The Limits of Quantum Computers (or: What We Can’t Do With Computers We Don’t Have)

1. NP-complete The Limits of Quantum Computers(or: What We Can’t Do With Computers We Don’t Have) Scott Aaronson (MIT)

2. So then why can’t we just ignore quantum computing, and get back to real work?

3. My picture of reality, as an eleven-year-old messing around with programming: + details (Also Stephen Wolfram’s current picture of reality) Because the universe isn’t classical  Fancier version: Extended Church-Turing Thesis

4. That’s why YOU should care about quantum computing  Shor’s factoring algorithm presents us with a choice Either • the Extended Church-Turing Thesis is false, • textbook quantum mechanics is false, or • there’s an efficient classical factoring algorithm. All three seem like crackpot speculations. At least one of them is true!

5. One-Slide Summary • Quantum computing is not a panacea—and that makes it more interesting rather than less! • On our current understanding, quantum computers could break RSA, simulate quantum dynamics, and do other important things, but not solve “generic” search problems exponentially faster than classical computers • 3. In this talk, I’ll tell you about some of what’s known about the capabilities and limits of quantum computers

6. Quantum Computing A quantum state of n “qubits” takes 2n complex numbers to describe: The goal of quantum computing is to exploit this exponentiality in our description of the world Idea: Get paths leading to wrong answers to “interfere destructively” and cancel each other out

7. Shor’s Result Quantum computers can factor integers in polynomial time (thereby break RSA, thereby swipe your credit card number…) To prove this, Shor had to exploit a special property of the factoring problem(namely its reducibility to period-finding) Ideas extend to computing discrete logarithms, solving Pell’s equation, breaking elliptic curve cryptography…

8. But these problems aren’t believed to be NP-complete So the question remains: can quantum computers solve NP-complete problems in polynomial time? Bennett et al. 1997: “Quantum magic” won’t be enough Suppose we throw away the problem structure, and just consider an abstract space of 2n possible solutions Then even a quantum computer will need ~2n/2 steps to find a correct solution Note: This square-root speedup is achievable, via “Grover’s algorithm”

9. The quantum adiabatic algorithm [Farhi. Goldstone, et al. 2000] does exploit problem structure Hi Hf Hamiltonian with easily-prepared ground state Ground state encodes solution to NP-complete problem But it suffers from provable limitations of its own… [van Dam, Mosca, Vazirani 2001]: Eigenvalue gap can be exponentially small

10. Another example of a “quantum black-box problem”: find a collision in a list of numbers 28 12 18 76 96 82 94 99 21 78 88 93 39 44 64 32 99 70 18 94 66 92 64 95 46 53 16 35 42 72 31 66 75 33 93 32 47 17 70 37 78 79 36 63 40 69 92 71 28 85 41 80 10 73 63 95 57 43 84 67 57 31 62 39 65 74 24 90 26 83 60 91 27 96 35 20 26 52 88 89 38 97 54 30 62 79 71 84 50 38 49 20 47 24 54 48 98 23 41 16 40 75 82 13 58 56 81 34 14 61 52 21 44 22 34 14 51 74 76 83 37 90 58 13 10 25 29 11 56 68 12 61 51 23 77 68 72 43 69 46 87 97 45 59 73 30 19 81 86 49 60 85 80 50 11 59 65 67 89 29 86 48 22 15 17 55 36 27 42 55 77 19 45 15 53 98 91 87 25 33

11. Break Crypto Hash Functions What Could We Do With A Fast Quantum Algorithm For the Collision Problem?  Solve Graph Isomorphismby finding a collision in [A. 2002]: Any quantum algorithm needs at least ~N1/5 queries to find a collision in a list of size N [Shi, Kutin, Ambainis, Midrijanis]: Improved to ~N1/3 (which is optimal)

12. Measure 2nd register What makes the problem so hard? Basically, that a quantum computer can almost find a collision after one query to f! “If only we could now measure twice!” Or: if only we could see the whole trajectory of a “hidden variable” coursing through the quantum system![A., Phys. Rev. A 2005] Previous techniques weren’t sensitive to the fact that quantum mechanics doesn’t allow these things

13. Cartoon Version of Proof Suppose it exists by way of contradiction… T-query quantum algorithm that finds collisions in 2-to-1 functions T-query quantum algorithm that distinguishes 1-to-1 from 2-to-1 functions [Beals et al. 1998] p(f) is a multilinear polynomial, of degree at most 2T, in Boolean indicator variables (f(x),y) Let p(f) = probability algorithm says f is 2-to-1 Crucial facts:q(k)  [0,1] for all k=1,2,3,…q(1)  1/3q(2)  2/3 Let q(k) = average of p(f) over all k-to-1 functions f

14. The magic step: q(k) itself is a univariate polynomial in k, of degree at most 2T Why? That’s why

15. Large derivative [A. A. Markov, 1889]: Hence the original quantum algorithm must have made (N1/5) queries Bounded in [0,1] at integer points 1 q(k) 0 . . . . . . . . . . 1 2 3 N2/5 k

16. Problem: We’re given black-box access to a function f:{0,1}nZ We want to find a local minimum of f, evaluating f as few times as possible 4 4 2 3 5 [Aldous 1983] Randomized algorithm making 2n/2n queries[A., STOC’04] Quantum algorithm making 2n/3n1/6 queries [Aldous 1983] Any randomized alg needs 2n/2-o(n) queries[A., STOC’04] Any quantum alg needs 2n/4/n queries My lower-bound proof uses Ambainis’s quantum adversary method, which upper-bounds how much the entanglement between algorithm and oracle can increase via a single query

17. Surprising part: “Quantum-inspired” argument also yields a better classical lower bound: 2n/2/n2 Also yields the first randomized or quantum lower bounds for local search on constant-dimensional grid graphs Quantum Generosity … Giving back because we careTM

18. “OK, so I accept that quantum computers have these limitations. Is there any physical means to solve (say) NP-complete problems in polynomial time?”

19. Famous proposal for how to solve NP-complete problems: Dip two glass plates with pegs between them into soapy water. Let the soap bubbles form a “minimum Steiner tree” connecting the pegs Other proposals with obvious scaling problems: protein folding, DNA computing, optical computing…

20. “Relativity Computing” Problem: Energy needed to accelerate to relativistic speed DONE Variant: Black hole computing

21. Abrams & Lloyd 1998: If the Schrödinger equation were nonlinear, one could exploit that fact to solve NP-complete problems in polynomial time One way to interpret this result: as additional evidence that the Schrödinger equation is linear… 1 solution to NP-complete problem No solutions

22. Do the first step of a computation in 1 second, the next in ½ second, the next in ¼ second, etc. “Zeno Computing” Problem: “Quantum foaminess” Below the Planck scale, our picture of space and time breaks down in not-yet-understood ways…

23. Scientific American, March 2008 www.scottaaronson.com