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Guided Hamiltonian Bloq
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‎dev_tools/qualtran_dev_tools/notebook_specs.py‎

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import qualtran.bloqs.gf_poly_arithmetic.gf2_poly_add
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import qualtran.bloqs.gf_poly_arithmetic.gf2_poly_add_k
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import qualtran.bloqs.gf_poly_arithmetic.gf_poly_split_and_join
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import qualtran.bloqs.hamiltonian_simulation.guided_hamiltonian
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import qualtran.bloqs.hamiltonian_simulation.hamiltonian_simulation_by_gqsp
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import qualtran.bloqs.mcmt.and_bloq
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import qualtran.bloqs.mcmt.controlled_via_and
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qualtran.bloqs.hamiltonian_simulation.hamiltonian_simulation_by_gqsp._Hamiltonian_Simulation_by_GQSP_DOC
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],
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),
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NotebookSpecV2(
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title='Guided Hamiltonian Problem',
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module=qualtran.bloqs.hamiltonian_simulation.guided_hamiltonian,
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bloq_specs=[
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qualtran.bloqs.hamiltonian_simulation.guided_hamiltonian._GUIDED_HAMILTONIAN_DOC,
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qualtran.bloqs.hamiltonian_simulation.guided_hamiltonian._GUIDED_HAMILTONIAN_PHASE_ESTIMATION_DOC,
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],
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),
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]
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NB_BY_SECTION = [

‎docs/bloqs/index.rst‎

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state_preparation/state_preparation_via_rotation.ipynb
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qsp/generalized_qsp.ipynb
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hamiltonian_simulation/hamiltonian_simulation_by_gqsp.ipynb
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hamiltonian_simulation/guided_hamiltonian.ipynb

‎qualtran/bloqs/hamiltonian_simulation/__init__.py‎

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# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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# See the License for the specific language governing permissions and
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# limitations under the License.
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from .guided_hamiltonian import GuidedHamiltonian, GuidedHamiltonianPhaseEstimation

‎qualtran/bloqs/hamiltonian_simulation/guided_hamiltonian.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": "a84f2ea4",
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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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"# Guided Hamiltonian Problem\n",
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"\n",
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"Section 4.4.2 Simulating the Kikuchi Hamiltonian\n",
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"\n",
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"This module contains oracles to implement the block-encoding of the Kikuchi\n",
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"Hamiltonian corresponding to an input k-XOR-SAT instance.\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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" Section 4.4.2 for algorithm. Section 2.4 for definitions and notation."
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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": "b7faaaff",
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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": "155994ae",
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"metadata": {
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"cq.autogen": "GuidedHamiltonian.bloq_doc.md"
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},
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"source": [
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"## `GuidedHamiltonian`\n",
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"Solve the Guided (Sparse) Hamiltonian Problem.\n",
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"\n",
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"Definition 4.8 (modified to accept any block-encoding):\n",
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"In the Guided Hamiltonian problem we are given the following as input:\n",
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"\n",
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"1. A $(\\sqrt{2} s, \\cdot, 0)$-block-encoding of a Hamiltonian $H$ such that $\\|H\\|_\\max \\le s$.\n",
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"2. A unitary program that prepares $|\\Psi\\rangle|0^A\\rangle$.\n",
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"3. Parameters $\\lambda \\in [-\\Lambda, \\Lambda]$, $\\alpha \\in (0, 1)$, $\\gamma \\in (0, 1]$.\n",
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"\n",
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"and we should output\n",
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"\n",
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"- YES (1) if $\\| \\Pi_{\\ge \\lambda} (H) |\\Psi\\rangle \\| \\ge \\gamma$\n",
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"- NO (0) if $\\|H\\| \\le (1 - \\alpha) \\lambda$\n",
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"\n",
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"Note that the above drops the sparsity requirement, and accepts any\n",
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"$(\\alpha_H, \\cdot, \\cdot)$-block-encoding of $H$.\n",
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"In the sparse Hamiltonian case, $\\alpha_H = s$ (where $s$ is the sparsity).\n",
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"\n",
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"Algorithm (Theorem 4.9):\n",
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" This uses phase estimation on the block-encoding of $e^{iHt}$, and then uses\n",
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" amplitude amplification to increase the success probability to $1 - o(1)$.\n",
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"\n",
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"We instead directly do phase-estimation on the qubitized walk operator for $H$.\n",
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"\n",
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"#### Parameters\n",
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" - `hamiltonian`: the block-encoding of $H$\n",
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" - `guiding_state`: the unitary that prepares $|\\Psi\\rangle$\n",
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" - `lambd`: parameter $\\lambda$\n",
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" - `alpha`: parameter $\\alpha$\n",
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" - `gamma`: parameter $\\gamma$ \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). Section 4.2 \"Guided Sparse Hamiltonian Problem\".\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": "d3d433e9",
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"metadata": {
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"cq.autogen": "GuidedHamiltonian.bloq_doc.py"
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},
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"outputs": [],
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"source": [
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"from qualtran.bloqs.hamiltonian_simulation import GuidedHamiltonian"
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]
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},
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{
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"cell_type": "markdown",
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"id": "b15c049e",
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"metadata": {
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"cq.autogen": "GuidedHamiltonian.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": "3f52ecab",
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"metadata": {
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"cq.autogen": "GuidedHamiltonian.guided_hamiltonian_symb"
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},
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"outputs": [],
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"source": [
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"import sympy\n",
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"\n",
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"from qualtran.bloqs.optimization.k_xor_sat import GuidingState, KikuchiHamiltonian, KXorInstance\n",
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"from qualtran.bloqs.state_preparation.black_box_prepare import BlackBoxPrepare\n",
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"from qualtran.symbolics import ceil, log2\n",
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"\n",
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"n, k, m, c = sympy.symbols(\"n k m c\", positive=True, integer=True)\n",
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"zeta = sympy.symbols(r\"\\zeta\", positive=True)\n",
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"\n",
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"inst_guide = KXorInstance.symbolic(n, (1 - zeta) * m, k, max_rhs=2)\n",
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"inst_solve = KXorInstance.symbolic(n, zeta * m, k, max_rhs=2)\n",
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"l = c * k\n",
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"s = l * ceil(log2(n))\n",
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"\n",
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"Psi = GuidingState(inst_guide, l)\n",
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"H = KikuchiHamiltonian(inst_solve, c * k, s)\n",
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"\n",
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"lambd, alpha, gamma = sympy.symbols(r\"\\lambda \\alpha \\gamma\", positive=True, real=True)\n",
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"guided_hamiltonian_symb = GuidedHamiltonian(H, BlackBoxPrepare(Psi), lambd, alpha, gamma)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "06377668",
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"metadata": {
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"cq.autogen": "GuidedHamiltonian.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": "e3c32914",
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"metadata": {
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"cq.autogen": "GuidedHamiltonian.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([guided_hamiltonian_symb],\n",
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" ['`guided_hamiltonian_symb`'])"
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]
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},
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{
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"cell_type": "markdown",
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"id": "7c4949b3",
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"metadata": {
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"cq.autogen": "GuidedHamiltonian.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": "85d0c98d",
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"metadata": {
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"cq.autogen": "GuidedHamiltonian.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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"guided_hamiltonian_symb_g, guided_hamiltonian_symb_sigma = guided_hamiltonian_symb.call_graph(max_depth=1, generalizer=ignore_split_join)\n",
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"show_call_graph(guided_hamiltonian_symb_g)\n",
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"show_counts_sigma(guided_hamiltonian_symb_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": "35fd7bde",
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"metadata": {
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"cq.autogen": "GuidedHamiltonianPhaseEstimation.bloq_doc.md"
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},
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"source": [
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"## `GuidedHamiltonianPhaseEstimation`\n",
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"Phase estimation subroutine for the Guided Hamiltonian Problem.\n",
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"\n",
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"Implements the algorithm $U_\\text{PE}$ used in the Guided Hamiltonian Problem.\n",
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"See the bloq `GuidedHamiltonian` for the full problem definition.\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). Section 4.2 \"Guided Sparse Hamiltonian Problem\".\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": "04b36a18",
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"metadata": {
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"cq.autogen": "GuidedHamiltonianPhaseEstimation.bloq_doc.py"
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},
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"outputs": [],
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"source": [
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"from qualtran.bloqs.hamiltonian_simulation import GuidedHamiltonianPhaseEstimation"
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]
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},
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{
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"cell_type": "markdown",
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"id": "79ebea83",
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"metadata": {
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"cq.autogen": "GuidedHamiltonianPhaseEstimation.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": "f01c6434",
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"metadata": {
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"cq.autogen": "GuidedHamiltonianPhaseEstimation.guided_phase_estimate_symb"
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},
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"outputs": [],
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"source": [
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"import sympy\n",
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"\n",
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"from qualtran.bloqs.optimization.k_xor_sat import GuidingState, KikuchiHamiltonian, KXorInstance\n",
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"from qualtran.bloqs.state_preparation.black_box_prepare import BlackBoxPrepare\n",
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"from qualtran.symbolics import ceil, log2\n",
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"\n",
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"n, k, c = sympy.symbols(\"n k c\", positive=True, integer=True)\n",
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"m_guide, m_solve = sympy.symbols(\"m_1 m_2\", positive=True, integer=True)\n",
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"\n",
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"inst_guide = KXorInstance.symbolic(n, m_guide, k, max_rhs=2)\n",
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"inst_solve = KXorInstance.symbolic(n, m_solve, k, max_rhs=2)\n",
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"l = c * k\n",
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"s = l * ceil(log2(n))\n",
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"\n",
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"Psi = GuidingState(inst_guide, l)\n",
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"H = KikuchiHamiltonian(inst_solve, c * k, s)\n",
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"\n",
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"eps, delta = sympy.symbols(r\"\\epsilon_{PE} \\delta_{PE}\", positive=True, real=True)\n",
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"guided_phase_estimate_symb = GuidedHamiltonianPhaseEstimation(\n",
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" H, BlackBoxPrepare(Psi), eps, delta\n",
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")\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": "aaff689f",
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"metadata": {
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"cq.autogen": "GuidedHamiltonianPhaseEstimation.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": "1913a0c3",
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"metadata": {
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"cq.autogen": "GuidedHamiltonianPhaseEstimation.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([guided_phase_estimate_symb],\n",
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" ['`guided_phase_estimate_symb`'])"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a72035ec",
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"metadata": {
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"cq.autogen": "GuidedHamiltonianPhaseEstimation.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": "0ce30cc4",
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"metadata": {
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"cq.autogen": "GuidedHamiltonianPhaseEstimation.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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"guided_phase_estimate_symb_g, guided_phase_estimate_symb_sigma = guided_phase_estimate_symb.call_graph(max_depth=1, generalizer=ignore_split_join)\n",
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"show_call_graph(guided_phase_estimate_symb_g)\n",
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"show_counts_sigma(guided_phase_estimate_symb_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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"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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}
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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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