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The Polynomial Method In Quantum and Classical Computing Scott Aaronson (MIT) OPEN PROBLEM

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Overview The polynomial method: Just an awesome tool that every CS theorist should know about Goes back to the prehistory of the field (1960s), but also plays a major role in current work [including at this FOCS] on machine learning, quantum computing, circuit lower bounds, communication complexity… Idea: Reduce CS questions to questions about the minimum degree of real polynomials Easy to learn! Look ma, no quantum

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This Talk: Just Some Basics 1. Polynomials in machine learning - Perceptrons 2. Polynomials in quantum computing - Optimality of Deutsch-Jozsa and Grover algorithms - Collision lower bound 3. Polynomials in circuit complexity - Linial-Mansour-Nisan and Bazzi 4. Polynomials everywhere! - Communication complexity, oracles, streaming… Stuff I wish I could cover but cant for lack of time - Polynomials over finite fields (Razborov-Smolensky) - Reduction of communication problems to polynomials - Sherstovs pattern matrix method - Deep connections to Fourier analysis

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Our story starts in St. Petersburg, around 1889… Dmitri Mendeleev (periodic table dude) A. A. Markov (inequality dude) привет! I proved a cool theorem: if p is a quadratic, And what if p has degree d? Uhh … youre on your own

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Markov did generalize Mendeleevs bound to arbitrary degree (about which more later) He thereby helped start a field called approximation theory Approximation theory is a proto-complexity theory! Real polynomials = Model of computation Degree = Complexity measure So, maybe not so surprising that it ends up being related to actual complexity theory…

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1. POLYNOMIALS IN MACHINE LEARNING

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Fast-forward to 1969… Bill Ayers was working for the McCain08 campaign And AI researchers were studying perceptrons A perceptron of order k is a Boolean function f:{0,1} n {0,1} thats a threshold of subfunctions on at most k variables each f1f1 fmfm f2f2 … f

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Minsky and Papert: Small perceptrons have serious limitations! Suppose f:{0,1} n {0,1} is represented by an order-k perceptron Then theres clearly a degree-k polynomial p:R n R such that for all x 1,…,x n {0,1}, Furthermore, without loss of generality p is multilinear: no variable raised to higher power than 1 Application: killed neural net research for a decade

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Example: The PARITY function Suppose for all x 1,…,x n {0,1}. Then what can we say about deg(p)? Key idea: Symmetrization Replace multivariate polynomials by univariate ones, which are easier to understand Theorem: deg(p) n

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Let Key Lemma: q(k) is itself a polynomial in k, of degree at most d How Symmetrization Works Let

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Proof: By linearity of expectation, which is a degree-|S| polynomial in k.

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So, suppose theres an order-k perceptron computing the parity of n bits Then theres a degree-k multilinear polynomial p such that Hence theres a degree-k univariate polynomial q such that for all k=0,…,n, Must have degree n

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2. POLYNOMIALS IN QUANTUM COMPUTING

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Quantum Query Model In One Slide Apply a unitary transformation What are the allowed operations? Initialize vector of amplitudes Measure Outcome i observed with probability | i | 2 Query the input bits Quantum state: Unit vector in C n One further detail: The quantum state can have more than n dimensions, with multiple components querying each x i, as well as components that dont make queries at all Complexity Measure: Q(f) = minimum number of queries needed to compute a Boolean function f with probability 2/3, on all inputs x=x 1 …x n

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Example: The Deutsch-Jozsa Algorithm Does something spectacular: Computes the XOR of two bits with one oracle call! By computing x 1 x 2, x 3 x 4, etc., can compute the parity of n bits with n/2 oracle calls Is that optimal?

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Lemma (Beals et al. 1998): If a quantum algorithm makes T queries, its probability of accepting is a degree-2T multilinear polynomial over the x i s Right-to-Left Proof: Entries are now degree-1 polynomials over the x i sStill degree-1 polynomialsDegree-2 polynomialsAfter T queries, degree-T polynomials Thenhas degree 2T Implication: If a quantum algorithm computed x 1 x n with

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Another Famous Quantum Algorithm: Grovers Computes the OR of n bits using O( n) queries Is Grovers algorithm optimal? BBBV 1994: Yes, by a quantum argument Well instead prove Grover is optimal using … wait for it …

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Theorem (Nisan-Szegedy 1994): Given a Boolean function f, let deg (f) be the minimum degree of a real polynomial p:R n R such that Observation: Is that lower bound tight? Yes, because of Grovers algorithm!

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To prove deg (OR)= ( n), we need to revisit our good friend Markov… Theorem (Markov): If p is a degree-d real polynomial, then Another convenient form: for all n>0,

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Markovs inequality is tight. The extremal cases are called the Chebyshev polynomials: Uhh … why is that a polynomial at all? which is a degree-d polynomial in cos x

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Let p satisfy We want to lower-bound deg(p) Symmetrize: 0 1

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0 1 One remaining problem: q(x) need not be bounded at non-integer x Solution: Notice So by Markovs inequality,

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Collision Problem Problem: Given f:[n] [n], decide whether f is 1-to-1 or 2-to-1, promised its one or the other [A. 2002]: Any quantum algorithm needs (n 1/5 ) queries. Improved to (n 1/3 ) by Shi Illustrates the amazing reach of the polynomial method By the Birthday Paradox, ~ n queries to f are necessary and sufficient classically [Brassard et al. 1997] gave a quantum algorithm making O(n 1/3 ) queries

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Lower bound by polynomial method Let Lemma (following Beals et al.): If a quantum algorithm makes T queries to f, the probability p(f) that it accepts is a degree-2T polynomial in the (x,h)s Now let be the expected acceptance probability on a random k-to-1 function

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The Miracle: q(k) is itself a polynomial in k, of degree at most 2T

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which is a degree-d polynomial in k. Thats why. Why? d3d3 d1d1 d2d2 d Technicality: Need to deal with k not dividing n

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Another Useful Hammernomial: Bernsteins Inequality Application: Any quantum algorithm to compute the MAJORITY of n bits requires (n) queries Ouch, that really hurts the degree!

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Oh, and dont forget the inequality of V. A. MarkovA. A.s younger brother! Application [A. 2004]: Direct product theorem for quantum search. After T queries, the probability that a quantum algorithm finds K marked items out of N is at most (cT 2 /N) K 01KN

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3. POLYNOMIALS IN CIRCUIT COMPLEXITY

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Linial-Mansour-Nisan 1993: If a Boolean function f is computable by an AC 0 circuit of size s and depth k, then we can find a degree-d real polynomial p such that Proof uses the Switching Lemma to upper-bound high-degree Fourier coefficients By Nisan-Szegedy, the above theorem would be false if we wanted |p(x)-f(x)| to be small for every x

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Bazzi 2007: Let F=C 1 C m be a DNF formula. Then we can find degree-d real polynomials p and q such that Implies that polylog-wise independent distributions fool small DNFs. The proof takes 64 pages [Razborov 2008]

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4. POLYNOMIALS EVERYWHERE

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Polynomials in Oracle-Building Beigel 1992: There exists an oracle relative to which P NP PP Use the following problem: Given exponentially-long integers x=x 1 …x N and y=y 1 …y N, is x y? Its in P NP, since we can use binary search to find the leftmost i such that x i y i But is there a low-degree polynomial p such that

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Sure: But by clever repeated use of Markovs inequality, one can show that any such polynomial must take on huge (doubly-exponentially-large) values This means the problem cant be in PP [A. 2006] generalized Beigels result to give an oracle relative to which PP has linear-size circuits Requires handling many polynomials simultaneously

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Slide of Guilt: The Polynomial Method in Communication Complexity Razborov 2002: Any quantum protocol for the Disjointness problem requires ( n) qubits of communication Razborov and Sherstov, this very FOCS: An AC 0 function with large unbounded-error communication complexity Sherstov, this very FOCS: Characterizes the unbounded- error communication complexity of symmetric functions Chattopadhyay-Ada, Lee-Shraibman 2008: Lower bounds for the k-party communication complexity of Disjointness in the Number-On-Forehead model And more!

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Some Positive Uses of Polynomials Harvey-Nelson-Onak, this very FOCS: Chebyshev polynomials used to give a streaming algorithm for approximating the Shannon entropy Beigel-Reingold-Spielman 1991: PP is closed under intersection

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Future Direction 1: Beyond Symmetrization Find better techniques to lower-bound the degrees of multivariate polynomials. OR AND n n Upper bound: O( n) (from quantum algorithm) Lower bound: (n 1/3 ) (can be proved using the n 1/3 collision lower bound) deg(f)=O(deg (f) 2 ) for all Boolean functions f? Best known relation: deg(f)=O(deg (f) 6 ) (Beals et al.)

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Future Direction 2: Understanding Bounded Real Polynomials Conjecture. Let p:R n [0,1] be a real polynomial of degree d. Suppose EX x,y [|p(x)-p(y)|]= (1). Then there exists an i [n] such that EX x [|p(x)-p(x i )|]= (1/poly(d)). Given a partial function f:S {0,1} (S {0,1} n ), let deg (f) be the minimum degree of a polynomial p such that (1) 0 p(x) 1 for all x {0,1} n, (2) |p(x)-f(x)| for all x S. Is there a partial f for which deg (f) is exponentially smaller than Q(f)? Would have major implications for quantum! e.g., for P vs. BQP relative to a random oracle

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Future Direction 3: Matrix- Valued Polynomials Conjecture. Suppose max (A(x)) [0,1] for all x {0,1} n max (A(x)) 2/3 for all x encoding a 1-to-1 function max (A(x)) 1/3 for all x encoding a 2-to-1 function Then d 2 (d+log m)= (n). What Boolean functions can we approximate as Would imply an oracle relative to which SZK QMA (i.e., there are no succinct quantum proofs for problems like graph non-isomorphism)

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Future Direction 4: Extending Bazzis Theorem to AC 0 (the Linial-Nisan Conjecture) Problem: Given f AC 0, construct polylog(n)-degree polynomials p,q:R n R such that If p,q have the further property that then we get an oracle relative to which BQP PH.

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The polynomial method: the choice of hardworking American lowerboundsmen OPEN PROBLEM I approve!

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