Author: Dr. Hakop Pashayan (14 May 2026) 

When we think of computation, we usually picture servers humming in massive 

data centers or the silicon chips embedded in our smartphones. We view 

computation as a purely human engineering endeavor—a tool we built to 

manipulate numbers and render graphics. But this view is fundamentally 

incomplete. Computation is not just an invention; it is a universal process woven 

into the very fabric of nature. 

In a recent exploration of the "Physics of Information," we are invited to radically 

shift our perspective. To truly understand the frontier of modern technology—

specifically, the enigma of quantum computing—we must take a journey that 

begins with abstract mathematics, collides with the harsh limits of logic, and 

ultimately reveals that the rules of computation are rigidly dictated by the laws of 

physics. 

The Illusion of Mathematics 

At its absolute core, computation is simply data processing. It is the application of 

a strict set of rules that maps input information to output information. We are 

familiar with basic logical operations, like AND or NOT gates, which can be strung 

together to perform addition or multiplication. But these mapping rules can also 

take highly geometric, visual forms. 

Consider Conway’s Game of Life, a "zero-player game" played on an infinite grid. 

Every cell on the grid is either alive (1) or dead (0). The grid updates in discrete 

time steps based on an incredibly simple, localized rule: a cell survives if it has two 

or three neighbors, dies if it is overcrowded or isolated, and springs to life if it is 

surrounded by exactly three living cells. 

When you apply these rigid, deterministic rules repeatedly, something profound 

happens: emergence. Out of the chaotic flashing of 1s and 0s, stable geometric 

structures form. Some structures, called "gliders," actually propel themselves 

across the grid. We see this same emergence in continuous mathematics. The 

famous Mandelbrot set is generated by a deceptively simple iterative equation 

($z_{n+1} = z_n^2 + c$). Yet, zooming into the boundary of this set reveals an 

infinitely complex, fractal universe born entirely from a single rule. 

To formalize computation and computing machines, mathematician Alan Turing 

introduced the Turing Machine in 1936. It is an abstract device featuring an 

infinite tape of 1s and 0s, a read/write head, and a basic list of internal states. 

Turing proved that this beautifully simple theoretical machine captures the 

absolute essence of computation. In fact, it revealed the principle of universality. 

The hardware doesn't actually matter. A standard Turing Machine, a 

supercomputer, and even the Conway's Game of Life are in some concrete 

mathematically sense equivalent. They can all simulate each other (Turing 

complete). 

Hitting the Wall: The Limits of Logic 

However, this mathematical utopia has boundaries. Since Turing Machines can 

compute anything that is computable, we are forced to ask: is there math they 

cannot do? 

The answer is a resounding yes. The most famous example is the Halting Problem, 

which proves it is fundamentally impossible to write a program that can flawlessly 

predict whether any other given program will eventually halt or loop forever. This 

uncomputability isn't just an abstract paradox; it shows up in simple counting 

exercises like the "Busy Beaver" game. If you ask what is the maximum number of 

1s an $N$-state Turing Machine can print before halting, the answers for 1, 2, 3, 

and 4 states are small, manageable numbers. But by the time you reach $N=6$, 

the number is so unfathomably large it dwarfs the number of atoms in the 

universe. 

For the problems we can compute, we run into a different wall: time and space. 

This is the domain of Computational Complexity. We classify problems by how 

drastically their difficulty scales as the input size grows. Problems in polynomial 

time (P) scale gently, while problems in NP might be exponentially hard to solve 

from scratch. Here, we leave the abstract realm of pure logic and are forced to 

confront reality. We don't have infinite tape, and we don't have infinite time. 

Computation is Physical 

This brings us to a crucial pivot in our understanding of information: doing actual 

computation requires a physical system subject to physical laws. 

Imagine you need to compute the square root of a number. You don't necessarily 

need a microchip. You could simply drop a ball from a specific height $h$ and 

measure the time it takes to hit the ground. Because $t = \sqrt{2h/g}$, the 

observable outcome of this physical system yields your mathematical answer. 

Every computation relies on physics. 

Because of this inescapable connection, computer scientists formulated the 

Extended Church-Turing Thesis. It essentially argues that classical physics 

fundamentally limits computational efficiency. No matter what bizarre classical 

physical system you build—whether it's powered by fluid dynamics, gravity, or 

complex chemical reactions—it cannot give you an exponential speed shortcut 

over a standard Turing Machine. Classical physics dictates the ultimate speed limit 

of classical computation. 

The Quantum Loophole 

But what happens if we build physical computation machines who’s behavior 

cannot be well approximated by the classical laws of physics? What if we compute 

using quantum mechanics? 

A classical bit is a switch, stubbornly restricted to exactly 0 or 1. A single quantum 

bit, or qubit, can exist in a state of superposition, holding a complex amplitude for 

being 0 and a complex amplitude for being 1. This can be described by a point on 

the 2D surface of a 3D sphere. But the true paradigm shift—the curse and blessing 

of dimensionality—happens when we scale up. 

If you have $n$ classical bits, you just have $n$ pieces of information. But if you 

have $n$ entangled qubits, they exist as a single vector in a massive, $2^n$-

dimensional complex space. To simulate just 300 fully entangled qubits, as far as 

we can tell, nature keeps track of approximately $2^{300}$ complex numbers—

more numbers than there are atoms in the observable universe. Quantum states 

have a physical embodiment, meaning nature handles this unimaginably vast 

math for free. The data is encoded in a physical system and we can use 

measurement to observe and query this vast data. 

Quantum evolution allows us to navigate this space using operations that have 

no classical equivalent. For example, there is a unitary quantum operation that, 

when applied twice, results in a perfect NOT gate. Classically, there is no such 

thing as a "square root of NOT." Applying it once moves the system into a 

computational dimension completely inaccessible to classical logic.

Computation is not just abstract mathematics hovering in the ether. It is physical 

data processing. By building machines that operate on the laws of quantum 

mechanics, we aren't just building faster calculators; we are unlocking a 

completely different physical sandbox. We are finally learning how to compute 

with the full fabric of reality.