Transcript Quantum Computing and the Limits of the Efficiently Computable Scott Aaronson MIT Things we never see… GOLDBACH CONJECTURE: TRUE NEXT QUESTION Warp drive Perpetuum mobile Übercomputer The (seeming) impossibility of the.

Quantum Computing and the Limits
of the Efficiently Computable
Scott Aaronson
MIT
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…
=
But Turing machines have fundamental limits—even more
so, if you need the answer in a reasonable amount of time!
P: Polynomial Time
Class of all “decision problems” (infinite sets of yes-or-no
questions) solvable by a Turing machine, using a number
of steps that scales at most like the size of the question
raised to some fixed power
Example: Given
this map, is there
a route from
Charlottesville to
Bartow?
NP: Nondeterministic Polynomial Time
Class of all decision problems for which a “yes” answer
can be verified in polynomial time, if you’re given a
witness or proof for it
Example: Does
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have a divisor ending in 7?
NP-hard: If you can solve it, then you
can solve every NP problem
NP-complete: NP-hard and in NP
Example:
Is there a tour
that visits each
city once?
Does P=NP?
The (literally) $1,000,000 question
If there actually were a machine with
[running time] ~Kn (or even only with ~Kn2),
this would have consequences of the
greatest magnitude.
—Gödel to von Neumann, 1956
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
n
a
i
i 1
i
we can multiply the vector of amplitudes
by a unitary matrix—one that preserves
n
a
i 1
i
2
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”
Quantum Computing
A general entangled state of n qubits requires ~2n amplitudes
Where we are: A QC has now factored 21 into
to specify:
37, with high probability x(Martín-López et al. 2012)
n
x


0
,
1

Scaling up is hard, because of decoherence! But
Presents
obvious
practical
problem
when
usingto be any
unlessan
QM
is wrong,
there
doesn’t
seem
conventional computers
to simulate
quantum mechanics
fundamental
obstacle
 
a
x
Interesting
Feynman 1981: So then why not turn things around, and
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
But factoring is not believed to be NP-complete!
And 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 on
Computational Complexity and Physics
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
3-4 photons…) by groups in Brisbane,
Oxford, Vienna, and Rome!
Computational Complexity of
Decoding Hawking Radiation
Firewall Paradox (2012): Hypothetical experiment
that involves waiting outside a black hole for
~1070 years, collecting all the Hawking photons it
emits, doing a quantum computation on them,
then jumping into the black hole to observe that
your computation “nonlocally destroyed” the
structure of spacetime inside the black hole
Harlow-Hayden (2013): Argument that the
requisite computation would take exponential
time (~210^70 years) even for a QC—by which time
the black hole has already fully evaporated!
Recently, I strengthened Harlow and Hayden’s argument, to
show that performing the computation is generically at least
as hard as inverting cryptographic “one-way functions”
Summary
Quantum computing is one of the most exciting things in
science—but the reasons are a little different from what the
press says
Even a quantum computers couldn’t solve all problems in an
instant (though they’d provide amazing speedups for a few problems,
like factoring and quantum simulation, and maybe broader speedups)
And building them is hard (though the real shock for physics would
be if they weren’t someday possible)
On the other hand, one thing quantum computing has already
done, is create a bridge between computer science and
physics, carrying amazing insights in both directions