{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "![Quantum Random Walks](/images/SymmetricQRW.png)\n", "\n", "# Quantum Random Walks\n", "\n", "Here we demonstrate some simple code to explore quantum random walks. To play with this notebook, click on:\n", "\n", "[![Binder](https://mybinder.org/badge.svg)](https://mybinder.org/v2/gh/mforbes/binder_quantum_random_walk/master?filepath=quantum-random-walks.ipynb)\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/javascript": [], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\n", "\n", "

This cell contains some definitions for equations and some CSS for styling\n", " the notebook. If things look a bit strange, please try the following:\n", "

    \n", "
  • Choose \"Trust Notebook\" from the \"File\" menu.
  • \n", "
  • Re-execute this cell.
  • \n", "
  • Reload the notebook.
  • \n", "
\n", "

\n", "
\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import mmf_setup; mmf_setup.nbinit()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Single Particle in a Box" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We start with a model of a single particle in a periodic box (Section III.A from [Kempe:2003]). Equation numbers correspond to [Kempe:2003.] The wavefunction is then simply a vector on our set of $N$ abscissa. This wavefunction lives in our original Hilbert space $\\mathcal{H}_P$ spanned by the basis of \"position\" eigenstates $\\ket{n}$ such that $\\abs{\\braket{\\psi|n}}^2$ is the probability that the particle is on the $n$th run of the ladder.\n", "$\n", " \\newcommand{\\ua}{\\uparrow}\n", " \\newcommand{\\da}{\\downarrow}\n", "$\n", "The full quantum random walk problem lives in an axuilliary Hilbert space $\\mathcal{H} = \\mathcal{H}_C\\otimes \\mathcal{H}_P$ where $\\mathcal{H}_C$ is a two-dimensional Hilbert space spanned by two spin states $\\ket{\\ua}$, $\\ket{\\da}$. The idea here is that motion up the ladder corresponds with being in the spin state $\\ket{\\ua}$ while motion down the ladder corresponds to $\\ket{\\da}$. The unitary evolution operator is thus:\n", "\n", "\\begin{gather}\n", " \\mat{S} = \\ket{\\ua}\\bra{\\ua}\\otimes \\sum_{n}\\ket{n+1}\\bra{n} + \\ket{\\da}\\bra{\\da}\\otimes \\sum_{n}\\ket{n-1}\\bra{n}.\n", " \\tag{12}\n", "\\end{gather}" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Populating the interactive namespace from numpy and matplotlib\n" ] } ], "source": [ "%pylab inline\n", "from collections import namedtuple\n", "\n", "def tprod(A, B):\n", " \"\"\"Return the tensor product of A and B.\"\"\"\n", " dim = len(A.shape)\n", " if dim == 1:\n", " AB = np.einsum('a,i->ai', A, B)\n", " elif len(A.shape) == 2:\n", " AB = np.einsum('ab,ij->aibj', A, B)\n", " else:\n", " raise NotImplementedError\n", " return AB.reshape(np.prod(AB.shape[:dim]),\n", " np.prod(AB.shape[dim:]))\n", "\n", "# Raise a matrix to a power\n", "mpow = np.linalg.matrix_power\n", "\n", "def apply(A, psi):\n", " \"\"\"Return `A(psi)`, applying the operator A to psi.\"\"\"\n", " return A.dot(psi.ravel()).reshape(psi.shape)\n", "\n", "def get_U(N=256):\n", " \"\"\"Return the unitary evolution operators for an N site lattice.\"\"\"\n", " x = np.arange(N) - N//2\n", " n = np.eye(N) # Position basis vectors\n", " n_plus_1 = np.hstack([n[:, 1:], n[:, :1]])\n", " n_minus_1 = np.hstack([n[:, -1:], n[:, :-1]])\n", "\n", " ua, da = s = np.eye(2) # Spin basis vectors\n", "\n", " S = (tprod(ua[:, None]*ua[None, :], n_plus_1.dot(n.T)) + \n", " tprod(da[:, None]*da[None, :], n_minus_1.dot(n.T)))\n", "\n", " # Hadamard coin (13)\n", " C = np.array([[1, 1], \n", " [1, -1]])/np.sqrt(2)\n", "\n", " # Unbiased coin (17)\n", " Y = np.array([[1, 1j], \n", " [1j, 1]])/np.sqrt(2)\n", " U_C = S.dot(tprod(C, np.eye(N)))\n", " U_Y = S.dot(tprod(Y, np.eye(N)))\n", " assert np.allclose(np.eye(2*N), S.dot(S.T.conj()))\n", " assert np.allclose(np.eye(2*N), U_C.dot(U_C.T.conj()))\n", " assert np.allclose(np.eye(2*N), U_Y.dot(U_Y.T.conj()))\n", " Results = namedtuple('Results', ['N', 'x', 'U_C', 'U_Y'])\n", " return Results(N=N, x=x, U_C=U_C, U_Y=U_Y)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we start from the state $\\ket{\\da}\\otimes\\ket{0}$ and evolve with the unitary evolution operator:\n", "\n", "$$\n", " \\mat{U} = \\mat{S}\\cdot(\\mat{C}\\otimes\\mat{1}).\n", "$$\n", "\n", "Doing this 100 times, we obtain Fig. 5 from [Kempe:2003]." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5,1,'Fig. 5 from [Kempte:2003]')" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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KUqjktQCoC0hRFCUzeS0AubmAvFoHoChKQZLnAmBbAFmygIp8Hl0PQFGUgiSvBcC5sWdq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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "res = get_U(N=200)\n", "psi = np.zeros((2, res.N))\n", "psi[1, res.N//2] = 1\n", "\n", "psi_100 = apply(mpow(res.U_C, 100), psi)\n", "\n", "# Plot only even indices as the odd sites are not occupied\n", "# Note: this depends on the value of N chosen.\n", "plt.plot(res.x[0::2], (abs(psi_100)**2).sum(axis=0)[0::2])\n", "plt.axis([-80, 80, 0, 0.1])\n", "plt.title(\"Fig. 5 from [Kempte:2003]\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Starting from a symmetric initial condition we obtain Fig. 6. (It appears they forgot to normalize the initial state properly.)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5,1,'Fig. 5 from [Kempte:2003]')" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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kMnS4WQvRUNDmGRhGE/FidpZaWh4TgwZS4iYyy8AwmkbCZxm0h51BmolBnprEQEQuEpGt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\nmyftde6Dg3GAQvM45twEXsl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k2bRvkKd2DbCnf5RXjsbZPxgnnszQEQ3SEQ3RFQuxdG4b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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "psi = np.zeros((2, res.N), dtype=complex)\n", "psi[0, res.N//2] = 1 #/np.sqrt(2) # Kempte: seems to miss this normalization\n", "psi[1, res.N//2] = 1j #/np.sqrt(2)\n", "\n", "psi = apply(mpow(res.U_C, 100), psi)\n", "\n", "# Plot only odd indices as the even sites are not occupied.\n", "plt.plot(res.x[0::2], (abs(psi)**2).sum(axis=0)[0::2])\n", "plt.axis([-100, 100, 0, 0.16])\n", "plt.title(\"Fig. 5 from [Kempte:2003]\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*MMF Note: Using the coin $Y$ from (17) does not seem to give symmetric evolution like they claim... not sure why.*" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[-80, 80, 0, 0.1]" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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xgJgFYBeCJR5TFEUZ6+SlADgWQK6FYGBlAqX6990tABUARVHyg7wWgFwLwZyxaRZAKMEC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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "psi = np.zeros((2, res.N), dtype=complex)\n", "psi[1, res.N//2] = 1.0\n", "\n", "psi_100 = apply(mpow(res.U_Y, 100), psi)\n", "\n", "# Plot only odd indices as the even sites are not occupied\n", "plt.plot(res.x[0::2], (abs(psi_100)**2).sum(axis=0)[0::2])\n", "plt.axis([-80, 80, 0, 0.1])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's look at the evolution now, anticipating a comparison with [Dadras:2018]." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5,1,'Fig. 1 from [Dadras:2018]')" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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Azz6/e63xD/z4Y7XGH3ruVrXGB9jl4nrXQtGus2uNzxNP1hsfYFb9vwfuntzL7fSRR0Q0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\nKu4nMSCGAAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "res = get_U(N=20)\n", "psi = np.zeros((2, res.N), dtype=complex)\n", "psi[0, res.N//2] = 1/np.sqrt(2)\n", "psi[1, res.N//2-1] = 1/np.sqrt(2)\n", "\n", "psis = [psi]\n", "Ts = np.arange(12)\n", "for T in Ts[1:]:\n", " psis.append(apply(res.U_C, psis[-1]))\n", "\n", "psis = np.asarray(psis)\n", "ns = (abs(psis)**2).sum(axis=1)\n", "plt.pcolormesh(Ts, res.x, ns.T)\n", "plt.colorbar()\n", "plt.title(\"Fig. 1 from [Dadras:2018]\")\n", "plt.savefig(\"images/SymmetricQRW.png\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One can get a biased quantum walk by starting from a different initial state, or by using a different coin. In [Dadras:2018] they use a different coin. Try to reproduce their Fig. 2." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5,1,'Not quite Fig. 2 from [Dadras:2018]')" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "res = get_U(N=20)\n", "psi = np.zeros((2, res.N), dtype=complex)\n", "psi[0, res.N//2] = 1\n", "psi[1, res.N//2-1] = 0\n", "\n", "psis = [psi]\n", "Ts = np.arange(12)\n", "for T in Ts[1:]:\n", " psis.append(apply(res.U_C, psis[-1]))\n", "\n", "psis = np.asarray(psis)\n", "ns = (abs(psis)**2).sum(axis=1)\n", "plt.pcolormesh(Ts, res.x, ns.T)\n", "plt.colorbar()\n", "plt.title(\"Not quite Fig. 2 from [Dadras:2018]\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# References" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* [**[Kempe:2008]**: Quantum random walks - an introductory overview](https://arxiv.org/abs/quant-ph/0303081)\n", "* [**[Dadras:2018]**: Realization of a quantum walk in momentum space with a Bose-Einstein condensate](https://arXiv.org/abs/1802.08160)" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.5.4" }, "nikola": { "date": "2018-03-01 17:41:34 UTC-08:00", "description": "", "link": "", "slug": "quantum-random-walks", "tags": "", "title": "Quantum Random Walks", "type": "text" } }, "nbformat": 4, "nbformat_minor": 2 }