text: """This module implements the U1Gate."""
  from __future__ import annotations
  
  import numpy as np
  import numpy.typing as npt
  
  from bqskit.ir.gates.qubitgate import QubitGate
  from bqskit.qis.unitary.differentiable import DifferentiableUnitary
  from bqskit.qis.unitary.optimizable import LocallyOptimizableUnitary
  from bqskit.qis.unitary.unitary import RealVector
  from bqskit.qis.unitary.unitarymatrix import UnitaryMatrix
  from bqskit.utils.cachedclass import CachedClass
  
  
  class U1Gate(
      QubitGate,
      DifferentiableUnitary,
      LocallyOptimizableUnitary,
      CachedClass,
  ):
      """
      The U1 single qubit gate.
  
      It is given by the following parameterized unitary:
  
      .. math::
  
          \\begin{pmatrix}
          1 & 0 \\\\
          0 & \\exp({i\\theta}) \\\\
          \\end{pmatrix}
      """
  
      _num_qudits = 1
      _num_params = 1
      _qasm_name = 'u1'
  
      def get_unitary(self, params: RealVector = []) -> UnitaryMatrix:
          """Return the unitary for this gate, see :class:`Unitary` for more."""
          self.check_parameters(params)
  
          exp = np.exp(1j * params[0])
  
          return UnitaryMatrix(
              [
                  [1, 0],
                  [0, exp],
              ],
          )
  
      def get_grad(self, params: RealVector = []) -> npt.NDArray[np.complex128]:
          """
          Return the gradient for this gate.
  
          See :class:`DifferentiableUnitary` for more info.
          """
          self.check_parameters(params)
  
          dexp = 1j * np.exp(1j * params[0])
  
          return np.array(
              [
                  [
                      [0, 0],
                      [0, dexp],
                  ],
              ], dtype=np.complex128,
          )
  
      def optimize(self, env_matrix: npt.NDArray[np.complex128]) -> list[float]:
          """
          Return the optimal parameters with respect to an environment matrix.
  
          See :class:`LocallyOptimizableUnitary` for more info.
          """
          self.check_env_matrix(env_matrix)
          a = np.real(env_matrix[1, 1])
          b = np.imag(env_matrix[1, 1])
          arctan = np.arctan(b / a)
  
          if a < 0 and b > 0:
              arctan += np.pi
          elif a < 0 and b < 0:
              arctan -= np.pi
  
          return [-arctan]
  