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The Polynomial Method In Quantum and Classical Computing y x Scott Aaronson (MIT) Overview The polynomial method: Just an awesome tool that every CS theorist should know about Goes back to the prehistory of the field (1960’s), 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” This Talk: Just Some Basics 1. Polynomials in machine learning - Perceptrons 2. Polynomials in quantum computing Stuff I -wish I could cover but can’t lack algorithms of time Optimality of Deutsch-Jozsa andfor Grover - Polynomials finite fields (Razborov-Smolensky) - Collisionover lower bound - Reduction of communication problems to polynomials 3.- Sherstov’s Polynomials in circuit complexity pattern matrix method - Deep - Linial-Mansour-Nisan connections to Fourier and analysis Bazzi 4. Polynomials everywhere! - Communication complexity, oracles, streaming… Our story starts in St. Petersburg, around 1889… привет! I proved a cool theorem: if pyou’re is a quadratic, Uhh … on max p' x 4 max px And what if p has degree d? your own 1 x1 1 x1 y x 1 Pr X kXˆ k Dmitri Mendeleev A. A. Markov (periodic table dude) (inequality dude) Markov did generalize Mendeleev’s 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… 1. POLYNOMIALS IN MACHINE LEARNING Fast-forward to 1969… And AI researchers were studying perceptrons f f1 f2 … m 1 working if i f i Bill Ayers was f i 1 for the McCain’08 0 otherwise campaign fm 1 , , m , A perceptron of order k is a Boolean function f:{0,1}n{0,1} that’s a threshold of subfunctions on at most k variables each Minsky and Papert: Small perceptrons have serious limitations! Application: “killed neural net research for a decade” Suppose f:{0,1}n{0,1} is represented by an order-k perceptron Then there’s clearly a degree-k polynomial p:RnR such that for all x1,…,xn{0,1}, sgn px1 ,, xn f x1 ,, xn Furthermore, without loss of generality p is multilinear: no variable raised to higher power than 1 Example: The PARITY function Suppose sgn px1 ,, xn x1 xn for all x1,…,xn{0,1}. Then what can we say about deg(p)? Theorem: deg(p)n Key idea: Symmetrization Replace multivariate polynomials by univariate ones, which are easier to understand How Symmetrization Works Let Let p x1 , , xn qk x S n S d S i iS px1 ,, xn x x k EX 1 n Key Lemma: q(k) is itself a polynomial in k, of degree at most d Proof: By linearity of expectation, qk S EX xi x1 xn k S n iS S d n S n EX xi k x1 xn k k S iS n S ! k!n k ! k S !n k ! n! n S ! k k 1 k S 1 n! which is a degree-|S| polynomial in k. So, suppose there’s an order-k perceptron computing the parity of n bits Then there’s a degree-k multilinear polynomial p such that sgn p x ,, x x x 1 n 1 n Hence there’s a degree-k univariate polynomial q such that for all k=0,…,n, q k 0 if k even q k 0 if k odd Must have degree n 2. POLYNOMIALS IN QUANTUM COMPUTING Quantum Query Model In One Slide What are the allowed operations? Initialize vector of amplitudes 1 n Quantum state: Unit vector in Cn Query the input bits 1 1 1 2 x1 1 2 x n n n One further detail: The quantum state can have more a than n dimensions, with multiple“Measure” components Apply unitary transformation querying each xi, as well as components that don’t 1 u11 u1n 1 make at all queries Outcome i observed 2 with probability | | Complexity Measure: Q(f) = minimum number of i u u nn n n n1 queries needed to compute a Boolean function f with probability 2/3, on all inputs x=x1…xn Example: The Deutsch-Jozsa Algorithm Does something spectacular: Computes the XOR of two bits with one oracle call! 1 x1 x2 x2 x1 1 2 1 2 1 1 2x 0 1 2 1 0 1 2 x 2 2 1 2 1 2 By computing x1x2, x3x4, etc., can compute the parity of n bits with n/2 oracle calls Is that optimal? Lemma (Beals et al. 1998): If a quantum algorithm makes T queries, its probability of accepting is a degree-2T multilinear polynomial over the xi’s Right-to-Left Proof: 0 u11 u1n 1 2 x1 0 1 1 2 x1 Implication: algorithm If a quantum computed x0 x 1with u queries, it would <n/2 lead to a 2 x u 0 1 2 x 1 n n n1 nn n n polynomial approximating PARITY with degree <n. Entries After are Deutsch-Jozsa now TStill queries, Degree-2 degree-1 degree-T polynomials polynomials polynomials polynomials over the xi’s Hence must be optimal! Then i accepting 2 i has degree 2T Another Famous Quantum Algorithm: Grover’s Computes the OR of n bits using O(n) queries Is Grover’s algorithm optimal? BBBV 1994: Yes, by a quantum argument We’ll instead prove Grover is optimal using … wait for it … Given a Boolean function f, let deg(f) be the minimum degree of a real polynomial p:RnR such that px f x x 0,1 n Theorem (Nisan-Szegedy 1994): deg ORn n Observation: Is that lower bound tight? Yes, because of Grover’s algorithm! To prove deg(OR)=(n), we need to revisit our good friend Markov… Theorem (Markov): If p is a degree-d real polynomial, then max p' x d max px 2 1 x1 1 x1 Another convenient form: for all n>0, deg p n max p' x 0 x n 2 max px 0 x n Markov’s inequality is tight. The extremal cases are called the Chebyshev polynomials: Td x cosd arccos x Uhh … why is that a polynomial at all? Td cos x cos dx Re eidx Re cos x i sin x d which is a degree-d polynomial in cos x y x y x y x y x y x y x y x y x y x y x Let p satisfy px OR x x 0,1 n We want to lower-bound deg(p) Symmetrize: qk EX px , degq deg p x k 1 0 1 qk 1 k 1,, n 1 1by q' xSo 2Markov’s for some 0 inequality, x 1 n max 1 2 ,2c 1 degq n c 0 q0 One remaining problem: q(x) need not be bounded at non-integer x Solution: Notice max q' x 2c 1, c max qx 0 x n 0 x n Collision Problem Illustrates the amazing reach of the polynomial method Problem: Given f:[n][n], decide whether f is 1-to-1 or 2-to-1, promised it’s one or the other By the Birthday Paradox, ~n queries to f are necessary and sufficient classically [Brassard et al. 1997] gave a quantum algorithm making O(n1/3) queries [A. 2002]: Any quantum algorithm needs (n1/5) queries. Improved to (n1/3) by Shi Lower bound by polynomial method 1 if f x h Let x, h 0 otherwise 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 qk p f k - to-1 functions f EX be the expected acceptance probability on a random k-to-1 function The Miracle: q(k) is itself a polynomial in k, of degree at most 2T Why? nr n d ! r dh n / k r EX xh , j , h r k - to-1 functions f n / k r h 1 j 1 k! k d h ! n n! n / k k!n / k h 1 n/k n r !n d ! k! n / k !n n / k ! r n!n! n / k r !n n / k !k!n / k r k d h ! d1 d2 h 1 d3 n r !n d ! n!n! n r !n d ! d n!n! h 1 n n n k k 1 k d 1 1 r 1 h k k k h 1 n r !n d ! n!n! k!r n / k ! r n / k r ! k d ! h r k 1 k d 1 nn k n rk k h r h 1 which is a degree-d polynomial in k. That’s why. Technicality: Need to deal with k not dividing n Another Useful Hammernomial: Bernstein’s Inequality deg p max p' x xn x 0 x n max p x 0 x n Application: Any quantum algorithm to compute the MAJORITY of n bits requires (n) queries Ouch, that really hurts the degree! Oh, and don’t forget the inequality of V. A. Markov—A. A.’s younger brother! p x d k p 2e deg p max 0 x n max k 0 x n dx nk 2 k 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 (cT2/N)K 0 1 K N 3. POLYNOMIALS IN CIRCUIT COMPLEXITY Linial-Mansour-Nisan 1993: If a Boolean function f is computable by an AC0 circuit of size s and depth k, then we can find a degree-d real polynomial p such that EX n px f x 2s 2 x0,1 2 d 1 / k / 20 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 Bazzi 2007: Let F=C1Cm be a DNF formula. Then we can find degree-d real polynomials p and q such that n px F x q x x 0,1 O 1 d EX qx px m 2 x0,1n Implies that polylog-wise independent distributions “fool” small DNFs. The proof takes 64 pages 4. POLYNOMIALS EVERYWHERE Polynomials in Oracle-Building Beigel 1992: There exists an oracle relative to which PNP PP Use the following problem: Given exponentially-long integers x=x1…xN and y=y1…yN, is xy? It’s in PNP, since we can use binary search to find the leftmost i such that xiyi But is there a low-degree polynomial p such that if x y 1 sgn px1 ,, xN , y1 ,, y N ? 1 otherwise Sure: 2 N 1 x1 2 N 2 x2 2 xN 2 0 N 1 y1 2 N 2 y2 2 y N 0 But by clever repeated use of Markov’s inequality, one can show that any such polynomial must take on huge (doubly-exponentially-large) values This means the problem can’t be in PP [A. 2006] generalized Beigel’s result to give an oracle relative to which PP has linear-size circuits Requires handling many polynomials simultaneously Slide of Guilt: The Polynomial Method in Communication Complexity Razborov 2002: Any quantum protocol for the Disjointness problem requires (n) qubits of Razborov and Sherstov, this very FOCS: communication An AC0 function with large unbounded-error communication complexity Sherstov, this very FOCS: Characterizes the unboundederror communication complexity Chattopadhyay-Ada, Lee-Shraibman 2008: Lower of symmetric functions bounds for the k-party communication complexity of Disjointness in the Number-On-Forehead And model more! 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 Future Direction 1: Beyond Symmetrization Find better techniques to lower-bound the degrees of multivariate polynomials. deg OR AND AND Upper bound: O(n) n (from quantum algorithm) AND ? Lower bound: (n1/3) n (can be proved using the n1/3 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.) Future Direction 2: Understanding Bounded Real Polynomials Conjecture. Let p:Rn[0,1] be a real polynomial of degree d. Suppose EXx,y[|p(x)-p(y)|]=(1). Then there exists an i[n] such that EXx[|p(x)-p(xi)|]=(1/poly(d)). Would have major implications for quantum! e.g., for P vs. BQP relative to a random oracle 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) 0p(x)1 for all x{0,1}n, (2) |p(x)-f(x)| for all xS. Is there a partial f for which deg(f) is exponentially smaller than Q(f)? Future Direction 3: Matrix- Valued Polynomials What Boolean functions can we approximate as Would imply an oracle relative to which p x , , x p x , , x 11 1 n 1m 1 n SZKQMA (i.e., “there are no succinct pij d ? max quantum for problems , deg proofs like graph p x ,, x non-isomorphism”) p x , , x n mm 1 n m1 1 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 d2(d+log m)=(n). Future Direction 4: Extending Bazzi’s Theorem to AC0 (the Linial-Nisan Conjecture) Problem: Given fAC0, construct polylog(n)-degree polynomials p,q:RnR such that px f x qx x 0,1 , EX n qx px n x0,1 If p,q have the further property that px qx C x C Clauses C C C x Clauses C C PrC On PrC O n 1/ 2 C C 1/ 2 C then we get an oracle relative to which BQPPH. The polynomial method: the choice of hardworking American lowerboundsmen I approve! y x OPEN PROBLEM