PYTHON · NUMPY

How cheap is computing allowed to be?

Compute the Landauer limit yourself and measure how far above it a real operation sits. The gap is the headroom the whole field is arguing about.

01Challenge

Calculate the thermodynamic floor for erasing a bit at room temperature, then express a modern multiply-accumulate as a multiple of it.

Predict firstBefore computing it: how many times the physical minimum do you think one multiply-accumulate costs?

02Model

In 1961 Rolf Landauer proved that the only step of computation which must dissipate energy is erasing information. Everything else is, in principle, free. That floor is kT ln 2 — about three zeptojoules at room temperature. A modern 8-bit multiply-accumulate costs a fraction of a picojoule. Those two numbers are not close: the arithmetic we consider cheap is already a hundred million times the price physics demands. Knowing that ratio changes what you believe is possible.

03Guided practice

1. The floor. Landauer's limit is kT ln 2, where k is Boltzmann's constant and T is temperature in kelvin. Compute it at 300 K.

2. A real operation. A modern 8-bit MAC costs roughly 0.1–1 pJ. Divide.

3. Read the ratio. You should land near 10⁸. Sit with that: the operation everyone calls cheap is a hundred million times the physical minimum. The efficiency conversation is not about shaving percentages.

04Feedback

PASS WHEN PASS when you print a floor near 2.87e-21 J and a ratio on the order of 1e8.

If your run fails, check:

If the floor looks wrong, check that you used ln 2 (math.log(2)) and not log base 10.
The ratio is meant to be shocking. If it is not roughly a hundred million, re-check your units — picojoules are 1e-12.

05Retrieve & space

What is the only step of computation that must dissipate energy?
Roughly how many times the Landauer floor does one multiply-accumulate cost?
Does the floor get lower or higher if you cool the machine?

06Mastery & project

You can compute the thermodynamic floor for computation and state how far above it real hardware sits.

Keep the ratio. Every efficiency claim you meet for the rest of the course gets measured against the floor, not against last year's chip.