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The Limits of Quantum Computers (or: What We Cant Do With Computers We Dont Have) Scott Aaronson University of Waterloo 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 QBASIC: + 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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My Spiel In One Slide 1. Ignoring quantum mechanics wont make it go away 2. Quantum computing is not a panaceaand that makes it more interesting rather than less! 3. On our current understanding, quantum computers could merely break RSA, simulate quantum physics, etc.not solve generic search problems exponentially faster 4. So then why do I worry about quantum computing? Because Im interested in fundamental limits on what can efficiently be computed in the physical world. That makes me professionally obligated to care!

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Where Do I Come In? My work, over the last seven years, has deepened our understanding of the limitations of quantum computers. Solved some of the fields notorious open problems: - Lower bound for finding collisions in hash functions - Direct product theorem for quantum search Made unexpected connections: - Classical lower bounds proved by quantum arguments - Quantum-state learning algorithm from a lower bound

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Plan of Talk The Gospel According to Shor Three Limitations of Quantum Computers - Finding collisions in hash functions - Solving NP-complete problems with advice - Finding local optima Turning Lemons into Lemonade - Approximately learning quantum states Summary of Contributions

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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 incorrect answers to interfere destructively and cancel each other out

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Plan of Talk The Gospel According to Shor Three Limitations of Quantum Computers - Finding collisions in hash functions - Solving NP-complete problems with advice - Finding local optima Turning Lemons into Lemonade - Approximately learning quantum states Summary of Contributions

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f x f(x) In this talk, Ill only care about the number of queries to a black box, not any other computational steps Example: Given a function f:{0,1} n {0,1}, suppose we want to decide if theres an x such that f(x)=1 Classically, ~2 n queries to f are needed Grover gave a quantum algorithm that uses only ~2 n/2 queries [BBBV 1997]: Grovers algorithm is optimal Yields black-box evidence that quantum computers cant solve NP-complete problems efficiently The Quantum Black-Box Model But why do black-box results tell us anything about the real world? Remember IP=PSPACE? You gotta start somewhere Almost all known quantum algorithms are black-box (no quantum IP=PSPACE yet) The proof of the pudding is in the proving

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Algorithms state: x: location to query w: workspace qubits After a query transformation: Between two queries, we can apply an arbitrary unitary matrix that doesnt depend on f Complexity = minimum number of queries needed to achieve for all oracles f

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Plan of Talk The Gospel According to Shor Three Limitations of Quantum Computers - Finding collisions in hash functions - Solving NP-complete problems with advice - Finding local optima Turning Lemons into Lemonade - Approximately learning quantum states Summary of Contributions

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Problem: Find 2 numbers that are the same (each number appears twice) By birthday paradox, a randomized algorithm must 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! 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 quantum query complexity of the collision problem [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… Trivial yet 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 [A. A. Markov, 1889]: Hence the original quantum algorithm must have made (N 1/5 ) queries

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Plan of Talk The Gospel According to Shor Three Limitations of Quantum Computers - Finding collisions in hash functions - Solving NP-complete problems with advice - Finding local optima Turning Lemons into Lemonade - Approximately learning quantum states Summary of Contributions

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Could there be a quantum state | left over from the Big Bang, such that given any 3SAT instance of size 1,000,000, we could quickly solve it by just measuring | in an appropriate basis? [A., CCC 2004] In the black-box model, no: there cannot exist any golden state for solving NP-complete problems in polynomial time The Hunt for the Golden State

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Efficient quantum algorithm to solve SAT using an m-qubit golden state Efficient quantum algorithm to solve (say) m 3 SAT instances, reusing the same golden state Algorithm to solve m 3 SAT instances with probability 2 -m Guess the golden state! Replace it by the maximally mixed state, i.e. a random m-bit string Suppose it exists by way of contradiction… To get a contradiction, I now need to prove a direct- product theorem for quantum search: If a quantum algorithm doesnt even have time to solve one search problem w.h.p., then the probability of its solving k search problems decreases exponentially with k

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How do I prove the direct-product theorem? Again using the polynomial method But this time I need a generalization of A. A. Markovs inequality due to [V. A. Markov 1892], which takes into account not just the first derivative but all higher derivatives [Klauck-Špalek-deWolf, FOCS04] tightened my direct product theorem, and also used it to prove the first quantum time-space tradeoffs n2n m3m3

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Plan of Talk The Gospel According to Shor Three Limitations of Quantum Computers - Finding collisions in hash functions - Solving NP-complete problems with advice - Finding local optima Turning Lemons into Lemonade - Approximately learning quantum states Summary of Contributions

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Problem: Were 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 [Aldous 1983] Randomized algorithm making 2 n/2 n queries [A., STOC04] Quantum algorithm making 2 n/3 n 1/6 queries [Aldous 1983] Any randomized alg needs 2 n/2-o(n) queries [A., STOC04] Any quantum alg needs 2 n/4 /n queries My lower-bound proof uses Ambainiss quantum adversary method, which upper-bounds how much the entanglement between algorithm and oracle can increase via a single query

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Quantum Generosity … Giving back because we care TM Surprising part: Quantum-inspired argument also yields a better classical lower bound: 2 n/2 /n 2 Also yields the first randomized or quantum lower bounds for local search on constant- dimensional grid graphs Subsequent improvements: [Santha-Szegedy, STOC04] [Zhang, STOC06] [Verhoeven, 2006] [Sun-Yao, FOCS06]

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Plan of Talk The Gospel According to Shor Three Limitations of Quantum Computers - Finding collisions in hash functions - Solving NP-complete problems with advice - Finding local optima Turning Lemons into Lemonade - Approximately learning quantum states Summary of Contributions

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[ANTV 1999]: | must have (n) qubitsno asymptotic savings over classical (Surprisingly, an n-qubit quantum state has no more independently accessible degrees of freedom than an n-bit classical string) The Lemon | n-bit string, x 1 …x n Any one bit x i of our choice, with high probability Quantum random access coding

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Upper bound on the sample complexity of PAC (Probably Approximately Correctly) learning a quantum state Informally: Can predict approximate expectation values of most measurements on an n-qubit state, after a number of sample measurements that increases only linearly with n The Lemonade | Quantum Occams Razor Theorem [A. 2006] By contrast, traditional quantum state tomography requires ~4 n measurements Record so far: n=8 Prohibitive for much larger n

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Plan of Talk The Gospel According to Shor Three Limitations of Quantum Computers - Finding collisions in hash functions - Solving NP-complete problems with advice - Finding local optima Turning Lemons into Lemonade - Approximately learning quantum states Summary of Contributions

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Solved several notorious open problems about the limitations of quantum computers Gave evidence that collision-resistant hash functions can still exist in a quantum world Proved the first direct product theorem for quantum search Gave evidence against golden states for NP-complete problems Solved open problems about classical local optimization using quantum techniques Used a quantum coding lower bound to propose a new learning algorithm, with possible experimental implications Summary of Contributions

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Ten Research Directions I Didnt Tell You About Today Addressing skepticism of quantum computing [A., STOC 2004] Grover search with finite speed of light [A.-Ambainis, FOCS 2003] Quantum versus classical proofs [A.-Kuperberg, CCC 2007] Need to uncompute garbage in quantum algorithms [A., QIC 2003] Practical simulation of stabilizer quantum circuits [A.-Gottesman, Phys Rev A 2004] Quantum software copy- protection [A., in preparation] Quantum computers with anthropic postselection [A., Proc. Roy. Soc. 2005] Quantum computers with closed timelike curves [A.-Watrous, in preparation] Provably-nonrelativizing circuit lower bounds [A., CCC 2006] Complexity of Bayesian agreement protocols [A., STOC 2005]

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