Quantum Horizons: Breaking the Code from Schrödinger’s Theories to Tackling TSP

Part 2 — Transforming classical concepts into the quantum realm.

In the intricate tapestry of our lives, computing has woven itself into the very fabric of our existence. At the core of a classical computer beats the heart of processing power: the Central Processing Unit (CPU). This marvel of technology, typically composed of a silicon wafer or intricate electric circuits, orchestrates the symphony of computations that underpin our digital world.

Unlike its classical counterpart, quantum computing harnesses the fundamental building blocks of nature itself: atoms. When using classical computers to forecast or simulate future possibilities, we face the challenge of dealing with exponentially increasing bits.

However, with quantum computers, we don’t have to stress as much; we simply observe atoms already in a state of superposition — a mixture of many possibilities that collapse to one when observed, leading us to the final answer. Thus, our processing capabilities exceed for certain questions, making us limitless. We would be able to simulate the universe, biological processes, etc.

The current state of quantum computers is imperfect, as they interact with the environment, potentially polluting the final answer. An ideal quantum computer would only need knowledge of computer science, but today’s quantum computers demand some understanding of underlying physics. Consequently, we must navigate through the classical model of statistical mechanics or the classical Ising model derived from the spin system.

Classical Ising Model

Imagine you have a bunch of tiny magnets, like little characters in a play. Each magnet has its personality, a unique quirk. Some magnets like to point north, some prefer to point south, and some even have a special angle they fancy. They have these built-in preferences because of the way their atoms are arranged inside. It’s like each magnet has its own favorite direction, and it just can’t help but follow its heart. imagine we bring these magnets close to each other on the stage. They start talking magnetically. If two magnets like each other’s direction, they’ll cuddle up, like two best friends holding hands. This magnetic attraction is what we call “coupling.

The Ising model is like a script for our magnetic play. It helps us understand what happens when we have a whole bunch of these magnet characters, each with its own bias and interacting with its neighbors. We use it to explore how they collectively behave, whether they form big groups or scatter apart.

Preference or bias can be introduced using an external magnetic field in a certain z-axis direction increasing the strength of some while decreasing of others.

Ernst Ising

The classical Ising Hamiltonian, named after the physicist Ernst Ising, describes the behavior of magnetic spins or binary variables in a system. It is a classical counterpart to the quantum Ising Hamiltonian used in quantum mechanics.

Given a set of binary variables σi, (Magnets for example) each taking values +1(maybe for north upwards) or -1, the classical Ising Hamiltonian H is expressed as:

The Ising model’s energy equation is: E_total = -∑(h_i σ_i) — J ∑(σ_i * σ_j), where the first summation is over all magnets and the second is over all pairs of neighboring magnets.

The goal in many problems involving the classical Ising Hamiltonian is to find the configuration of σi variables that minimize the energy of the system, i.e., finding the values of σi that minimize H(σ).

Transverse Field

So far, we’ve been considering magnets that naturally prefer to align with an external magnetic field (the bias). Now, imagine we have another influence on our magnets — a field that acts perpendicular to their natural alignment, like a gentle breeze nudging them sideways.

The transverse field is like an external force that can flip a magnet’s orientation, regardless of its natural bias. We represent this influence using another parameter, denoted as Γ.

Now, let’s add the transverse field term to our Ising model’s energy equation:

E_total = -∑(h_i σ_i) — J ∑(σ_i σ_j) — Γ ∑σ_i

Here, the third term, -Γ * ∑σ_i, represents the influence of the transverse field. It sums up the contributions of the transverse field acting on each individual magnet σ_i.

The presence of the transverse field can lead to fascinating behaviors in the Ising model.

It can induce quantum effects and phase transitions, making our magnetic ensemble dance in unexpected ways. Indeed, finding the optimal value of Γ is often a critical part of studying the Ising model with a transverse field. It depends on various factors, including the strength of the natural biases (h_i) and the coupling strength (J) between magnets. Making it a Np-hard problem, impossible to be solved by classical computers.

Conversion of Classical Ising to Quantum realm

So, our magnetic friends were quite set in their ways in the classical world, always pointing either up or down, right? But now, let’s introduce them to a quantum realm where they can do something truly enchanting: exist in multiple states at once!

Instead of just having two choices (+1 or -1), our quantum magnets can exist in a superposition of states. It’s as if they can simultaneously be a bit up and a bit down, and everything in between. We represent this magical property using quantum spin operators, like σ_x, σ_y, and σ_z.

Bloch sphere (visual representation of qubits which are in actuality wave functions)

We keep the bias, represented by h_i, but now it becomes a quantum bias. The quantum magnets can still have a preference to point up or down along the z-axis.

For the coupling between neighboring magnets, we use quantum operators as well. The interaction term is similar to the classical one, but now it’s described using quantum spin operators, like σ_zⱼ and σ_zⱼ’.

The transverse field acts on our quantum magnets, but instead of pushing them up or down, it nudges them sideways, along the x-axis. This field can make our quantum magnets flip and dance between different states. In the quantum Ising model, our magnets become like graceful ballet dancers, twirling and spinning through various quantum states, thanks to the transverse field’s enchantment.

In terms of mathematics, we replace the classical spins with quantum spin operators and adjust the Ising Hamiltonian accordingly. We use operators like σ_x, σ_y, and σ_z to describe the quantum spins, and the Hamiltonian includes terms for bias, coupling, and the transverse field, just like in the classical model.

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Next part we will deal with the quantum complexity theory and some optimization algorithm tools.