ANDRIS AMBAINISUNIVERSITY OF

LATVIA

Quantum algorithms vs. polynomials and the maximum quantum-classical

gap in the query model

Query model

Function f(x1, ..., xN), xi{0,1}.

xi given by a black box:

i xi

Complexity = number of queries

Quantum query model

Fixed starting state. U0, U1, …, UT – independent of x1, …, xN. Q – queries:

U0 Q Qstart U1 UT…

i

xi

ii iaia i)1(

Reasons to study query model Encompasses many quantum

algorithms (Grover’s search, quantum part of factoring, etc.).

Provable quantum-vs-classical gaps.

Quantum vs. classical

1 query quantumly

How many queries classically?

Period finding

x1, x2, ..., xN - periodic

i xi

Find period r

1 query quantumly

Period-finding

Quantum algorithm works if N r2. T classical queries – can test T2

possible periods.

4 Nc

i xi

queries classically

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.

Fourier checking/Forrelation

Forrelation Input: (x1, ..., xN, y1, ..., yN) {-1, 1}2N. Are vectors

N

x

x

x

x

...2

1

N

Ny

y

y

y

F...2

1

highly correlated? FN – Fourier transform over ZN.

More precisely... Is the inner product

3/5 or 1/100?

ji

jijiyx yxFN ,

,

1

Quantum algorithm

1. Generate states

in parallel (1 query).

2. Apply FN to 2nd state.

3. Test if states equal (SWAP test).

,1

N

iix ix

N

iiy iy

1

Classical lower bound Theorem Any classical algorithm for

FORRELATION uses

queries.

N

N

log

REAL FORRELATION

Distinguish between random (xi’s - Gaussian);

random, .

Nx

x

x

x...2

1

Ny

y

y

y...2

1

xFy N

x

yx,

Real-valued vectors

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 yj uncorrelated random variables.

N

N

log

xFy N

Reduction

Proof idea: Replace xi sgn(xi) to achieve xi{-1, 1}.

T query algorithm for FORRELATION

T query algorithm for REAL FORRELATION

Simulating 1 query quantum algorithms

Simulation

Theorem Any 1 query quantum algorithm can be simulated probabilistically using O(N) queries.

Analyzing query algorithms

Q Qstart Q UT…U1

1,1|1,1+ 1,2|1, 2+ … + N, M|N, M

1,1 is actually 1,1(x1, ..., xN)

Polynomials method

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

are polynomials in x1, ..., xN of degree k.

21, ...,, Nji xxMeasurement:

Polynomial of degree 2k

Nji xx ...,,1,

Our task

Pr[A outputs 1] = p(x1, ..., xN), deg p =2.

0 p(x1, ..., xN) 1.

Task: estimate p(x1, ..., xN) with precision .

Solution: random sampling.

Pre-processing

Problem: large error if sampling omits xi with large influence in p(x1, ..., xN).

Solution: replace influential xi’s by several variables with smaller influence.

Sampling 1

ji

jijiN xxaxxxp,

,21 ,,,

sampledji

jiji xxa),(

,

Good if we sample N of N2 terms independently.

Estimator:

Requires sampling N variables xi!

Sampling 2

jijijiN xxaxxxp

,,21 ,,,

NNN

Sampling N terms ai,jxixj

Sampling N variables xi

Extension to k queries

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?

K-fold forrelation

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

K-fold forrelation: given f1(x), ..., fk(x), estimate

yx

yx ygxfF,

, )()(

kxx

kkxxxx xfFxfFxf,...,

,22,11

1

3221)(...)()(

Results

Theorem k-fold forrelation can be solved with k/2 quantum queries.

Conjecture k-fold forrelation requires (N1-1/k) queries classically.

From polynomials to quantum algorithms

(with Scott Aaronson, Jānis Iraids, Mārtiņš Kokainis, Juris Smotrovs)

Quantum algorithm with t queries

Polynomials of degree 2t

??

Quantum algorithm with 1 query

Polynomials of degree 2

Our result

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.

N

jijijiN xxaxxp

1,,1 ...,,

N

jijijiNN yxayyxxq

1,,11 ...,,,...,,

Standard polynomial representation

Block-multilinearrepresentation

Step 1

Requirements

q(x1, ..., xN, x1, ..., xN) f(x1, ..., xN);

q(x1, ..., xN, y1, ..., yN) [-1, 1] for all xi, yj {0, 1}.

N

jijijiNN yxayyxxq

1,,11 ...,,,...,,

Step 2: evaluating q

U = (Nai,j) – unitary.

SWAP test on |x and U|y:

N

jijijiNN yxayyxxq

1,,11 ...,,,...,,

,1

1

N

iix ix

N.

1

1

N

iiy iy

N

Still works if ||U|| C!

Two norms

N

jijijiNN yxayyxxq

1,,11 ...,,,...,, )( , jiaA

Have: 11

A|q|1

Need: CAN

Step 3: variable splitting

Replace xi by , - new variables.

N

jijijiNN yxayyxxq

1,,11 ...,,,...,,

mii xx

m ...

11

jix

1. |q|1 preserved;

2. Influential variables - eliminated.

Result

1:1

AA KANA '':'

Variable-splitting

K – Groethendieck’s constant

Summary

1 quantum query = (N) classical queries.

k quantum queries can be simulated with O(N1-1/2k) classical queries.

1 quantum query = polynomials of degree 2.

Open problem 1

Does k-fold FORRELATION require (N1-

1/2k) queries classically? Plausible but looks quite difficult

matematically.

Open problem 2

Best quantum-classical gaps:1 quantum query - (N) classical queries;2 quantum queries - (N) classical queries;...log N quantum queries - classical

queries. NN log

Any problem that requires O(log N) queries quantumly, (Nc), c>1/2 classically?

Open problem 3

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

2 queries polynomials of degree 4? Polynomials of degree 3 2 query

algorithms?