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Exploring the Limits of the
Efficiently Computable
Research Directions I Like In Complexity and Physics
Scott Aaronson (MIT)
Papers and slides at www.scottaaronson.com
Quantum Mechanics in One Slide
Probability Theory:
Quantum Mechanics:
 s11  s1n   p1   q1 

   
          
s  s  p  q 
nn   n 
 n1
 n
 u11  u1n  1   1 

   
          
 u  u      
nn   n 
 n1
 n
pi  0,
n
p
i 1
i
1
Linear transformations
that conserve 1-norm of
probability vectors:
Stochastic matrices
 i  C,
n

i 1
2
i
1
Linear transformations
that conserve 2-norm of
amplitude vectors:
Unitary matrices
Quantum Computing
A general entangled state of n qubits requires ~2n amplitudes
to specify:
x
x0,1n
 

x
Presents an obvious practical problem when using
conventional computers to simulate quantum mechanics
Feynman 1981: So then why not turn things around, and
build computers that themselves exploit superposition?
Could such a machine get any advantage over a classical
computer with a random number generator? If so, it would
have to come from interference between amplitudes
BQP (Bounded-Error Quantum Polynomial-Time): The class
of problems solvable efficiently by aInteresting
quantum computer,
defined by Bernstein and Vazirani in 1993
Shor 1994: Factoring integers is in BQP
NP-complete
NP
BQP
Factoring
P
This Talk: Three Recent Directions
1. Meeting Experimentalists Halfway
Using complexity theory to find quantum advantage in systems
of current experimental interest (e.g. linear-optical networks),
which fall short of universal quantum computers
2. Publicly-Verifiable Quantum Money
First scheme based on a “standard” crypto assumption
3. Rise and Fall of Complexity in Thermodynamic Systems
Resource-bounded sophistication and coffee cups
1. Meeting Experimentalists
Halfway
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
i,
i
S
nn submatrix of A
corresponding to S
Amazing connection between permanents and physics, which
even leads to a simpler proof of Valiant’s famous result that
the permanent is #P-complete [A. 2011]
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
#P=BPPNP
solvable
in
classical
polynomial
time.
Then
P
Upshot: Compared to (say) Shor’s factoring
algorithm,
we get
different/stronger
evidence
Better
Theorem:
Suppose
we can sample
DA eventhat a
weaker system
can dopolynomial
somethingtime.
classically
approximately
in classical
Thenhard
in
BPPNP, it’s possible to estimate Per(X), with high
nn
probability over a Gaussian random matrix X ~ Ν 0,1C
We conjecture that the above problem is already
#P-complete. If it is, then a fast classical
algorithm for approximate BosonSampling would
already have the consequence that P#P=BPPNP
Challenges
Prove #P-completeness for natural average-case
approximation problems (like permanents of Gaussians)—
thereby yielding hardness for approximate BosonSampling
As a first step, understand the distribution of Per(X), X Gaussian
Early experimental implementations have been done (Rome,
Brisbane, Vienna, Oxford)! But so far with just 3-4 photons.
For scaling, will be crucial to understand the complexity of
BosonSampling when a constant fraction of photons are lost
Can the BosonSampling model solve hard “conventional”
problems? How do we verify that a BosonSampling device is
working correctly? [A.-Arkhipov 2014, A.-Nguyen 2014]
BosonSampling with thermal states: fast classical algorithm to
approximate Per(X) when X is positive semidefinite?
BQP vs. the Polynomial Hierarchy
Can a quantum computer solve problems for which a classical
computer can’t even efficiently verify the answers? Or better
yet: that are still classically hard even if P=NP?
Boils down to: are there problems in BQP but not in PH?
BosonSampling: A candidate for such a problem. If it’s
solvable anywhere in BPPPH, then PH collapses.
A. 2009: Unconditionally, there’s a black-box sampling
problem (Fourier Sampling) solvable in BQP but not in BPPPH
Given a Boolean function
output
z{0,1}n
f : 0,1   1,1
n
2
ˆ
with probability f  z  
 fˆ z  : 1
n

2

  1
x z
x0 ,1n

f  x 


What’s the largest possible
quantum speedup?
“Forrelation”: Given two Boolean functions f,g:{0,1}n{-1,1},
estimate how correlated g is with the Fourier transform of f:
1
2
3n / 2
 f x  1
x y
x , y0 ,1n
gy
 0.01?
 0.6 ?
A.-Ambainis 2014: This problem is solvable using only 1
quantum query, but requires at least ~2n/2/n queries classically
Furthermore, this separation is essentially the largest
possible! Any N-bit problem that’s solvable with k quantum
queries, is also solvable with ~N1-1/2k classical queries
Conjecture (A. 2009): Forrelation  Polynomial Hierarchy
2. Publicly-Verifiable Quantum
Money
Quantum Money
Idea: Quantum states that can be created
by a bank, traded as currency, and
verified as legitimate, but can’t be cloned
by counterfeiters, because of quantum
mechanics’ No-Cloning Theorem

 
Wiesner ca. 1970: First quantum money scheme, but only
the bank could verify the bills. If anyone can verify a bill, then
computational assumptions clearly needed, in addition to QM
A.-Christiano 2012: First quantum money scheme where
anyone can verify a bill, and whose security is based on a
“conventional” crypto assumption
Our Hidden Subspace Scheme
Quantum money state:
A :
1
2
n/4
x
xA
A  R GF 2 
n
n
dim  A 
2
Mint can easily choose a random A and prepare |A
Corresponding “serial number” s: Somehow describes
how to check membership in A and in A (the dual
subspace of A), yet doesn’t reveal A or A
Our proposal: Random low-degree polynomials p1,…,pm
and q1,…,qm that vanish on A and A respectively
Procedure to Verify Money State
(assuming ability to decide membership in A and A)
1. Project onto A elements
A
(reject if this fails)
2. Hadamard all n qubits to
map |A to |A
3. Project onto A elements
A
(reject if this fails)
4. Hadamard all n qubits to
return state to |A
Theorem: The above just implements a projection onto
|A—i.e., it accepts | with probability ||A|2
Security
Theorem: There’s no efficient counterfeiting procedure,
assuming there’s no an efficient quantum algorithm to
learn a basis for A with 2-O(n) probability, given p1,…,pm
and q1,…,qm. [Recently: Attack on noiseless version of scheme]
Theorem: If the A and A membership tests are black
boxes, then any counterfeiting procedure requires Ω(2n/2)
queries to them.
3. Rise and Fall of Complexity in
Thermodynamic Systems
How to Measure Interesting Structure?
Can define structure and in many other ways
One simpleminded measure: the Kolmogorov complexity of
a coarse-grained description of our cellular automaton or
other system
Sean Carroll’s example:
The Coffee Automaton
A., Carroll, Mohan, Ouellette, Werness 2015: A probabilistic
nn reversible system that starts half “coffee” and half
“cream.” At each time step, we randomly “shear” half the
coffee cup horizontally or vertically (assuming a toroidal cup)
Compressed File Size
We prove that the apparent
complexity of this image has
a rising-falling pattern, with a
maximum of at least ~n1/6
500
450
400
350
300
250
200
150
100
50
0
-100
100
300
500
Time Steps
700
900
Other Work
Quantum lower bound for the collision problem [A. 2002]
Quantum (+classical!) lower bound for local search [A. 2004]
First direct product theorem for quantum search [A. 2004]
PostBQP = PP [A. 2004]
BQP/qpoly  QMA/poly, learnability of quantum states [A.Drucker 2010, A. 2004, A. 2006]
Algebrization [A.-Wigderson 2008]
Some Future Directions
Quantum copy-protected software
Complexity theory of quantum states and unitary
transformations
Classification of quantum gate sets
Noisy BosonSampling
The power of quantum proofs
See also my talk at Perimeter on Wednesday at
10:30, for complexity and quantum gravity!