computation, quantum theory, and you scott aaronson, uc berkeley qualifying exam may 13, 2002
TRANSCRIPT
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 Scientist’s Idea of Physics
+ details
What Does Our World Have That Conway’s Doesn’t?
• 3 or more spatial dimensions
• Continuity?
• Relativistic covariance
• Quantum theory
• And more?
Quantum theory
My Own View…
What weexperience
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
1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994
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 computableBlack-box model: No such restriction
1/2 -1/2
1/2 1/2(|0+ |1)/2 = |1
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:
3. Collision Lower Bound
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)
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
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
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?
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
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
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.
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,
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
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:
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
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
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
4. Dynamical Models
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
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/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
BPP
BQP SZK
DQP
5. Current and Future Work
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
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)
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
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?)
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?