Optimize gate count of Subtract to n-1 Toffolis

qualtran/bloqs/arithmetic/addition.py

@@ -129,8 +129,19 @@ def on_classical_vals(
     ) -> Dict[str, 'ClassicalValT']:
         unsigned = isinstance(self.a_dtype, (QUInt, QMontgomeryUInt))
         b_bitsize = self.b_dtype.bitsize
-        N = 2**b_bitsize if unsigned else 2 ** (b_bitsize - 1)
-        return {'a': a, 'b': int(math.fmod(a + b, N))}
+        N = 2**b_bitsize
+        if unsigned:
+            return {'a': a, 'b': int((a + b) % N)}
+
+        # Addition of signed integers can result in overflow. In most classical programming languages (e.g. C++)
+        # what happens when an overflow happens is left as an implementation detail for compiler designers.
+        # However for quantum subtraction the operation shoudl be unitary and that means that the unitary of
+        # the bloq should be a permutation matrix.
+        # If we hold `a` constant then the valid range of values of `b` [-N/2, N/2) gets shifted forward or backwards
+        # by `a`. to keep the operation unitary overflowing values wrap around. this is the same as moving the range [0, N)
+        # by the same amount modulu $N$. that is add N/2 before addition and then remove it.
+        half_n = N >> 1
+        return {'a': a, 'b': int(a + b + half_n) % N - half_n}
 
     def _circuit_diagram_info_(self, _) -> cirq.CircuitDiagramInfo:
         wire_symbols = ["In(x)"] * int(self.a_dtype.bitsize)
@@ -188,6 +199,9 @@ def decompose_from_registers(
         # reverse the order of qubits for big endian-ness.
         input_bits = quregs['a'][::-1]
         output_bits = quregs['b'][::-1]
+        if self.b_dtype.bitsize == 1:
+            yield CNOT().on(input_bits[0], output_bits[0])
+            return
         ancillas = context.qubit_manager.qalloc(self.b_dtype.bitsize - 1)[::-1]
         # Start off the addition by anding into the ancilla
         yield And().on(input_bits[0], output_bits[0], ancillas[0])

qualtran/bloqs/arithmetic/addition_test.py

@@ -316,6 +316,13 @@ def test_classical_add_k_unsigned(bitsize, k, x, cvs, ctrls, result):
     assert bloq_classical[-1] == result
 
 
+@pytest.mark.parametrize('bitsize', range(2, 5))
+def test_classical_add_signed_overflow(bitsize):
+    bloq = Add(QInt(bitsize))
+    mx = 2 ** (bitsize - 1) - 1
+    assert bloq.call_classically(a=mx, b=mx) == (mx, -2)
+
+
 # TODO: write tests for signed integer addition (subtraction)
 # https://github.com/quantumlib/Qualtran/issues/606
 @pytest.mark.parametrize('bitsize,k,x,cvs,ctrls,result', [(5, 2, 0, (1, 0), (1, 0), 2)])

qualtran/bloqs/arithmetic/subtraction.ipynb

@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "5e1ac363",
+   "id": "3f62c768",
    "metadata": {
     "cq.autogen": "title_cell"
    },
@@ -13,7 +13,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "3a2de513",
+   "id": "1e50eeb1",
    "metadata": {
     "cq.autogen": "top_imports"
    },
@@ -30,31 +30,36 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b91ab938",
+   "id": "9c974476",
    "metadata": {
     "cq.autogen": "Subtract.bloq_doc.md"
    },
    "source": [
     "## `Subtract`\n",
     "An n-bit subtraction gate.\n",
     "\n",
-    "Implements $U|a\\rangle|b\\rangle \\rightarrow |a\\rangle|a-b\\rangle$ using $4n - 4 T$ gates.\n",
+    "Implements $U|a\\rangle|b\\rangle \\rightarrow |a\\rangle|a-b\\rangle$ using $4n - 4$ T gates.\n",
     "\n",
-    "This first negates $b$ (assuming a two's complement representation), and then adds $a$ into it.\n",
+    "This construction uses the relation `a - b = ~(~a + b)` to turn the operation into addition. This relation is used in\n",
+    "[Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization](https://arxiv.org/pdf/2007.07391)\n",
+    "to turn addition into subtraction conditioned on a qubit.\n",
     "\n",
     "#### Parameters\n",
     " - `a_dtype`: Quantum datatype used to represent the integer a.\n",
     " - `b_dtype`: Quantum datatype used to represent the integer b. Must be large enough to hold the result in the output register of a - b, or else it simply drops the most significant bits. If not specified, b_dtype is set to a_dtype. \n",
     "\n",
     "#### Registers\n",
     " - `a`: A a_dtype.bitsize-sized input register (register a above).\n",
-    " - `b`: A b_dtype.bitsize-sized input/output register (register b above).\n"
+    " - `b`: A b_dtype.bitsize-sized input/output register (register b above). \n",
+    "\n",
+    "#### References\n",
+    " - [Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization, page 9](https://arxiv.org/pdf/2007.07391). \n"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "64dd238f",
+   "id": "0af034cf",
    "metadata": {
     "cq.autogen": "Subtract.bloq_doc.py"
    },
@@ -65,7 +70,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4f77ad26",
+   "id": "9156f4e7",
    "metadata": {
     "cq.autogen": "Subtract.example_instances.md"
    },
@@ -76,7 +81,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "2cd50aa6",
+   "id": "135553b1",
    "metadata": {
     "cq.autogen": "Subtract.sub_symb"
    },
@@ -89,7 +94,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "e2862687",
+   "id": "84c7207a",
    "metadata": {
     "cq.autogen": "Subtract.sub_small"
    },
@@ -101,7 +106,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "8b6584fc",
+   "id": "e98b3206",
    "metadata": {
     "cq.autogen": "Subtract.sub_large"
    },
@@ -113,7 +118,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "27312995",
+   "id": "e003f5ad",
    "metadata": {
     "cq.autogen": "Subtract.sub_diff_size_regs"
    },
@@ -122,9 +127,22 @@
     "sub_diff_size_regs = Subtract(QInt(bitsize=4), QInt(bitsize=16))"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "4aabfa98",
+   "metadata": {
+    "cq.autogen": "Subtract.sub_symp_decomposition"
+   },
+   "outputs": [],
+   "source": [
+    "n = sympy.Symbol('n')\n",
+    "sub_symp_decomposition = Subtract(QInt(bitsize=n)).decompose_bloq()"
+   ]
+  },
   {
    "cell_type": "markdown",
-   "id": "82c580af",
+   "id": "9b580c5b",
    "metadata": {
     "cq.autogen": "Subtract.graphical_signature.md"
    },
@@ -135,20 +153,20 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "4e508f72",
+   "id": "aa412d69",
    "metadata": {
     "cq.autogen": "Subtract.graphical_signature.py"
    },
    "outputs": [],
    "source": [
     "from qualtran.drawing import show_bloqs\n",
-    "show_bloqs([sub_symb, sub_small, sub_large, sub_diff_size_regs],\n",
-    "           ['`sub_symb`', '`sub_small`', '`sub_large`', '`sub_diff_size_regs`'])"
+    "show_bloqs([sub_symb, sub_small, sub_large, sub_diff_size_regs, sub_symp_decomposition],\n",
+    "           ['`sub_symb`', '`sub_small`', '`sub_large`', '`sub_diff_size_regs`', '`sub_symp_decomposition`'])"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "a282a369",
+   "id": "6fad45f3",
    "metadata": {
     "cq.autogen": "Subtract.call_graph.md"
    },
@@ -159,7 +177,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "43f1884d",
+   "id": "6a3d1dbc",
    "metadata": {
     "cq.autogen": "Subtract.call_graph.py"
    },
@@ -173,7 +191,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "12650fb1",
+   "id": "c4da55d6",
    "metadata": {
     "cq.autogen": "SubtractFrom.bloq_doc.md"
    },
@@ -200,7 +218,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "746c25eb",
+   "id": "ccc21662",
    "metadata": {
     "cq.autogen": "SubtractFrom.bloq_doc.py"
    },
@@ -211,7 +229,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "75ea545f",
+   "id": "b8182c9f",
    "metadata": {
     "cq.autogen": "SubtractFrom.example_instances.md"
    },
@@ -222,7 +240,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "e7ee57a8",
+   "id": "4b56cde9",
    "metadata": {
     "cq.autogen": "SubtractFrom.sub_from_symb"
    },
@@ -235,7 +253,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "8123a738",
+   "id": "cc45ae49",
    "metadata": {
     "cq.autogen": "SubtractFrom.sub_from_small"
    },
@@ -247,7 +265,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "e469b17a",
+   "id": "8e40b0fb",
    "metadata": {
     "cq.autogen": "SubtractFrom.sub_from_large"
    },
@@ -258,7 +276,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a32af1e3",
+   "id": "6d965b01",
    "metadata": {
     "cq.autogen": "SubtractFrom.graphical_signature.md"
    },
@@ -269,7 +287,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "0fd2dbd7",
+   "id": "093d2a0d",
    "metadata": {
     "cq.autogen": "SubtractFrom.graphical_signature.py"
    },
@@ -282,7 +300,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "47e50576",
+   "id": "420fd950",
    "metadata": {
     "cq.autogen": "SubtractFrom.call_graph.md"
    },
@@ -293,7 +311,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "4564d561",
+   "id": "7d1b9b0e",
    "metadata": {
     "cq.autogen": "SubtractFrom.call_graph.py"
    },