Tutorial on the adversary method for quantum and classical lower bounds

23 Pages·2005·1.43 MB·English

Tutorial on
the adversary method for quantum
and classical lower bounds
Sophie Laplante
LRI, Université Paris-Sud Orsay
Adversary method lower bounds
Semidefinite Quantum
Exact
Programming
complexity query complexity
[BSS03]
Strong
lower
bound
Kolmogorov Polynomial
method method
[LM04] [BBCMW01]
Weighted
Spectral
adversary
method
[A03]
[BSS03]
Adversary
Weak
method [A02]
lower
bound
2
Summary of this talk
Deterministic method
|R|
DT(f) ≥
max |R |
i i
- General statement
!
m m
≥ max{ , }
!
l l
- Combinatorial bound
Quantum method
c |R|
ε
Q (f) ≥
ε
Progress (i)
i t
- General statement
! !
c mm
ε
≥
2 " ll!
- Combinatorial bound [A02]
! !
p(x)p(y)p (y)p (x)
Weighted method [A03, LM04] cε ! x,i y,i
Q (f) ≥ min
ε
2 x,y,i q(x,y)
c "Γ"
ε
Spectral method [BSS03] Q (f) ≥
ε
2 max "Γ "
i i
3
Classical decision tree model
To compute a boolean function
x =?
1
f : {0,1}n → {0,1},
x = 0 x = 1
1 1
Model : decision tree
x =? x =?
2 3
x = 0 x = 1 x = 0 x = 1
Cost : Number of queries to 2 2 3 3
input x =?
3
x = 0 x = 1
3 3
Query complexity of f :
x =? x =?
5 5
depth of shallowest decision x = 0 x = 0
5 5
x = 1 x = 1
5 5
tree for f
ACC REJ REJ ACC
4
Quantum query model
ancilla
• O
Query: Unitary transformation that
U x
Computation 0
maps | i , b ! to | i , b ⊕ x "
i
O
Query x
•
Computation :
= 0
|ψ ! U O · · · O U | !
U T T x x 0
Computation 1
O
Query x • Output: Measure 1st qubit of | ψ T !
•
Error probability at most 1/3
O •
Query x Same model with stochastic matrices for
randomized query complexity
U
Computation T
output
5
Estimating number of pieces by their size
6
Adversary method, deterministic setting
{x : f(x)=1} R {y : f(y)=0}
Goal: separate {x : f(x)=1}
from {y : f(y)=0} with
minimum queries to input
bits
• x y
Subrelation R : pairs for i≠ i
i
which query i is useful
y
x
•
Queries ≥ number of R
i
needed to cover R.
R
|R| i
DT(f) ≥
max |R |
i i
7
Deterministic lower bound
{x : f(x)=1} R {y : f(y)=0}
•
left degree of R ≥ m
right degree ≥ m’
•
left degree of all R ≤ l
i
x y
right degree ≤ l’ i≠ i
•
|R| ≥ m|X|, m’|Y|
y
x
•
|R | ≤ l|X|, l’|Y|
i
|R|
DT(f) ≥
R
max |R | i
i i
!
m m
≥ max{ , }
!
l l
8
Progress on x, y, i after a query
(cid:15464)
At time t, progress towards distinguishing (x,y) R
i
by making one query :
Progress x,y(i)
• t
Deterministic case
i
1 if query(x,t) =
DProgress x,y(i) =
t
! 0 otherwise
•
Randomized case
RProgress x,y(i) = x y
2 min{p (i), p (i)}
t
t t
•
Quantum case ? (next slide)
9
Quantum progress
State of the system after query
t, on input x is written x
|ψ !
t
(cid:15464)
Fix (x,y) R.
i
Amplitude2 of | ! x
|ψ !
in query t, input x 0
•
At beginning, x y = 1 y
!ψ |ψ " |ψ !
0 0 0
•
At each time step,
|!ψx|ψy" − !ψx |ψy "| ≤ 2 px(i)py(i)
t t t+1 t+1 ! " t t
i:x !=y
i i
•
At the end, x y
|!ψ |ψ "| ≤ 2!ε(1 − ε)
T T
c
ε
T · 2 px(i)py(i) ≥ 1−2 ε(1−ε)
"
! t t #
i
Progresst (i)
x,y
10