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planted noisy kXOR algorithm
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‎dev_tools/qualtran_dev_tools/notebook_specs.py‎

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qualtran.bloqs.optimization.k_xor_sat.kikuchi_block_encoding._KIKUCHI_HAMILTONIAN_DOC
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],
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),
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NotebookSpecV2(
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title='Algorithm: Planted Noisy kXOR',
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module=qualtran.bloqs.optimization.k_xor_sat.planted_noisy_kxor,
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bloq_specs=[
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qualtran.bloqs.optimization.k_xor_sat.planted_noisy_kxor._PLANTED_NOISY_KXOR_DOC
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],
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),
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]
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# --------------------------------------------------------------------------

‎docs/bloqs/index.rst‎

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optimization/k_xor_sat/kikuchi_adjacency_list.ipynb
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optimization/k_xor_sat/kikuchi_adjacency_matrix.ipynb
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optimization/k_xor_sat/kikuchi_block_encoding.ipynb
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optimization/k_xor_sat/planted_noisy_kxor.ipynb
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.. toctree::
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:maxdepth: 2

‎qualtran/bloqs/optimization/k_xor_sat/__init__.py‎

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from .kikuchi_guiding_state import GuidingState, SimpleGuidingState
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from .kxor_instance import Constraint, KXorInstance
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from .load_kxor_instance import LoadConstraintScopes, LoadUniqueScopeIndex, PRGAUniqueConstraintRHS
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from .planted_noisy_kxor import AliceTheorem, PlantedNoisyKXOR

‎qualtran/bloqs/optimization/k_xor_sat/planted_noisy_kxor.ipynb‎

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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "ee412c69",
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"metadata": {
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"cq.autogen": "title_cell"
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},
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"source": [
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"# Algorithm: Planted Noisy kXOR"
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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": "df5a7866",
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"metadata": {
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"cq.autogen": "top_imports"
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},
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"outputs": [],
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"source": [
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"from qualtran import Bloq, CompositeBloq, BloqBuilder, Signature, Register\n",
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"from qualtran import QBit, QInt, QUInt, QAny\n",
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"from qualtran.drawing import show_bloq, show_call_graph, show_counts_sigma\n",
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"from typing import *\n",
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"import numpy as np\n",
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"import sympy\n",
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"import cirq"
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]
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},
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{
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"cell_type": "markdown",
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"id": "36559e8d",
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"metadata": {
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"cq.autogen": "PlantedNoisyKXOR.bloq_doc.md"
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},
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"source": [
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"## `PlantedNoisyKXOR`\n",
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"Algorithm for Planted Noisy kXOR.\n",
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"\n",
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"Problem (Problem 2.6 of Ref [1]):\n",
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"\n",
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"Given a noisy-kXOR instance $\\hat{\\mathcal{I}}$ which is drawn either:\n",
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"\n",
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"1. with planted advantage $\\rho$, from $\\tilde\\mathcal{P}^{z}_{n, k}(m, \\rho)$.\n",
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"2. at random, from $\\tilde\\mathcal{R}_{n, k}(m)$.\n",
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"\n",
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"output a single bit such that it is whp `1` in case 1, and `0` in case 2.\n",
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"\n",
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"Algorithm (Section 4.4, Theorem 4.18):\n",
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"We first split the instance into $\\hat{\\mathcal{I}} = \\mathcal{I} \\cup \\mathcal{I}_\\text{guide}$,\n",
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"by placing each constraint independently in $\\mathcal{I}$ with prob. $1 - \\zeta$,\n",
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"otherwise in $\\mathcal{I}_\\text{guide}$.\n",
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"$\\zeta$ is picked to be $1 / \\ln n$.\n",
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"\n",
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"#### Parameters\n",
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" - `inst_guide`: The subset of contraints $\\mathcal{I}_\\text{guide}$ for the guided state.\n",
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" - `inst_solve`: The subset of constraints $\\mathcal{I}$ for eigenvalue estimation.\n",
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" - `ell`: Kikuchi parameter $\\ell$.\n",
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" - `rho`: the planted advantage $\\rho$ in the planted case. \n",
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"\n",
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"#### References\n",
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" - [Quartic quantum speedups for planted inference](https://arxiv.org/abs/2406.19378v1). \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": "f930da5f",
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"metadata": {
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"cq.autogen": "PlantedNoisyKXOR.bloq_doc.py"
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},
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"outputs": [],
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"source": [
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"from qualtran.bloqs.optimization.k_xor_sat import PlantedNoisyKXOR"
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]
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},
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{
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"cell_type": "markdown",
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"id": "18f43abd",
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"metadata": {
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"cq.autogen": "PlantedNoisyKXOR.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": "1a28aa9c",
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"metadata": {
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"cq.autogen": "PlantedNoisyKXOR.solve_planted_symbolic"
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},
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"outputs": [],
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"source": [
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"from qualtran.bloqs.optimization.k_xor_sat import KXorInstance\n",
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"from qualtran.symbolics import HasLength\n",
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"\n",
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"n, m = sympy.symbols(\"n m\", positive=True, integer=True)\n",
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"k = sympy.symbols(\"k\", positive=True, integer=True, even=True)\n",
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"c = sympy.symbols(\"c\", positive=True, integer=True)\n",
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"ell = c * k\n",
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"rho = sympy.Symbol(r\"\\rho\", positive=True, real=True)\n",
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"\n",
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"inst = KXorInstance.symbolic(n, m, k)\n",
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"zeta = 1 / ln(n)\n",
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"solve_planted_symbolic = PlantedNoisyKXOR(\n",
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" inst_guide=inst.subset(HasLength((1 - zeta) * m)),\n",
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" inst_solve=inst.subset(HasLength(zeta * m)),\n",
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" ell=ell,\n",
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" rho=rho,\n",
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")"
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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": "2df9b4ea",
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"metadata": {
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"cq.autogen": "PlantedNoisyKXOR.solve_planted"
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},
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"outputs": [],
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"source": [
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"from qualtran.bloqs.optimization.k_xor_sat import KXorInstance\n",
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"\n",
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"rng = np.random.default_rng(42)\n",
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"n, m, k = 50, 1000, 4\n",
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"ell = k\n",
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"rho = 0.8\n",
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"\n",
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"inst = KXorInstance.random_instance(n, m, k, planted_advantage=rho, rng=rng)\n",
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"solve_planted = PlantedNoisyKXOR.from_inst(inst, ell=ell, rho=rho, zeta=0.1, rng=rng)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "505c1e9c",
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"metadata": {
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"cq.autogen": "PlantedNoisyKXOR.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": "2ead66fe",
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"metadata": {
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"cq.autogen": "PlantedNoisyKXOR.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([solve_planted_symbolic, solve_planted],\n",
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" ['`solve_planted_symbolic`', '`solve_planted`'])"
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]
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},
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{
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"cell_type": "markdown",
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"id": "bd068fb3",
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"metadata": {
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"cq.autogen": "PlantedNoisyKXOR.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": "946fca15",
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"metadata": {
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"cq.autogen": "PlantedNoisyKXOR.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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"solve_planted_symbolic_g, solve_planted_symbolic_sigma = solve_planted_symbolic.call_graph(max_depth=1, generalizer=ignore_split_join)\n",
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"show_call_graph(solve_planted_symbolic_g)\n",
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"show_counts_sigma(solve_planted_symbolic_sigma)"
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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": 13,
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"id": "2c8e9045-060a-4386-99be-5af7ee653fd6",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/markdown": [
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"#### Counts totals:\n",
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" - `Adjoint(subbloq=GuidedHamiltonianPhaseEstimation)`: $\\displaystyle \\left\\lceil{\\frac{202.020202020202 c^{0.5} k^{0.5} \\left(c k\\right)^{\\frac{c k}{2}}}{Part_{k}(\\ell)^{0.5} \\rho^{0.5} \\left(\\frac{\\left(\\frac{m \\left(1 - \\frac{1}{\\operatorname{log}_{2}{\\left(n \\right)}}\\right) + \\frac{m}{\\operatorname{log}_{2}{\\left(n \\right)}}}{{\\binom{n}{k}}}\\right)^{c} \\left(\\frac{\\rho^{2} m}{\\left(m \\left(1 - \\frac{1}{\\operatorname{log}_{2}{\\left(n \\right)}}\\right) + \\frac{m}{\\operatorname{log}_{2}{\\left(n \\right)}}\\right) \\operatorname{log}_{2}{\\left(n \\right)}}\\right)^{c}}{\\log{\\left(n \\right)}^{2}}\\right)^{0.5}}}\\right\\rceil$\n",
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" - `GuidedHamiltonianPhaseEstimation`: $\\displaystyle \\left\\lceil{\\frac{202.020202020202 c^{0.5} k^{0.5} \\left(c k\\right)^{\\frac{c k}{2}}}{Part_{k}(\\ell)^{0.5} \\rho^{0.5} \\left(\\frac{\\left(\\frac{m \\left(1 - \\frac{1}{\\operatorname{log}_{2}{\\left(n \\right)}}\\right) + \\frac{m}{\\operatorname{log}_{2}{\\left(n \\right)}}}{{\\binom{n}{k}}}\\right)^{c} \\left(\\frac{\\rho^{2} m}{\\left(m \\left(1 - \\frac{1}{\\operatorname{log}_{2}{\\left(n \\right)}}\\right) + \\frac{m}{\\operatorname{log}_{2}{\\left(n \\right)}}\\right) \\operatorname{log}_{2}{\\left(n \\right)}}\\right)^{c}}{\\log{\\left(n \\right)}^{2}}\\right)^{0.5}}}\\right\\rceil + 1$\n",
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" - `MultiControlZ`: $\\displaystyle \\left\\lceil{\\frac{202.020202020202 c^{0.5} k^{0.5} \\left(c k\\right)^{\\frac{c k}{2}}}{Part_{k}(\\ell)^{0.5} \\rho^{0.5} \\left(\\frac{\\left(\\frac{m \\left(1 - \\frac{1}{\\operatorname{log}_{2}{\\left(n \\right)}}\\right) + \\frac{m}{\\operatorname{log}_{2}{\\left(n \\right)}}}{{\\binom{n}{k}}}\\right)^{c} \\left(\\frac{\\rho^{2} m}{\\left(m \\left(1 - \\frac{1}{\\operatorname{log}_{2}{\\left(n \\right)}}\\right) + \\frac{m}{\\operatorname{log}_{2}{\\left(n \\right)}}\\right) \\operatorname{log}_{2}{\\left(n \\right)}}\\right)^{c}}{\\log{\\left(n \\right)}^{2}}\\right)^{0.5}}}\\right\\rceil$\n",
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" - `ZGate`: $\\displaystyle \\left\\lceil{\\frac{202.020202020202 c^{0.5} k^{0.5} \\left(c k\\right)^{\\frac{c k}{2}}}{Part_{k}(\\ell)^{0.5} \\rho^{0.5} \\left(\\frac{\\left(\\frac{m \\left(1 - \\frac{1}{\\operatorname{log}_{2}{\\left(n \\right)}}\\right) + \\frac{m}{\\operatorname{log}_{2}{\\left(n \\right)}}}{{\\binom{n}{k}}}\\right)^{c} \\left(\\frac{\\rho^{2} m}{\\left(m \\left(1 - \\frac{1}{\\operatorname{log}_{2}{\\left(n \\right)}}\\right) + \\frac{m}{\\operatorname{log}_{2}{\\left(n \\right)}}\\right) \\operatorname{log}_{2}{\\left(n \\right)}}\\right)^{c}}{\\log{\\left(n \\right)}^{2}}\\right)^{0.5}}}\\right\\rceil$"
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],
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"text/plain": [
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"<IPython.core.display.Markdown object>"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"_, sigma = solve_planted_symbolic.call_graph(max_depth=2, generalizer=ignore_split_join)\n",
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"show_counts_sigma(sigma) # inverse of Eq. 150"
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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 (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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"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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"nbformat_minor": 5
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}

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