Talk Outline
1. Sermon
2. Quantum Computing Overview
3. Collision Lower Bound
4. Dynamical Models
5. Current and Future Work
What Does Our World Have That Conway’s Doesn’t?
• 3 or more spatial dimensions
• Continuity?
• Relativistic covariance
• Quantum theory
• And more?
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.)
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:
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
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
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?)
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