标签：#maths

Introduction Having laid out some mathematical foundations in the [previous part][1], we will proceed to discuss quantum gates and circuits in depth here. As usual, we will drive home the theories with worked examples from [@Rieffel2011] and check their accuracy with qiskit and/or cirq code. Prerequisite Understanding quantum gates and circuits will be greatly aided by the knowledge of classical gates (AND, OR, NOT, XOR and the universal gates: NAND and NOR). Also, some familiarity with the Python programming language will help you understand qiskit and/or cirq code. Classical & Quantum gates A classical computer, possibly the one you're reading this with, is a sophisticated engineering piece, no doubt. However, the underlying "magic" comprises logic gates. The National Institute of Standards and Technology (NIST) gives this incredible analogical description of classical computers in relation to logic gates [@NISTQGATE2026]: Traditional computers are like microscopic cities. The roads of these cities are wires with electricity coursing through them. These roads have lots of gates, known as logic gates, which enable computers to do their job. Like physical gates that allow or block cars, logic gates allow or block electricity. Electricity that goes through the gates represents a “1” of digital data, and blocked electricity is a “0.” When you pick up a motherboard, for instance, you may not see the said gates as they are made of tiny transistors (in modern systems, MOSFETs) in specific combinations. Also, the $0$ and $1$ referred to in the analogy mean low and high voltage ranges, respectively. Their actual voltage values depend on the hardware. :::note All schematics were written in and rendered by @schemd/core . Check it out. ::: Logic Gates refresher Since we will be realizing some classical circuits using quantum gates, it's necessary to get familiar with common logic gates. NOT ($\neg$) Gate This gate takes a single classical bit and flips it. For instance, if $0

@kungfufk Since the dawn of computing, we have built Artificial Intelligence on a flawed premise: perfect rationality. We brute-force algorithms to find the optimal solution, assuming infinite time and infinite capacity. But humans don't work like that. As Herbert Simon famously coined, we operate on Bounded Rationality. We make decisions based on limited time, limited cognitive capacity, and limited information. What if, instead of forcing AI to be perfectly rational, we created a mathematical equivalent for human processing? What if we modeled human cognition using the laws of physics — wave theory, thermodynamics, and mechanical energy equations — to build a heavy, complex, but highly probabilistic AI engine? Here is a blueprint for a new field of research: Computational Cognitive Mechanics . 1. The Core Equations of Cognitive Processing To model bounded rationality mathematically, we first need to define the relationship between Knowledge ($K$), Cognitive Capacity ($C$), and Processing Time ($T$). Based on human observation, we can establish these foundational proportions: Knowledge vs. Time — The more knowledge you possess, the faster you can generate a decision. $$T \propto \frac{1}{K}$$ Capacity vs. Time — High cognitive capacity (skills, processing power) inversely relates to the time required to solve a problem. $$T \propto \frac{1}{C}$$ Knowledge vs. Capacity — This is the most fascinating limit. Knowledge does not scale linearly with capacity. Gaining true knowledge requires exponential capacity (effort/skill). Therefore, knowledge is roughly proportional to the square root of capacity. $$K \propto \sqrt{C}$$ By integrating these, we can build a baseline processing algorithm for an AI. Instead of giving an AI unlimited time to compute, we cap its computing time based on a synthetic "Knowledge and Capacity" matrix, forcing it to use heuristics — just like a human. 2. Cognitive Wave Theory & FFT: Information as Interference In physics, waves interact throug