Andris - PowerPoint Presentation

Slide1

Andris AmBAINISUNIVERSITY OF LATVIA

Quantum algorithms vs. polynomials and the maximum quantum-classical gap in the query modelSlide2

Query model

Function f(x

1

, ..., xN), xi{0,1}.xi given by a black box:

i

x

i

Complexity = number of queriesSlide3

Quantum query model

Fixed starting state.

U

0, U1, …, UT – independent of x1, …, xN.

Q – queries:

U0

Q

Q

U

1

U

T

…Slide4

Reasons to study query modelEncompasses many quantum algorithms (Grover’s search, quantum part of factoring, etc.).

Provable quantum-vs-classical gaps.Slide5

Quantum vs. classical

1 query quantumly

How many queries

classically?Slide6

Period finding

x

1

, x

2, ..., xN - periodic

i

x

i

Find period r

1 query quantumlySlide7

Period-finding

Quantum algorithm works if N

 r

2.T classical queries – can test T2 possible periods.

i

x

i

queries classicallySlide8

Our result [Aaronson, A]

Task that requires 1 query quantumly,

(N) classically.

1 query quantum algorithms can be simulated by O(N) query probabilistic algorithms.Slide9

Fourier checking/ForrelationSlide10

Forrelation

Input: (x

1

, ..., xN, y1, ..., yN)

{-1, 1}2N.Are vectors

highly correlated?

F

N

– Fourier transform over Z

N

.Slide11

More precisely...

Is the inner product

 3/5 or  1/100?Slide12

Quantum algorithm

Generate states

in parallel (1 query).

Apply FN to 2nd state.

Test if states equal (SWAP test).Slide13

Classical lower bound

Theorem

Any classical algorithm for FORRELATION uses

queries. Slide14

REAL FORRELATIONDistinguish between

random (x

i

’s - Gaussian); random, .

Real-valued vectorsSlide15

Lower bound

Claim

REAL FORRELATION requires queries.

Intuition: if , each variable – Gaussian, correlations between xi’s and yj’s - weak.

o(N) values xi and y

j  uncorrelated random variables. Slide16

Reduction

Proof idea

: Replace x

i  sgn(xi) to achieve xi{-1, 1}.

T query algorithm for FORRELATION

T query algorithm for REAL

FORRELATIONSlide17

Simulating 1 query quantum algorithmsSlide18

Simulation

Theorem

Any 1 query quantum algorithm can be simulated probabilistically using O(

N) queries.Slide19

Analyzing query algorithms

Q

Q

Q

U

T

…

U

1



1,1

|1

,1



+ 

1,2

|

1,

2

+

…

+ 

N, M

|N

, M





1,1

is actually



1,1

(x

1

, ..., x

N

)Slide20

Polynomials method

Lemma

[Beals et al., 1998] After k queries, the amplitudes

are polynomials in x

1, ..., xN of degree  k.

Measurement:

Polynomial of degree

 2kSlide21

Our task

Pr[A outputs 1] = p(x

1

, ..., xN), deg p =2.0  p(x1

, ..., xN)  1.Task: estimate p

(x1, ..., xN) with precision

.

Solution: random sampling.Slide22

Pre-processingProblem

: large error if sampling omits x

i

with large influence in p(x1, ..., xN).Solution: replace influential x

i’s by several variables with smaller influence.Slide23

Sampling 1

Good if we s

ample N of N

2

terms independently.

Estimator:

Requires sampling N variables x

i

!Slide24

Sampling 2

Sampling N terms a

i,j

x

i

x

j

Sampling



N variables x

i

Slide25

Extension to k queries

Theorem

k query quantum algorithms can be simulated probabilistically with O(N

1-1/2k) queries. Proof: Algorithm

 polynomial of degree 2k; Random sampling.

Question: Is this optimal?Slide26

K-fold forrelationSlide27

Forrelation

: given black box functions f(x) and g(y), estimate

K-fold forrelation

: given f1(x), ..., fk(x), estimateSlide28

ResultsTheorem k-fold forrelation can be solved with



k/2

 quantum queries.Conjecture k-fold forrelation requires (N

1-1/k) queries classically.Slide29

From polynomials to quantum algorithms(with Scott Aaronson, Jānis Iraids, Mārtiņš Kokainis, Juris Smotrovs)Slide30

Quantum algorithm with t queries

Polynomials of degree 2t

??Slide31

Quantum algorithm with 1 query

Polynomials of degree 2



Our resultSlide32

More precisely...Polynomial p represents f with error

 if:

f = 0  p  [0, ];

f = 1  p  [1- , 1];f – undefined  p  [0, 1].Theorem

Q(f)=1 for some <1/2 iff f can be represented by p: deg p=2 with error  <1/2.Slide33

Standard polynomial

representation

Block-multilinear

representation

Step 1Slide34

Requirements q(x

1

, ..., x

N, x1, ..., xN)  f(x1, ..., xN);

q(x1, ..., xN, y1, ..., yN)

 [-1, 1] for all xi, y

j  {0, 1}.Slide35

Step 2: evaluating qU = (N

a

i,j

) – unitary. SWAP test on |x and U|

y:

Still works if ||U||

 C!Slide36

Two norms

Have:

|q|

1

Need:Slide37

Step 3: variable splitting

Replace x

i

by , - new variables.

|q|

1 preserved;

Influential variables - eliminated.Slide38

Result

Variable-splitting

K – Groethendieck’s constantSlide39

Summary 1 quantum query =

(N) classical queries.

k quantum queries can be simulated with O(N

1-1/2k) classical queries.

1 quantum query = polynomials of degree 2.Slide40

Open problem 1Does k-fold FORRELATION require

(N

1-1/2k

) queries classically?Plausible but looks quite difficult matematically.Slide41

Open problem 2Best quantum-classical gaps:

1 quantum query -

(N) classical queries;

2 quantum queries - (N) classical queries;...log N quantum queries - classical queries.

Any problem that requires O(log N) queries

quantumly,

(N

c), c>1/2 classically?Slide42

Open problem 3Characterize quantum algorithms with 2, 3, ..., queries?

2 queries

 polynomials of degree 4?

Polynomials of degree 3  2 query algorithms?