None: self.form = form if nqubits is not None: self.nqubits = nqubits @property def dense(self) -> "MatrixHamiltonian": """Creates the equivalent Hamiltonian matrix.""" if self._dense is None: log.warning( "Calculating the dense form of a symbolic Hamiltonian. " "This operation is memory inefficient." ) self.dense = self.calculate_dense() return self._dense @dense.setter def dense(self, hamiltonian): assert isinstance(hamiltonian, Hamiltonian) self._dense = hamiltonian self._eigenvalues = hamiltonian._eigenvalues self._eigenvectors = hamiltonian._eigenvectors self._exp = hamiltonian._exp @property def form(self): return self._form @form.setter def form(self, form): # Check that given form is a ``sympy`` expression if not isinstance(form, sympy.Expr): raise_error( TypeError, f"Symbolic Hamiltonian should be a ``sympy`` expression but is {type(form)}.", ) # Calculate number of qubits in the system described by the given # Hamiltonian formula nqubits = 0 for symbol in form.free_symbols: if isinstance(symbol, self._qiboSymbol): q = symbol.target_qubit elif isinstance(symbol, sympy.Expr): if symbol not in self.symbol_map: raise_error(ValueError, f"Symbol {symbol} is not in symbol map.") q, matrix = self.symbol_map.get(symbol) if not isinstance(matrix, self.backend.tensor_types): # ignore symbols that do not correspond to quantum operators # for example parameters in the MaxCut Hamiltonian q = 0 if q > nqubits: nqubits = q self._form = form self.nqubits = nqubits + 1 @property def terms(self): """List of terms of which the Hamiltonian is a sum of. Terms will be objects of type :class:`qibo.core.terms.HamiltonianTerm`. """ if self._terms is None: # Calculate terms based on ``self.form`` from qibo.hamiltonians.terms import ( # pylint: disable=import-outside-toplevel SymbolicTerm, ) form = sympy.expand(self.form) terms = [] for f, c in form.as_coefficients_dict().items(): term = SymbolicTerm(c, f, self.symbol_map) if term.target_qubits: terms.append(term) else: self.constant += term.coefficient self._terms = terms return self._terms @terms.setter def terms(self, terms): self._terms = terms self.nqubits = max(q for term in self._terms for q in term.target_qubits) + 1 @property def matrix(self): """Returns the full matrix representation. Consisting of :math:`2^{n} \\times 2^{n}`` elements. """ return self.dense.matrix
[docs] def eigenvalues(self, k=6): return self.dense.eigenvalues(k)
[docs] def eigenvectors(self, k=6): return self.dense.eigenvectors(k)
[docs] def ground_state(self): return self.eigenvectors()[:, 0]
[docs] def exp(self, a): return self.dense.exp(a)
def _get_symbol_matrix(self, term): """Calculates numerical matrix corresponding to symbolic expression. This is partly equivalent to sympy's ``.subs``, which does not work in our case as it does not allow us to substitute ``sympy.Symbol`` with numpy arrays and there are different complication when switching to ``sympy.MatrixSymbol``. Here we calculate the full numerical matrix given the symbolic expression using recursion. Helper method for ``_calculate_dense_from_form``. Args: term (sympy.Expr): Symbolic expression containing local operators. Returns: ndarray: matrix corresponding to the given expression as an array of shape ``(2 ** self.nqubits, 2 ** self.nqubits)``. """ if isinstance(term, sympy.Add): # symbolic op for addition result = sum( self._get_symbol_matrix(subterm) for subterm in term.as_ordered_terms() ) elif isinstance(term, sympy.Mul): # symbolic op for multiplication # note that we need to use matrix multiplication even though # we use scalar symbols for convenience factors = term.as_ordered_factors() result = self._get_symbol_matrix(factors[0]) for subterm in factors[1:]: result = result @ self._get_symbol_matrix(subterm) elif isinstance(term, sympy.Pow): # symbolic op for power base, exponent = term.as_base_exp() matrix = self._get_symbol_matrix(base) # multiply ``base`` matrix ``exponent`` times to itself result = matrix for _ in range(exponent - 1): result = result @ matrix elif isinstance(term, sympy.Symbol): # if the term is a ``Symbol`` then it corresponds to a quantum # operator for which we can construct the full matrix directly if isinstance(term, self._qiboSymbol): # if we have a Qibo symbol the matrix construction is # implemented in :meth:`qibo.core.terms.SymbolicTerm.full_matrix`. result = term.full_matrix(self.nqubits) else: q, matrix = self.symbol_map.get(term) if not isinstance(matrix, self.backend.tensor_types): # symbols that do not correspond to quantum operators # for example parameters in the MaxCut Hamiltonian result = complex(matrix) * np.eye(2**self.nqubits) else: # if we do not have a Qibo symbol we construct one and use # :meth:`qibo.core.terms.SymbolicTerm.full_matrix`. result = self._qiboSymbol(q, matrix).full_matrix(self.nqubits) elif term.is_number: # if the term is number we should return in the form of identity # matrix because in expressions like `1 + Z`, `1` is not correspond # to the float 1 but the identity operator (matrix) result = complex(term) * np.eye(2**self.nqubits) else: raise_error( TypeError, f"Cannot calculate matrix for symbolic term of type {type(term)}.", ) return result def _calculate_dense_from_form(self) -> Hamiltonian: """Calculates equivalent Hamiltonian using symbolic form. Useful when the term representation is not available. """ matrix = self._get_symbol_matrix(self.form) return Hamiltonian(self.nqubits, matrix, backend=self.backend) def _calculate_dense_from_terms(self) -> Hamiltonian: """Calculates equivalent Hamiltonian using the term representation.""" if 2 * self.nqubits > len(EINSUM_CHARS): # pragma: no cover # case not tested because it only happens in large examples raise_error(NotImplementedError, "Not enough einsum characters.") matrix = 0 chars = EINSUM_CHARS[: 2 * self.nqubits] for term in self.terms: ntargets = len(term.target_qubits) tmat = np.reshape(term.matrix, 2 * ntargets * (2,)) n = self.nqubits - ntargets emat = np.reshape(np.eye(2**n, dtype=tmat.dtype), 2 * n * (2,)) gen = lambda x: (chars[i + x] for i in term.target_qubits) tc = "".join(chain(gen(0), gen(self.nqubits))) ec = "".join(c for c in chars if c not in tc) matrix += np.einsum(f"{tc},{ec}->{chars}", tmat, emat) matrix = np.reshape(matrix, 2 * (2**self.nqubits,)) return Hamiltonian(self.nqubits, matrix, backend=self.backend) + self.constant def calculate_dense(self): if self._terms is None: # calculate dense matrix directly using the form to avoid the # costly ``sympy.expand`` call return self._calculate_dense_from_form() return self._calculate_dense_from_terms()
[docs] def expectation(self, state, normalize=False): return Hamiltonian.expectation(self, state, normalize)
[docs] def expectation_from_samples(self, freq, qubit_map=None): terms = self.terms for term in terms: # pylint: disable=E1101 for factor in term.factors: if not isinstance(factor, Z): raise_error( NotImplementedError, "Observable is not a Z Pauli string." ) if len(term.factors) != len(set(term.factors)): raise_error(NotImplementedError, "Z^k is not implemented since Z^2=I.") keys = list(freq.keys()) counts = self.backend.cast(list(freq.values()), self.backend.precision) / sum( freq.values() ) qubits = [] for term in terms: qubits_term = [] for k in term.target_qubits: qubits_term.append(k) qubits.append(qubits_term) if qubit_map is None: qubit_map = list(range(len(keys[0]))) expval = 0 for j, q in enumerate(qubits): subk = [] expval_q = 0 for i, k in enumerate(keys): subk = [int(k[qubit_map.index(s)]) for s in q] expval_k = 1 if subk.count(1) % 2 == 1: expval_k = -1 expval_q += expval_k * counts[i] expval += expval_q * self.terms[j].coefficient.real return expval + self.constant.real
def __add__(self, o): if isinstance(o, self.__class__): if self.nqubits != o.nqubits: raise_error( RuntimeError, "Only hamiltonians with the same number of qubits can be added.", ) new_ham = self.__class__( symbol_map=dict(self.symbol_map), backend=self.backend ) if self._form is not None and o._form is not None: new_ham.form = self.form + o.form new_ham.symbol_map.update(o.symbol_map) if self._terms is not None and o._terms is not None: new_ham.terms = self.terms + o.terms new_ham.constant = self.constant + o.constant if self._dense is not None and o._dense is not None: new_ham.dense = self.dense + o.dense elif isinstance(o, self.backend.numeric_types): new_ham = self.__class__( symbol_map=dict(self.symbol_map), backend=self.backend ) if self._form is not None: new_ham.form = self.form + o if self._terms is not None: new_ham.terms = self.terms new_ham.constant = self.constant + o if self._dense is not None: new_ham.dense = self.dense + o else: raise_error( NotImplementedError, f"SymbolicHamiltonian addition to {type(o)} not implemented.", ) return new_ham def __sub__(self, o): if isinstance(o, self.__class__): if self.nqubits != o.nqubits: raise_error( RuntimeError, "Only hamiltonians with the same number of qubits can be subtracted.", ) new_ham = self.__class__( symbol_map=dict(self.symbol_map), backend=self.backend ) if self._form is not None and o._form is not None: new_ham.form = self.form - o.form new_ham.symbol_map.update(o.symbol_map) if self._terms is not None and o._terms is not None: new_ham.terms = self.terms + [-1 * x for x in o.terms] new_ham.constant = self.constant - o.constant if self._dense is not None and o._dense is not None: new_ham.dense = self.dense - o.dense elif isinstance(o, self.backend.numeric_types): new_ham = self.__class__( symbol_map=dict(self.symbol_map), backend=self.backend ) if self._form is not None: new_ham.form = self.form - o if self._terms is not None: new_ham.terms = self.terms new_ham.constant = self.constant - o if self._dense is not None: new_ham.dense = self.dense - o else: raise_error( NotImplementedError, f"Hamiltonian subtraction to {type(o)} " "not implemented.", ) return new_ham def __rsub__(self, o): if isinstance(o, self.backend.numeric_types): new_ham = self.__class__( symbol_map=dict(self.symbol_map), backend=self.backend ) if self._form is not None: new_ham.form = o - self.form if self._terms is not None: new_ham.terms = [-1 * x for x in self.terms] new_ham.constant = o - self.constant if self._dense is not None: new_ham.dense = o - self.dense else: raise_error( NotImplementedError, f"Hamiltonian subtraction to {type(o)} not implemented.", ) return new_ham def __mul__(self, o): if not isinstance(o, (self.backend.numeric_types, self.backend.tensor_types)): raise_error( NotImplementedError, f"Hamiltonian multiplication to {type(o)} not implemented.", ) o = complex(o) new_ham = self.__class__(symbol_map=dict(self.symbol_map), backend=self.backend) if self._form is not None: new_ham.form = o * self.form if self._terms is not None: new_ham.terms = [o * x for x in self.terms] new_ham.constant = self.constant * o if self._dense is not None: new_ham.dense = o * self._dense return new_ham
[docs] def apply_gates(self, state, density_matrix=False): """Applies gates corresponding to the Hamiltonian terms. Gates are applied to the given state. Helper method for :meth:`qibo.hamiltonians.SymbolicHamiltonian.__matmul__`. """ total = 0 for term in self.terms: total += term( self.backend, self.backend.cast(state, copy=True), self.nqubits, density_matrix=density_matrix, ) if self.constant: # pragma: no cover total += self.constant * state return total
def __matmul__(self, o): """Matrix multiplication with other Hamiltonians or state vectors.""" if isinstance(o, self.__class__): if self._form is None or o._form is None: raise_error( NotImplementedError, "Multiplication of symbolic Hamiltonians " "without symbolic form is not implemented.", ) new_form = self.form * o.form new_symbol_map = dict(self.symbol_map) new_symbol_map.update(o.symbol_map) new_ham = self.__class__( new_form, symbol_map=new_symbol_map, backend=self.backend ) if self._dense is not None and o._dense is not None: new_ham.dense = self.dense @ o.dense return new_ham if isinstance(o, self.backend.tensor_types): rank = len(tuple(o.shape)) if rank not in (1, 2): raise_error( NotImplementedError, f"Cannot multiply Hamiltonian with rank-{rank} tensor.", ) state_qubits = int(np.log2(int(o.shape[0]))) if state_qubits != self.nqubits: raise_error( ValueError, f"Cannot multiply Hamiltonian on {self.nqubits} qubits to " + f"state of {state_qubits} qubits.", ) if rank == 1: # state vector return self.apply_gates(o) return self.apply_gates(o, density_matrix=True) raise_error( NotImplementedError, f"Hamiltonian matmul to {type(o)} not implemented.", )
[docs] def circuit(self, dt, accelerators=None): """Circuit that implements a Trotter step of this Hamiltonian. Args: dt (float): Time step used for Trotterization. accelerators (dict, optional): Dictionary with accelerators for distributed circuits. Defaults to ``None``. """ from qibo import Circuit # pylint: disable=import-outside-toplevel from qibo.hamiltonians.terms import ( # pylint: disable=import-outside-toplevel TermGroup, ) groups = TermGroup.from_terms(self.terms) circuit = Circuit(self.nqubits, accelerators=accelerators) circuit.add( group.term.expgate(dt / 2.0) for group in chain(groups, groups[::-1]) ) return circuit
class TrotterHamiltonian: """""" def __init__(self, *parts): raise_error( NotImplementedError, "`TrotterHamiltonian` is substituted by `SymbolicHamiltonian` " + "and is no longer supported. Please check the documentation " + "of `SymbolicHamiltonian` for more details.", ) @classmethod def from_symbolic(cls, symbolic_hamiltonian, symbol_map): return cls()