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NoureldinYosriNoureldinYosri
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Add bloq for Binary Polynomial Multiplication (#1554)
The bloqs `BinaryPolynomialMultiplication` and `MultiplyPolyByOnePlusXk` implement karatsuba's algorithm for binary polynomial multiplication with toffoli count of $n^{\log_2{3}}$ and clifford count upper bounded by $(10 + \frac{1}{3}) n^{\log_2{3}}$. The construction of each recursively calls the other. These two bloqs are algorithms 3 and 2 from https://arxiv.org/pdf/1910.02849v2 respectively. In a final PR, I will implement algorithm 4 which implements galois multiplication in GF($2^n$) $\mod m(x)$ where $m(x)$ is an irreducible binary polynomial.
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‎dev_tools/qualtran_dev_tools/notebook_specs.py‎

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bloq_specs=[
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qualtran.bloqs.gf_arithmetic.gf2_multiplication._GF2_MULTIPLICATION_DOC,
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qualtran.bloqs.gf_arithmetic.gf2_multiplication._MULTIPLY_BY_CONSTANT_MOD_DOC,
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qualtran.bloqs.gf_arithmetic.gf2_multiplication._MULTIPLY_POLY_BY_ONE_PLUS_XK_DOC,
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qualtran.bloqs.gf_arithmetic.gf2_multiplication._BINARY_POLYNOMIAL_MULTIPLICATION_DOC,
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],
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),
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NotebookSpecV2(

‎qualtran/bloqs/gf_arithmetic/__init__.py‎

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from qualtran.bloqs.gf_arithmetic.gf2_addition import GF2Addition
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from qualtran.bloqs.gf_arithmetic.gf2_inverse import GF2Inverse
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from qualtran.bloqs.gf_arithmetic.gf2_multiplication import (
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BinaryPolynomialMultiplication,
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GF2Multiplication,
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GF2MultiplyByConstantMod,
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MultiplyPolyByOnePlusXk,
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)
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from qualtran.bloqs.gf_arithmetic.gf2_square import GF2Square

‎qualtran/bloqs/gf_arithmetic/gf2_multiplication.ipynb‎

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"show_call_graph(gf2_multiply_by_constant_modulu_g)\n",
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"show_counts_sigma(gf2_multiply_by_constant_modulu_sigma)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "0b276213",
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"metadata": {
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"cq.autogen": "MultiplyPolyByOnePlusXk.bloq_doc.md"
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},
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"source": [
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"## `MultiplyPolyByOnePlusXk`\n",
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"Out of place multiplication of $(1 + x^k) fg$\n",
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"\n",
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"Applies the transformation\n",
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"$$\n",
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"\\ket{f}\\ket{g}\\ket{h} \\rightarrow \\ket{f}{\\ket{g}}\\ket{h \\oplus (1+x^k)fg}\n",
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"$$\n",
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"\n",
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"Note: While this construction follows Algorithm2 of https://arxiv.org/abs/1910.02849v2,\n",
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"it has a slight modification. Namely that the original construction doesn't work in\n",
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"some cases where $k < n$. However reversing the order of the first set of CNOTs (line 2)\n",
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"makes the construction work for all $k \\leq n+1$.\n",
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"\n",
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"#### Parameters\n",
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" - `n`: The degree of the polynomial ($2^n$ is the size of the galois field).\n",
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" - `k`: An integer specifing the shift $1 + x^k$ (or $1 + 2^k$ for galois fields.) \n",
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"\n",
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"#### Registers\n",
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" - `f`: The first polynomial.\n",
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" - `g`: The second polyonmial.\n",
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" - `h`: The target polynomial. \n",
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"\n",
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"#### References\n",
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" - [Space-efficient quantum multiplication of polynomials for binary finite fields with sub-quadratic Toffoli gate count](https://arxiv.org/abs/1910.02849v2). Algorithm 2\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "9949a24e",
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"metadata": {
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"cq.autogen": "MultiplyPolyByOnePlusXk.bloq_doc.py"
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},
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"outputs": [],
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"source": [
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"from qualtran.bloqs.gf_arithmetic import MultiplyPolyByOnePlusXk"
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]
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},
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{
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"cell_type": "markdown",
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"id": "dd7c9a74",
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"metadata": {
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"cq.autogen": "MultiplyPolyByOnePlusXk.example_instances.md"
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},
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"source": [
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"### Example Instances"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "ad918c9f",
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"metadata": {
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"cq.autogen": "MultiplyPolyByOnePlusXk.multiplypolybyoneplusxk"
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},
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"outputs": [],
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"source": [
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"n = 5\n",
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"k = 3\n",
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"multiplypolybyoneplusxk = MultiplyPolyByOnePlusXk(n, k)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "aa939469",
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"metadata": {
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"cq.autogen": "MultiplyPolyByOnePlusXk.graphical_signature.md"
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},
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"source": [
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"#### Graphical Signature"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "648d128f",
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"metadata": {
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"cq.autogen": "MultiplyPolyByOnePlusXk.graphical_signature.py"
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},
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"outputs": [],
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"source": [
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"from qualtran.drawing import show_bloqs\n",
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"show_bloqs([multiplypolybyoneplusxk],\n",
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" ['`multiplypolybyoneplusxk`'])"
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]
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},
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{
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"cell_type": "markdown",
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"id": "4bb8dcef",
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"metadata": {
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"cq.autogen": "MultiplyPolyByOnePlusXk.call_graph.md"
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},
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"source": [
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"### Call Graph"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "0890198e",
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"metadata": {
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"cq.autogen": "MultiplyPolyByOnePlusXk.call_graph.py"
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},
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"outputs": [],
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"source": [
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"from qualtran.resource_counting.generalizers import ignore_split_join\n",
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"multiplypolybyoneplusxk_g, multiplypolybyoneplusxk_sigma = multiplypolybyoneplusxk.call_graph(max_depth=1, generalizer=ignore_split_join)\n",
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"show_call_graph(multiplypolybyoneplusxk_g)\n",
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"show_counts_sigma(multiplypolybyoneplusxk_sigma)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "9d9d1540",
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"metadata": {
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"cq.autogen": "BinaryPolynomialMultiplication.bloq_doc.md"
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},
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"source": [
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"## `BinaryPolynomialMultiplication`\n",
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"Out of place multiplication of binary polynomial multiplication.\n",
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"\n",
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"Applies the transformation\n",
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"$$\n",
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"\\ket{f}\\ket{g}\\ket{h} \\rightarrow \\ket{f}{\\ket{g}}\\ket{h \\oplus fg}\n",
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"$$\n",
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"\n",
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"The toffoli cost of this construction is $n^{\\log_2{3}}$, while CNOT count is\n",
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"upper bounded by $(10 + \\frac{1}{3}) n^{\\log_2{3}}$.\n",
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"\n",
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"#### Parameters\n",
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" - `n`: The degree of the polynomial ($2^n$ is the size of the galois field). \n",
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"\n",
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"#### Registers\n",
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" - `f`: The first polynomial.\n",
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" - `g`: The second polyonmial.\n",
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" - `h`: The target polynomial. \n",
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"\n",
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"#### References\n",
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" - [Space-efficient quantum multiplication of polynomials for binary finite fields with sub-quadratic Toffoli gate count](https://arxiv.org/abs/1910.02849v2). Algorithm 3\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "1ba3a057",
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"metadata": {
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"cq.autogen": "BinaryPolynomialMultiplication.bloq_doc.py"
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},
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"outputs": [],
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"source": [
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"from qualtran.bloqs.gf_arithmetic import BinaryPolynomialMultiplication"
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]
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},
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{
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"cell_type": "markdown",
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"id": "296d193b",
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"metadata": {
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"cq.autogen": "BinaryPolynomialMultiplication.example_instances.md"
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},
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"source": [
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"### Example Instances"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "907fcd25",
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"metadata": {
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"cq.autogen": "BinaryPolynomialMultiplication.binarypolynomialmultiplication"
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},
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"outputs": [],
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"source": [
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"n = 5\n",
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"binarypolynomialmultiplication = BinaryPolynomialMultiplication(n)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a100a193",
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"metadata": {
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"cq.autogen": "BinaryPolynomialMultiplication.graphical_signature.md"
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},
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"source": [
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"#### Graphical Signature"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "aad80309",
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"metadata": {
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"cq.autogen": "BinaryPolynomialMultiplication.graphical_signature.py"
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},
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"outputs": [],
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"source": [
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"from qualtran.drawing import show_bloqs\n",
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"show_bloqs([binarypolynomialmultiplication],\n",
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" ['`binarypolynomialmultiplication`'])"
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]
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},
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{
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"cell_type": "markdown",
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"id": "c1d23614",
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"metadata": {
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"cq.autogen": "BinaryPolynomialMultiplication.call_graph.md"
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},
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"source": [
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"### Call Graph"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "8dc799ef",
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"metadata": {
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"cq.autogen": "BinaryPolynomialMultiplication.call_graph.py"
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},
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"outputs": [],
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"source": [
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"from qualtran.resource_counting.generalizers import ignore_split_join\n",
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"binarypolynomialmultiplication_g, binarypolynomialmultiplication_sigma = binarypolynomialmultiplication.call_graph(max_depth=1, generalizer=ignore_split_join)\n",
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"show_call_graph(binarypolynomialmultiplication_g)\n",
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"show_counts_sigma(binarypolynomialmultiplication_sigma)"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"name": "python"
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.9"
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}
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},
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"nbformat": 4,

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