Ising

 Wondering, this morning, whether I should continue with the Adder code.

The first trial did not work out, and a second one wth a simpler set of numbers

had to be used.

Episode 4 of coding with qiskit seems rather complex, but it introduces Primitives

ie the simplest elements one can work with at a given level of abstraction. The example

is work on an Ising system.

Gemini:

Absolutely, the Ising model is a fascinating concept in statistical mechanics. Here's a breakdown of the key ideas:

**What it is:**

* The Ising model is a mathematical model used to describe ferromagnetism. 

* Ferromagnetism is a property of certain materials where their atomic magnetic moments (spins) tend to align in the same direction, creating a net magnetic field.

**The Model:**

* Imagine a lattice of atomic sites, like a grid. 

* Each site can be in one of two states, often represented by up (+1) or down (-1) spins, signifying the magnetic orientation.

* Spins interact with their nearest neighbors on the lattice.

**The Interactions:**

* The interaction energy favors alignment. Neighboring spins that are aligned (both up or both down) have lower energy.

* In contrast, anti-aligned spins (one up, one down) have higher energy.

* This interaction is captured by a parameter J in the Ising Hamiltonian (energy equation).

**Thermal Fluctuations:**

* At a finite temperature, the system isn't perfect. Thermal fluctuations can cause a spin to flip, even if it increases the energy slightly.

* A parameter β, related to temperature, determines the probability of such flips.

**Why it's Interesting:**

* The Ising model, despite its simplicity, exhibits a phase transition. 

* At high temperatures (low β), the spins are random (disordered phase). 

* As temperature lowers (β increases), the interaction becomes more dominant, and the spins tend to align (ordered phase, ferromagnetic behavior).

**Applications:**

* The Ising model goes beyond magnetism. 

* It can be applied to study various systems with binary states, like Ising glasses in metallurgy or protein folding in biology.

**Further Exploration:**

* For a deeper dive, you can explore the Ising Hamiltonian equation, which describes the system's energy. 

* The Ising model is exactly solvable in two dimensions, but solving it in higher dimensions requires computational methods like Monte Carlo simulations.

 

There are also resources online that can provide a more visual explanation, including simulations: 

* Try searching for "[YouTube] Ising Model" to see an animation of the model in action.

Ising model - Wikipedia, the free encyclopedia

Evidence for the utility of quantum computing before fault tolerance | Nature

Bing/Chat:

Copilot

A Monte Carlo simulation is a powerful modeling technique used to predict the probability of various outcomes when dealing with random variables. Here’s how it works:

1. Problem Context: Imagine you have a process with uncertain or stochastic elements (like financial investments, project timelines, or physical phenomena).
2. Random Sampling: Monte Carlo simulations involve generating random inputs based on probability distributions for these uncertain factors.
3. Computation: For each set of random inputs, perform a deterministic computation to estimate the output.
4. Aggregation: Repeat the process many times (thousands or more) to aggregate the results and understand the overall behavior of the system.

For example, consider estimating the value of π using a Monte Carlo method:

1. Draw a square and inscribe a quadrant within it.
2. Scatter points uniformly over the square.
3. Count the number of points inside the quadrant (within a distance of 1 from the origin).
4. The ratio of points inside the quadrant to the total sample count approximates the ratio of areas: π/4.
5. Multiply this result by 4 to estimate π¹.

Monte Carlo simulations are widely used in finance, engineering, and other fields to evaluate risk and uncertainty. They’ve enabled breakthroughs in science and technology, but they also come with challenges like computational cost and reliability of random number generators¹²³. 🎲📊

Copilot

Stochastic elements refer to components in a system that exhibit randomness or uncertainty. These elements are characterized by their probabilistic behavior rather than deterministic outcomes. Here are some examples:

1. Random Variables: These are quantities whose values are determined by chance. For instance, the outcome of rolling a die or the stock price at a future time can be modeled as random variables.

2. Noise: In various scientific and engineering models, noise represents unpredictable disturbances or fluctuations. For example, electronic circuits experience thermal noise due to random electron motion.

3. Environmental Factors: Weather conditions, market fluctuations, and natural phenomena (like earthquakes) are stochastic elements because they are influenced by complex, unpredictable processes.

4. Quantum Mechanics: At the subatomic level, particles exhibit probabilistic behavior. Quantum mechanics relies heavily on stochastic principles.

5. Simulation Inputs: When using Monte Carlo simulations (as mentioned earlier), the input parameters (e.g., interest rates, project completion times) are often treated as stochastic elements.

In summary, stochastic elements introduce randomness into systems, and understanding their behavior requires statistical methods and probability theory. 🎲🌟

2of10

**Trotterized time-evolution** is a technique used in quantum computing and quantum simulations. Let me break it down for you:

1. **Quantum Simulations**:

   - Quantum systems are notoriously difficult to simulate on classical computers due to their exponential complexity.

   - Quantum simulators aim to model the behavior of quantum systems to understand their properties and dynamics.

2. **Time Evolution**:

   - In quantum mechanics, the state of a system evolves over time according to a unitary operator (a quantum gate).

   - The time evolution operator $$U(t)$$ describes how a quantum state changes from time $$t=0$$ to $$t$$.

3. **Trotterization**:

   - Trotterization is an approximation method to simulate quantum time evolution.

   - It breaks down the overall evolution into smaller steps by approximating the unitary operator as a product of simpler operators.

   - For example, if we want to evolve a system for time $$t$$, we divide it into $$N$$ small time steps: $$t/N$$.

   - At each step, we apply a simplified version of the evolution operator.

4. **Why Trotterization?**:

   - Quantum computers have limitations (noise, decoherence, etc.), making exact simulations challenging.

   - Trotterization allows us to approximate the quantum evolution while mitigating noise effects.

   - It's widely used in quantum algorithms and simulations.

5. **Effective Noise Model**:

   - When simulating quantum circuits on noisy quantum computers, we consider an **effective noise model**.

   - This model incorporates noise as static Lindblad terms, modifying the original unitary dynamics.

   - The noisy algorithm model approximates the actual dynamics of the noisy quantum circuit¹².

In summary, Trotterization simplifies quantum time evolution for practical simulations, especially on noisy quantum devices. 🚀🔬

                                                       *     *     *

Started work on Coding with Qiskit, Episode4; and I'm in the queue.

We build the code described in the Ising paper, and its mirror image. If

things do not return to zero for all our qubits, then we will have a view of the

noise in the system.

                                                                   

The circuit:

                                                                                                                          

The circuit and its mirror image: