cosmetic changes

Author: merav-aharoni

aer/matrix_product_state.ipynb

@@ -47,7 +47,7 @@
     "\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 entaglement of the system, and therefore determine the performance of the circuit.\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",
@@ -85,7 +85,7 @@
     "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.\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)"
    ]