Quantum Horizons: Breaking the Code from Schrödinger’s Theories to Tackling TSP
Part 1: Quantum Foundations — Navigating the Quantum Landscape
We don’t have enough parallel universes to allow all uses of all junction types in the absence of quantum computing the combinatorics are not in our favor … — Larry Wall
Ever wondered how the fundamental principles of the cosmos could be harnessed to revolutionize the world of computing?
Perhaps you were captivated by the genius of Stephen Hawking, and physics was more than just a subject — it was a portal to the unknown. If you’ve ever dreamt of quantum leaps and parallel computations, you’re in for a treat. This series is your backstage pass to the quantum stage, where the rules of classical computing are rewritten.
Whether you’re an aspiring quantum coder ready to plunge into this exciting realm or a curious mind eager for a deeper understanding of the quantum fabric that underlies our universe. Welcome aboard!
Since you don’t essentially need to master the grand quantum theories to work on the billion-dollar quantum computer nor do you need to be an experienced coder to write and run your first few lines of code — you may directly skip to later parts of the series where we discuss code.
What lies ahead?
Since this is a vast field, I have divided the content into parts, the whole series shall be covering the below parts in general
- Quantum Theories and the mathematical references
- Types of quantum computers and the latest development
- How to start with qiskit and d-wave, Solving TSP using a quantum computer
Let’s Start with part 1 of the series ...
Quantum World
From plank discoveries to Einstein mirabilis, from double slit experiments to accepting wave-particle duality, and today’s quantum computing, the world came a long way.
Quantum computing is like something straight out of a science fiction novel, yet it’s very much a reality. At its heart, it’s all about tapping into the weird and wonderful laws that govern the tiniest particles in the universe — quantum mechanics.
The quantum world stands on the three pillars:
- Superposition
- Entanglement
- Interference
In the world of classical computing, we deal with bits that can either be 0 or 1, like flipping a coin. But in the quantum realm, we have these magical entities called qubits. These qubits, unlike boring bits, can be in a state of 0, 1, or both 0 and 1 at the same time, thanks to a mind-boggling concept called superposition. But it collapses to a classical state once observed.
Qubits can also become entangled, like two particles sharing a cosmic connection. When you manipulate one qubit, its entangled partner instantly reacts, no matter how far apart they are. It’s like having a telepathic conversation with your twin on the other side of the galaxy!
But that’s not all — quantum interference comes into play. Imagine waves in the ocean colliding, sometimes making bigger waves and other times canceling each other out. Quantum computers use this interference to process information in a way that classical computers can only dream of. They’re like the Jedi knights of computing, performing complex calculations with grace and speed.
But, there’s a catch. When you measure a quantum state, it collapses into a definite 0 or 1. It’s like trying to catch a quantum butterfly — as soon as you touch it, it decides whether to be a 0 or a 1, and that can be a bit unpredictable.
So, quantum computers use these principles to perform calculations that would take classical computers eons to complete. They can solve complex problems like cracking encryption codes, simulating molecules for drug discovery, and optimizing supply chains with unprecedented efficiency.
Let’s head back to the basics of class 11th, where we studied Werner Heisenberg’s Uncertainty principle. Before we observe a particle it can exist anywhere in space with some probability.
In the everyday world, one might describe a ball by saying it’s red, round, and bouncing. In the quantum world, we don’t have such simple descriptions. Instead, we have quantum states, and these are like the secret ingredients that make up the particle’s identity. They can describe not just where a particle is or what it’s doing, but also its spin, polarization, and more.
Here’s where the concept of Hilbert space comes in. Think of Hilbert space as this vast, almost infinite, mathematical playground where these quantum states live. We say the quantum state or the final energy of the system is the superposition or combination of an infinite or huge number of states/energies with their probability distribution taken into consideration.
Infinity is not a number, it is a concept because if we treat infinity like a number it is almost impossible to wrap our heads around the fact that the sum of 1+ 2… to infinity is -1/12, a negative fraction.
It seems absurd for the sum of linear numbers to add up to become e^x. When one goes adding up to the infinity, you get something that is out of the input space (for example, integer numbers), something (for example, e^x) absurd sometimes. So, imagine there is a quantum vector space and outside, then our answer lies on the boundary. It is difficult to make something out of it. That is why Hilbert space rules say that the outcome must also be a quantum state.
See, in classical physics, one can describe the position of a ball using simple coordinates like x, y, and z. But in the quantum world, it’s more like describing a particle in an abstract, multi-dimensional space — that’s Hilbert space! Each dimension represents a different aspect of the particle’s state. It’s like saying, ‘Hey, this particle isn’t just here or there; it’s also spinning this way, and it’s got this kind of energy.’
The need for Hilbert space becomes crystal clear when we deal with quantum systems that have many particles, like atoms or molecules. These systems can have a mind-boggling number of possible quantum states. Hilbert space allows us to elegantly describe and manipulate these states, making it an indispensable tool in quantum physics and quantum computing.
So, in the weird and wonderful world of quantum states, Hilbert space is our mathematical stage where particles perform their intricate dance of possibilities. It’s where the magic happens, where we unlock the secrets of the quantum universe, and it’s an essential tool for exploring the mysteries of the quantum realm.
Notations:
Bra-ket notation serves as the efficient and compact language of quantum mechanics, for describing quantum states and operations. The ‘bra’ represents the complex conjugate of a quantum state, and it looks like this: ⟨ψ|. Think of it as the starting point, where you’ve got a quantum state ready to roll.
In physics, a quantum state is defined by a column vector similar to a vector in space. Unlike normal people, physicists treat everything that is generic with a combination of elements as space like a space of possibilities, etc.
On the other hand, the ‘ket’ is like the ending point, and it represents the quantum state itself, like this: |φ⟩. Imagine it as where you want to end up or the state you’re aiming to measure or manipulate.
⟨ψ|φ⟩: This can tell you things like how likely it is to measure the state |φ⟩ when you start with |ψ⟩ or how a quantum operator transforms |ψ⟩ into |φ⟩.
While Bra-ket together gives something similar to scalar/dot product(inner product), ket-bra gives the density matrix that helps in decoherence properties and noisy system. Decoherence is a fundamental concept in quantum computing, and it refers to the loss or degradation of quantum information stored in a quantum system due to interactions with the environment. Understanding decoherence is critical because it poses a significant challenge to building practical and robust quantum computers.
The unitary operation(only operation allowed to be performed) U∣ψ⟩=∣φ⟩ symbolizes a norm-preserving and reversible transformation for the sake that outcome remains in hilbert space.
Schrödinger Equation & Hamiltonian
In the quantum realm, particles are described by mysterious wavefunctions — complex-valued functions of space and time, acting as vectors in a complex vector space.
Schrödinger equation is a partial differential equation that guides the evolution of these wavefunctions over time. In other words, it tells us how the quantum states change and dance as time ticks on.
The solutions to this equation are the eigenvectors (wavefunctions) of the Hamiltonian operator, and the associated eigenvalues are like the energy levels themselves.
Hamiltonian operator (H) represents the total energy of our quantum system. This operator often takes the form of a differential operator, which means it acts on the wave function to calculate the system’s energy.
Picture it like this: Imagine you have a puzzle, and the Hamiltonian operator is the key piece. When you solve the puzzle by finding its eigenvectors and eigenvalues, you unlock the quantum system’s energy states. Each eigenvector tells you how the system behaves in a particular energy state, and the eigenvalue is the precise energy of that state.
The Schrödinger equation, written as iħ∂Ψ/∂t = HΨ, describes how the wavefunction Ψ of a quantum system changes over time. On the left side, ∂Ψ/∂t represents this change rate. On the right side, HΨ represents the energy of the system in that state, calculated by the Hamiltonian operator H. In simple terms, it connects the system’s energy to how its quantum state evolves with time, making it a foundational equation in quantum mechanics. Solving this equation allows us to predict and understand the behavior of quantum systems, making it a cornerstone of quantum mechanics.
Preview
That is all for part 1. In part 2 we will discuss a few complex classical concepts and their transformation into the quantum realm, before delving into quantum complexity theory and basics of optimization algorithms required for solving TSP.
You may explore the below for more detailed information:
If you want to have an idealist talk and get an overview of possibilities of what quantum computers are capable of do, try this.
