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cleanup notebook
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‎qualtran/bloqs/optimization/k_xor_sat/tutorial_planted_noisy_kxor.ipynb‎

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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "0",
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np"
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]
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},
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{
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"cell_type": "markdown",
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"id": "1",
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"metadata": {},
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"source": [
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"### Implementing [Quartic quantum speedups for planted inference](https://arxiv.org/abs/2406.19378v1)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "2",
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"metadata": {},
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"source": [
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"### Problem Definition"
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]
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},
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{
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"cell_type": "markdown",
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"id": "3",
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"metadata": {},
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"source": [
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"**Definition 2.2 (simplified):**\n",
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"A kXOR instance $\\mathcal{I}$ on $n$ variables in ${0, 1}$ is a collection of $m$ constraints, each of the form\n",
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"$$ x_{c_1} \\oplus x_{c_2} \\oplus ... x_{c_k} = b $$ "
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]
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},
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{
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"cell_type": "markdown",
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"id": "4",
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"metadata": {},
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"source": [
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"**Notation 2.3**\n",
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"Random Instance: Pick each clause independently:\n",
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"- Pick $C$, a $k$ subset of $[n] = \\{1, ... n\\}$ uniformly at random.\n",
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"- Pick $b \\in {0, 1}$ uniformly at random."
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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": "5",
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"metadata": {},
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"outputs": [],
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"source": [
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"from qualtran.bloqs.optimization.k_xor_sat import KXorInstance, Constraint\n",
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"\n",
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"n, k = 10, 4\n",
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"cs = (\n",
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" Constraint((0, 1, 2, 3), -1), # read: x_0 ^ x_1 ^ x_2 ^ x_3 = 0\n",
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" Constraint((0, 2, 4, 5), 1),\n",
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" Constraint((0, 3, 4, 5), 1),\n",
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" Constraint((0, 3, 4, 5), 1),\n",
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" Constraint((1, 2, 3, 4), -1),\n",
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" Constraint((1, 3, 4, 5), -1),\n",
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" Constraint((1, 3, 4, 5), -1),\n",
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" Constraint((2, 3, 4, 5), 1),\n",
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")\n",
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"simple_inst = KXorInstance(n, k, cs)\n",
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"simple_inst"
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]
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},
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{
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"cell_type": "markdown",
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"id": "6",
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"metadata": {},
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"source": [
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"**Notation 2.4 (simplified)** Planted Instance:\n",
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"Given $\\rho \\in [0, 1]$ (the _planted advantage_),\n",
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"\n",
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"first pick a secret assignment $z \\in \\{0, 1\\}^n$.\n",
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"Now pick each clause independently by: \n",
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"- Pick $C$, a $k$ subset of $[n]$ uniformly at random.\n",
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"- Pick noise $\\eta \\in {0, 1}$, s.t. $\\eta = 0$ with probability $(1 + \\rho)/2$\n",
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"- Set $b = C(z) \\oplus \\eta$\n",
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"\n",
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"Note: when $\\rho = 0$, the noise is random, and when $\\rho = 1$, there is no noise."
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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": "7",
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"metadata": {},
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"outputs": [],
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"source": [
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"random_inst = KXorInstance.random_instance(\n",
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" n=10, \n",
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" m=20, \n",
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" k=4,\n",
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" planted_advantage=0.8,\n",
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" rng=np.random.default_rng(42),\n",
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")\n",
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"random_inst"
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]
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},
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{
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"cell_type": "markdown",
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"id": "8",
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"metadata": {},
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"source": [
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"## Problem\n",
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"\n",
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"**Problem 2.6 (Planted Noisy kXOR)**\n",
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"Given $\\rho \\in (0, 1)$, and an instance $\\mathcal{I}$ that is promised to be either drawn from the random distribution or planted distribution (with $\\rho$), distinguish which case it is."
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]
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},
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{
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"cell_type": "markdown",
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"id": "9",
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"metadata": {},
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"source": [
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"## Kikuchi Method\n",
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"This is a technique to reduce $k$XOR problems to $2$XOR problems, on an exponentially larger set of variables (say, $O(n^k)$).\n",
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"The 2XOR is known to be efficiently solvable by some spectral analysis.\n",
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"\n",
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"For this, we pick our new variables as subsets of $[n]$ of size $k$, call them $X_S$ for each subset $S$.\n",
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"There are ${n \\choose k}$ variables now, and for $k \\ll n$, this is about $O(n^k)$.\n",
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"\n",
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"The equations are of the form $X_S \\oplus X_T = b(S, T)$ for every $S, T$ with $|S \\Delta T| = k$.\n",
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"Here $b(S, T)$ is the xor of all variables in S and T (common ones cancel out, leaving only the $k$ as above)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "10",
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"metadata": {},
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"source": [
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"## Quantum Algorithm\n",
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"\n",
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"**Theorem 4.18 (simplified)**\n",
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"Let $k$ (even) and $\\rho \\in (0, 1)$ be known constants.\n",
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"\n",
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"We are given an instance $\\mathcal{I}$ which is either random or planted (with advantage $\\rho$),\n",
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"where the number of constraints $m$ is picked above a given threshold (see Alice Theorem).\n",
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"\n",
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"For a parameter $\\ell$, if we have a classical _Kikuchi style_ algorithm with complexity $\\tilde{O}(n^\\ell)$,\n",
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"then there is a quantum algorithm with $\\tilde{O}(n^{\\ell/4} m \\ell^{O{\\ell}} \\log^{\\ell/2k}n)$."
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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": "11",
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"metadata": {},
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"outputs": [],
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"source": [
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"from qualtran.bloqs.optimization.k_xor_sat.planted_noisy_kxor import PlantedNoisyKXOR\n",
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"from qualtran.drawing import show_bloq, show_call_graph, show_counts_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": null,
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"id": "12",
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"metadata": {},
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"outputs": [],
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"source": [
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"def make_algo_example():\n",
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" k = 4\n",
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" n, m = 100, 1000\n",
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" rho = 0.8\n",
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" \n",
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" c = 2 # Kikuchi param: ell = c * k\n",
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" \n",
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" # generate instance\n",
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" rng = np.random.default_rng(142)\n",
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" ell = c * k\n",
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" inst = KXorInstance.random_instance(n=n, m=m, k=k, planted_advantage=rho, rng=rng)\n",
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" algo_bloq = PlantedNoisyKXOR.from_inst(inst=inst, ell=ell, rho=rho, zeta=1 / np.log(n), rng=rng)\n",
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"\n",
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" expected_complexity = n ** (ell/4) * m * ell**ell * np.log(n)**(c//2)\n",
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"\n",
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" return algo_bloq, expected_complexity"
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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": "13",
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"metadata": {},
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"outputs": [],
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"source": [
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"bloq, cost_O_tilde = make_algo_example()"
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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": "14",
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"metadata": {},
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"outputs": [],
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"source": [
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"show_bloq(bloq)"
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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": "15",
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"metadata": {},
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"outputs": [],
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"source": [
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"g, sigma = bloq.call_graph(max_depth=6)\n",
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"show_call_graph(g)"
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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": "16",
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"metadata": {},
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"outputs": [],
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"source": [
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"from qualtran.resource_counting import get_cost_value, QECGatesCost\n",
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"\n",
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"gc = get_cost_value(bloq, QECGatesCost())\n",
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"gc.asdict()"
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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": "17",
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"metadata": {},
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"outputs": [],
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"source": [
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"(gc * (1/cost_O_tilde)).asdict()"
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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": "18",
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"metadata": {},
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"outputs": [],
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"source": [
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"f\"{cost_O_tilde:e}\""
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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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