Library for rotation operations (#2040)

This is a Q# library for rotation operations. So far it includes an
implementation of the Hamming weight phasing operation. I intend to
include other rotation operations at a later point.

Author: msoeken

library/rotations/README.md

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+# Rotations
+
+The `rotations` library defines operations for applying rotations to qubits and
+registers.

library/rotations/qsharp.json

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+{
+  "author": "Microsoft",
+  "license": "MIT",
+  "files": [
+    "src/HammingWeightPhasing.qs",
+    "src/Main.qs",
+    "src/Tests.qs"
+  ]
+}

library/rotations/src/HammingWeightPhasing.qs

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+// Copyright (c) Microsoft Corporation. All rights reserved.
+// Licensed under the MIT License.
+
+import Std.Arrays.Enumerated, Std.Arrays.Most, Std.Arrays.Partitioned, Std.Arrays.Tail;
+import Std.Convert.IntAsDouble;
+import Std.Diagnostics.Fact;
+import Std.Math.BitSizeI, Std.Math.Floor, Std.Math.Lg, Std.Math.MaxI, Std.Math.MinI;
+
+/// # Summary
+/// Applies a Z-rotation (`Rz`) with given angle to each qubit in qs.
+///
+/// # Description
+/// This implementation is based on Hamming-weight phasing to reduce the number
+/// of rotation gates.  The technique was first presented in [1], and further
+/// improved in [2] based on results in [3, 4].  Note, that the reduction of
+/// rotation gates comes at a cost of additional qubits and additional quantum
+/// operations to compute the Hamming-weight.
+///
+/// # Reference
+/// - [1](https://arxiv.org/abs/1709.06648) "Halving the cost of quantum
+///   addition", Craig Gidney.
+/// - [2](https://arxiv.org/abs/2012.09238) "Early fault-tolerant simulations of
+///   the Hubbard model", Earl T. Campbell.
+/// - [3](https://cpsc.yale.edu/sites/default/files/files/tr1260.pdf) "The exact
+///   multiplicative complexity of the Hamming weight function", Joan Boyar and
+///   René Peralta.
+/// - [4](https://arxiv.org/abs/1908.01609) "The role of multiplicative
+///   complexity in compiling low T-count oracle circuits", Giulia Meuli,
+///   Mathias Soeken, Earl Campbell, Martin Roetteler, Giovanni De Micheli.
+operation HammingWeightPhasing(angle : Double, qs : Qubit[]) : Unit {
+    WithHammingWeight(qs, (sum) => {
+        for (i, sumQubit) in Enumerated(sum) {
+            Rz(IntAsDouble(2^i) * angle, sumQubit);
+        }
+    });
+}
+
+internal operation WithHammingWeight(qs : Qubit[], action : Qubit[] => Unit) : Unit {
+    let n = Length(qs);
+
+    if n <= 1 {
+        action(qs);
+    } elif n == 2 {
+        use sum = Qubit();
+
+        within {
+            AND(qs[0], qs[1], sum);
+            CNOT(qs[0], qs[1]);
+        } apply {
+            action([qs[1], sum]);
+        }
+    } elif n == 3 {
+        WithSum(qs[0], qs[1..1], qs[2..2], action);
+    } else {
+        let splitSize = 2^(BitSizeI(n - 1) - 1);
+        let (leftLen, rightLen) = (n - splitSize, splitSize - 1);
+        // handle corner case if n is power of 2; in that case the first
+        // partition is longer than the second one, and we want to avoid that.
+        let split = Partitioned([MinI(leftLen, rightLen), MaxI(leftLen, rightLen)], qs);
+        Fact(Length(split) == 3 and Length(split[2]) == 1, $"Unexpected split for n = {n}");
+
+        WithHammingWeight(split[0], (leftHW) => {
+            WithHammingWeight(split[1], (rightHW) => {
+                WithSum(split[2][0], leftHW, rightHW, action);
+            });
+        });
+    }
+}
+
+internal operation Carry(carryIn : Qubit, x : Qubit, y : Qubit, carryOut : Qubit) : Unit is Adj {
+    CNOT(carryIn, x);
+    CNOT(carryIn, y);
+    AND(x, y, carryOut);
+    CNOT(carryIn, x);
+    CNOT(x, y);
+    CNOT(carryIn, carryOut);
+}
+
+internal operation WithSum(carry : Qubit, xs : Qubit[], ys : Qubit[], action : Qubit[] => Unit) : Unit {
+    let n = Length(ys);
+    Fact(Length(xs) <= n, "Length of xs must be less or equal to length of ys");
+    Fact(n > 0, "Length must be at least 1");
+
+    use carryOut = Qubit[n];
+    let carryIn = [carry] + Most(carryOut);
+
+    within {
+        for i in 0..n-1 {
+            if i < Length(xs) {
+                Carry(carryIn[i], xs[i], ys[i], carryOut[i]);
+            } else {
+                // there is no corresponding bit in xs; this is a version of
+                // Carry in which x == 0.
+                CNOT(carryIn[i], ys[i]);
+                AND(carryIn[i], ys[i], carryOut[i]);
+                CNOT(carryIn[i], carryOut[i]);
+            }
+        }
+    } apply {
+        action(ys + [Tail(carryOut)]);
+    }
+}

library/rotations/src/Main.qs

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+// Copyright (c) Microsoft Corporation. All rights reserved.
+// Licensed under the MIT License.
+
+export HammingWeightPhasing.HammingWeightPhasing;

library/rotations/src/Tests.qs

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+// Copyright (c) Microsoft Corporation. All rights reserved.
+// Licensed under the MIT License.
+
+import Std.Diagnostics.CheckOperationsAreEqual, Std.Diagnostics.Fact;
+import Std.Math.HammingWeightI, Std.Math.PI;
+
+import HammingWeightPhasing.HammingWeightPhasing, HammingWeightPhasing.WithHammingWeight;
+
+operation Main() : Unit {
+    TestHammingWeight();
+    TestPhasing();
+}
+
+operation TestHammingWeight() : Unit {
+    // exhaustive
+    use qs = Qubit[4];
+
+    for x in 0..2^Length(qs) - 1 {
+        ApplyXorInPlace(x, qs);
+        WithHammingWeight(qs, sum => {
+            Fact(MeasureInteger(sum) == HammingWeightI(x), $"wrong Hamming weight computed for x = {x}");
+        });
+        ResetAll(qs);
+    }
+
+    // some explicit cases
+    for (width, number) in [
+        (1, 1),
+        (2, 0),
+        (2, 3),
+        (8, 10),
+        (7, 99)
+    ] {
+        use qs = Qubit[width];
+
+        ApplyXorInPlace(number, qs);
+        WithHammingWeight(qs, sum => {
+            Fact(MeasureInteger(sum) == HammingWeightI(number), $"wrong Hamming weight computed for number = {number}");
+        });
+        ResetAll(qs);
+    }
+}
+
+operation TestPhasing() : Unit {
+    for theta in [1.0, 2.0, 0.0, -0.5, 5.0 * PI()] {
+        for numQubits in 1..6 {
+            Fact(CheckOperationsAreEqual(numQubits, qs => HammingWeightPhasing(theta, qs), qs => ApplyToEachA(Rz(theta, _), qs)), "Operations are not equal");
+        }
+    }
+}