Quantum Computing and the Limits of the Efficiently Computable

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Quantum Computing and the Limits of the
Efficiently Computable

• Scott Aaronson
• MIT

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Things we never see
Warp drive
Übercomputer
Perpetuum mobile
The (seeming) impossibility of the first two
machines reflects fundamental principles of
physicsSpecial Relativity and the Second Law
respectively
So what about the third one? What are the
ultimate physical limits on what can be feasibly
computed? And do those limits have any
implications for physics?
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NP-hardAll NP problems are efficiently reducible
to these
NP-complete
NPEfficiently verifiable
OUR STANDARD MODEL
PEfficiently solvable
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Does PNP?
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
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An important presupposition underlying P vs. NP
is the
The Extended Church-Turing Thesis (ECT) Any
physically-realistic computing device can be
simulated by a deterministic or probabilistic
Turing machine, with at most polynomial overhead
in time and memory
But how sure are we of this thesis?What would a
challenge to it look like?
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The LHC Computer?
Jordan-Lee-Preskill 2012 Simple interacting
quantum field theories (e.g., ?4 theory) can be
simulated efficiently using a garden-variety
quantum computer
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Old proposal Dip two glass plates with pegs
between them into soapy water. Let the soap
bubbles form a minimum Steiner tree connecting
the pegsthereby solving a known NP-hard problem
instantaneously
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Relativity Computer
DONE
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Zenos Computer
Time (seconds)
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Time Travel Computer
S. Aaronson and J. Watrous. Closed Timelike
Curves Make Quantum and Classical Computing
Equivalent, Proceedings of the Royal Society A
465631-647, 2009. arXiv0808.2669.
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Quantum Computing
A general entangled state of n qubits requires
2n amplitudes to specify
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?
Shor 1994 Such a computer could do more than
simulate QMe.g., it could factor integers in
polynomial time
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But Can QCs Actually Be Built?
Where we are now A quantum computer has factored
21 into 3?7, with high probability (Martín-López
et al. 2012)
Why is scaling up so hard? Because of
decoherence unwanted interaction between a QC
and its external environment, prematurely
measuring the quantum state
A few skeptics, in CS and physics, even argue
that building a QC will be fundamentally
impossible
I dont expect them to be right, but I hope they
are! If so, it would be a revolution in physics
And for me, putting quantum mechanics to the test
is the biggest reason to build QCsthe
applications are icing!
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Key point factoring is not believed to be
NP-complete! And today, we dont believe quantum
computers can solve NP-complete problems in
polynomial time in general (though not
surprisingly, we cant prove it)
Bennett et al. 1997 Quantum magic wont 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 Grovers algorithm!)
If theres a fast quantum algorithm for
NP-complete problems, it will have to exploit
their structure somehow
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Quantum Adiabatic Algorithm(Farhi et al. 2000)
Hi
Hf
Hamiltonian with easily-prepared ground state
Ground state encodes solution to NP-complete
problem
Problem Eigenvalue gap can be exponentially
small
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Includes P?NP as a special case, but is
stronger No longer a purely mathematical
conjecture, but also a claim about the laws of
physics Could be invoked to explain why
adiabatic systems have small spectral gaps, why
protein folding gets stuck in metastable states,
why the Schrödinger equation is linear, why time
only flows in one direction
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OK, but can computational complexity engage even
more deeply with the content of modern physics?
What other new insights has it given the
physicists?
Thanks for asking! Ill give several examples,
drawn from my own work and others
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Quantum Computing and the Interpretation of
Quantum Mechanics?
David Deutschs argument for Many Worlds
To those who still cling to a single-universe
world-view, I issue this challenge explain how
Shor's algorithm works When Shor's algorithm
has factorized a number, using 105 or so times
the computational resources that can be seen to
be present, where was the number factorized?
How, and where, was the computation performed?
Possible response To those who cling to a
many-universe world-view, explain why the
NP-complete problems still seem to be hard
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Schrödinger vs. Heisenberg vs. Feynman?
Schrödinger and Heisenberg pictures of quantum
mechanics Require exponential time and
exponential space to simulate using a classical
computer
Feynman picture Still exponential time, but only
polynomial space
Bohmian mechanics?
Postulates real trajectories for particles,
which are guided along by the quantum state to
reproduce the predictions of quantum mechanics
A. 2005 Calculating Bohmian trajectories is
probably intractable even for a quantum
computer!If we could do it, then we could also
solve Graph Isomorphism in polynomial time, and
break arbitrary collision-resistant hash functions
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Two of my favorite functions
Easily computable
P-complete Valiant
Free fermions can be simulated easily by a
classical computerValiant, Terhal-DiVincenzo
Free bosons probably cant be easily simulated by
a classical computerA.-Arkhipov
BOSONS
FERMIONS
Two Basic Types of Particle in Nature
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Computational Complexity and the Black-Hole
Information Loss ProblemMaybe the single most
striking application so far of complexity to
fundamental physics
Hawking 1970s Black holes radiate! The radiation
seems thermal (uncorrelated with whatever fell
in)but if quantum mechanics is true, then it
cant be
Susskind et al. 1990s Black-hole
complementarity. In string theory / quantum
gravity, the Hawking radiation should just be a
scrambled re-encoding of the same quantum states
that are also inside the black hole
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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 cant we do something to those Hawking qubits
(after waiting 1070 years for enough to come
out), then dive into the black hole, and see that
weve completely destroyed the spacetime geometry
in the interior?
Entanglement among Hawking photons detected!
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Harlow-Hayden 2013 Sure, theres some unitary
transformation that Alice could apply to the
Hawking radiation, that would generate a
firewall inside the event horizon. But how
long would it take her to apply it?
Plausible answer Exponential in the number of
qubits inside the black hole! Or for an
astrophysical black hole,
years
She wouldnt have made a dent before the black
hole had already evaporated anyway! So problem
solved?
HHs argument If Alice could achieve (a
plausible formalization of) her decoding task,
then she could also break collision-resistant
hash functionsbeyond what even QCs seem able to
do
Recently, I strengthened the HH argument, to show
that Alice could even invert arbitrary one-way
functions
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Conclusion
Any one of these would make me as happy as
low-energy SUSY would make you Prove P?NP Prove
that not even quantum computers can solve
NP-complete problems Build a scalable quantum
computer (or even more interesting, show that
its impossible) Clarify whether all of known
physics can be simulated by a quantum
computer Use No-SuperSearch or related
impossibility principles to make progress in
quantum gravity