Work around lmfit limiting RBAnalysis to 10,000 Cliffords (#1595)

lmfit (through its asteval dependnecy) limits exponentiation in its
expression models to 10,000. The limit is there to avoid denial of
service attacks where user input could be substituted into a model and
Python integers which are unbounded and could take up exponential
amounts of memory. In qiskit-experiments, this case is not a concern.
The workaround implemented here is to divide the exponential in the
curve by 10 and then exponentiate the result by 10. This change allows
up to 100,000 Cliffords which should be enough. The alternative
workaround is to change `asteval.astutils.MAX_EXPONENT` to a
sufficiently large value.

Closes
#1594

Author: wshanks

qiskit_experiments/library/randomized_benchmarking/rb_analysis.py

@@ -50,6 +50,14 @@ class RBAnalysis(curve.CurveAnalysis):
 
             F(x) = a \alpha^x + b
 
+        .. note::
+
+            The string expression model used by the class is implemented using
+            :math:`(\alpha^{(x / 10)})^{10}` in order to allow :math:`x` to vary up
+            to 100,000 without hitting an exponentiation limit of 10,000
+            imposed by the `LMFIT
+            <https://lmfit.github.io/lmfit-py/intro.html>`__ fitting framework.
+
     # section: fit_parameters
         defpar a:
             desc: Height of decay curve.
@@ -73,7 +81,14 @@ def __init__(self):
         super().__init__(
             models=[
                 lmfit.models.ExpressionModel(
-                    expr="a * alpha ** x + b",
+                    # NOTE: the `/10` and then `** 10` are done because asteval
+                    # (used by lmfit to evaluate the model expression) limits
+                    # exponents to 10,000 as a security precaution. For gates
+                    # with errors around 1e-4, it is desirable to go beyond
+                    # 10,000 Cliffords. By splitting out the factors of 10 like
+                    # this, we can have `x` go up to 100,000 Cliffords without
+                    # hitting the asteval limit.
+                    expr="a * (alpha ** (x/10)) ** 10 + b",
                     name="rb_decay",
                 )
             ]

releasenotes/notes/rb-clifford-10k-3429e74f4a82848e.yaml

@@ -0,0 +1,8 @@
+---
+fixes:
+  - |
+    A workaround was added for a limit on exponentiation imposed by `LMFIT
+    <https://lmfit.github.io/lmfit-py/intro.html>`__ which prevented
+    :class:`.RBAnalysis` from working on experiments with Clifford lengths
+    longer than 10,000. See `#1594
+    <https://github.com/qiskit-community/qiskit-experiments/issues/1594>`__.