What More Than Turing- Universality Do You Want? Scott Aaronson (MIT) Papers and slides at +?

The Pervasiveness of Universality Almost any programming language or cellular automaton you can think to invent, provided it’s “sufficiently complicated,” will be able to simulate a Turing machine For n large enough, almost any n-bit logic gate will be capable of expressing all Boolean functions Almost any 2-qubit unitary transformation can be used to approximate any unitary transformation on any number of qubits, to any desired precision Yet precisely because universality is “common as dirt,” it’s not useful for distinguishing among candidate physical theories

versus

What We Could Ask of Physical Laws “Beyond just Turing-universality” Simplicity Symmetry Relativity (at least Galilean) Quantum Mechanics (but why?) Robustness (i.e., fault-tolerant universality) Physical Universality (cf. constructor theory) Interesting Structure Formation in “Generic” Cases

Symmetry

Classical Reversible Gates Flip second bit iff first bit is 1 Not universal (affine) Fredkin Toffoli CNOT Flip third bit iff first two bits are both 1 Universal; can generate all permutations of n- bit strings Swap second and third bits iff first bit is 1 Computationally universal, but has a symmetry (preserves Hamming weight)

A.-Grier-Schaefer 2015: Classified all sets of reversible gates in terms of which n-bit reversible transformations they generate (assuming swaps and ancilla bits are free)

Physical Universality

Schaeffer 2014: The first known “physically- universal” cellular automaton (able to implement any transformation in any bounded region, by suitably initializing the complement of that region) Solved open problem of Janzing 2010

One of My Favorite Open Questions For every n-qubit unitary transformation U, is there a Boolean function f such that U can be realized by a polynomial-time quantum algorithm with an oracle for f? (I’m giving you any computational capability f you could possibly want—but it’s still far from obvious how to get the physical capability U!) Can show: For every n-qubit state | , there’s a Boolean function f such that |  can be prepared by a polynomial-time quantum algorithm with an oracle for f

Interesting Structure Formation

How to Measure Interesting Structure? Many people have studied this; Jim Crutchfield will tell you about how to define structure in terms of predictability One simpleminded measure: the Kolmogorov-Chaitin complexity of a coarse-grained description of our cellular automaton or other system Sean Carroll’s example:

The Coffee Automaton A., Carroll, Mohan, Ouellette, Werness 2015: A probabilistic n  n reversible system that starts half “coffee” and half “cream.” At each time step, we randomly “shear” half the coffee cup horizontally or vertically (assuming a toroidal cup) We prove that the apparent complexity of this image has a rising-falling pattern, with a maximum of at least ~n 1/6

Interesting Computations Should Be Not Merely Expressible, But Succinctly Expressible? BB(n) = the maximum number of steps that a 1-tape, 2- symbol, n-state Turing machine can take on an initially blank tape before halting BB(1)=1 BB(2)=6 BB(3)=21 BB(4)=107 BB(5)  47,176,870 BB(6)  7.4  Gödel  beyond some finite point, the values of BB(n) are not even provable in ZF set theory! (assuming ZF is consistent) Yedidia 2015 (building on Harvey Friedman): This happens at n  533,482 Also, ~10,000 states suffice to test Goldbach (Famous uncomputably- rapidly growing function)