Threshold for Life

MIT Media Laboratory

Prof. Isaac ChuangProf. Isaac Chuang

Objective

• Claim:

Life can be constructed from faulty components

• Question:

How Faulty?

An Amazing Theorem

Reliable computers can be constructed from faulty

components• A circuit containing N (error-free) gates can

be simulated with probability of error at most , using N log(N/) faulty gates, which fail with probability p, so long as p<pth.von Neumann

(1956)Quantum version: Preskill, Shor, Aharonov, Ben-Or, Gottesman, Chuang*,…

* D. Gottesman and I. Chuang, Nature, vol. 402, p. 390, 1999

Fault-Tolerant Circuits

Fault-Tolerant Circuits

The Fault-Tolerance ThresholdA circuit containing N (error-free) gates can be simulated with probability of error < , using cdN faulty gates, which fail with probability p, if p<pth.Proof: Assume 3-input gates. N=1 easy.

Non-Fault-Tolerant

Circuit 1: Circuit 2: Circuit 3:

p

Fails with probability 3+p < 4

Circuit 1: Circuit 2: Circuit 3:

pCircuit 1: Circuit 2: Circuit 3:

pp

Circuit 1: Circuit 2: Circuit 3:

pCircuit 1: Circuit 2: Circuit 3:

pp

Circuit 1: Circuit 2: Circuit 3:

pCircuit 1: Circuit 2: Circuit 3:

pp

MAJ

Fails with probability < 3(42 + p < 492 If p<2 and 492 < p<

Fault-Tolerant

Efficient Fault ToleranceA circuit containing N (error-free) gates can be simulated with prob. of error < , using O(N log N) faulty gates, which fail with prob. p, if p<pth.• Key concept: concatenation (encoding at multiple

levels)

• Status: pth ¼ 1/6 (Hajek & Weller ’91)

Fault-Tolerant Life

• Nature does not seem to use such constructions

How can life be made fault-tolerant?

What is Life?

• Desirable characteristics:• Self repair• Self reproduction• Growth• Evolution

Schrodinger

• von Neumann (1966) Theory of Self-Reproducing Automata: elementary machine capable of reproduction and universal computation

Von Neumann’s Automaton

• Model based on 29 state Cellular Automaton

(simulation from H-H Chou)

Langton’s 8-state replicator (not universal)

• Universal Turing machine• Universal constructor

Visions of Fault-Tolerant Life

• Two interesting questions:

What is the fault-tolerance threshold for Life?

What would fault-tolerant creatures be like?

Rad-hard biology Immortal Biosystems …Reproduction?