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The Limits of Quantum Computers (or: What We Cant Do With Computers We Dont Have) Scott Aaronson (MIT) BQP NP- complete SZK

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So then why cant we just ignore quantum computing, and get back to real work?

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Because the universe isnt classical My picture of reality, as an eleven-year-old messing around with BASIC: + details Fancier version: Extended Church-Turing Thesis (Also Stephen Wolframs current picture of reality)

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Shors factoring algorithm presents us with a choice 1. the Extended Church-Turing Thesis is false, 2. textbook quantum mechanics is false, or 3. theres an efficient classical factoring algorithm. All three seem like crackpot speculations. At least one of them is true! Thats why YOU should care about quantum computing Either

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One-Slide Summary 1.Quantum computing is not a panaceaand 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, Ill tell you about some of the known limits of quantum computers 4. Ill also discuss a more general question: can NP- complete problems be solved efficiently by any physical means?

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What Quantum Mechanics Says If we observe, we see |0 with probability | | 2 |1 with probability | | 2 Also, the object collapses to whichever outcome we see 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

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To modify a state we can multiply vector of amplitudes by a unitary matrixone that preserves

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Were seeing interference of amplitudesthe source of all quantum weirdness

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A quantum state of n qubits takes 2 n complex numbers to describe: Quantum Computing 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

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Shors 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 Pells equation, breaking elliptic curve cryptography…

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But these problems arent 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 wont be enough Suppose we throw away the problem structure, and just consider an abstract space of 2 n possible solutions Then even a quantum computer will need ~2 n/2 steps to find a correct solution The quantum adiabatic algorithm (Farhi et al. 2000) does exploit problem structure. But it suffers from provable limitations of its own… Note: This square-root speedup is achievable, via Grovers algorithm

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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) By the birthday paradox, a randomized algorithm has to examine N of the N numbers [Brassard-Høyer-Tapp 1997] Quantum algorithm based on Grover that uses only N 1/3 queries Is that optimal? Proving a lower bound better than constant was open for 5 years

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Motivation for the Collision Problem Graph Isomorphism: find a collision in Statistical Zero Knowledge (SZK) protocols ? Cryptographic Hash Functions

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What makes the problem so hard? Basically, that a quantum computer can almost find a collision after one query to f! Measure 2 nd register Or: if only we could see the whole trajectory of a hidden variable coursing through the quantum system! [A., Phys. Rev. A 2005] If only we could now measure twice! Previous techniques werent sensitive to the fact that quantum mechanics doesnt allow these things

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[A., STOC02]N 1/5 lower bound on number of queries needed by a quantum computer to find collisions [Shi, FOCS02] [A.-Shi, J. ACM 2004] Improved to N 1/3 ; also N 2/3 lower bound for element distinctness [Kutin 2003] [Ambainis 2003] [Midrijanis 2003] Simplifications and generalizations

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Cartoon Version of Proof 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 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) Suppose it exists by way of contradiction… Crucial facts: q(k) [0,1] for all k=1,2,3,… q(1) 1/3 q(2) 2/3

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Thats why The magic step: q(k) itself is a univariate polynomial in k, of degree at most 2T Why?

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

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

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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

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Relativity Computing DONE Variant: Black hole computing Problem: Energy needed to accelerate to relativistic speed

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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 No solutions 1 solution to NP-complete problem One way to interpret this result: as additional evidence that the Schrödinger equation is linear…

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

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Quantum Advice Could there be a fixed quantum state thats been sitting around since the Big Bangand 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 couldnt solve NP-complete problems in polynomial time without exploiting the problem structure

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