Fix #511 for fixed point amplitude amplification

Standard/src/AmplitudeAmplification/Convert.qs

@@ -31,6 +31,11 @@ namespace Microsoft.Quantum.AmplitudeAmplification {
 
         mutable phasesTarget = [0.0, size = nPhasesRef];
         mutable phasesStart = [0.0, size = nPhasesRef];
+
+        if nPhasesRot == 1 {
+            return ReflectionPhases(phasesStart, phasesTarget);
+        }
+
         set phasesTarget w/= 0 <- ((rotPhases!)[0] - (rotPhases!)[1]) - PI();
         set phasesStart w/= 0 <- -(rotPhases!)[0] + 0.5 * PI();
 

Standard/src/AmplitudeAmplification/StandardAlgorithms.qs

@@ -2,8 +2,9 @@
 // Licensed under the MIT License.
 
 namespace Microsoft.Quantum.AmplitudeAmplification {
-    open Microsoft.Quantum.Convert;
     open Microsoft.Quantum.Arrays;
+    open Microsoft.Quantum.Convert;
+    open Microsoft.Quantum.Diagnostics;
     open Microsoft.Quantum.Math;
 
     /// # Summary
@@ -52,13 +53,27 @@ namespace Microsoft.Quantum.AmplitudeAmplification {
     /// for phases in the `RotationPhases` format.
     function FixedPointReflectionPhases(nQueries : Int, successMin : Double)
     : ReflectionPhases {
-        let twoPi = 2.0 * PI();
+        // In this implementation `nQueries` corresponds to $L$ in
+        // arXiv:1409.3305.
+        Fact(nQueries % 2 == 1, "nQueries must be odd");
+
+        // Initializes L rotation phases, this also initializes the first
+        // rotation phase with 0.0.
         mutable phasesRot = [0.0, size = nQueries];
         let nQueriesDouble = IntAsDouble(nQueries);
-        set phasesRot w/= 0 <- 0.0;
-        let beta = Cosh((1.0 / nQueriesDouble) * ArcCosh(Sqrt(successMin)));
-        let alpha = Sqrt(1.0 - beta * beta);
 
+        // The success probability `successMin` is $1 - \delta^2$ in
+        // arXiv:1409.3305. Variable `beta` corresponds to $\gamma^{-1}$ in
+        // arXiv:1409.3305, right below Eq. (11)
+        let beta = Cosh((1.0 / nQueriesDouble) * ArcCosh(Sqrt(1.0 / (1.0 - successMin))));
+
+        // `alpha` is $\sqrt(1 - \gamma^2)$ in Eq. (11) in arXiv:1409.3305,
+        // therefore it is $\sqrt(1 - (1 / \beta^2))$
+        let alpha = Sqrt(1.0 - 1.0 / (beta * beta));
+
+        // Iterative computation of rotation phases is described in Eq. (30) in
+        // arXiv:1603.03996.  In there, we can set $j = 1$.
+        let twoPi = 2.0 * PI();
         for idxPhases in 1 .. nQueries - 1 {
             set phasesRot w/= idxPhases <-
                 phasesRot[idxPhases - 1] +

Standard/tests/AmplitudeAmplificationTests.qs

@@ -2,13 +2,15 @@
 // Licensed under the MIT License.
 
 namespace Microsoft.Quantum.Tests {
-    open Microsoft.Quantum.Intrinsic;
+    open Microsoft.Quantum.AmplitudeAmplification;
+    open Microsoft.Quantum.Arrays;
     open Microsoft.Quantum.Canon;
     open Microsoft.Quantum.Convert;
     open Microsoft.Quantum.Diagnostics;
-    open Microsoft.Quantum.AmplitudeAmplification;
-    open Microsoft.Quantum.Oracles;
+    open Microsoft.Quantum.Intrinsic;
+    open Microsoft.Quantum.Logical;
     open Microsoft.Quantum.Math;
+    open Microsoft.Quantum.Oracles;
 
     ///Here we consider the smallest example of amplitude amplification
     ///Suppose we have a single-qubit oracle that prepares the state
@@ -94,6 +96,18 @@ namespace Microsoft.Quantum.Tests {
         }
     }
 
+    @Test("QuantumSimulator")
+    operation TestRotationPhasesAsReflectionPhases() : Unit {
+        let rotationPhases = RotationPhases([0.1, 0.2, 0.3, 0.4, 0.5]);
+        let reflectionPhases = RotationPhasesAsReflectionPhases(rotationPhases);
+
+        EqualityFactI(Length(reflectionPhases::AboutStart), 3, "Unexpected length of reflection phases");
+        EqualityFactI(Length(reflectionPhases::AboutTarget), 3, "Unexpected length of reflection phases");
+
+        Fact(All(NearlyEqualD, Zipped(reflectionPhases::AboutStart, [1.4707963267948965,3.041592653589793,3.041592653589793])), "Unexpected reflection phases");
+        Fact(All(NearlyEqualD, Zipped(reflectionPhases::AboutTarget, [-3.241592653589793,-3.241592653589793,-1.0707963267948966])), "Unexpected reflection phases");
+    }
+
 }