Download presentation

Presentation is loading. Please wait.

Published byRyan Faulkner Modified over 5 years ago

1
The Polynomial Method In Quantum and Classical Computing Scott Aaronson (MIT) OPEN PROBLEM

2
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

3
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

4
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

5
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…

6
1. POLYNOMIALS IN MACHINE LEARNING

7
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

8
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

9
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

10
Let Key Lemma: q(k) is itself a polynomial in k, of degree at most d How Symmetrization Works Let

11
Proof: By linearity of expectation, which is a degree-|S| polynomial in k.

12
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

13
2. POLYNOMIALS IN QUANTUM COMPUTING

14
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

15
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?

16
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 <n/2 queries, it would lead to a polynomial approximating PARITY with degree <n. Hence Deutsch-Jozsa must be optimal!

17
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 …

18
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!

19
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,

20
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

31
Let p satisfy We want to lower-bound deg(p) Symmetrize: 0 1

32
0 1 One remaining problem: q(x) need not be bounded at non-integer x Solution: Notice So by Markovs inequality,

33
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

34
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

35
The Miracle: q(k) is itself a polynomial in k, of degree at most 2T

36
which is a degree-d polynomial in k. Thats why. Why? d3d3 d1d1 d2d2 d Technicality: Need to deal with k not dividing n

37
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!

38
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

39
3. POLYNOMIALS IN CIRCUIT COMPLEXITY

40
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

41
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]

42
4. POLYNOMIALS EVERYWHERE

43
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

44
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

45
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!

46
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

47
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.)

48
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

49
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)

50
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.

51
The polynomial method: the choice of hardworking American lowerboundsmen OPEN PROBLEM I approve!

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google