Transcript Slide 1
The Limits of Quantum Computers (or: What We Can’t Do With Computers We Don’t Have) Scott Aaronson (MIT) NPcomplete SZK BQP So then why can’t we just ignore quantum computing, and get back to real work? Because the universe isn’t classical My picture of reality, as an eleven-year-old messing around with BASIC: + details (Also Stephen Wolfram’s current picture of reality) Fancier version: Extended Church-Turing Thesis Shor’s factoring algorithm presents us with a choice Either 1. the Extended Church-Turing Thesis is false, 2. textbook quantum mechanics is false, or That’s why YOU should care 3. there’s an efficient classical factoring algorithm. about quantum computing All three seem like crackpot speculations. At least one of them is true! One-Slide Summary 1. Quantum computing is not a panacea—and that makes it more interesting rather than less! 2. On our current understanding, quantum computers could “merely” break RSA, simulate quantum physics, etc.—not solve generic search problems exponentially faster 3. In this talk, I’ll tell you about some of the known limits of quantum computers 4. I’ll also discuss a more general question: can NPcomplete problems be solved efficiently by any physical means? What Quantum Mechanics Says If an object can be in two distinguishable states |0 or |1, then it can also be in a superposition |0 + |1 Here and are complex amplitudes satisfying ||2+||2=1 If we observe, we see |0 with probability ||2 |1 with probability ||2 Also, the object collapses to whichever outcome we see 1 0 1 2 0 To modify a state n i 1 i i we can multiply vector of amplitudes by a unitary matrix—one that preserves n i 1 2 i 1 0 1 1 2 1 2 1 1 1 2 12 20 1 01 11 2 2 2 1 0 1 2 We’re seeing interference of amplitudes—the source of all “quantum weirdness” 2 0 Quantum Computing A quantum state of n “qubits” takes 2n complex numbers to describe: x0,1 x x n 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 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… 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” The quantum adiabatic algorithm (Farhi et al. 2000) does exploit problem structure. But it suffers from provable limitations of its own… Another example of a “quantum black-box problem”: given a two-to-one function f:[N][N], find any x,y pair such that f(x)=f(y) 28 12 18 76 96 82 94 99 21 78 88 93 39 44 64 32 99 94 66 92 64 95 46a 53 16 35 42 72 By70 the18 “birthday paradox”, randomized 31 algorithm 66 75 33has 93 to32examine 47 17 70 78 N 79numbers 36 63 40 N 37 of the 69 92 71 28 85 41 80 10 73 63 95 57 43 84 67 57[Brassard-Høyer-Tapp 31 62 39 65 74 24 1997] 90 26 Quantum 83 60 91algorithm 27 96 35 20 26 52 88 89 38 97 30 only 62 79 84 50 38 based on Grover that54 uses N1/371queries 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 bound better 37 Is 90that 58 optimal? 13 10 25Proving 29 11 a 56lower 68 12 61 51 23 77 than for30 5 years 68 72 43 69 constant 46 87 97was 45 open 59 73 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 Motivation for the Collision Problem Cryptographic Hash Functions Graph Isomorphism: find a collision in 1 G , , n ! G , 1 H , , n! H Statistical Zero Knowledge (SZK) protocols What makes the problem so hard? Basically, that a quantum computer can almost find a collision after one query to f! 1 N x y N x f x x 1 2nd Measure register f x 2 “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 [A., STOC’02] N1/5 lower bound on number of queries needed by a quantum computer to find collisions [Shi, FOCS’02] Improved to N1/3; also [A.-Shi, J. ACM 2004] N2/3 lower bound for element distinctness [Kutin 2003] Simplifications and [Ambainis 2003] generalizations [Midrijanis 2003] Cartoon Version of Proof T-query quantum algorithm that finds collisions in 2-to-1 functions Suppose it exists by way of contradiction… T-query quantum algorithm that distinguishes 1-to-1 from 2-to-1 functions Let p(f) = probability algorithm says f is 2-to-1 Let q(k) = average of p(f) over all k-to-1 functions f [Beals et al. 1998] p(f) is a multilinear polynomial, of degree at most 2T, in Boolean indicator variables (f(x),y) Crucial facts: q(k) [0,1] for all k=1,2,3,… q(1) 1/3 q(2) 2/3 The magic step: q(k) itself is a univariate polynomial in k, of degree at most 2T Why? That’s why Bounded in [0,1] at integer points 1 q(k) Large derivative 0 1 2 3 . . . . . . . . . . k [A. A. Markov, 1889]: degq N 2 / 5 max2 / 5 dqx / dx 0 x N 2 max2 / 5 qx N 1/ 5 0 x N Hence the original quantum algorithm must have made (N1/5) queries N2/5 “OK, so I accept that quantum computers have these limitations. Is there any physical means to solve (say) NP-complete problems in polynomial time?” 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… For the latest, please see Slashdot “Relativity Computing” DONE Problem: Energy needed to accelerate to relativistic speed Variant: Black hole computing Abrams & Lloyd 1998: If the Schrödinger equation governing quantum mechanics 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 “Zeno Computing” Do the first step of a computation in 1 second, the next in ½ second, the next in ¼ second, etc. Problem: “Quantum foaminess” Below the Planck scale (10-33 cm or 10-43 sec), our usual picture of space and time breaks down in not-yet-understood ways… Quantum Advice Could there be a fixed quantum state that’s been sitting around since the Big Bang—and that if found, would be a “magic key” to performing quantum computations that were otherwise infeasible? [A. 2004]: Even under such a strange assumption, we still couldn’t solve NP-complete problems in polynomial time without exploiting the problem structure www.scottaaronson.com/papers