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estimation prob workflow
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‎Estimation problem workflow.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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"metadata": {},
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"source": [
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"# Estimation problem workflow\n",
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"\n",
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"For a multivariate random variable $X$ and a real-valued function $f$ the goal is to compute \n",
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"\n",
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"$$\n",
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" \\mathbb{E}[f(X)], \\text{ where } f: \\mathbb{R}^n \\mapsto I \\subset \\mathbb R\n",
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"$$\n",
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"\n",
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"Here, this is achieved using quantum amplitude estimation (QAE), based on a $\\mathcal{A}$ operator in the format\n",
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"\n",
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"$$\n",
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" \\mathcal{A}|0\\rangle = \\sqrt{1 - a}|\\Psi_0\\rangle + \\sqrt{a}|\\Psi_1\\rangle\n",
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" = \\sum_{\\hat x = 0}^{2^n - 1} \\sqrt{p_{\\phi(\\hat x)} (1 - \\hat f(\\phi(\\hat x)))} |x\\rangle |0\\rangle\n",
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" + \\sum_{\\hat x = 0}^{2^n - 1} \\sqrt{p_{\\phi(\\hat x)} \\hat f(\\phi(\\hat x))} |x\\rangle |1\\rangle\n",
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"$$\n",
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"\n",
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"To revert the scalings from $f$ to $\\hat f$ and possible post-processing from the function approximation, the result of QAE, $\\tilde a$, is mapped to the result via a post-processing function $\\eta$\n",
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"\n",
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"$$\n",
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"\\mathbb{E}[f(X)] = \\eta(a) \\approx \\eta(\\tilde a)\n",
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"$$\n",
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"\n",
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"**Notes:**\n",
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"\n",
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"An amplitude function $F$ of a function $\\hat f$ is a mapping\n",
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"\n",
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"$$\n",
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"F: |x\\rangle|0\\rangle \\mapsto \\sqrt{1 - \\hat f(x)}|x\\rangle|0\\rangle + \\sqrt{\\hat f(x)}|x\\rangle|1\\rangle\n",
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"$$\n",
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"\n",
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"where\n",
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"\n",
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"$$\n",
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"\\hat f: \\mathbb{N}_0 \\rightarrow [0, 1].\n",
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"$$\n",
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"\n",
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"If we have a function $f$ that does not act on these spaces we must normalize the image and apply an affine transformation from the input domain to $\\mathbb{N}_0$. Let the function $f$ be defined on $2^n$ equidistant points in $[a, b]$:\n",
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"\n",
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"$$\n",
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"f: [a, b] \\rightarrow [c, d]\n",
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"$$\n",
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"\n",
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"then the rescaled version is\n",
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"\n",
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"$$\n",
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"\\hat f(x) = \\frac{f(\\phi(x)) - c}{d - c}\n",
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"$$\n",
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"\n",
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"where $\\phi(x) = a + (b - a) x / 2^n$ is the affine transformation.\n",
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"\n",
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"The normalized function $\\hat f$ can be implemented in different manners. In general this requires a post-processing step, $\\zeta$:\n",
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"\n",
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"$$\n",
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"\\hat f \\approx \\zeta(T\\hat f)\n",
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"$$\n",
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"\n",
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"where $T\\hat f$ is the amplitude we can map onto the qubits.\n",
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"One example is a Taylor approximation of $\\sin^2$ which can be mapped to qubits using RY gates.\n",
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"\n",
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"Examples:\n",
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"\n",
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"* quadratic terms can be approximated with\n",
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" $$\n",
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" ax^2 \\approx \\sin^2(c\\sqrt{a}x) / c^2\n",
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" $$\n",
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" where the sine part is estimated with amplitude estimation and the rescaling is $\\zeta(x) = x/c^2$\n",
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" \n",
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"* linear terms can be approximated as\n",
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" $$\n",
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" ax \\approx \\zeta\\left( \\sin^2\\left(\\frac{\\pi}{4} + \\frac{\\pi c}{2} \\left(f(x) - \\frac{1}{2}\\right)\\right) \\right) \n",
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" $$\n",
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" with\n",
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" $$\n",
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" \\zeta(x) = \\frac{2}{\\pi c}\\left(x - \\frac{1}{2}\\right) + \\frac{1}{2}\n",
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" $$\n",
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"\n",
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"Let $\\tilde a$ be the output of amplitude estimation, then in general we have to apply a post-processing in the form of\n",
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"\n",
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"$$\n",
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"c + (d - c) \\zeta(\\tilde a)\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Defining the random variable\n",
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"\n",
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"The distribution of the random variable is loaded from the circuit library. The function will be evaluated on the x-values ``[0, ..., 2 ** num_qubits - 1]``. "
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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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"metadata": {},
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"outputs": [],
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"source": [
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"from qiskit.circuit.library import NormalDistribution\n",
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"X = NormalDistribution(num_qubits=3, mu=1, sigma=0.5, bounds=(0, 2))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Defining the objective function"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Workflow #1\n",
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"\n",
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"Similar to the oracle compiler: take a Python function and synthesize the circuit automatically."
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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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"metadata": {},
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"outputs": [],
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"source": [
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"def classical_f(x):\n",
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" if x > 1:\n",
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" return x ** 2\n",
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" return x\n",
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"\n",
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"\n",
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"from qiskit.circuit.library import AmplitudeFunction\n",
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"f = AmplitudeFunction(classical_f, domain=(0, 2)) # domain must be the same as the bounds of the probability dist\n",
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"post_processing = f.post_processing # c + (d - c) zeta"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Workflow #2\n",
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"\n",
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"Similar to the `UnivariatePiecewiseLinearObjective`, specify the function and construct the circuit from this information. Includes rescalings."
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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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"metadata": {},
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"outputs": [],
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"source": [
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"from qiskit.circuit.library import LinearAmplitudeFunction\n",
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"f = LinearAmplitudeFunction(num_state_qubits=3, \n",
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" slope=[-1, 1],\n",
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" offset=[1, 0],\n",
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" breakpoints=[0, 1],\n",
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" domain=(0, 2))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Workflow #3\n",
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"\n",
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"Manual."
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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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"metadata": {},
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"outputs": [],
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"source": [
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"from qiskit.circuit.library import PiecewiseLinearPauliRotations\n",
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"\n",
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"slopes = np.array([-1, 1])\n",
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"offsets = np.array([1, 0])\n",
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"breakpoints = np.array([0, 1])\n",
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"domain = (0, 2)\n",
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"num_state_qubits = 3\n",
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"N = 2 ** num_state_qubits\n",
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"\n",
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"# apply rescaling to the right domain \n",
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"phi_inv = lambda x: (x - domain[0]) / (domain[1] - domain[0]) * N\n",
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"breakpoints = phi_inv(breakpoints)\n",
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"slopes = (domain[1] - domain[0]) / (N - 1) * slopes\n",
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"# offsets remain the same\n",
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"\n",
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"# apply normalization of the image space\n",
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"offsets = (offsets - fmin) / (fmax - fmin)\n",
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"slopes = slopes / (fmax - fmin)\n",
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"\n",
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"# apply taylor \n",
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"offsets *= c * np.pi / 2\n",
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"slopes *= c * np.pi / 2\n",
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"\n",
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"f = PiecewiseLinearPauliRotations(num_state_qubits, \n",
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" breakpoints=breakpoints,\n",
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" slopes=2 * slopes, \n",
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" offsets=2 * offsets) "
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Conclusion\n",
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"\n",
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"Option #2 makes sense for now. This means porting most of the `UnivariatePiecewiseLinearObjective` to the circuit library."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Defining the A operator\n",
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"\n",
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"### Workflow #1\n",
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"\n",
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"Stack together the function circuit and probability distribution circuit."
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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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"metadata": {},
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"outputs": [],
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"source": [
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"from qiskit.circuit import QuantumCircuit\n",
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"\n",
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"A = QuantumCircuit(f.num_qubits)\n",
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"A.compose(X, inplace=True)\n",
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"A.compose(f, inplace=True)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Workflow #2\n",
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"\n",
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"Can introduce a class for that but I don't really think the work it does justifies a class, see rather the section about encapsulating the entire workflow."
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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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"metadata": {},
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"outputs": [],
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"source": [
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"from qiskit.circuit.library import EstimationProblem # or aqua algorithms\n",
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"\n",
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"A = EstimationProblem(f, X) # but here f is not the circuit but a function! and we do all the synthesisa"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Run QAE\n",
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"\n",
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"QAE in the basic form takes the $\\mathcal{A}$ operator and a post processing function."
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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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"metadata": {},
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"outputs": [],
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"source": [
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"from qiskit import Aer\n",
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"from qiskit.aqua.algorithms import AmplitudeEstimation\n",
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"\n",
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"ae = AmplitudeEstimation(num_eval_qubits=4, state_in=A, post_processing=f.post_processing)\n",
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"result = ae.run(Aer.get_backend('qasm_simulator'))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Encapsulating the workflow\n",
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"\n",
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"Add a class that is fully equivalent to `UnivariatePiecewiseLineaObjective`? Essentially this means accepting the probabilit"
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]
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}
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],
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"kernelspec": {
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"display_name": "Python 3.7.7 64-bit ('py377': conda)",
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