QUANTUM MIND (Neuroscience: Mind, Brain and Behavior/ Consciousness)

Quantum Mechanics

Before diving into the complex world of quantum mechanics, the significance of getting acquainted with certain terminologies is immense!

Bloch Sphere

In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit) , named after the physicist Felix Bloch.

Bit vs Qubit

A bit (binary digit or Classical bit or Coin bit) is the basic unit of data, representing a value of either 0 or 1 (“off” or “on”). Bits are independent of each other which means one state at a time. Whereas a qubit (quantum bit) is the basic unit of data which uses quantum mechanics to represent a 0, a 1, or both at the same time. Qubits are in a combined state together which signifies that there can be many states at the same time.

P:S: Consider qubit as a unit of information and a quantum particle as the physical hardware used to store that information.

Bit (a) Qubit (b) Figure 1. Bit vs. Qubit (a) Bit system. (b) Qubit system.

Superposition

Qubit can be thought of as an arrow in a 3D space; superposition is when the arrow can point in any direction and not in just up or down position as in classical computing. This superposition is a combination state of 0 and 1. Probability is set by the direction of the arrow. For example, when the arrow is at the equator the probability of getting 0 or 1 is 50:50. When the arrow is pointing upward, say at a 45 degree angle from the equator, probability of 0 is more likely than 1 ( 85% and 15% respectively). When the arrow is pointing downward, it is more likely to get 1 than 0 (15% and 85% respectively).

The Hadamard gate is a fundamental single-qubit operation in quantum computing. Its primary purpose is to create a quantum superposition, allowing a qubit to exist in multiple states simultaneously.

Entanglement

Bits remain separated and do not interfere with each other, whereas qubits can be entangled with each other and become a part of one large quantum state together. According to Classical physics by Sir Isaac Newton, every object is different, having specific properties and separated by time and space. Everything is predictable if we have enough information based on the laws of motion. Reality as we experience everyday and understand at present is stable and understandable. But this concept is dismantled by Quantum Mechanics.

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Refer to the spin qubits depicted in the above figure.

A superposition of the four states of two electron spin qubits: 1st electron is spin up, 2nd is spin up (00 state) + 1st electron is spin up, 2nd is spin down (01 state) + 1st electron is spin down, 2nd is spin up (10 state) + 1st electron is spin down, 2nd is spin down (11 state). The probability distribution of the two spin qubits therefore are 00, 01, 10,11.

Number of Qubits	Number of States
1	2
2	4
3	8
4	16
5	32
…	….
n	2n

Quantum Algorithm

A quantum algorithm is a step-by-step procedure or set of instructions designed specifically to be executed on a quantum computer. Quantum algorithms are designed using the quantum wave-function (interference) to amplify the correct answers (constructive interference) while canceling out the incorrect ones (destructive interference) to arrive at the final solution.

7177 x 3001= FACTORS
21538177 = FACTORISATION

There is a fast, efficient and classical algorithm to find the answer. But if we start with the answer and ask, what are the original numbers to multiply to make this number, it is more difficult. This is known as factorization, and these numbers are called factors. Finding the original numbers is difficult and the reason is the search space of possible factors is so large (massive number of potential combinations of numbers that would need to be checked to find the correct factors of a large integer) and there is no efficient classical algorithm for finding the factors of large numbers. For this reason we use this mathematical property for internet encryption; secure websites, emails and bank accounts. If we know these factors, we can decrypt the information, but if we don’t we need to find them first which is intractable on the world’s most powerful computer.

Quantum Complexity Theory

It is a subfield of the world of computational complexity theory which deals with the categorization of algorithms. Categorization depends upon how much harder it is to solve the problem as the problem gets larger . How hard is it to factorize a number where N=9 compared to when N=8 (what is the trend when we add more and more digits) ? It is called its complexity or scaling. For factorization it is exponential. Therefore, it would be a great benefit if we can come up with better algorithms to solve harder problems.
There are many quantum algorithms. We are focussing on the Shor’s and Grover's Algorithms for now.

Shor’s Algorithm

In 1994, Peter Shor published a fast quantum algorithm that can find the factors of large integers.This was the first application to a real world problem with potentially huge real world
security implications.

Question: Factorize a number with N digits.
N=8 (21538177)
N=9 (215381778)

Scaling of factorization =

The best classical algorithm is exponential (Anything with the “N” in the exponent is hard).

Whereas Shor’s algorithm is polynomial

log (N); Log N (usually base 2 in computer science, written as log2N) essentially represents the number of bits (or digits) needed to write out the number N in binary.
With Shor’s algorithm, intractable problems can be solved but only if you have a working quantum computer.

Grover's Algorithm: This algorithm provides a massive speed boost for searching through unstructured data or finding a specific item in an unsorted database.

Interference

Bloch sphere is just the visualization of a state of qubit. In reality it is described as a quantum wave function. When many qubits are entangled together all of their wave functions are added together which transform into an overall wave function describing the state of the qubits. This adding together of wave functions is the interference. It is because when we add the waves together they can constructively interfere and add together to make a bigger wave, or destructively interfere to cancel each other out. The overall wave function of the qubits is what sets the different probabilities of the different states, and by changing the states of different qubits we can change the probabilities that different states will come out when we make the measurements. Wave functions are the fundamental mathematical description of everything in quantum mechanics. In a nutshell, interference is a fundamental phenomenon where the wave-like properties of particles (such as electrons and photons) interact with one another, causing their probabilities of existing in certain states or locations to amplify or cancel each other out. The easiest way to visualize quantum interference is through the famous double-slit experiment.

According to Classical thinking, the world is a stable, predictable and object-based reality whereas in quantum mechanics reality is disturbed; not stable, neither predictable or understandable.

In quantum mechanics, the particles exist in superposition which function as one shared system and are described by a joint wave function as mentioned above (Interference, probability distribution). When you measure one, the other responds instantly. If one particle is noticed to have a specific property such as spin, the other will immediately take the corresponding opposite state even if it is separated by vast distances and there’s no delay to this. It’s not a signal traveling from one particle to another through space. The correlation appears to happen with no time passing. This is where the structure of reality begins to fracture because this behavior violates classical thinking: local realism, objects having definite properties and any influence between them must travel through space over time. Entanglement rejects the idea of classical thinking. According to quantum physics, particles do not have definite properties until they are measured and the outcome of one measurement can be linked to another across any distance without any observable transmission of information between the particles. It does not follow Newton’s law which describes the motion of every object and the forces that cause them. At the deepest level reality is not made up of things but of relationships. If two particles can form a system across space then space itself begins to lose its meaning as a separator of objects. From this perspective, the universe doesn’t consist of isolated objects interacting across empty space. It consists of interconnected systems that only appear separate when observed at a certain scale. The correlation between the system is instantaneous, it can’t be controlled. When you measure a particle, the outcome is particularly random, you can’t choose whether the particle is up or down. As you cannot control the result, you cannot use it to send information. The universe allows for perfect correlation across distance but denies any ability to use that correlation for communication. It connects outcomes but not intentions. The result being a system that appears to violate the very structure of space while still preserving the rule of causality. Therefore it suggests that the world that we experience is stable, predictable,
object-based reality is not a foundation of existence but an illusion, where properties do not exist until they are measured, and where two points in space can behave as if they are not separate at all!

The whole universe is built on probabilities, non-local connections and undefined states giving rise to solid objects, stable identities, and predictable cause and effect relationships. It is because of decoherence which simply means dilution of quantum entanglement. It describes what happens when a delicate quantum system interacts with its environment. Instead of remaining isolated, the system becomes entangled with innumerable surrounding particles, air molecules, thermal radiation, and electromagnetic noise. Each interaction spreads the quantum information outward, distributing across a vast number of degrees of freedom. What once was a clean isolated superb position becomes entangled with an environment that can no longer be observed as a coherent whole. The system does not lose its quantum nature. It just becomes too entangled to track. Entanglement rather being rare is actually ubiquitous. Everything is constantly interacting, constantly exchanging information, constantly becoming entangled with everything else. But because this entanglement spreads so quickly and so widely , it loses the sharp correlations that make quantum effects noticeable. At large scales, the chaotic web of quantum interactions averages out into stable classical behavior. Objects do appear to have definite positions. Systems behave independently from one another, and cause and effect seems local and continuous. That is why physicists can observe entanglement in controlled laboratory environments. By isolating systems from environmental noise, they temporarily prevent decoherence from washing out the correlations.

Reality is a byproduct of entanglement spreading beyond our ability to observe it.

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