A ndris Ambainis Uni v ersity of Latvia Joint w o r k with Scott Aaronson Kaspars Balodi s Aleksandrs Belovs T r o y Le e Miklos Santha and J ID: 612088
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Slide1
The largest possible gaps between quantum and classical algorithms
A
ndris
Ambainis
Uni
v
ersity
of
Latvia
Joint
w
o
r
k with:
Scott Aaronson,
Kaspars
Balodi
s
,
Aleksandrs Belovs,
T
r
o
y Le
e
,
Miklos
Santha
,
and
J
u
r
is
Smotr
o
vsSlide2
k quantum steps
How many
steps classically?Quantum vs. classicalSlide3
2 / 39
Models of
Computation
Deterministic
Randomized
|
QuantumSlide4
Query algorithms
Task: compute
f(x
1
, ..., xN)
.Input variables xi accessed via queries.Complexity = number of queries.
i
x
iSlide5
Does there exist
i
:xi=1? Classically, N queries required.Quantum: O(
N) [Grover, 1996].
0
1
0
0
...
x
1
x
2
x
N
x
3
Grover’s searchSlide6
Reasons to study query model
Encompasses many quantum algorithms (Grover’s search, quantum part of factoring, etc.).
Provable quantum-vs-classical gaps.Slide7
Computation
Models
D
:
Dete
r
ministic
(Decision
T
ree)
x
1
x
2
x
3
x
3
0
1
0
1
0
1
Compl
e
xity
:
on
input:
in
total:
Number
of
que
r
ies
W
orst
input
(length
(depth
of
the
of
the
path)
tree)
2
or
3
3
•
•
x
2Slide8
Computation
Models
R:
Randomized
(
Probability
dist
r
i
b
ution
on
decision
trees
)
x
1
x
2
x
3
x
3
0
1
0
1
0
1
x
2
0
1
0
1
Compl
e
xity
:
on
input:
in
total:
Expected
n
umber
of
que
r
ies
W
orst
input
2
or
8/
3
8/
3
•
•Slide9
Quantum query modelU
0, U1, …, UT
– independent of x1, …, xN.Q – queries:|i (-1)xi|i.
U
0
Q
Q
U
1
U
T
…
Computation
Models
Q:
Quantum
(
Quantum query algorithms
)Slide10
Computation
Models
Q
2
vs R2 vs D?D – deterministic (decision tree)R – randomizedR0
– zero error;R1 – one sided error;
R2 – bounded error;Q
– quantumQE – exact;
Q2 – bounded error;Slide11
Settings
Partial functions:
For some inputs
(x
1
, ..., xN), the algorithm is allowed to output anything.Huge quantum speedups.Total functions:Prescribed anwer f(x1, ..., xN) for every (x1, ..., x
N).Biggest quantum speedup: Grover.Slide12
Partial functionsSlide13
x
1, x
2, ..., xN - periodici
xi
Find period r1 query quantumly
Period finding
queries classicallySlide14
Task that requires 1 query quantumly, (N/log N)
classically. 1 query quantum algorithms can be simulated by
O(N) query probabilistic algorithms.
Our result [Aaronson, A]Slide15
FORRELATION =
F
ourier C
ORRELATIONSlide16
High level idea
Quantum
Fourier
transform
– hard to simulate classically.Task
Input/output should be classical.Slide17
Input: (x1
, ..., xN, y1, ..., yN
) {-1, 1}2N.Are vectors
highly correlated?
FN – Fourier transform over ZN.
ForrelationSlide18
Is the inner product
at least 3/5
or at most 1/100?
More precisely ...Slide19
Generate a superposition of
(1 query).
Apply FN to 2nd state.Test if states equal (SWAP test).
Quantum algorithmSlide20
Theorem Any classical algorithm for FORRELATION
usesqueries.
Classical lower boundSlide21
Simulating 1 query quantum algorithmsSlide22
Theorem Any 1 query quantum algorithm can be simulated probabilistically with
O(N) queries.
SimulationSlide23
Overview
|
x
1x2+4x2x4-x3x4+x3x5
SamplingSlide24
Q
Q
Q
U
T
…
U
1
1,1
|1
,1
+
1,2
|
1,
2
+
…
+
N, M
|N
, M
i,j
is actually
i,j
(x
1
, ..., x
N
)
Analyzing query algorithmsSlide25
Lemma [Beals et al., 1998] After
k queries, the amplitudes
are polynomials in x1, ..., xN of degree k.
Measurement:
Polynomial of degree 2k
Polynomials methodSlide26
Pr[A outputs
1] = p(x1, ..., xN
), deg p =2.0 p(x1, ..., xN) 1.Task: estimate p(x1, ..., x
N) with precision
. Solution: random sampling.
Our taskSlide27
Pre-processing
Problem: some xi’s in p(x
1, ..., xN) may be more influential than others.influential xi
...
s
everal lessinfluential xiSlide28
Good if we sample
N
of
N2 terms independently.
Estimator:Requires sampling N variables xi!
SamplingSlide29
x
1
x
2
x3x4x5
x6x7x5x6
x7
x4
x3
x2x
1
N variables
N
N
N = N
terms
Sampling 2Slide30
Theorem k query quantum algorithms can be simulated probabilistically with
O(N1-1/2k) queries. Proof:
Algorithm polynomial of degree 2k; Random sampling.Question: Is this optimal?
Extension to k queriesSlide31
k-fold forrelationSlide32
Forrelation: given black box functions f(x)
and g(y), estimate
k-fold forrelation: given f1(x), ..., fk(x), estimateSlide33
Theorem k-fold forrelation can be solved with
k/2 quantum queries.
Conjecture k-fold forrelation requires (N1-1/k) queries classically.
ResultsSlide34
Does k-fold FORRELATION require
(N1-1/2k) queries classically?Plausible but looks quite difficult matematically.
Open problem
1Slide35
Best quantum-classical gaps:1 quantum query -
(N/log N) classical queries;2 quantum queries - (N/log N)
classical;...log N quantum queries - classical queries.
Any problem that requires
O(log N) queries quantumly, (Nc), c>1/2 classically?
Open problem
2Slide36
Total functionsSlide37
Partial functions:huge quantum advantages ...
achieved by ignoring the inputs where quantum algorithm does not provide a conclusive answer.
Why total functions?
What if the algorithm has to output a conclusive answer for every
(x
1, ..., x
N)?Slide38
The biggest known speedup:Grover’s search on N elements (1996);
Q2=O(N), R
2=D=N.
Quantum vs Classical
Our result
*
:
Q2
=O(
), D=N
.
* up to log N factorsSlide39
The biggest known gap:Binary AND-OR tree (Snir, 1996).R
0=O(N0.7537...
), D=N.
Randomized vs Deterministic
Our result
*
:R0
=O(N),
D=N.
* up to log N factorsSlide40
4th power gap between D and
Q:D=N, Q
2=O().Quantum-vs-randomized gap still quadratic (Grover).[Aaronson, Ben-David, Kothari, 2016]:
Q2
=O().
Two notesSlide41
G
o
¨
o
¨s-Pitassi-
WatsonSlide42
P
aperSlide43
Goal
Communication vs. Partition number
.
f
with following properties:
D
– large;
f=1
can be certified by values for a small number of variables.
Certificates are unambiguous.Slide44
D
ve
r
sus
1-ce
r
tificates
Function
o
f
nm
v
a
r
ia
b
les
n
sho
r
t
1-ce
r
tificates
B
UT
not
unambiguou
s
.
1
0
1
1
0
1
0
1
0
0
0
1
1
1
0
f=1
iff there exists unique all-1 column
m
D=nmSlide45
D
ve
r
sus
1-ce
r
tificates
Function
o
f
nm
v
a
r
ia
b
les
n
1
0
1
1
0
1
0
1
0
0
0
1
1
1
0
f=1
iff there exists unique all-1 column
m
D=nm
Should specify which 0 to choose from each columnSlide46
P
ointe
r
s
f=1
iff
1
0
1
1
0
1
0
1
0
0
0
1
1
1
0
t
here is an all-1 column
b
,
in
b
there is a unique
r
with non-zero pointer,
f
ollowing the pointers from
r
, we traverse
exactly one zero in all other columns.Slide47
P
ointe
r
s
D
=
nm
and
un
ambiguous
short
1-ce
r
tificates
.
1
0
1
1
0
1
0
1
0
0
0
1
1
1
0Slide48
Features
Highly
elusi
v
e(fl
exible)
Still
t
raversab
le(if
know
where to
star
t).Slide49
Our
ModificationsSlide50
Bina
r
y
T
ree
Instead
of
a
list
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
More
elusi
v
e
Random
access
we use a balanced binary treeSlide51
Definition
f=1
iff
There
is
a
(unique)
all-1
column
b
;
in
b
,
there
is
a
unique
element
r
with
non-
z
ero
pointers;
f
or
each
j
≠
b
,
f
oll
o
wing
a
path
T
(
j
)
from
r
gi
v
es
a
z
ero
in
the
j
th
column.
1
1
1
1
0
1
0
0
0
1
0
0
1
0
1Slide52
Definition
Ba
c
k
pointers
to
column
s
.
F=1
iff
all
the
le
a
v
es
ba
c
k
point
to
the
all-1
column
b
.
1
1
1
1
0
1
0
0
0
1
0
0
1
0
1Slide53
Q
2
/
R
0
versus DSlide54
Summary
1
1
1
1
0
1
0
0
0
1
0
0
1
0
1
Let
n=2m
;
Theorem
D =
(nm).
R
0
= O(m);
Q
2
=
.
Slide55
Deterministic algorithms
1
1
1
1
0
1
0
0
0
1
0
0
1
0
1
All-1 column might be the last column to be queried.Slide56
Fooling strategy
1
.
0
1
1
1
1
0
0
0
0
1
1
1
1
Let
n=2m
.
If the
k-
th element is queried
in a column:
If
k≤m
, return
Otherwise, return with back
pointer to column
k-m
.
1
0
At the end, the column contains
m
and
m
with back pointers to all columns
1, 2, ..., m
.Slide57
L
o
wer
Bound
1
1
1
1
0
0
0
0
1
1
1
1
The algorithm does not know
the value of the function until
it has queried
>m
elements in
each of
m
columns.
Lower bound:
Slide58
Randomized algorithms
Each
column
contains
a
ba
c
k
pointer
to
the
all-1
column
1
1
1
1
0
1
0
0
0
1
0
0
1
0
1
B
UT
there can be several back pointers
Which is the right one?Slide59
1
1
1
1
0
0
0
0
Upper
Bound:
In
f
ormal
W
e
t
r
y
each
ba
c
k
pointer
b
y
que
r
ing
a
f
e
w
elements
in
the
column,
and
proceed
to
a
column
where
no
z
eroes
w
ere
f
ound.
E
v
en
if
this
is
not
the
all-1
column,
w
e
can
find a column with fewer 0s,
with a high probability.Slide60
1
1
1
1
0
0
0
0
Upper
Bound:
In
f
ormal
Column with
M
zeroes
Column with
M/2
zeroes
Column with
M/4
zeroes
...Slide61
Summa
r
y
a
D algorithm is Ω(
nm )
.
Low
erUpper
bound
bound
for
for
(n +
m).
Quad
ratic
sepa
ration between
R0 and
D.
a R
0 algo
rithm is
O
Grover:
Q2 algorithm with
queries.
4th power
s
epa
r
ation
between Q2
and D.Slide62
Other resultsSlide63
Exact quantum algorithm: outputs correct answer with certainty.Our
result 1:QE = O(N), R
0 = D = N.Our result 2:QE = O(N2/3), R2
=N.
Quantum exact vs. classicalSlide64
Our result: R
2 = O(N), R0 = N
.The first separation between two types of randomized (with error and no error).
Classical resultSlide65
Open
P
r
o
blemsCan we
resolv
e R
2
↔ D?
Kn
own:
R2 =
Ω(D
1/3)
and R
2=
O(D1/2
).
Can
we
separ
ate R
2 from
R1
?
The
same about
Q
↔ D
Kn
own:
Q =
Ω(D1/
6) and
Q =
O(
D1/4
)
and
QE ↔
D?
Kno
wn: QE
=
Ω(D1/
3) and
QE
= O
(D1/
2).Slide66
A
n
y
questions?