The densest computer physics allows is a black hole

Any thought that needs the whole mind to weigh in has to wait for information to travel from one side to the other. That crossing time works like a clock speed for the whole system, and it grows with size, so the same light-speed wiring that crosses a 6-centimeter chip in 0.2 nanoseconds (one tick of a 5-gigahertz clock) needs about 8.3 minutes to span the distance from the Earth to the Sun and about 100,000 years to cross the Milky Way.

Your brain plays by the same rule at a much lower speed. A signal crossing its roughly 15 centimeters at 120 meters per second takes about 1.25 milliseconds, and since its neurons peak at around 200 signals per second, that crossing fits comfortably inside one neural tick.

A computer the size of the Earth has it much worse. Each of its signals moves millions of times faster than a nerve impulse, yet light still needs about 42 milliseconds to get across, so the whole machine can coordinate itself only about 24 times a second, and stretching it to the Earth-Moon system pushes each crossing to about 1.3 seconds. That's why the first pressure on any fast mind is to stay compact.

Gravity, as general relativity describes it, sets a hard limit on how much mass you can cram into a region. Every mass has a size called its Schwarzschild radius, and once you squeeze the mass inside that radius, nothing can stop it collapsing into a black hole. The physicist Raphael Bousso puts the consequence for information bluntly: a system's "mass must be less than the mass M of a black hole of the same surface area. Otherwise, the system could not be gravitationally stable, and from the outside point of view it would already be a black hole" 1.

Jacob Bekenstein, who first found the limit on how much information a region can hold, described what goes wrong if you try to beat it by piling up more and more memory chips: "what fails is the pile itself: it would collapse under its own gravity and form a black hole before that impasse was reached" 2.

That leads to one of the strangest results in physics. The most information you can fit inside any region is set by the area of its boundary rather than its volume. Bousso gives the number: "the area of any surface limits the information content of adjacent spacetime regions, at 1.4 × 10⁶⁹ bits per square meter" 1. So if you double the width of a memory bank, its volume grows eightfold while the ceiling on what it can hold grows only fourfold.

To get a feel for the scale, take a ball with a one-meter radius. The holographic bound, as this area rule is called, says it can hold at most about 1.7×10⁷⁰ bits. Reaching that limit takes a black hole with a one-meter radius, which means cramming about 6.7×10²⁶ kilograms into the ball, roughly a third of the mass of Jupiter.

Seth Lloyd, the physicist who worked out the speed limit for a one-kilogram "ultimate laptop", took the idea to its end. He imagined compressing his one-kilogram computer down to its Schwarzschild radius, which is 1.485×10⁻²⁷ meters, far smaller than a proton. In his words, it "can perform 5.4258 × 10⁵⁰ operations per second" on about 3.8×10¹⁶ bits 3.

Two things about that result surprised me. First, the black-hole version stores less than the ordinary ultimate laptop, about 4×10¹⁶ bits against 2×10³¹, because a one-kilogram black hole is absurdly tiny and its capacity depends on its surface area. Capacity grows with the square of the mass, though, so a black hole as heavy as the Sun could hold around 10⁷⁷ bits.

Second, it doesn't last. Black holes slowly radiate away their mass, and small ones go fast. Lloyd estimates "a lifetime for a 1 kilogram black hole of ≈ 10⁻¹⁹ seconds, during which time the hole can perform ≈ 10³² operations on its ≈ 10¹⁶ bits" 3, which works out to each bit being flipped about once every 7×10⁻³⁵ seconds. How you would actually read the answer back out of the radiation is, as Lloyd admits, a question "whose answer will have to await a unified theory of quantum gravity" 3.