Download - Quantum algorithms vs. polynomials and the maximum quantum-classical gap in the query model
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?