The Limits of Efficient Computation

1. The Limits of Efficient Computation Scott Aaronson (University of Texas at Austin) Bernays Lecture, ETH Zurich, Sep. 3, 2019

2. 1950s: Actual Computers Available Great! Let’s ask them to solve some computable problems … like, say, first-order reasoning about real numbers Oops. Will take longer than the age of the universe! And still would, even for Google in 2019!  Computability is not enough

3. Asymptotics 2n Key distinction: polynomial vs. exponential “What about n1000? 21000n? 1.00000001n?” n2 n

4. Problem: Modeled as a set of binary strings, L{0,1}*. Given an input x{0,1}*, the task is to decide if xL P The class of problems for which there’s an algorithm, for a deterministic digital computer, that always correctly decides if xL, after a number of steps upper-bounded by some polynomial in |x| (the length of x) Examples: Primality Testing Connectivity Linear Programming

5. (Nondeterministic Polynomial-Time) NP Informally, the class of problems for which there’s a polynomial-time algorithm to recognize valid solutions (but the solutions might be exponentially hard to find) Factoring Examples: NP-complete Traveling Salesperson Bin Packing Theorem Proving

6. Hamilton cycleSteiner treeGraph 3-coloringSatisfiabilityMaximum clique… Matrix permanentHalting problem… Graph isomorphismFactoring… Graph connectivityPrimality testingMatrix determinantLinear programming… NP-hard NP-complete NPNondeterministic Polynomial Time PPolynomial Time

7. P=NP? The (literally) $1,000,000 question If there actually were a machine with [running time] ~Kn (or even only with ~Kn2), this would have consequences of the greatest magnitude.—Gödel to von Neumann, 1956 For masochists: (122 pages)

8. Suppose we believe PNP. Is there any physical means—going beyond Turing machines—to solve NP-complete problems in polynomial time? The Extended Church-Turing Thesis (ECT) Feasibly computable in the physical world = In P (feasibly computable by Turing machine) What sort of thing would overthrow the ECT, while still upholding the “original” Church-Turing Thesis?

9. Soap Bubble Computer

10. Protein Folding “If NP is so easy, why aren’t you rich? Why haven’t you proved the Riemann Hypothesis?”

11. Quantum Computing: The One Serious Challenge to the ECT Quantum mechanics: “Probability with minus signs” “Obviously,” p =p1+p2 The true laws:  = 1 + 2 p = ||2 p1 p1 1C  p When you try: p p1+p2 !! 2C p2 Unless you look!

12. If we observe, we see |0 with probability |a|2 |1 with probability |b|2 Also, the object collapses to whichever outcome we see A bit more precisely: the key claim of quantum mechanics is that, if an object can be in two distinguishable states, call them |0 or |1, then it can also be in a superpositiona|0 + b|1 Here a and b are complex numbers called amplitudes satisfying |a|2+|b|2=1

13. Quantum Computing What if instead of just a superposition over 2 states, we had 21000? How to arrange such a thing? Entanglement 1000 quantum bits (qubits) 21000 states, each with its own complex amplitude ~21000 time to simulate on a computer?? Yikes!

14. Interesting Feynman 1981: Why not turn this lemon into lemonade, by making the computer itself quantum? But supposing we built one, what would such a “quantum computer” be good for? Feynman’s answer: Simulating quantum mechanics itself! Shor 1994: Also factoring, discrete log  breaking today’s public-key cryptography (!!)

15. Why Can’t QCs Do Even More—Like NP-Complete Problems? The Fundamental Misconception: “A QC could just try all the 2n possible answers in parallel!” HERE IT IS! The entire question: how do you use interference to quickly concentrate a large amplitude, and hence large probability, on the answer you want? Shor did it for factoring, by exploiting special structure in that problem—but it’s subtle!

16. Black-Box Quantum Search Grover’s Algorithm (1996): Searches any list of N possible solutions in only ~N steps, rather than N MARKED ITEM Bennett, Bernstein, Brassard, Vazirani 1994: But even for a QC, this is the fastest possible—unless one exploits additional structure

17. Attempt to exploit that additional structure: the Quantum Adiabatic Algorithm (Farhi et al. 2000) Hi Hf Hamiltonian with easily-prepared ground state Ground state encodes solution to NP-complete problem Problem: “Eigenvalue gap” can be exponentially small

18. Conjectured Picture NP-complete Bounded-Error Quantum Polynomial Time Post-quantum crypto? NP Factoring BQP P Can all physics be simulated in BQP (“Quantum ECT”)? Are there any physical means to solve NP-complete problems in polynomial time?

19. By adding the concept of efficiency to computability, the Extended Church-Turing Thesis formed an intellectual underpinning for today’s computer science, and for the still-unsolved P vs. NP problem Summary Quantum computing poses a severe challenge to the ECT—perhaps the only challenge rooted in the known laws of physics Contrary to myth, even quantum computers will be limited. But for a few problems, they’ll extend the frontier of the feasible … enough to motivate the current worldwide race to build them!

20. Can we actually build small quantum computers—good enough to refute the Extended Church-Turing Thesis experimentally? How soon? And what else will they be good for? Next Lecture