diff --git a/qiskit_experiments/curve_analysis/curve_fit.py b/qiskit_experiments/curve_analysis/curve_fit.py
index 41d654e308..13045c21e2 100644
--- a/qiskit_experiments/curve_analysis/curve_fit.py
+++ b/qiskit_experiments/curve_analysis/curve_fit.py
@@ -85,7 +85,7 @@ def curve_fit(
             upper = [bounds[key][1] for key in param_keys]
             param_bounds = (lower, upper)
         else:
-            param_bounds = None
+            param_bounds = ([-np.inf] * len(param_keys), [np.inf] * len(param_keys))
 
         # Convert fit function
         def fit_func(x, *params):
@@ -94,7 +94,10 @@ def fit_func(x, *params):
     else:
         param_keys = None
         param_p0 = p0
-        param_bounds = bounds
+        if bounds:
+            param_bounds = bounds
+        else:
+            param_bounds = ([-np.inf] * len(p0), [np.inf] * len(p0))
         fit_func = func
 
     # Check the degrees of freedom is greater than 0
diff --git a/qiskit_experiments/library/characterization/t1_analysis.py b/qiskit_experiments/library/characterization/t1_analysis.py
index 12103883c3..d7addafccc 100644
--- a/qiskit_experiments/library/characterization/t1_analysis.py
+++ b/qiskit_experiments/library/characterization/t1_analysis.py
@@ -47,11 +47,6 @@ class T1Analysis(BaseAnalysis):
         - :math:`amplitude\_guess`: Determined by :math:`(y_0 - offset\_guess)`
         - :math:`offset\_guess`: Determined by the last :math:`y`
         - :math:`t1\_guess`: Determined by the mean of the data points
-
-    Bounds
-        - :math:`amplitude\_bounds`: [0, 1]
-        - :math:`offset\_bounds`: [0, 1]
-        - :math:`t1\_bounds`: [0, infinity]
     """
 
     @classmethod
@@ -60,9 +55,6 @@ def _default_options(cls):
             t1_guess=None,
             amplitude_guess=None,
             offset_guess=None,
-            t1_bounds=None,
-            amplitude_bounds=None,
-            offset_bounds=None,
         )
 
     # pylint: disable=arguments-differ
@@ -72,9 +64,6 @@ def _run_analysis(
         t1_guess=None,
         amplitude_guess=None,
         offset_guess=None,
-        t1_bounds=None,
-        amplitude_bounds=None,
-        offset_bounds=None,
         plot=True,
         ax=None,
     ) -> Tuple[List[CurveAnalysisResultData], List["matplotlib.figure.Figure"]]:
@@ -87,11 +76,6 @@ def _run_analysis(
             amplitude_guess (float): Optional, an initial guess of the coefficient
                                      of the exponent
             offset_guess (float): Optional, an initial guess of the offset
-            t1_bounds (list of two floats): Optional, lower bound and upper bound to T1
-            amplitude_bounds (list of two floats): Optional, lower bound and upper
-                                                   bound to the amplitude
-            offset_bounds (list of two floats): Optional, lower bound and upper
-                                                bound to the offset
             plot (bool): Generator plot of exponential fit.
             ax (AxesSubplot): Optional, axes to add figure to.
 
@@ -120,20 +104,13 @@ def _run_analysis(
             offset_guess = ydata[-1]
         if amplitude_guess is None:
             amplitude_guess = ydata[0] - offset_guess
-        if t1_bounds is None:
-            t1_bounds = [0, np.inf]
-        if amplitude_bounds is None:
-            amplitude_bounds = [0, 1]
-        if offset_bounds is None:
-            offset_bounds = [0, 1]
 
         # Perform fit
         def fit_fun(x, a, tau, c):
             return a * np.exp(-x / tau) + c
 
         init = {"a": amplitude_guess, "tau": t1_guess, "c": offset_guess}
-        bounds = {"a": amplitude_bounds, "tau": t1_bounds, "c": offset_bounds}
-        fit_result = curve_fit(fit_fun, xdata, ydata, init, sigma=sigma, bounds=bounds)
+        fit_result = curve_fit(fit_fun, xdata, ydata, init, sigma=sigma)
 
         result_data = {
             "value": fit_result["popt"][1],
diff --git a/test/test_t1.py b/test/test_t1.py
index 0227a576b2..68f81e7f19 100644
--- a/test/test_t1.py
+++ b/test/test_t1.py
@@ -87,9 +87,9 @@ def test_t1_parallel_different_analysis_options(self):
         delays = list(range(1, 40, 3))
 
         exp0 = T1(0, delays)
-        exp0.set_analysis_options(t1_bounds=[10, 30])
+        exp0.set_analysis_options(t1_guess=30)
         exp1 = T1(1, delays)
-        exp1.set_analysis_options(t1_bounds=[100, 200])
+        exp1.set_analysis_options(t1_guess=1000000)
 
         par_exp = ParallelExperiment([exp0, exp1])
         res = par_exp.run(T1Backend([t1, t1]))
@@ -101,7 +101,7 @@ def test_t1_parallel_different_analysis_options(self):
 
         self.assertEqual(sub_res[0].quality, "good")
         self.assertAlmostEqual(sub_res[0].data()["value"], t1, delta=3)
-        self.assertFalse(sub_res[1].data()["success"])
+        self.assertEqual(sub_res[1].quality, "bad")
 
     def test_t1_analysis(self):
         """