diff --git a/qiskit/algorithms/amplitude_amplifiers/amplitude_amplifier.py b/qiskit/algorithms/amplitude_amplifiers/amplitude_amplifier.py
index eb3d830a7c57..a94481593f76 100644
--- a/qiskit/algorithms/amplitude_amplifiers/amplitude_amplifier.py
+++ b/qiskit/algorithms/amplitude_amplifiers/amplitude_amplifier.py
@@ -47,8 +47,8 @@ def __init__(self) -> None:
         self._top_measurement: str | None = None
         self._assignment = None
         self._oracle_evaluation: bool | None = None
-        self._circuit_results = None
-        self._max_probability = None
+        self._circuit_results: list[np.ndarray] | list[dict[str, int]] | None = None
+        self._max_probability: float | None = None
 
     @property
     def top_measurement(self) -> Optional[str]:
diff --git a/qiskit/algorithms/amplitude_amplifiers/grover.py b/qiskit/algorithms/amplitude_amplifiers/grover.py
index 4a0f81511f55..c8bfd051401a 100644
--- a/qiskit/algorithms/amplitude_amplifiers/grover.py
+++ b/qiskit/algorithms/amplitude_amplifiers/grover.py
@@ -14,7 +14,7 @@
 
 import itertools
 import operator
-from typing import Iterator, List, Optional, Union
+from typing import Iterator, List, Optional, Union, Mapping
 import warnings
 
 import numpy as np
@@ -156,7 +156,9 @@ def __init__(
 
         if growth_rate is not None:
             # yield iterations ** 1, iterations ** 2, etc. and casts to int
-            self._iterations = map(lambda x: int(growth_rate**x), itertools.count(1))
+            self._iterations: Optional[
+                Union[Mapping[int, float], List[int], Iterator[int], int]
+            ] = map(lambda x: int(growth_rate**x), itertools.count(1))
         elif isinstance(iterations, int):
             self._iterations = [iterations]
         else:
diff --git a/qiskit/algorithms/amplitude_estimators/ae.py b/qiskit/algorithms/amplitude_estimators/ae.py
index 186be13d0655..8aa6fed8633c 100644
--- a/qiskit/algorithms/amplitude_estimators/ae.py
+++ b/qiskit/algorithms/amplitude_estimators/ae.py
@@ -419,7 +419,7 @@ def estimate(self, estimation_problem: EstimationProblem) -> "AmplitudeEstimatio
         # store the number of oracle queries
         result.num_oracle_queries = result.shots * (self._M - 1)
 
-        # run the MLE post processing
+        # run the MLE post-processing
         mle = self.compute_mle(result)
         result.mle = mle
         result.mle_processed = estimation_problem.post_processing(mle)
@@ -585,7 +585,7 @@ def integrand(x):
 
 def _fisher_confint(
     result: AmplitudeEstimationResult, alpha: float, observed: bool = False
-) -> Tuple[float, float]:
+) -> tuple[float, float]:
     """Compute the Fisher information confidence interval for the MLE of the previous run.
 
     Args:
@@ -607,7 +607,7 @@ def _fisher_confint(
 
 def _likelihood_ratio_confint(
     result: AmplitudeEstimationResult, alpha: float
-) -> Tuple[float, float]:
+) -> tuple[float, float]:
     """Compute the likelihood ratio confidence interval for the MLE of the previous run.
 
     Args:
diff --git a/qiskit/algorithms/amplitude_estimators/amplitude_estimator.py b/qiskit/algorithms/amplitude_estimators/amplitude_estimator.py
index 3a0d196d30f0..f5cc7c15c188 100644
--- a/qiskit/algorithms/amplitude_estimators/amplitude_estimator.py
+++ b/qiskit/algorithms/amplitude_estimators/amplitude_estimator.py
@@ -40,14 +40,14 @@ class AmplitudeEstimatorResult(AlgorithmResult):
 
     def __init__(self) -> None:
         super().__init__()
-        self._circuit_results: Optional[Union[np.ndarray, Dict[str, int]]] = None
-        self._shots: Optional[int] = None
-        self._estimation: Optional[float] = None
-        self._estimation_processed: Optional[float] = None
-        self._num_oracle_queries: Optional[int] = None
-        self._post_processing: Optional[Callable[[float], float]] = None
-        self._confidence_interval: Optional[Tuple[float, float]] = None
-        self._confidence_interval_processed: Optional[Tuple[float, float]] = None
+        self._circuit_results: np.ndarray | dict[str, int] | None = None
+        self._shots: int | None = None
+        self._estimation: float | None = None
+        self._estimation_processed: float | None = None
+        self._num_oracle_queries: int | None = None
+        self._post_processing: Callable[[float], float] | None = None
+        self._confidence_interval: tuple[float, float] | None = None
+        self._confidence_interval_processed: tuple[float, float] | None = None
 
     @property
     def circuit_results(self) -> np.ndarray | dict[str, int] | None:
diff --git a/qiskit/algorithms/amplitude_estimators/fae.py b/qiskit/algorithms/amplitude_estimators/fae.py
index e510f53edead..15836db165ad 100644
--- a/qiskit/algorithms/amplitude_estimators/fae.py
+++ b/qiskit/algorithms/amplitude_estimators/fae.py
@@ -343,10 +343,10 @@ class FasterAmplitudeEstimationResult(AmplitudeEstimatorResult):
 
     def __init__(self) -> None:
         super().__init__()
-        self._success_probability: Optional[int] = None
-        self._num_steps: Optional[int] = None
-        self._num_first_state_steps: Optional[int] = None
-        self._theta_intervals: Optional[List[List[float]]] = None
+        self._success_probability: int | None = None
+        self._num_steps: int | None = None
+        self._num_first_state_steps: int | None = None
+        self._theta_intervals: list[list[float]] | None = None
 
     @property
     def success_probability(self) -> int:
diff --git a/qiskit/algorithms/amplitude_estimators/iae.py b/qiskit/algorithms/amplitude_estimators/iae.py
index af7d8357b5fc..c781abc30f3b 100644
--- a/qiskit/algorithms/amplitude_estimators/iae.py
+++ b/qiskit/algorithms/amplitude_estimators/iae.py
@@ -536,15 +536,15 @@ class IterativeAmplitudeEstimationResult(AmplitudeEstimatorResult):
 
     def __init__(self) -> None:
         super().__init__()
-        self._alpha: Optional[float] = None
-        self._epsilon_target: Optional[float] = None
-        self._epsilon_estimated: Optional[float] = None
-        self._epsilon_estimated_processed: Optional[float] = None
-        self._estimate_intervals: Optional[List[List[float]]] = None
-        self._theta_intervals: Optional[List[List[float]]] = None
-        self._powers: Optional[List[int]] = None
-        self._ratios: Optional[List[float]] = None
-        self._confidence_interval_processed: Optional[Tuple[float, float]] = None
+        self._alpha: float | None = None
+        self._epsilon_target: float | None = None
+        self._epsilon_estimated: float | None = None
+        self._epsilon_estimated_processed: float | None = None
+        self._estimate_intervals: list[list[float]] | None = None
+        self._theta_intervals: list[list[float]] | None = None
+        self._powers: list[int] | None = None
+        self._ratios: list[float] | None = None
+        self._confidence_interval_processed: tuple[float, float] | None = None
 
     @property
     def alpha(self) -> float:
diff --git a/qiskit/algorithms/amplitude_estimators/mlae.py b/qiskit/algorithms/amplitude_estimators/mlae.py
index f6d7344883a2..f414356679b0 100644
--- a/qiskit/algorithms/amplitude_estimators/mlae.py
+++ b/qiskit/algorithms/amplitude_estimators/mlae.py
@@ -361,7 +361,7 @@ def estimate(
                 except Exception as exc:
                     raise AlgorithmError("The job was not completed successfully. ") from exc
 
-                result.circuit_results = []
+                result.circuit_results: list[dict] = []
                 shots = ret.metadata[0].get("shots")
                 if shots is None:
                     for i, quasi_dist in enumerate(ret.quasi_dists):
@@ -416,7 +416,7 @@ class MaximumLikelihoodAmplitudeEstimationResult(AmplitudeEstimatorResult):
     def __init__(self) -> None:
         super().__init__()
         self._theta: float | None = None
-        self._minimizer: callable | None = None
+        self._minimizer: typing.Callable | None = None
         self._good_counts: list[float] | None = None
         self._evaluation_schedule: list[int] | None = None
         self._fisher_information: float | None = None
@@ -437,7 +437,7 @@ def minimizer(self) -> callable:
         return self._minimizer
 
     @minimizer.setter
-    def minimizer(self, value: callable) -> None:
+    def minimizer(self, value: typing.Callable) -> None:
         """Set the number minimizer used for the search of the likelihood function."""
         self._minimizer = value
 
@@ -579,7 +579,7 @@ def _likelihood_ratio_confint(
     result: MaximumLikelihoodAmplitudeEstimationResult,
     alpha: float = 0.05,
     nevals: int | None = None,
-) -> typing.Tuple[float, float]:
+) -> tuple[float, float]:
     """Compute the likelihood-ratio confidence interval.
 
     Args:
@@ -640,7 +640,9 @@ def _get_counts(
         AlgorithmError: If self.run() has not been called yet.
     """
     one_hits = []  # h_k: how often 1 has been measured, for a power Q^(m_k)
-    all_hits: np.ndarray | list[float] = []  # shots_k: how often has been measured at a power Q^(m_k)
+    all_hits: np.ndarray | list[
+        float
+    ] = []  # shots_k: how often has been measured at a power Q^(m_k)
     if all(isinstance(data, (list, np.ndarray)) for data in circuit_results):
         probabilities = []
         num_qubits = int(np.log2(len(circuit_results[0])))  # the total number of qubits
diff --git a/qiskit/algorithms/aux_ops_evaluator.py b/qiskit/algorithms/aux_ops_evaluator.py
index 9ce9349a7de8..b50eb131c2c8 100644
--- a/qiskit/algorithms/aux_ops_evaluator.py
+++ b/qiskit/algorithms/aux_ops_evaluator.py
@@ -160,7 +160,7 @@ def _prepare_result(
         A list or a dictionary of tuples (mean, standard deviation).
     """
     if isinstance(observables, list):
-        observables_eigenvalues = [None] * len(observables)
+        observables_eigenvalues: ListOrDict[Tuple[complex, complex]] = [None] * len(observables)
         key_value_iterator = enumerate(observables_results)
     else:
         observables_eigenvalues = {}
diff --git a/qiskit/algorithms/eigen_solvers/vqd.py b/qiskit/algorithms/eigen_solvers/vqd.py
index 1ffce32ab580..b8762221f635 100644
--- a/qiskit/algorithms/eigen_solvers/vqd.py
+++ b/qiskit/algorithms/eigen_solvers/vqd.py
@@ -495,7 +495,9 @@ def _eval_aux_ops(
         # None operators are already dropped in compute_minimum_eigenvalue if aux_operators is a
         # dict.
         if isinstance(aux_operators, list):
-            aux_operator_eigenvalues = [None] * len(aux_operators)
+            aux_operator_eigenvalues: ListOrDict[Tuple[complex, complex]] = [None] * len(
+                aux_operators
+            )
             key_value_iterator = enumerate(aux_op_results)
         else:
             aux_operator_eigenvalues = {}
diff --git a/qiskit/algorithms/eigensolvers/vqd.py b/qiskit/algorithms/eigensolvers/vqd.py
index 6b48a30090ed..0540be1dfa6f 100644
--- a/qiskit/algorithms/eigensolvers/vqd.py
+++ b/qiskit/algorithms/eigensolvers/vqd.py
@@ -205,7 +205,7 @@ def compute_eigenvalues(
             # Drop None and convert zero values when aux_operators is a dict.
             if isinstance(aux_operators, list):
                 key_op_iterator = enumerate(aux_operators)
-                converted = [zero_op] * len(aux_operators)
+                converted: ListOrDict[BaseOperator | PauliSumOp] = [zero_op] * len(aux_operators)
             else:
                 key_op_iterator = aux_operators.items()
                 converted = {}
@@ -410,13 +410,13 @@ class VQDResult(EigensolverResult):
 
     def __init__(self) -> None:
         super().__init__()
-        self._cost_function_evals = None
-        self._optimizer_times = None
-        self._optimal_values = None
-        self._optimal_points = None
-        self._optimal_parameters = None
-        self._optimizer_results = None
-        self._optimal_circuits = None
+        self._cost_function_evals: Sequence[int] | None = None
+        self._optimizer_times: Sequence[float] | None = None
+        self._optimal_values: Sequence[float] | None = None
+        self._optimal_points: Sequence[np.ndarray] | None = None
+        self._optimal_parameters: Sequence[dict] | None = None
+        self._optimizer_results: Sequence[OptimizerResult] | None = None
+        self._optimal_circuits: list[QuantumCircuit] | None = None
 
     @property
     def cost_function_evals(self) -> Sequence[int] | None:
@@ -479,7 +479,7 @@ def optimizer_results(self, value: Sequence[OptimizerResult]) -> None:
         self._optimizer_results = value
 
     @property
-    def optimal_circuits(self) -> list[QuantumCircuit]:
+    def optimal_circuits(self) -> list[QuantumCircuit] | None:
         """The optimal circuits. Along with the optimal parameters,
         these can be used to retrieve the different eigenstates."""
         return self._optimal_circuits
diff --git a/qiskit/algorithms/gradients/utils.py b/qiskit/algorithms/gradients/utils.py
index 5c536fdf2173..5cdac74ee783 100644
--- a/qiskit/algorithms/gradients/utils.py
+++ b/qiskit/algorithms/gradients/utils.py
@@ -21,6 +21,7 @@
 from collections import defaultdict
 from copy import deepcopy
 from dataclasses import dataclass
+from typing import Sequence
 
 import numpy as np
 
@@ -215,7 +216,7 @@ def _param_shift_preprocessing(circuit: QuantumCircuit) -> ParameterShiftGradien
 def _make_param_shift_parameter_values(
     gradient_circuit_data: ParameterShiftGradientCircuit,
     base_parameter_values: list[np.ndarray],
-    parameter_values: np.ndarray,
+    parameter_values: Sequence[float],
     param_set: set[Parameter],
 ) -> list[np.ndarray]:
     """Makes parameter values for the parameter shift method. Each parameter value will be added to
diff --git a/qiskit/algorithms/phase_estimators/phase_estimation_result.py b/qiskit/algorithms/phase_estimators/phase_estimation_result.py
index f860d21cc960..a0342cb1b5c0 100644
--- a/qiskit/algorithms/phase_estimators/phase_estimation_result.py
+++ b/qiskit/algorithms/phase_estimators/phase_estimation_result.py
@@ -133,8 +133,8 @@ def filter_phases(self, cutoff: float = 0.0, as_float: bool = True) -> dict:
                     # But, we chose to apply the unitaries such that the phase is recorded
                     # in reverse order. So, we reverse the bitstrings here.
                     binary_phase_string = numpy.binary_repr(idx, self._num_evaluation_qubits)[::-1]
-                    if as_float:
-                        _key = _bit_string_to_phase(binary_phase_string)
+                    if as_float:  # TODO: should be not as_float?
+                        _key: str | float = _bit_string_to_phase(binary_phase_string)
                     else:
                         _key = binary_phase_string
                     phases[_key] = amplitude
diff --git a/qiskit/algorithms/phase_estimators/phase_estimation_scale.py b/qiskit/algorithms/phase_estimators/phase_estimation_scale.py
index 268f1a9aae31..307e50cf8e9c 100644
--- a/qiskit/algorithms/phase_estimators/phase_estimation_scale.py
+++ b/qiskit/algorithms/phase_estimators/phase_estimation_scale.py
@@ -141,7 +141,7 @@ def from_pauli_sum(
             bound = sum(abs(coeff) for coeff in pauli_sum.coeffs)
             return PhaseEstimationScale(bound)
         elif isinstance(pauli_sum, Operator):
-            bound = np.sum(np.abs(np.eigvalsh(pauli_sum)))
+            bound = np.sum(np.abs(np.eigvalsh(pauli_sum)))  # TODO: should be np.linalg.eigvalsh?
             return PhaseEstimationScale(bound)
         elif isinstance(pauli_sum, BaseOperator):
             raise ValueError(
diff --git a/qiskit/algorithms/time_evolvers/pvqd/pvqd.py b/qiskit/algorithms/time_evolvers/pvqd/pvqd.py
index bb2418640e8d..0e412d7d7c20 100644
--- a/qiskit/algorithms/time_evolvers/pvqd/pvqd.py
+++ b/qiskit/algorithms/time_evolvers/pvqd/pvqd.py
@@ -368,7 +368,7 @@ def evolve(self, evolution_problem: TimeEvolutionProblem) -> TimeEvolutionResult
             )
             observable_values = [evaluate_observables(self.initial_parameters)]
 
-        fidelities = [1]
+        fidelities = [1]  # TODO: should be float?
         parameters = [self.initial_parameters]
         times = np.linspace(0, time, num_timesteps + 1).tolist()  # +1 to include initial time 0
 
diff --git a/qiskit/algorithms/utils/validate_bounds.py b/qiskit/algorithms/utils/validate_bounds.py
index 65f00cfcbd79..108aed56746a 100644
--- a/qiskit/algorithms/utils/validate_bounds.py
+++ b/qiskit/algorithms/utils/validate_bounds.py
@@ -17,7 +17,7 @@
 from qiskit.circuit import QuantumCircuit
 
 
-def validate_bounds(circuit: QuantumCircuit) -> list[tuple(float | None, float | None)]:
+def validate_bounds(circuit: QuantumCircuit) -> list[tuple[float | None, float | None]]:
     """
     Validate the bounds provided by a quantum circuit against its number of parameters.
     If no bounds are obtained, return ``None`` for all lower and upper bounds.
@@ -31,8 +31,11 @@ def validate_bounds(circuit: QuantumCircuit) -> list[tuple(float | None, float |
     Raises:
         ValueError: If the number of bounds does not the match the number of circuit parameters.
     """
-    if hasattr(circuit, "parameter_bounds") and circuit.parameter_bounds is not None:
-        bounds = circuit.parameter_bounds
+    if (
+        hasattr(circuit, "parameter_bounds")
+        and circuit.parameter_bounds is not None  # type: ignore[attr-defined]
+    ):
+        bounds = circuit.parameter_bounds  # type: ignore[attr-defined]
         if len(bounds) != circuit.num_parameters:
             raise ValueError(
                 f"The number of bounds ({len(bounds)}) does not match the number of "
diff --git a/qiskit/algorithms/variational_algorithm.py b/qiskit/algorithms/variational_algorithm.py
index 0f1661b1a35e..95b1d2c62dd4 100644
--- a/qiskit/algorithms/variational_algorithm.py
+++ b/qiskit/algorithms/variational_algorithm.py
@@ -62,8 +62,8 @@ def __init__(self) -> None:
         self._optimal_value: float | None = None
         self._optimal_point: np.ndarray | None = None
         self._optimal_parameters: Dict | None = None
-        self._optimizer_result = None
-        self._optimal_circuit = None
+        self._optimizer_result: OptimizerResult | None = None
+        self._optimal_circuit: QuantumCircuit | None = None
 
     @property
     def optimizer_evals(self) -> Optional[int]: