fix random statevector distribution (Qiskit#10866)

* fix random statevector distribution

* add citation and release note

* mention Haar measure in doc and use link in release note

* cite better reference

* Format reference

---------

Co-authored-by: Julien Gacon <[email protected]>

qiskit/quantum_info/states/random.py

@@ -31,7 +31,8 @@ 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. This is the measure
+    induced by the Haar measure on unitary matrices.
 
     Args:
         dims (int or tuple): the dimensions of the state.
@@ -40,6 +41,10 @@ def random_statevector(
 
     Returns:
         Statevector: the random statevector.
+
+    Reference:
+        K. Zyczkowski and H. Sommers (2001), "Induced measures in the space of mixed quantum states",
+        `J. Phys. A: Math. Gen. 34 7111 <https://arxiv.org/abs/quant-ph/0012101>`__.
     """
     if seed is None:
         rng = np.random.default_rng()
@@ -49,14 +54,10 @@ 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 :func:`.random_statevector` so that it samples from the uniform distribution.