SCOTT AARONSON (MIT), ANDRIS AMBAINIS (UNIV. OF

LATVIA)

Forrelation: A Problem that Optimally Separates Quantum from Classical

Computing

Quantum vs. classical

1 query quantumly

How many queries classically?

Total vs. partial functions Total functions f(x1, ..., xN):

R = O(Q2.5) [previous talk];D = O(Q4) [yesterday];

Partial functions f(x1, ..., xN):Much bigger gaps possible.

Period findingx1, x2, ..., xN - periodic

i xi

Find period r

[Shor, 1994]: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 Task that requires 1 query

quantumly, (N/log N) classically.

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

FORRELATION =Fourier CORRELATION

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

N

x

x

xx

...2

1

N

Ny

y

yy

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 algorithm1. Generate a superposition of

(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.

NN

log

REAL FORRELATION

Distinguish between random (xi’s, yi’s - Gaussian); random, .

Nx

xx

x...2

1

Ny

yy

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/log N) values xi and yj uncorrelated random variables.

NN

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 with O(N) queries.

Analyzing query algorithms

Q Qstart Q UT…U1

1,1|1,1+ 1,2|1, 2+ … + N, M|N, M

i,j is actually i,j(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

jijijiN xxaxxxp

,,21 ,,,

sampledji

jiji xxa),(

,

Good if we sample N of N2 terms independently.

Estimator:

Requires sampling N variables xi!

Sampling 2x1

x2

x3

x4

x5

x6

x7x5 x6

x7

x4x3x2x1

N variables

N

N N = Nterms

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,...,

,22,111

3221)(...)()(

Results Theorem k-fold forrelation can be

solved with k/2 quantum queries.

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

Open problem 1 FORRELATION - the biggest gap

between quantum from probabilistic. Provides a precise meaning for «QFT

is hard to simulate classically».

Can we find an application for it?

Open problem 2 Does k-fold FORRELATION require

(N1-1/2k) queries classically? Plausible but looks quite difficult

matematically.

Open problem 3 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. NN log

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