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

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

1. Sermon

The Computer Scientists Idea of Physics + details

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

My Own View… What we experience Quantum theory

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.)

2. Quantum Computing

Some Milestones

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

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

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:

3. Collision Lower Bound

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)

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

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

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?

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

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

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.

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

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

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:

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

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

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

4. Dynamical Models

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?

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

Results (submitted to PRL, quant-ph/ ) 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

BPP BQPSZK DQP

5. Current and Future Work

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

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)

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

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?)

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?