“Computational Complexity
and Physics”
Scott Aaronson (MIT)
New Insights Into Computational Intractability
Oxford University, October 3, 2013
Title is too broad!
So, I’ll just give two “tales from the trenches”:
- One about BosonSampling, one about black holes
What you should take away:
1990s:
Today:
Computational
Complexity
Computational
Complexity
Shor &
Grover
Quantum
Physics
Quantum
Physics
BosonSampling (A.-Arkhipov 2011)
A rudimentary type of quantum computing, involving
only non-interacting photons
Classical counterpart:
Galton’s Board
Replacing the balls by photons leads to
famously counterintuitive phenomena,
like the Hong-Ou-Mandel dip
In general, we consider a network of
beamsplitters, with n input “modes” (locations)
and m>>n output modes
n identical photons enter, one per input mode
Assume for simplicity they all leave in
different modes—there are  m  possibilities
n
 
The beamsplitter network defines a column-orthonormal
matrix ACmn, such that Pr outcome S  Per  A  2

where
Per X  

n
x  


S n i 1
is the matrix permanent
i,
i
S
nn submatrix of A
corresponding to S
For simplicity, I’m ignoring outputs
with ≥2 photons per mode
Example
For Hong-Ou-Mandel experiment,


Proutput 1,1   Per



1
2
1
2
2
1 
2

2   11 0
1 
2 2


2
In general, an nn complex permanent is a sum of n!
terms, almost all of which cancel
How hard is it to estimate the “tiny residue” left over?
Answer: #P-complete, even for constant-factor approx
(Contrast with nonnegative permanents!)
So, Can We Use Quantum Optics to
Solve a #P-Complete Problem?
That sounds way too good to be true…
Explanation: If X is sub-unitary, then |Per(X)|2
will usually be exponentially small. So to get a
reasonable estimate of |Per(X)|2 for a given X,
we’d generally need to repeat the optical
experiment exponentially many times
Better idea: Given ACmn as input, let BosonSampling
be the problem of merely sampling from the same
distribution DA that the beamsplitter network samples
from—the one defined by Pr[S]=|Per(AS)|2
Theorem (A.-Arkhipov 2011): Suppose BosonSampling is
solvable in classical polynomial time. Then P#P=BPPNP
Better Theorem: Suppose we can sample DA even
approximately in classical polynomial time. Then in
BPPNP, it’s possible to estimate Per(X), with high
nn


probability over a Gaussian random matrix X ~ Ν 0,1 C
Upshot: Compared to (say) Shor’s factoring
algorithm, we get different/stronger evidence that a
weaker system can do something classically hard
Experiments
Last year, groups in Brisbane,
Oxford, Rome, and Vienna
reported the first 3-photon
BosonSampling experiments,
confirming that the amplitudes
were given by 3x3 permanents
# of experiments > # of photons!
Obvious Challenges for Scaling Up:
- Reliable single-photon sources (optical multiplexing?)
- Minimizing losses
- Getting high probability of n-photon coincidence
Goal (in our view): Scale to 10-30 photons
Don’t want to scale much beyond that—both because
(1) you probably can’t without fault-tolerance, and
(2) a classical computer probably couldn’t even verify
the results!
Theoretical Challenge: Argue that, even with photon
losses and messier initial states, you’re still solving a
classically-intractable sampling problem
Recent Criticisms of Gogolin et al.
(arXiv:1306.3995)
Suppose you ignore which actual photodetectors light
up, and count only the number of times each output
configuration occurs. In that case, the BosonSampling
distribution DA is exponentially-close to the uniform
distribution U
Response: Why would you ignore which
detectors light up??
The output of almost any algorithm is also
gobbledygook if you ignore the order of the
output bits…
Recent Criticisms of Gogolin et al.
(arXiv:1306.3995)
OK, so maybe DA isn’t close to uniform. Still, the very
same arguments we gave for why polynomial-time
classical algorithms can’t sample DA, suggest that
they can’t even distinguish DA from U!
Response: That’s why we said to focus on 10-30
photons—a range where a classical computer can
verify a BosonSampling device’s output, but the
BosonSampling device might be “faster”!
(And 10-30 photons is probably the best you can do anyway,
without quantum fault-tolerance)
More Decisive Responses
(A.-Arkhipov, arXiv:1309.7460)
Theorem: Let ACmn be a Haar-random
BosonSampling matrix, where m≥n5.1/.
Then with 1-O() probability over A, the
BosonSampling distribution DA has Ω(1)
variation distance from the uniform
distribution U
Histogram of (normalized)
probabilities under DA
Under U
Necessary, though not sufficient, for
approximately sampling DA to be hard
Theorem (A. 2013): Let ACmn be Haar-random, where
m>>n. Then there is a classical polynomial-time algorithm
C(A) that distinguishes DA from U (with high probability over A
and constant bias, and using only O(1) samples)
Strategy: Let AS be the nn submatrix of A corresponding to
output S. Let P be the product of squared 2-norms of AS’s
rows. If P>E[P], then guess S was drawn from DA; otherwise
guess S was drawn from U
P under uniform distribution
(a lognormal random variable)
AS
P under a BosonSampling
distribution
A
n
P  v1  vn
2
2
n
  ?
m
Using Quantum Optics to Prove
that the Permanent is #P-Complete
[A., Proc. Roy. Soc. 2011]
Valiant showed that the permanent is #P-complete—but
his proof required strange, custom-made gadgets
We gave a new, arguably more transparent proof by
combining three facts:
(1) n-photon amplitudes correspond to nn permanents
(2) Postselected quantum optics can simulate universal
quantum computation [Knill-Laflamme-Milburn 2001]
(3) Quantum computations can encode #P-complete
quantities in their amplitudes
Black Holes and Computational
Complexity??
QAM
AM
QSZK
SZK
BQP
BPP
YES!!
Amazing connection made this year by Harlow & Hayden
But first, we need to review 40 years of black hole history
Bekenstein 1972, Hawking 1975: Black holes have entropy
and temperature! They emit radiation
The Information Loss Problem: Calculations suggest that
Hawking radiation is thermal—uncorrelated with whatever
fell in. So, is infalling information lost forever? Would violate
the unitarity / reversibility of QM
OK then, assume the information somehow gets out!
The Xeroxing Problem: How could the same qubit | fall
inexorably toward the singularity, and emerge in Hawking
radiation? Would violate the No-Cloning Theorem
Black Hole Complementarity (Susskind, ‘t Hooft): An external
observer can describe everything unitarily without including
the interior at all! Interior should be seen as “just a
scrambled re-encoding” of the exterior degrees of freedom
The Firewall Paradox (AMPS 2012)
R = Faraway Hawking Radiation
H = Just-Emitted Hawking Radiation
Near-maximal
entanglement
B = Interior
of “Old”
Black Hole
Also near-maximal
entanglement
Violates “monogamy of entanglement”! The same
qubit can’t be maximally entangled with 2 things
Harlow-Hayden 2013 (arXiv:1301.4504): Striking argument
that Alice’s decoding task would require exponential time
Complexity theory to the rescue of quantum field theory??
The HH decoding problem:
Given an n-qubit pure state |BHR produced by a known,
poly-size quantum circuit. Promised that, by acting only
on R (the “Hawking radiation”), it’s possible to distill an
EPR pair (|00+|11)/√2 between R and B (the “black hole
interior”). Distill such a pair, by applying a unitary
transformation UR to the qubits in R.
Theorem (HH): This decoding problem is QSZK-hard.
So presumably intractable, unless QSZK=BQP (which we have
oracle evidence against)!
Improvement (A. 2013): Suppose there are poly-size quantum
circuits for the HH decoding problem. Then any OWF
f:{0,1}n{0,1}p(n) can be inverted in quantum polynomial time.
Proof: Assume for simplicity that f is injective. Consider

RBH

1
2
n 1

x0,1
n
 x0
p  n  n
,0
R
0
B

 f x ,1 R 1 B x
H
Suppose applying UR to R decodes an EPR pair between R and
B. Then for some {|x}x, {|x}x, we must have
U R x0 p n  n ,0   x 0 , U R f  x ,1   x 1
Generalizing to arbitrary OWFs requires
Furthermore, to get perfect entanglement, we need |x=|x
techniques
similar
to
those
used
in
HILL
Theorem!
for all x! So from U , we can get unitaries V,W such that
R
V x0 p n  n   x , W f  x    x
 V 1W f  x   x0 p n  n
Other Exciting Recent
Complexity/Physics Connections
Quantum PCP Conjecture. “Is approximate quantum
MAX-k-SAT hard for quantum NP?” If so, will require
the construction exotic condensed matter systems,
exhibiting room-temperature entanglement!
Beautiful recent Isurvey
by Aharonov
et al.many
(arXiv:1309.7495)
hope for,
and expect,
more such
striking connections!
Using Entanglement to Steer Quantum Systems. Can
Indeed,
makingfor
these
connections possible
use Bell inequality
violation
guaranteed
might be(Vazirani-Vidick),
the most important
result
randomness expansion
blind
andof
if useful
authenticatedquantum
QC withcomputing
a classicalresearch—even
polynomial-time
QCs eventually get built!
verifier, QMIP=MIP* (Reichardt-Unger-Vazirani)…