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Computation, Quantum Theory, and You Scott Aaronson, UC Berkeley Qualifying Exam May 13, 2002

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Talk Outline 1.Sermon 2.Quantum Computing Overview 3.Collision Lower Bound 4.Dynamical Models 5.Current and Future Work

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1. Sermon

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The Computer Scientists Idea of Physics + details

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What Does Our World Have That Conways Doesnt? 3 or more spatial dimensions Continuity? Relativistic covariance Quantum theory And more? Quantum theory

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My Own View… What we experience Quantum theory

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Research Goal Prove complexity results, focusing on quantum computing, that are motivated by this gap between physics and what we experience. (Disclaimer: I will not bridge the gap in my thesis.)

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2. Quantum Computing

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Some Milestones 1982 1983 1984 19851986 1987 1988 1989 1990 1991 1992 1993 1994

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The Quantum Model State of computer: superposition over binary strings To each string Y, associate complex amplitude Y Y | Y | 2 = 1 On measuring, see Y with probability | Y | 2 Dirac ket notation: State written | = Y Y |Y Each |Y is called a basis state

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Unitary Evolution Quantum state changes by multiplying amplitude vector with unitary matrix: | (t+1) = U| (t) U is unitary iff U -1 =U, conjugate transpose (Linear transformation that preserves norm=1) Example: Circuit model: U must be efficiently computable Black-box model: No such restriction 1/ 2 -1/ 2 1/ 2 ( |0 + |1 )/ 2 = |1

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Quantum Query Model State after t queries: : workbits i: index to query z: output Query: |,i,z | x i,i,z Arbitrary unitaries that dont depend on X By end:

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3. Collision Lower Bound

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Collision Problem Given Promised: (1) X is one-to-one (permutation) or (2) X is two-to-one Problem: Decide which w.h.p., using few queries to the x i Randomized alg: ( n)

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Result Any quantum algorithm for the collision problem uses (n 1/5 ) queries (A, STOC2002) Previously no lower bound better than (1). Open since 1997 Shi improved to (n 1/4 ) (n 1/3 ) when |range| >> n

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Implications Oracle A for which SZK A BQP A –SZK: Statistical Zero Knowledge No trivial polytime quantum algorithms for –graph isomorphism –nonabelian hidden subgroup –breaking cryptographic hash functions

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Brassard-Høyer-Tapp (1997) (n 1/3 ) quantum alg for collision problem n 1/3 x i s, queried classically, sorted for fast lookup Grovers algorithm over n 2/3 x i s Do I collide with any of the pink x i s?

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Previous Lower Bound Techniques Block sensitivity (Beals et al. 1998): Q 2 (f) = ( bs(f)) Quantum adversary method (Ambainis 2000) Problem: Every 1-1 input differs in at least n/2 places from every 2-1 input

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Lemma (follows Beals et al. 1998): Let (x i,h)=1 if x i =h, 0 otherwise. Then P(X) is poly of deg 2T over the (x i,h). Proof: Let t,X,,i,z = amplitude of |,i,z after t queries. t,X,,i,z is poly of deg t, by induction. Base case (t=0) trivial. Unitaries cant increase degree. Query replaces t,X,,i,z by

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Input Distribution D(g): Uniform distribution over g-1 inputs Technicality: g might not divide n But assume for simplicity that it does Let Exercise: Show that, if T=O( n), then P(g) is a polynomial of degree 2T in g for integers 1 g n.

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Monomials of P(X) I(X) = product of r variables (x i,h) Let Then for some I,

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Calculating (I,g): #1 Range of I: Y.w=|Y|. (I,g) = 0 unless Y S (range of X) So since

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Calculating (I,g): #2 Given an S containing Y, # of g-1 inputs of size n: n!/(g!) n/g Let {y 1,…,y w } be distinct values in Y –r i = # of times y i appears in Y –r 1 + … + r w = r # of g-1 inputs X with range S s.t. I(X)=1:

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Becomes ~polynomial(g) Polynomial in g of degree w + (r-w) = r 2T

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Markovs Inequality Let P(x) be a poly with b 1 P(x) b 2 for all a 1 x a 2 and |dP(x*)/dx| c for some a 1 x* a 2. Then Long Short Large derivative

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Lower Bound 0 P(g) 1 for all 0 g n P(1) 1/10 and P(2) 9/10 So dP/dg 4/5 somewhere (n 1/4 ) lower bound would follow if g always divided n Can fix to obtain an (n 1/5 ) bound Shi found a better way to fix

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4. Dynamical Models

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A Puzzle Let|O R = you seeing a red dot |O B = you seeing a blue dot What is the probability that you see the dot change color?

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Why Is This An Issue? Quantum theory says nothing about multiple-time or transition probabilities But then what is a prediction, or the output of a computation, or the utility of a decision? Reply: But we have no direct knowledge of the past anyway, just records

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Results (submitted to PRL, quant-ph/0205059) What if you could examine an observers entire history? Defined class DQP Showed SZK DQP. Combined with collision bound, implies oracle A for which BQP A DQP A Can search an N-element list in order N 1/3 steps, though not fewer

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BPP BQPSZK DQP

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5. Current and Future Work

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BQP versus PH Almost-complete (?!) joint work with Umesh Conjecture: BQP A PH A for an oracle A (Best known: BQP A ( 2 ) A ) Use Recursive Fourier Sampling Have reduced problem to generalizing the Razborov-Smolensky circuit lower bound Need to show replacer gates dont help us compute sum modulo 3

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BPP A vs. BQP A for random A Conjecture: If BPP=BQP, then BPP A =BQP A with probability 1 What I can show: If BPP=BQP then BPTime[polylog]=BQTime[polylog] Whats missing: Extend the result of Beals et al. (1998) that D(f)=O(Q 2 (f) 6 ) for all total f to almost-total f Does the same hold for BPP vs. SZK, or even P vs. NP coNP? (cf. Rudichs thesis)

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Limitations of Shor-like algorithms Defined a class BPP BQP shor BQP Subclass of quantum algorithms that prepare a state x |x |f(x), then ignore |f(x) and do something simple to |x Conjecture 1: BQP shor AM. Implies that if NP BQP shor then PH= 2 Conjecture 2: Shor-like query algorithms yield no asymptotic speedup for any total function

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Physics Modulo Complexity Assumptions Can some version of M-theory decide SAT? (cf. Preskills talk) If so, move on to the next version! Anthropic computer for solving NP-complete problems efficiently Stupid question: Why cant I just will myself to solve NP-complete problems? (Or generate truly random sequences?)

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Postulate: No matter who you are, someone can give you a 3SAT instance that you cant decide with probability ½+. What constraints does that impose?

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