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**March 12, 2012**

# Ultimate physical limits to computation

## Lloyd, Seth. 2000. Ultimate physical limits to computation. Nature 406:1047–1054.

I just re-read part of this classic CS paper (PDF), and the figure captions at the back stood out to me as being particularly hilarious:

Figure 1: The Ultimate LaptopThe ‘ultimate laptop’ is a computer with a mass of one kilogram and a volume of one liter, operating at the fundamental limits of speed and memory capacity ﬁxed by physics. [...] Although its computational machinery is in fact in a highly speciﬁed physical state with zero entropy, while it performs a computation that uses all its resources of energy and memory space it appears to an outside observer to be in a thermal state at approx. \( 10^9 \) degrees Kelvin. The ultimate laptop looks like a small piece of the Big Bang.

Figure 2: Computing at the Black-Hole LimitThe rate at which the components of a computer can communicate is limited by the speed of light. In the ultimate laptop, each bit can ﬂip approx. \( 10^{19} \) times per second, while the time to communicate from one side of the one liter computer to the other is on the order of 10^9 seconds: the ultimate laptop is highly parallel. The computation can be sped up and made more serial by compressing the computer. But no computer can be compressed to smaller than its Schwarzschild radius without becoming a black hole. A one-kilogram computer that has been compressed to the black hole limit of \( R_S = \frac{2Gm}{c^2} = 1.485 \times 10^{−27} \) meters can perform \( 5.4258 \times 10^{50} \) operations per second on its \( I = 4\pi\frac{Gm2}{ln(2hc)} = 3.827 \times 10^{16} \) bits. At the black-hole limit, computation is fully serial: the time it takes to ﬂip a bit and the time it takes a signal to communicate around the horizon of the hole are the same.

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