typo fix

Author: fedimser

research/Incrementer.ipynb

@@ -11,27 +11,27 @@
     "\n",
     "### Summary\n",
     "\n",
-    "Here I present a circuit for incrementing n-qubit integer register using only O(log n) ancillas.\n",
+    "Here I present a circuit for incrementing an n-qubit integer register using only O(log n) ancillas.\n",
     "\n",
     "### Motivation\n",
     "\n",
-    "The practical use case for low-ancilla incrementer is constant addition when the constant is small (let's say bit size of constant is $n_c$). Then we can use constant adder circuit with carry on first $n_c$ qubits. For the rest of the register, we just need to add a carry qubit to it, which can be done by applying incrementer controlled by the carry bit.\n",
+    "The practical use case for a low-ancilla incrementer is in constant addition when the constant is small (let's say the bit size of the constant is $n_c$). Then we can use a constant adder circuit with carry on the first $n_c$ qubits. For the rest of the register, we just need to add a carry bit to it, which can be done by applying an incrementer controlled by the carry bit.\n",
     "\n",
     "### Baseline\n",
     "\n",
-    "As baseline, I am using incrementer from [this paper](https://www.worldscientific.com/doi/abs/10.1142/S0217979213501919) which I generalized to a constant adder (https://arxiv.org/pdf/2501.07060) and which is implemented in [ConstAdder.qs](../lib/src/QuantumArithmetic/ConstAdder.qs).\n",
+    "As a baseline, I am using the incrementer from [this paper](https://www.worldscientific.com/doi/abs/10.1142/S0217979213501919), which I generalized to a constant adder (https://arxiv.org/pdf/2501.07060) and which is implemented in [ConstAdder.qs](../lib/src/QuantumArithmetic/ConstAdder.qs).\n",
     "\n",
     "### Implementation idea\n",
     "\n",
-    "Define CTO(x) - \"count trailing ones\", i.e. number of least significant bits in x equal to 1 before first 0 bit.\n",
+    "Define CTO(x) - \"count trailing ones\", i.e., the number of least significant bits in x equal to 1 before the first 0 bit.\n",
     "\n",
-    "Then to increment x we need to flip first CTO(x)+1 bits in x.\n",
+    "Then to increment x we need to flip the first CTO(x)+1 bits in x.\n",
     "\n",
     "So, incrementing is reduced to implementing two operations:\n",
     "* CountTrailingOnes(x, ans) - computes ans:=CTO(x)\n",
     "* FlipFirst(target, ctr) - flips first `ctr` bits in `target`.\n",
     "\n",
-    "Both CountTrailingOnes and FlipFirst can be implemented recursively by splitting input in 2 parts, first of them having length equal to a power of 2. Both of them use $O(\\log n)$ ancilla, adding one ancilla for each level of recursion.\n",
+    "Both CountTrailingOnes and FlipFirst can be implemented recursively by splitting the input in 2 parts, the first of them having a length equal to a power of 2. Both of them use $O(\\log n)$ ancilla, adding one ancilla for each level of recursion.\n",
     "\n",
     "The incrementer works like this:\n",
     "* Allocate counter register and carry qubit.\n",
@@ -45,21 +45,21 @@
     "\n",
     "### Version with carry\n",
     "\n",
-    "To turn presented incremented in incrementer with carry:\n",
+    "To turn the presented incrementer into an incrementer with carry:\n",
     "* Instead of using ancilla for carry, make it input qubit.\n",
     "* Do not uncompute the carry qubit.\n",
     "\n",
     "### Cost\n",
     "\n",
     "Baseline incrementer uses $n-3$ ancillary qubits.\n",
     "\n",
-    "The presented incrementer uses exactly $2 \\lceil \\log_2(n+2) \\rceil -1$ ancillary qubits which becomes less than base starting from n=11.\n",
+    "The presented incrementer uses exactly $2 \\lceil \\log_2(n+2) \\rceil -1$ ancillary qubits, which becomes less than the baseline starting from n=11.\n",
     "\n",
-    "On depth, the proposed circuit uses ~10n CCZ gates while base circuit uses ~1n CCZ gates.\n",
+    "In terms of depth, the proposed circuit uses ~10n CCZ gates while the baseline circuit uses ~1n CCZ gates.\n",
     "\n",
     "So it's much more expensive in depth, but might be worth it if it can reduce overall space requirement of an algorithm.\n",
     "\n",
-    "The table below compares ancilla count and CCZ coutn between the baseline and proposed incrementer."
+    "The table below compares ancilla count and CCZ count between the baseline and proposed incrementer."
    ]
   },
   {