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Boson-Sampling in the light of sample complexity
Boson-Sampling in the light of sample complexity:
a review
Christian Gogolin, Martin Kliesch, Leandro Aolita, and Jens Eisert
Dahlem Center for Complex Quantum Systems, Freie Universit¨
at Berlin
DPG Spring Meeting 2014, Berlin
1 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
Why BosonSampling?
As building a quantum computer is difficult, can we
demonstrate quantum supremacy by simpler means?
2 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
Physical BosonSampling
m modes
n bosons
Input:
Fock state | 1n i = |1i |1i |1i |0i |0i |0i |0i
U and n define
an instance of
BosonSampling.
U
∈ U (m)
Output: sample S = ( s1 , s2 , s3 , s4 , s5 , s6 , s7 )
3 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
4 / 15
The abstract sampling problem
BosonSampling
Given n and a fixed m × m unitary matrix U , sample from
X
{S = (s1 , . . . , sm ) : sj ≥ 0 ∧
sj = n}
j
according to
Pr [S] := | Perm(US )|2 /
DU
S
m
Y
US
(sj !).
j=1
U=
n
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
5 / 15
Approximate BosonSampling is hard
Ref. [2] provides strong evidence that:
Hardness of BosonSampling
If 1-norm approximate classical efficient BosonSampling were possible for
Haar random U and m ∈ Ω(n5 ) then the polynomial hierarchy would
collapse to the third level.
[2] S. Aaronson and A. Arkhipov, in: Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC ’11, New
York: ACM Press, 2011, p. 333
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
5 / 15
Approximate BosonSampling is hard
Ref. [2] provides strong evidence that:
Hardness of BosonSampling
If 1-norm approximate classical efficient BosonSampling were possible for
Haar random U and m ∈ Ω(n5 ) then the polynomial hierarchy would
collapse to the third level.
Remarks:
Is almost as unlikely as P=NP
[2] S. Aaronson and A. Arkhipov, in: Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC ’11, New
York: ACM Press, 2011, p. 333
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
5 / 15
Approximate BosonSampling is hard
Ref. [2] provides strong evidence that:
Hardness of BosonSampling
If 1-norm approximate classical efficient BosonSampling were possible for
Haar random U and m ∈ Ω(n5 ) then the polynomial hierarchy would
collapse to the third level.
Remarks:
Is almost as unlikely as P=NP
Based on hardness of approximating permanents
[2] S. Aaronson and A. Arkhipov, in: Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC ’11, New
York: ACM Press, 2011, p. 333
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
5 / 15
Approximate BosonSampling is hard
Ref. [2] provides strong evidence that:
Hardness of BosonSampling
If 1-norm approximate classical efficient BosonSampling were possible for
Haar random U and m ∈ Ω(n5 ) then the polynomial hierarchy would
collapse to the third level.
Remarks:
Is almost as unlikely as P=NP
Based on hardness of approximating permanents
Still holds after post selection on bit string outcomes
[2] S. Aaronson and A. Arkhipov, in: Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC ’11, New
York: ACM Press, 2011, p. 333
es 1 and 3 at mome
withntum
R. Fickler for help
Faculty
–p/4
andAward
2.
D. Aharono
a TIBCOv, A. Ambainis,
grant;
J. Kempe, U.
Vazirani,
We thank
ments:
perfec
t transm
“Quantu
100, of Acknowledg
m walks on graphs,” Proceed
ission betwe|enBosonSampling
Rev. Lett.
and
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Photonic Boson Samplin
g
in a Tunable Circuit
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
But:
!
BosonSampling is a sampling problem, hence not in NP.
7 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
But:
!
BosonSampling is a sampling problem, hence not in NP.
+
Calculating PrDU [S] for fixed S from U is in general hard.
7 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
But:
!
BosonSampling is a sampling problem, hence not in NP.
+
Calculating PrDU [S] for fixed S from U is in general hard.
⇒
There is no obvious way to certify an implementation.
7 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
But:
!
BosonSampling is a sampling problem, hence not in NP.
+
Calculating PrDU [S] for fixed S from U is in general hard.
⇒
There is no obvious way to certify an implementation.
Particularly high standards should be applied
as BosonSampling has no other use!
7 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
What is certification?
“hard”
+
8 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
What is certification?
naive
+
8 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
What is certification?
impossible
+
8 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
What is certification?
impossible
ideal
+
+
8 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
8 / 15
What is certification?
impossible
ideal
good enough
+
+
+
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
First step: distinguishing from uniform distribution
Simpler task:
Distinguishing from uniform
!
Promise: Device either does BosonSampling or samples from the
corresponding uniform distribution.
?
Question: Decide which is the case.
9 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
9 / 15
First step: distinguishing from uniform distribution
Simpler task:
Distinguishing from uniform
!
Promise: Device either does BosonSampling or samples from the
corresponding uniform distribution.
?
Question: Decide which is the case.
Uniform
BosonSampling
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
9 / 15
First step: distinguishing from uniform distribution
Simpler task:
Distinguishing from uniform
!
Promise: Device either does BosonSampling or samples from the
corresponding uniform distribution.
?
Question: Decide which is the case.
Uniform
Now look at this in two settings. . .
BosonSampling
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
State discrimination setting
!
Assume DU is completely known.
10 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
State discrimination setting
!
Assume DU is completely known.
!
Note that this is unrealistic.
10 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
10 / 15
State discrimination setting
!
Assume DU is completely known.
!
Note that this is unrealistic.
⇒
Let m ∈ O(poly(n)), then any instance of BosonSampling that
could be potentially hard to sample 1-norm approximately classically
can be distinguished from the uniform distribution from O(n2+ )
samples.
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
Black box setting
!
It is unclear how/whether U can be used in a computationally
efficient way for certification.
11 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
Black box setting
!
It is unclear how/whether U can be used in a computationally
efficient way for certification.
?
What can one do without using U ?
11 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
Black box setting
!
It is unclear how/whether U can be used in a computationally
efficient way for certification.
?
What can one do without using U ?
⇒
If U Haar random and m ∈ Ω(nν ) with ν > 3 (ν > 2 with
post selection), then with probability supra-exponentially
close to one in n no symmetric probabilistic algorithm can
distinguish DU from the uniform distribution from fewer than
Ω(en/2 ) many samples.
11 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
12 / 15
Corroboration/partial certification
One can look at:
1
Boson bunching [6, 9]
2
Moments
3
Row norm estimators [1]
⇒
Allows to efficiently distinguish
from uniform distribution.
⇒
This yields no certification.
All algorithms can be fooled!
...
[1] S. Aaronson and A. Arkhipov, ECCC, 135.135 (2013), TR13–135, arXiv: 1309.7460
[6] C. Gogolin, M. Kliesch, L. Aolita, and J. Eisert (2013), arXiv: 1306.3995
[9] M. C. Tichy, M. Tiersch, F. Mintert, and A. Buchleitner, New Journal of Physics, 14.9 (2012), 093015
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
This is no coincidence
!
DU typically has a high min-entropy [6].
[1] S. Aaronson and A. Arkhipov, ECCC, 135.135 (2013), TR13–135, arXiv: 1309.7460
[6] C. Gogolin, M. Kliesch, L. Aolita, and J. Eisert (2013), arXiv: 1306.3995
[11] L. Trevisan, M. Tulsiani, and S. Vadhan, ECCC, (2010), TR08–103
13 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
13 / 15
This is no coincidence
!
⇒
DU typically has a high min-entropy [6].
Observation of F. Brandao (published in [1] based on [11]):
Existence of an efficiently samplable cheating distribution
For every instance of BosonSampling with high min-entropy and every
circuit length T ∈ O(poly(n)) there is a classically efficiently samplable
distribution indistinguishable from DU by all circuits of length T .
[1] S. Aaronson and A. Arkhipov, ECCC, 135.135 (2013), TR13–135, arXiv: 1309.7460
[6] C. Gogolin, M. Kliesch, L. Aolita, and J. Eisert (2013), arXiv: 1306.3995
[11] L. Trevisan, M. Tulsiani, and S. Vadhan, ECCC, (2010), TR08–103
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
Conclusions
!
Loud claims seem inappropriate at present.
14 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
Conclusions
!
Loud claims seem inappropriate at present.
!
Classical certification of BosonSampling is likely difficult.
14 / 15
Boson-Sampling in the light of sample complexity | BosonSampling in a nutshell
Conclusions
!
Loud claims seem inappropriate at present.
!
Classical certification of BosonSampling is likely difficult.
!
BosonSampling is both close and far from uniform.
14 / 15
Boson-Sampling in the light of sample complexity | References
15 / 15
References
Thank you for your attention!
−→ slides: www.cgogolin.de
[1]
S. Aaronson and A. Arkhipov, ECCC, 135.135
(2013), TR13–135, arXiv: 1309.7460.
[2]
S. Aaronson and A. Arkhipov, in: Proceedings of
the 43rd annual ACM symposium on Theory of
computing - STOC ’11, New York: ACM Press,
2011, p. 333.
[3]
M. a. Broome, A. Fedrizzi, S. Rahimi-Keshari,
J. Dove, S. Aaronson, T. C. Ralph, and
A. G. White, Science (New York, N.Y.), 339.6121
(2013), 794–8.
[4]
J. Carolan, J. D. A. Meinecke, P. Shadbolt,
N. J. Russell, N. Ismail, K. W¨
orhoff, T. Rudolph,
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and A. Laing (2013), arXiv: 1311.2913.
[5]
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E. F. Galv˜
ao, N. Spagnolo, C. Vitelli, E. Maiorino,
P. Mataloni, and F. Sciarrino, Nature Photonics,
7.7 (2013), 545–549.
[6]
C. Gogolin, M. Kliesch, L. Aolita, and J. Eisert
(2013), arXiv: 1306.3995.
[7]
N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod,
A. Crespi, F. Flamini, S. Giacomini, G. Milani,
R. Ramponi, P. Mataloni, R. Osellame,
E. F. Galvao, and F. Sciarrino (2013), arXiv:
1311.1622.
[8]
J. B. Spring, B. J. Metcalf, P. C. Humphreys,
W. S. Kolthammer, X.-M. Jin, M. Barbieri,
A. Datta, N. Thomas-Peter, N. K. Langford,
D. Kundys, J. C. Gates, B. J. Smith,
P. G. R. Smith, and I. A. Walmsley, Science (New
York, N.Y.), 339.6121 (2013), 798–801.
[9]
M. C. Tichy, M. Tiersch, F. Mintert, and
A. Buchleitner, New Journal of Physics, 14.9
(2012), 093015.
[10]
M. Tillmann, B. Daki´
c, R. Heilmann, S. Nolte,
A. Szameit, and P. Walther, Nature Photonics, 7.7
(2013), 540–544.
[11]
L. Trevisan, M. Tulsiani, and S. Vadhan, ECCC,
(2010), TR08–103.