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

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Computation, Quantum Theory, and You. Scott Aaronson, UC Berkeley Qualifying Exam May 13, 2002. Talk Outline. Sermon 2.Quantum Computing Overview Collision Lower Bound Dynamical Models 5.Current and Future Work. 1. Sermon. The Computer Scientist’s Idea of Physics. + details. - PowerPoint PPT Presentation

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Page 1: Computation, Quantum Theory, and You

Computation, Quantum Theory, and You

Scott Aaronson, UC Berkeley

Qualifying Exam

May 13, 2002

Page 2: Computation, Quantum Theory, and You

Talk Outline

1. Sermon

2. Quantum Computing Overview

3. Collision Lower Bound

4. Dynamical Models

5. Current and Future Work

Page 3: Computation, Quantum Theory, and You

1. Sermon

Page 4: Computation, Quantum Theory, and You

The Computer Scientist’s Idea of Physics

+ details

Page 5: Computation, Quantum Theory, and You

What Does Our World Have That Conway’s Doesn’t?

• 3 or more spatial dimensions

• Continuity?

• Relativistic covariance

• Quantum theory

• And more?

Quantum theory

Page 6: Computation, Quantum Theory, and You

My Own View…

What weexperience

Quantum theory

Page 7: Computation, Quantum Theory, and You

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.)

Page 8: Computation, Quantum Theory, and You

2. Quantum Computing

Page 9: Computation, Quantum Theory, and You

Some Milestones

1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

Page 10: Computation, Quantum Theory, and You

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

Page 11: Computation, Quantum Theory, and You

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 computableBlack-box model: No such restriction

1/2 -1/2

1/2 1/2(|0+ |1)/2 = |1

Page 12: Computation, Quantum Theory, and You

Quantum Query Model• State after

t queries:: workbits i: index to query z: output

, , ,, ,

, ,t i zi z

i z

•Query: |,i,z |xi,i,z

•Arbitrary unitaries that don’t depend on X

2

, , ,1,

1( ) , ( )

10T ii

P X P X f X

•By end:

Page 13: Computation, Quantum Theory, and You

3. Collision Lower Bound

Page 14: Computation, Quantum Theory, and You

Collision Problem• Given 1 : 1, , 1, ,nX x x n n

• 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 xi

• Randomized alg: (n)

Page 15: Computation, Quantum Theory, and You

Result• Any quantum algorithm for the

collision problem uses (n1/5) queries (A, STOC’2002)

• Previously no lower bound better than (1). Open since 1997

• Shi improved to (n1/4)

(n1/3) when |range| >> n

Page 16: Computation, Quantum Theory, and You

Implications

• Oracle A for which SZKA BQPA

– SZK: Statistical Zero Knowledge

• No “trivial” polytime quantum algorithms for

– graph isomorphism

– nonabelian hidden subgroup

– breaking cryptographic hash functions

Page 17: Computation, Quantum Theory, and You

Brassard-Høyer-Tapp (1997)(n1/3) quantum alg for collision problem

n1/3 xi’s, queried classically,

sorted for fast lookup

Grover’s algorithm over n2/3 xi’s

Do I collide with any of the pink xi’s?

Page 18: Computation, Quantum Theory, and You

Previous Lower Bound Techniques

• Block sensitivity (Beals et al. 1998):Q2(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

Page 19: Computation, Quantum Theory, and You

Lemma (follows Beals et al. 1998): Let (xi,h)=1 if xi=h, 0 otherwise. Then P(X) is poly of deg 2T over the (xi,h).

, , , ,1

, .t X h i z ih n

x h

Proof: Let t,X,,i,z = amplitude of |,i,z after t queries. t,X,,i,z is poly of degt, by induction.

Base case (t=0) trivial. Unitaries can’t increase degree.

Query replaces t,X,,i,z by

Page 20: Computation, Quantum Theory, and You

Input Distribution• D(g): Uniform distribution over g-1 inputs

•Technicality: g might not divide n

But assume for simplicity that it does

X D gP g EX P X•Let

• Exercise: Show that, if T=O(n), then P(g) is a polynomial of degree 2T in g for integers 1gn.

Page 21: Computation, Quantum Theory, and You

Monomials of P(X)

• I(X) = product of r variables (xi,h)

, .X D gI g EX I X •Let

: 2

, .II r T

P g I g

•Then for some I,

Page 22: Computation, Quantum Theory, and You

Calculating (I,g): #1

•“Range” of I: Y. w=|Y|.

(I,g) = 0 unless YS (“range” of X)

2 .n n

S T rg n

/Pr

/

n w

n g wY S

n

n g

•So

since

Page 23: Computation, Quantum Theory, and You

Calculating (I,g): #2

• Given an S containing Y,

# of g-1 inputs of size n: n!/(g!)n/g

•Let {y1,…,yw} be distinct values in Y

–ri = # of times yi appears in Y

–r1 + … + rw = r

/

1

!

! !w

n g w

ii

n r

g g r

•# of g-1 inputs X with range S s.t. I(X)=1:

Page 24: Computation, Quantum Theory, and You

Becomes ~polynomial(g)

11

20 1 1

! !,

!

irw w

i i j

n w n rI g n gi g j

n

Polynomial in g of degree

w + (r-w) = r 2T

Page 25: Computation, Quantum Theory, and You

Markov’s InequalityLet P(x) be a poly with b1P(x)b2 for all

a1xa2 and |dP(x*)/dx|c for some a1x*a2. Then

2 1

2 1

deg .c a a

Pb b

Long

Short

Large derivative

Page 26: Computation, Quantum Theory, and You

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

(n1/4) lower bound would follow if g always divided n

• Can fix to obtain an (n1/5) bound

Shi found a better way to fix

Page 27: Computation, Quantum Theory, and You

4. Dynamical Models

Page 28: Computation, Quantum Theory, and You

A Puzzle• Let |OR = you seeing a red dot

|OB = you seeing a blue dot

• What is the probability that you see the dot change color?

( )

R R B B

R R B B

O O

H

O O

Page 29: Computation, Quantum Theory, and You

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”

Page 30: Computation, Quantum Theory, and You

Results(submitted to PRL, quant-ph/0205059)

• What if you could examine an observer’s entire history? Defined class DQP

• Showed SZK DQP. Combined with collision bound, implies oracle A for which BQPA DQPA

• Can search an N-element list in order N1/3 steps, though not fewer

Page 31: Computation, Quantum Theory, and You

BPP

BQP SZK

DQP

Page 32: Computation, Quantum Theory, and You

5. Current and Future Work

Page 33: Computation, Quantum Theory, and You

BQP versus PH• Almost-complete (?!) joint work with Umesh

• Conjecture: BQPA PHA for an oracle A

(Best known: BQPA (2)A)

• Use Recursive Fourier Sampling

• Have reduced problem to generalizing the Razborov-Smolensky circuit lower bound

• Need to show “replacer gates” don’t help us compute sum modulo 3

Page 34: Computation, Quantum Theory, and You

BPPA vs. BQPA for random A

• Conjecture: If BPP=BQP, then BPPA=BQPA with probability 1

• What I can show: If BPP=BQP then BPTime[polylog]=BQTime[polylog]

• What’s missing: Extend the result of Beals et al. (1998) that D(f)=O(Q2(f)6) for all total f to almost-total f

• Does the same hold for BPP vs. SZK, or even P vs. NPcoNP? (cf. Rudich’s thesis)

Page 35: Computation, Quantum Theory, and You

Limitations of Shor-like algorithms

• Defined a class BPPBQPshorBQP

• Subclass of quantum algorithms that prepare a state x|x|f(x), then ignore |f(x) and do something “simple” to |x

• Conjecture 1: BQPshorAM. Implies that if NPBQPshor then PH=2

• Conjecture 2: Shor-like query algorithms yield no asymptotic speedup for any total function

Page 36: Computation, Quantum Theory, and You

Physics Modulo Complexity Assumptions

• Can some version of M-theory decide SAT? (cf. Preskill’s talk)

If so, move on to the next version!

• “Anthropic computer” for solving NP-complete problems efficiently

• Stupid question: Why can’t I just “will” myself to solve NP-complete problems? (Or generate truly random sequences?)

Page 37: Computation, Quantum Theory, and You

Postulate: No matter who you are, someone can give you a 3SAT instance that you can’t decide with probability ½+.

What constraints does that impose?