Transcript The Learnability of Quantum States

The Computational Complexity of
Linear Optics
vs
Scott Aaronson and Alex Arkhipov
MIT
The Extended ChurchTuring Thesis (ECT)
Everything feasibly
computable in the physical
world is feasibly computable
by a (probabilistic) Turing
machine
Shor’s Theorem: QUANTUM SIMULATION has no efficient
classical algorithm, unless FACTORING does also
So the ECT is false … what more
evidence could anyone want?
Building a QC able to factor large numbers is damn
hard! After 16 years, no fundamental obstacle has
been found (or even seriously proposed), but who knows?
Can’t we “meet the physicists halfway,” and show
computational hardness for quantum systems closer to
what they actually work with now?
FACTORING might be in BPP! At any rate, it’s an
extremely “special” problem
Wouldn’t it be great to show that if BPP=BQP, then (say)
the polynomial hierarchy collapses?
Today: “A New Attack on the ECT”
We define a model of computation based on linear
optics: n identical photons traveling through a
network of poly(n) beamsplitters, phase-shifters, etc.,
then a measurement of where the photons ended up
Crucial point: No entangling interactions
between pairs of photons needed!
Our model is contained in BQP, but seems unlikely to
be BQP-complete. We don’t know if it solves any
decision problems that are hard classically.
But for sampling and search problems, the situation is
completely different…
Theorem 1. Suppose that for every linear-optics
network, the probability distribution over
measurement outcomes can be sampled in classical
polynomial time. Then P#P=BPPNP (so PH collapses)
More generally, let O be any oracle that simulates a
“OK, network
but isn’tA,
the
realaquestion
theof A and a
linear-optics
given
description
of approximate sampling?O
randomhardness
string r. Then
#P are noisy, and
NP
After all, experiments
P  BPP
not even the linear-optics network
itselfoptics
can sample
exactly!” in BPPPH, that
So even if linear
can be simulated
still collapses PH! (New evidence that QCs have capabilities
beyond PH, complementing [A’10],[FU’10])
Theorem 2. Suppose two plausible conjectures are
true: the permanent of a Gaussian random matrix is
(1) #P-hard to approximate, and
(2) not too concentrated around 0.
Let O be any oracle takes as input a description of a
linear-optics network A, a random string r, and 01/,
and that samples from a distribution -close to A’s in
variation distance. Then
#P
NPO
P
 BPP
In other words: if our conjectures hold, then even
simulating noisy linear-optics experiments is
classically intractable, unless PH collapses
Particle Physics In One Slide
There are two basic types of particle in the universe…
All I can say is, the bosons
got the harder job
Indeed, [Valiant 2002, Terhal-DiVincenzo 2002]
BOSONS
FERMIONS
showed
that noninteracting fermion
systems can
be simulated in BPP. But, confirming Avi’s joke,
we’ll argue that the analogous problem for bosons
Their transition
givenharder…
respectively by…
(such amplitudes
as photons)are
is much
Per A 
n
a  


S n i 1
i,
i
Det A 
 1


sgn  
S n
n
a  
i,
i 1
i
Linear Optics for Dummies
We’ll be considering a special kind of quantum
computer, which is not based on qubits
The basis states have the form |S=|s1,…,sm, where
si is the number of photons in the ith “mode”
We’ll never create or destroy photons. So if there
are n photons, then s1,…,sm are nonnegative integers
summing to n
For us, m=poly(n)
Initial state: |I=|1,…,1,0,……,0 
You get to apply any mm unitary matrix U
If n=1 (i.e., there’s only one photon, in a superposition over the m
modes), U acts on that photon in the obvious way
 m  n  1

M : 
In general, there are
 n 
ways to
distribute n identical photons into m modes
U induces an MM unitary (U) on the n-photon
states as follows:
Per U
 US ,T 

S ,T

s1! sm!t1!tm!
Here US,T is an nn submatrix of U (possibly with repeated
rows and columns), obtained by taking si copies of the ith
row of U and tj copies of the jth column for all i,j
Example: The “Hong-Ou-Mandel Dip”
1 1 1 
Suppose U 

.
2 1  1
U
Then Pr[the two photons land in different modes] is
Per U   0
2
Pr[they both land in the first mode] is
2
 1 1 1 
1
1

  
Per
2
2!  2 1 1 
Beautiful Alternate Perspective
The “state” of our computer, at any time, is a degree-n
polynomial over the variables x=(x1,…,xm) (n<<m)
Initial state: p(x) := x1xn
We can apply any mm unitary transformation U to x,
to obtain a new degree-n polynomial
p' x  pUx 
 x



S  s1 ,, sm
s1
S 1
x
Then on “measuring,” we see the monomial
2
with probability
 S s1! sm !
sm
m
s1
1
sm
m
x x
OK, so why is it hard to sample the
distribution over photon numbers classically?
Given any matrix ACnn, we can construct an mm
unitary U (where m2n) as follows:
 A B 

U  
 C D
Suppose we start with |I=|1,…,1,0,…,0 (one photon
in each of the first n modes), apply U, and measure.
Then the probability of observing |I again is
p : I  U  I
2

2n
Per  A
2
Claim 1: p is #P-complete to
estimate (up to a constant factor)
Idea: Valiant proved that the
PERMANENT is #P-complete.
Can use a classical reduction
to go from a multiplicative
approximation of |Per(A)|2
to Per(A) itself.
 
Claim 2: Suppose we had a
fast classical algorithm for
linear-optics sampling. Then
we could estimate p in BPPNP
Idea: Let M be our classical
sampling algorithm, and let r
be its randomness. Use
approximate counting to
estimate Pr M r  outputs I
2
r

 
Conclusion:
p : Suppose
I  U weI had a fast
 classical
Per Aalgorithm
for linear-optics sampling. Then P#P=BPPNP.
2n
2

As I said before, I find this result unsatisfying, since it
only talks about the classical hardness of exactly
sampling the distribution over photon numbers
What about sampling a distribution that’s 1/poly(n)close in variation distance?
Difficulty: The sampler might adversarially refuse to
output the one submatrix whose permanent we care
about! That changes the output distribution by only
exp(-n), so we still have an excellent sampler … but
we can no longer use it to estimate |Per(A)|2 in BPPNP
To get around this difficulty, it seems we need to
“smuggle in” the matrix A that we about as a random
submatrix of U
Main Result
Consider applying a Haar-random mm unitary matrix
U, to n photons in m=poly(n) modes:
Distribution DU over
photon numbers
Main technical lemma used in proof:
Suppose there’s
a classical algorithm to sample a
6
Let mn . Then an nn submatrix of an mm
distribution -close to DU in poly(n,1/) time. Then for
Haar unitary matrix is Õ(1/n)-close
in variation
NP
all ,1/poly(n), there’s also a BPP algorithm to
distance to a matrix
of independent Gaussians.
2
estimate |Per(X)| to within additive error n!, with
probability 1- over a Gaussian random matrix
U
X
nn


~ N 0,1 C
So the question boils down to this: how hard is it to
additively estimate |Per(X)|2, with high probability
nn
over a Gaussian random matrix
 C
X ~ N 0,1
?
We conjecture that it’s #P-hard—in which case, even
approximate classical simulation of our linear-optics
experiment would imply P#P=BPPNP
We can decompose this conjecture into two
plausible sub-conjectures: that multiplicatively
estimating Per(X) is #P-hard for Gaussian X, and that
Per(X) is “not too concentrated around 0”
The Permanent-of-Gaussians
Conjecture (PGC)
The following problem is #P-hard. Given a matrix
XCnn of i.i.d. Gaussian entries, together with 01/ and
01/, output an approximation z such that
Pr
X ~ N 0,1Cnn
 z  Per X    Per X    1   .
We can prove #P-hardness if =0 or =0. So what
makes the PGC nontrivial is really the combination
of average-case with approximation
The Permanent Anti-Concentration
Conjecture (PACC)
There exist constants C,D and >0 such that for all n and >0,
Pr
X ~ N 0,1Cnn
PerX    n! Cn 
D

Empirically true!
Also, we can prove it with
determinant in place of
permanent
Experimental Prospects
It seems well within current technology to do our
experiment with (say) n=4 photons and m=20 modes
(Current record: n=2 photons)
Physicists we consulted: “Sounds hard! But
If you can scale to n photons and error  in variation
not as hard as building a universal QC”
distance, using “poly(n,1/) experimental effort,” then
modulo our complexity conjectures, the ECT is false
What Remark:
would it take
No point
to scale
in scaling
to (say)this
n=20
experiment
photons and
m=500
much
modes?
beyond 20 or 30 photons, since then a
classical
computer
can’t(standard
even verify
thegood
answers!
- Reliable
single-photon
sources
laser isn’t
enough!)
- Reliable photodetector arrays
- Stable apparatus to ensure that w.h.p., all n photons arrive at
photodetector arrays at exactly the same time
Open Problems
Prove the PGC ($200) and PACC ($100)!
Can our linear-optics model solve classically-intractable
decision problems? What about problems for which a
classical computer can verify the answers?
Do BPP=BQP or PromiseBPP=PromiseBQP have
interesting structural complexity consequences?
Similar hardness results for other natural quantum
systems (besides linear optics)?
Bremner, Jozsa, Shepherd 2010: Another system for
which exact classical simulation would collapse PH