Merge pull request #12 from merav-aharoni/mps-community-tutorial

Matrix product state - community tutorial

Author: Abraham Asfaw

aer/graph_of_random_circuits.jpg

@@ -0,0 +1,115 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Matrix product state overview\n",
+    "\n",
+    "### Contributors\n",
+    "\n",
+    "Merav Aharoni$^{1}$, Elad Goldman$^{1}$, Yehuda Naveh$^{1}$ and Yotam Vaknin$^{1}$\n",
+    "\n",
+    "1. IBM Research Haifa, Haifa University Campus, Mount Carmel Haifa, Israel\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Introduction\n",
+    "`Tensor networks` are used as an alternate representation for a network of qubits. This representation consists of a network of tensors with connections among them. The `matrix product state` (MPS) is the simplest type of tensor network, where we have a one dimensional set of tensors, with connections between every two consecutive tensors. This representation can be used to represent a system of qubits, where each qubit is represented by one tensor. The MPS structure is often depicted graphically, as follows: \n",
+    "![](mps.jpg)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The concept of `matrix product state`was initially proposed in [1], to the best of our knowledge.  There are other papers that describe the structure in more detail, for example [2].\n",
+    "\n",
+    "A pure quantum state is usually described as a state vector, by the expression $|\\psi\\rangle =  \\sum_{i_1=0}^1 {\\ldots} \\sum_{i_n=0}^1 c_{i_1 \\ldots i_n} |i_i\\rangle {\\otimes} {\\ldots} {\\otimes} |i_n\\rangle$.\n",
+    "\n",
+    "The state vector representation implies an exponential size representation, regardless of the actual circuit. Every quantum gate operating on this representation requires exponential time and memory.\n",
+    "\n",
+    "The matrix product state reprentation offers a local representation, in the form:\n",
+    "$\\Gamma^{[1]} \\lambda^{[1]} \\Gamma^{[2]} \\lambda^{[2]}\\ldots \\Gamma^{[1]} \\lambda^{[n-1]} \\Gamma^{[n]}$. The $c_{i_1 \\ldots i_n}$ can be computed from this structure.  \n",
+    "\n",
+    "Every $\\Gamma^{[i]}$ is a tensor of complex numbers that represents qubit $i$. Every $\\lambda^{[i]}$ is a matrix of real numbers that is used to normalize the amplitudes of qubits $i$ and $i+1$. \n",
+    "\n",
+    "Gate operations can be performed by local operations on the relevant $\\Gamma$s. Computations such as expectation value can also be obtained by operations on the relevant $\\Gamma$s, without need to compute the full state vector. Measurement on a single qubit can also be performed locally on the respective $\\Gamma$, but its effect must be propagated to the remaining qubits to preserve the effects of entanglement.\n",
+    "\n",
+    "We implemented MPS as a simulation method in the qasm simulator. In our implementation, every tensor $\\Gamma^{[i]}$ is represented as a pair of matrices $\\Gamma^{[i]}[0]$ and $\\Gamma^{[i]}[1]$, that are used to represent the amplitudes of $|0\\rangle$ and $|1\\rangle$ respectively.\n",
+    "\n",
+    "Single-qubit gates operate only on the relevant tensor. For example, the Pauli-$X$ gate is implemented by simply swapping between $\\Gamma^{[i]}[0]$ and $\\Gamma^{[i]}[1]$\n",
+    "\n",
+    "Two-qubit gates operate on consecutive qubits $i$ and $i+1$. This involves a contract operation over $\\lambda^{[i-1]}$, $\\Gamma^{[i-1]}$, $\\lambda^{[i]}$, $\\Gamma^{[i+1]}$ and  $\\lambda^{[i+1]}$, that creates a single tensor. In our implementation, this tensor consists of four matrices: $\\Gamma^{[i, i+1]}[00]$, $\\Gamma^{[i, i+1]}[01]$, $\\Gamma^{[i, i+1]}[10]$ and $\\Gamma^{[i, i+1]}[11]$.\n",
+    "\n",
+    "We apply the gate to this tensor, and then decompose back to the original structure, using singular value decomposition (SVD). SVD decomposes the tensor into three matrices $U S V^{\\dagger}$, such that $U$ and $V$ are complex unitary matrices, and $S$ is a diagonal matrix of real numbers, where some of the entries on the diagonal may be $0$. The number of non-zero entries on this diagonal is named the `Schmidt coefficient`. Following a normalization step by dividing into $\\lambda^{[i-1]}$ and $\\lambda^{[i+1]}$ respectively, $U$ becomes $\\Gamma^{[i]}$, $V$ becomes $\\Gamma^{[i+1]}$. $S$ becomes the new $\\lambda^{[i]}$.\n",
+    "\n",
+    "Note that two-qubit operations may increase the size of the respective tensors. The sizes are determined by the Schmidt coefficient. Intuitively, the Schmidt coefficients provide a measurement of the entanglement of the system, and therefore determine the performance of the circuit.\n",
+    "\n",
+    "Gates that involve two qubits that are not consecutive, require a series of swap gates to bring the two qubits next to each other and then the reverse swaps, to bring the qubits back to their original positions. \n",
+    "\n",
+    "In the worst case, the tensors may grow exponentially. However, the size of the overall structure remains 'small' for circuits that do not have 'many' two-qubit gates. This allows much more efficient operations in circuits with relatively 'low' entanglement. Characterizing when to use this method over other methods is a subject of current research.\n",
+    "\n",
+    "We referred to several additional papers, for example [3], [4], [5] to provide further insight into understanding and implementing MPS as a simulation method in Qiskit."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## References\n",
+    "\n",
+    "1. G. Vidal, *Efficient classical simulation of slightly entangled quantum computations*, Phys. Rev. Lett. 91 (2003) 147902, https://arxiv.org/abs/quant-ph/0301063.\n",
+    "\n",
+    "2. U. Schollwöck, *The density-matrix renormalization group in the age of matrix product states*, Annals of Physics 326 (1) (2011) 96 - 192, january 2011, Special Issue, https://arxiv.org/abs/1008.3477.\n",
+    "\n",
+    "3. R. Orús, *A practical introduction to tensor networks: Matrix product states and projected entangled pair states*, Annals of Physics 349 (2014) 117 - 158, https://arxiv.org/abs/1306.2164.\n",
+    "\n",
+    "4. J. Biamonte and V. Bergholm, *Tensor networks in a nutshell*, https://arxiv.org/abs/1306.2164.\n",
+    "\n",
+    "5. D. Jaschke, M. L. Walla, L. D. Carr, *Open source Matrix Product States: Opening ways to simulate entangled many-body quantum systems in one dimension*, Computer Physics Communications 225, 59-91, https://arxiv.org/abs/1703.00387."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Using the matrix product state simulation method\n",
+    "The matrix product state simulation method is invoked in the qasm simulator by setting the `simulation_method` in the qasm_simulator to 'matrix_product_state'.\n",
+    "We demonstrate this using simple use-cases in the tutorial `matrix_product_state_simulation_method`.\n",
+    "\n",
+    "As we show there, performance of circuits using the MPS simulation method can be much more efficient than using the Statevector simulation method, as long as:\n",
+    "1. The internal data structures don't grow too much, i.e., entanglement is not too high.\n",
+    "2. We do not wish to compute the state vector, nor all the amplitudes, because this computation is always exponential in the number of qubits.\n",
+    "\n",
+    "We demonstrate this in the following graph. We ran these two simulation methods on a set of randomly generated circuits, where the percentage of two-qubit gates is 0.1. The depth of the circuits is kept constant at 120 gates. The final computation of the circuit is the expectation value of random Pauli gates on 5 random qubits.\n",
+    "\n",
+    "![](graph_of_random_circuits.jpg)"
+   ]
+  }
+ ],
+ "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": "3.6.8"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}