Random walk phase estimation example in Python #1579 (#2060)

* Random walk phase estimation example in Python #1579

* Formatting and spelling

* Missed one formatting change

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Co-authored-by: Bettina Heim <[email protected]>

docs/sphinx/examples/python/random_walk_qpe.py

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+# Compile and run with:
+# ```
+# nvq++ random_walk_qpe.cpp -o qpe.x && ./qpe.x
+# ```
+
+import cudaq
+from typing import List
+from cudaq import spin
+
+# Here we demonstrate an algorithm expressed as a CUDA-Q kernel
+# that incorporates non-trivial control flow and conditional
+# quantum instruction invocation.
+
+
+# Define the random walk phase estimation kernel
+@cudaq.kernel
+def rwpe_kernel(n_iter: int, mu: float, sigma: float) -> float:
+    iteration = 0
+
+    # Allocate the qubits
+    number_of_qubits = 2
+    qubits = cudaq.qvector(number_of_qubits)
+
+    # Alias them
+    aux = qubits[0]
+
+    target = qubits[number_of_qubits - 1]
+
+    x(target)
+
+    while iteration < n_iter:
+        h(aux)
+        rz(1.0 - (mu / sigma), aux)
+        rz(.25 / sigma, target)
+        x.ctrl(aux, target)
+        rz(-.25 / sigma, target)
+        x.ctrl(aux, target)
+        h(aux)
+        if mz(aux):
+            x(aux)
+            mu = mu + sigma * .6065
+        else:
+            mu = mu - sigma * .6065
+
+        sigma *= .7951
+        iteration += 1
+
+    return 2.0 * mu
+
+
+# Main function to execute the kernel
+def main():
+    n_iterations = 24
+    mu = 0.7951
+    sigma = 0.6065
+
+    phase = rwpe_kernel(n_iterations, mu, sigma)
+    print(f"Phase = {phase:.6f}")
+
+
+if __name__ == "__main__":
+    main()