fix random statevector distribution

qiskit/quantum_info/states/random.py

@@ -31,7 +31,7 @@ def random_statevector(
 ) -> Statevector:
     """Generator a random Statevector.
 
-    The statevector is sampled from the uniform (Haar) measure.
+    The statevector is sampled from the uniform distribution.
 
     Args:
         dims (int or tuple): the dimensions of the state.
@@ -40,6 +40,10 @@ def random_statevector(
 
     Returns:
         Statevector: the random statevector.
+
+    Reference:
+        "Choosing a Point from the Surface of a Sphere" by George Marsaglia
+        http://doi.org/10.1214/aoms/1177692644
     """
     if seed is None:
         rng = np.random.default_rng()
@@ -49,14 +53,10 @@ def random_statevector(
         rng = default_rng(seed)
 
     dim = np.prod(dims)
-
-    # Random array over interval (0, 1]
-    x = rng.random(dim)
-    x += x == 0
-    x = -np.log(x)
-    sumx = sum(x)
-    phases = rng.random(dim) * 2.0 * np.pi
-    return Statevector(np.sqrt(x / sumx) * np.exp(1j * phases), dims=dims)
+    vec = rng.standard_normal(dim).astype(complex)
+    vec += 1j * rng.standard_normal(dim)
+    vec /= np.linalg.norm(vec)
+    return Statevector(vec, dims=dims)
 
 
 def random_density_matrix(

releasenotes/notes/fix-random-statevector-a8dbf991cbbdee8e.yaml

@@ -0,0 +1,4 @@
+---
+fixes:
+  - |
+    Fixes the implementation of qiskit.quantum_info.random_statevector so that it samples from the uniform distribution.