Corrections: terminology and technical correctness

Author: augustehirth

docs/time_crystals/time_crystal_circuit_generation.ipynb

@@ -30,13 +30,13 @@
    },
    "source": [
     "# Time Crystal Circuit Generation\n",
-    "This notebook covers how Many Body Local Discrete Time Crystal circuit lists are created, from the paper: Observation of Time-Crystalline Eigenstate Order on a Quantum Processor ([arxiv:2107.13571](https://arxiv.org/abs/2107.13571)). \n",
+    "This notebook covers how Many Body Local Discrete Time Crystal circuit lists are created, from the paper: Observation of Time-Crystalline Eigenstate Order on a Quantum Processor ([Nature](https://www.nature.com/articles/s41586-021-04257-w)). \n",
     "\n",
-    "Quantum computers, and gate-based quantum circuits turn out to be well suited for crafting systems that exhibit time-crystalline behavior. Behavior is crystalline with respect to time if it has some consistent and stable pattern over time. This system's pattern must be resilient against perturbation, including the introduction of random noise and heat, in the same way that a space-crystalline object, like a diamond, stays in shape if moved or heated. \n",
+    "Quantum computers and gate-based quantum circuits turn out to be well suited for crafting systems that exhibit time-crystalline behavior. Behavior is crystalline with respect to time if it has some consistent and stable pattern over time. This system's pattern must be resilient against perturbation in the same way that a space-crystalline object, like a diamond, is still a diamond if moved or heated. \n",
     "\n",
-    "The implementation of a time-crystalline system on a quantum computer starts with a set of qubits that are connected in a single chain. This serves as a many body local system which, at the low superconducting temperature that the qubits are run at, becomes resistant to changes cause by small amounts of introduced heat. The heat's energy is spread across the qubits, thin enough that it is not sufficient to change the qubits' states. The amount of heat dissipation in the DTC circuits is modeled with the thermalization constant `g`. \n",
+    "The quantum computer supplies a system of many qubits, locally connected to each other in a chain. A many-body local system like this is critical for the existence of a time crystal. Without an MBL system, it is expected that the system's state would decay into a maximum entropy state that is incompatible with the goal of stable and consistent time structure. \n",
     "\n",
-    "The actual time-crystalline behavior that the DTC circuits implement is perhaps the simplest possible kind of time-structured behavior, oscillation. Each circuit is built with some number of identical $U$-cycles. Time is represented by a circuit list where each circuit is ordered with increasingly many $U$-cycles; each cycle is a discrete time step. The eventual effect of these $U$-cycles, as demonstrated in the paper and in the [Time Crystal Data Analysis](time_crystal_data_analysis.ipynb) notebook, is consistent oscillations of each qubits' polarizations. Each $U$-cycle includes a component to model to model the influence of thermalization, with the constant `g`,  and random noise, with the `local_fields` variables. The behavior of the DTC circuits is shown to be time-crystalline in spite of the consistently introduced randomness. "
+    "The time-crystalline behavior that the DTC circuits demonstrate is perhaps the simplest kind of time-structured behavior, oscillation. Each circuit is built with some number of identical $U$-cycles. Time is represented by a circuit list where each circuit is ordered with increasingly many $U$-cycles; each cycle is a discrete time step. The eventual effect of these $U$-cycles is consistent oscillations of each qubits' polarizations. The experiments performed demonstrate that this time-crystalline oscillation behavior is stable in spite of different initial states and introduced random potentials. "
    ]
   },
   {
@@ -436,11 +436,11 @@
     "id": "e13de6da1fc6"
    },
    "source": [
-    "The construction of each circuit is surprisingly short. \n",
+    "The construction of each circuit is surprisingly succinct. \n",
     "\n",
     "The circuit expects the quantum computer to be in the all-zeros state, and starts with a sequence of `cirq.Y` gates conditioned on the provided `initial state` parameter, after initializing the necessary symbolic variables. \n",
     "\n",
-    "Each $U$-cycle consists of three moments. First, a moment of `cirq.PhasedXZGate`s, with one for each qubit. Each `cirq.PhasedXZGate` takes the thermalization constant `g` as its X-exponent, and the random variance provided by `local_fields` for its Y-exponent. This moment collectively models the disorder caused by heat and random noise. \n",
+    "Each $U$-cycle consists of three moments. First, a moment of `cirq.PhasedXZGate`s, with one for each qubit. Each `cirq.PhasedXZGate` takes the control parameter `g` as its X-exponent, and the random potentials necessary for many-body localization provided by `local_fields` for its Y-exponent.\n",
     "\n",
     "The second and third moments both cause the oscillation behavior and compensate for the first disorder moment. The qubits are connected in a chain, and each qubit pair connection in that chain is coupled with a `cirq.PhasedFSimGate` that uses the parameters `[theta, zetas, chi, gamma, phi]`. To keep gates from overlapping on the same qubit, this chain of gates is split into the second and third moments, such that no two gates share a qubit within each moment. \n",
     "\n",