fixed typos in krylov, shor, and qft module

learning/courses/fundamentals-of-quantum-algorithms/phase-estimation-and-factoring/shor-algorithm.ipynb

@@ -233,16 +233,16 @@
     "\n",
     "$$\n",
     "M_a \\vert \\psi_1 \\rangle\n",
-    "= \\sum_{k = 0}^{r-1} \\omega_r^{-k} M_a\\vert a^k \\rangle\n",
-    "= \\sum_{k = 0}^{r-1} \\omega_r^{-k} \\vert a^{k+1} \\rangle\n",
-    "= \\sum_{k = 1}^{r} \\omega_r^{-(k - 1)} \\vert a^{k} \\rangle\n",
-    "= \\omega_r \\sum_{k = 1}^{r} \\omega_r^{-k} \\vert a^{k} \\rangle\n",
+    "= \\frac{1}{\\sqrt{r}}\\sum_{k = 0}^{r-1} \\omega_r^{-k} M_a\\vert a^k \\rangle\n",
+    "= \\frac{1}{\\sqrt{r}}\\sum_{k = 0}^{r-1} \\omega_r^{-k} \\vert a^{k+1} \\rangle\n",
+    "= \\frac{1}{\\sqrt{r}}\\sum_{k = 1}^{r} \\omega_r^{-(k - 1)} \\vert a^{k} \\rangle\n",
+    "= \\frac{1}{\\sqrt{r}}\\omega_r \\sum_{k = 1}^{r} \\omega_r^{-k} \\vert a^{k} \\rangle\n",
     "$$\n",
     "\n",
     "Then, because $\\omega_r^{-r} = 1 = \\omega_r^0$ and $\\vert a^r \\rangle = \\vert 1\\rangle = \\vert a^0\\rangle,$ we see that\n",
     "\n",
     "$$\n",
-    "\\sum_{k = 1}^{r} \\omega_r^{-k} \\vert a^{k} \\rangle = \\sum_{k = 0}^{r-1} \\omega_r^{-k} \\vert a^k \\rangle\n",
+    "\\frac{1}{\\sqrt{r}}\\sum_{k = 1}^{r} \\omega_r^{-k} \\vert a^{k} \\rangle = \\frac{1}{\\sqrt{r}}\\sum_{k = 0}^{r-1} \\omega_r^{-k} \\vert a^k \\rangle\n",
     "= \\vert\\psi_1\\rangle,\n",
     "$$\n",
     "\n",

learning/courses/quantum-diagonalization-algorithms/krylov.ipynb

@@ -248,7 +248,7 @@
     "\\det(\\tilde{A}_2-\\lambda I)=0\n",
     "$$\n",
     "$$\n",
-    "\\begin{vmatrix}4-\\lambda&1&0\\\\1&4-\\lambda&1\\\\0&1&4-\\lambda\\end{vmatrix} = (4-\\lambda)((4-\\lambda)^3-1)-(4-\\lambda)=0\\\\\n",
+    "\\begin{vmatrix}4-\\lambda&1&0\\\\1&4-\\lambda&1\\\\0&1&4-\\lambda\\end{vmatrix} = (4-\\lambda)((4-\\lambda)^2-1)-(4-\\lambda)=0\\\\\n",
     "$$\n",
     "$$\n",
     "4-\\lambda=0,4-\\lambda=±2^{1/2}→\\lambda=4-2^{1/2},4,4+2^{1/2}≈2.59,4,5.41\n",

learning/modules/computer-science/qft.ipynb

@@ -112,7 +112,7 @@
     "\n",
     "$$ | \\phi_y \\rangle =  \\frac{1}{\\sqrt{N}}\\sum_{x=0}^{N-1}\\omega_N^{y x} \\vert x \\rangle$$\n",
     "\n",
-    "with $\\omega_N^{yx} = e^{\\frac{2\\pi y x}{N}}$ as above, and $N$ is the number of basis states in your quantum system. Note that, since we're working with qubits now, $m$ qubits gives you $2^m$ basis states, so $N=2^m$. Here, the basis states are written as a single number $|x\\rangle$ where $x$ ranges from $0$ to $N-1$, but you might more typically see the basis states expressed as $|00...00\\rangle$, $|00...01\\rangle$, $|00...11\\rangle$, ..., $|11...11\\rangle$, where each binary digit represents the state of qubit 0 through $m-1$, from right to left. There's an easy way to convert these binary states to a single number: simply treat them like binary numbers! So, $|00...00\\rangle = |0\\rangle$, $|00...01\\rangle = |1\\rangle$, $|00...10\\rangle = |2\\rangle$, $|00...11\\rangle = |3\\rangle$, and so on, all the way up to  $|11...11\\rangle = |2^m -1\\rangle = |N-1\\rangle$."
+    "with $\\omega_N^{yx} = e^{\\frac{2\\pi i y x}{N}}$ as above, and $N$ is the number of basis states in your quantum system. Note that, since we're working with qubits now, $m$ qubits gives you $2^m$ basis states, so $N=2^m$. Here, the basis states are written as a single number $|x\\rangle$ where $x$ ranges from $0$ to $N-1$, but you might more typically see the basis states expressed as $|00...00\\rangle$, $|00...01\\rangle$, $|00...11\\rangle$, ..., $|11...11\\rangle$, where each binary digit represents the state of qubit 0 through $m-1$, from right to left. There's an easy way to convert these binary states to a single number: simply treat them like binary numbers! So, $|00...00\\rangle = |0\\rangle$, $|00...01\\rangle = |1\\rangle$, $|00...10\\rangle = |2\\rangle$, $|00...11\\rangle = |3\\rangle$, and so on, all the way up to  $|11...11\\rangle = |2^m -1\\rangle = |N-1\\rangle$."
    ]
   },
   {
@@ -159,7 +159,7 @@
     "\n",
     "\n",
     "\n",
-    "In general, for an $n$-qubit state, there will be $2^n$ Fourier basis states, whose frequency in phase variation ranges from constant, for $|\\phi_0\\rangle$, to rapidly varying for $|\\phi_{2^n-1}\\rangle$, completing $2^n-1$ windings around $2\\pi$ over the superposition of states. So, when we take a QFT of a quantum state, we're essentially doing the same basic analysis that we did for the musical waveform in the Intro. We're determining the Fourier frequency components that contribute to creating the quantum state of interest.\n",
+    "In general, for an $m$-qubit state, there will be $2^m$ Fourier basis states, whose frequency in phase variation ranges from constant, for $|\\phi_0\\rangle$, to rapidly varying for $|\\phi_{2^m-1}\\rangle$, completing $2^m-1$ windings around $2\\pi$ over the superposition of states. So, when we take a QFT of a quantum state, we're essentially doing the same basic analysis that we did for the musical waveform in the Intro. We're determining the Fourier frequency components that contribute to creating the quantum state of interest.\n",
     "\n",
     "### Try some example QFTs\n",
     "\n",