Download The Future of Computer Science

Quantum Computing and the Limits
of the Efficiently Computable
Scott Aaronson (MIT  UT Austin)
NYSC, West Virginia, June 24, 2016
Things we never see…
GOLDBACH
CONJECTURE:
TRUE
NEXT QUESTION
Warp drive
Perpetuum mobile
Übercomputer
The (seeming) impossibility of the first two machines
reflects fundamental principles of physics—Special
Relativity and the Second Law respectively
So what about the third one?
Moore’s Law
Extrapolating: Robot uprising?
But even a killer robot would still be
“merely” a Turing machine, operating on
principles laid down in the 1930s…
=
And Turing machines have limitations—on
what they can compute at all, and certainly
on what they can compute efficiently
NP-hard
All NP problems are efficiently
reducible to these
Steiner tree
Coin balancing
Maximum cut
Satisfiability
Maximum clique
…
NPcomplete
NP
Efficiently
verifiable
Graph connectivity
Primality testing
Matrix determinant
Linear programming
…
P
Efficiently
solvable
Matrix permanent
Halting problem
…
Factoring
…
As Dana discussed, most computer scientists
believe that PNP…
But if so, there’s a further question: is there
any way to solve NP-complete problems in
polynomial time, consistent with the laws of
physics?
Old proposal: Dip two glass plates with pegs between them
into soapy water.
Let the soap bubbles form a minimum Steiner tree
connecting the pegs—thereby solving a known NP-hard
problem “instantaneously”
Relativity Computer
DONE
Zeno’s Computer
Time (seconds)
STEP 1
STEP 2
STEP 3
STEP 4
STEP 5
Time Travel Computer
S. Aaronson and J. Watrous. Closed Timelike
Curves Make Quantum and Classical
Computing Equivalent, Proceedings of the Royal
Society A 465:631-647, 2009. arXiv:0808.2669.
Answer
Polynomial
Size Circuit
C
“Closed
Timelike
Curve
Register”
R CTC
R CR
0 0 0
“CausalityRespecting
Register”
Ah, but what about
quantum computing?
(you knew it was coming)
Quantum mechanics: “Probability
theory with minus signs”
(Nature seems to prefer it that way)
The Famous Double-Slit Experiment
Probability of landing in “dark patch” = |amplitude|2 =
|amplitudeSlit1 + amplitudeSlit2|2 = 0
Yet if you close one of the slits, the photon can appear in
that previously dark patch!!
A bit more precisely: the key claim of quantum mechanics
is that, if an object can be in two distinguishable states,
call them |0 or |1, then it can also be in a superposition
a|0 + b|1
1
Here a and b are complex
numbers called amplitudes
satisfying |a|2+|b|2=1
If we observe, we see
|0 with probability |a|2
|1 with probability |b|2
Also, the object collapses to
whichever outcome we see
0 1
2
0
To modify a state
a1 1   aN N
we can multiply the vector of amplitudes
by a unitary matrix—one that preserves
2
2
a1    a N  1
0 1





1
2
1
2
1  1  1 

2  1 2   20
      
1  01   11 
2   2   2 
2
1
0 1
2
0
We’re seeing interference
of amplitudes—the source
of “quantum weirdness”
Two qubits:
a 0 0 b 0 1 c1 0 d 1 1
|a|2+|b|2+|c|2+|d|2=1
What happens when you measure? |a|2, |b|2, |c|2, |d|2
What if you measure (say) the first qubit only?
Separable state:
a 0
 b 1 c 0  d 1 
 ac 0 0  ad 0 1  bc 1 0  bd 1 1
Example:
0 0  0 1 1 0 1 1
2
Is this state separable?
0 0 1 1
2
 ac 0 0  ad 0 1  bc 1 0  bd 1 1
No—we call it “entangled”
“Spooky Action at a Distance”?
No-Communication Theorem vs. The Bell Inequality
The No-Cloning Theorem:
No physical procedure can copy an unknown quantum state

 0
 
  1  0   0   1  0   1 
  2 0 0   0 1   1 0   2 1 1
Quantum Computing
A general entangled state of n qubits requires ~2n amplitudes
to specify:
x factored 21 into 37,
Where we are: A QC has
n
x


0
,
1

with high probability (Martín-López et al. 2012)
 
a
x
Presents an obvious practical problem when using
Scaling up
is hard,tobecause
decoherence!
conventional
computers
simulateofquantum
mechanicsBut
unless QM is wrong, there doesn’t
seem to be any
Interesting
Feynman 1981: So then
why not turn
things around, and
fundamental
obstacle
build computers that themselves exploit superposition?
Shor 1994: Such a computer could do more than simulate
QM—e.g., it could factor integers in polynomial time
NP-complete
Bounded-Error
Quantum
Polynomial-Time
BQP
NP
Factoring
P
Factoring is in BQP, but not believed to be NP-complete!
Today, we don’t believe quantum computers can solve
NP-complete problems in polynomial time in general
(though not surprisingly, we can’t prove it)
Bennett et al. 1997: “Quantum magic” won’t be enough
If you throw away the problem structure, and just consider an
abstract “landscape” of 2n possible solutions, then even a
quantum computer needs ~2n/2 steps to find the correct one
(That bound is actually achievable, using Grover’s algorithm!)
If there’s a fast quantum algorithm for NP-complete problems,
it will have to exploit their structure somehow
The “Adiabatic
Optimization” Approach to
Solving NP-Hard Problems
with a Quantum Computer
Hi
Operation with easilyprepared lowest energy state
Hf
Operation whose lowest-energy state
encodes solution to NP-hard problem
Hope: “Quantum tunneling” could give
speedups over classical optimization
methods for finding local optima
Remains unclear
whether you can get a
practical speedup this
way over the best
classical algorithms.
We might just have to
build QCs and test it!
Problem: “Eigenvalue gap”
can be exponentially small
Some Examples of My Research…
BosonSampling (with Alex Arkhipov):
A proposal for a rudimentary photonic
quantum computer, which doesn’t
seem useful for anything (e.g. breaking
codes), but does seem hard to
simulate using classical computers
(We showed that a fast, exact classical simulation would “collapse the
polynomial hierarchy to the third level”)
Experimentally demonstrated with 6
photons by a group in Bristol, UK
Quantum Computing and Black Holes
Hawking 1970s: Black holes radiate
The radiation seems thermal (uncorrelated with whatever
fell in). But if quantum mechanics is true, then it can’t be!
Susskind, ‘t Hooft 1990s: “Black-hole complementarity.”
Idea that quantum states emerging from black hole are
somehow “the same states” as the ones trapped inside,
just measured in a different way
The Firewall Paradox [Almheiri et al. 2012]
If the black hole interior is “built”
out of the same qubits coming out as
Hawking radiation, then why can’t
we do something to those Hawking
qubits (after waiting ~1067 years for
enough to come out), then dive into
the black hole, and see that we’ve
completely destroyed the spacetime
geometry in the interior?
Entanglement among
Hawking photons detected!
Harlow-Hayden 2013: Argued that, to do the experiment
on the Hawking radiation that would produce a “firewall”
in the interior, would require an amount of processing time
exponential in the number of qubits—meaning
1067
~2
for a black hole the mass of our sun! In which case, long
before one had made a dent in the problem, the black hole
would’ve already evaporated…
Their evidence used a theorem I proved as a grad student
in 2002: given a “black box” function with N outputs and
>>N inputs, any quantum algorithm needs at least ~N1/5
steps to find two inputs that both map to the same output
(improved to ~N1/3 by Yaoyun Shi, which is optimal)
Summary
Quantum computers are the most powerful kind of computer
allowed by the currently-known laws of physics
There’s a realistic prospect of building them
Contrary to what you read, even quantum computers would
have limits
But those limits might help protect the geometry of
spacetime!