Computational Phenomena in
Physics
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? What are the ultimate
physical limits on what can be feasibly computed? And
do those limits have any implications for physics?
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?
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”
What About Quantum Mechanics?
“Like probability, but with minus signs”
Probability Theory:
Quantum Mechanics:
 s11  s1n   p1   q1 

   
          
s  s  p  q 
 n
nn   n 
 n1
 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
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 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 don’t 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 QCs—the applications are icing!
Key point: 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
Quantum Adiabatic Algorithm
(Farhi et al. 2000)
Hi
Hamiltonian with easilyprepared ground state
Hf
Ground state encodes solution
to NP-complete problem
Problem: “Eigenvalue gap”
can be exponentially small
“The No-SuperSearch Postulate”
There is no physical means to solve NP-complete
problems in polynomial time.
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…
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! I’ll give several
examples, drawn from my own work
and others’
Quantum Computing and the
Interpretation of Quantum Mechanics?
David Deutsch’s argument for Many Worlds:
“To those who still cling to a single-universe worldview, I issue this challenge: explain how Shor's
algorithm works … When Shor's algorithm has
factorized a number, using 10⁵⁰⁰ 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”
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
The bosons got the harder job!
Two of Avi’s Favorite Functions
Easily computable
Det  A 
 1


sgn 
S n
#P-complete [Valiant]
n
a  
i,
i
Per  A 
n
a  


i,
1
2012:i Experimental
S n i 1
demonstrations of
Free fermions “BosonSampling”
can be
If free bosons could
simulated in
P 3-4 photons! be simulated in P, it
with
[Valiant, Terhalwould collapse PH
DiVincenzo]
[A.-Arkhipov]
FERMIONS
BOSONS
Two Basic Types of Particle in Nature
i
Computational Complexity and the BlackHole Information Loss Problem
Maybe 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 can’t 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
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 ~1070 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: Sure, there’s 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,
1070
years
She wouldn’t have made a dent before the black hole had
already evaporated anyway! So … problem solved?
~2
HH’s argument: If Alice could achieve (a plausible formalization of) her
decoding task, then she could also break collision-resistant
hash functions—beyond what even QCs seem able to do
Recently, I strengthened the HH argument, to show that Alice
could even invert arbitrary one-way functions
I haven’t even touched on the huge interplay between
computational complexity and condensed-matter physics!
“Quantum Hamiltonian Complexity”
2D spin lattices with
energy gap?
2D spin lattices
1D spin chains
[Oliveira-Terhal]
Theoretical
science
notions (polynomial(2D Area
Law Conjecture)
with energy
gap computer
approximation schemes, QMA-completeness)
[Hastings,time
AharonovArad-Landau-Vazirani, Arbitrary spin networks 1D spin chains
are used even to define what it means for[Aharonova
Landau-Vazirani-Vidick…]
thermal
ground statein
toabe
“simple”bath?
or “complicated”
Gottesman-Irani(Quantum PCP Conjecture)
SIMPLE, LOW-ENTANGLEMENT
GROUND STATES
(always)
Kempe]
GNARLY, ENTANGLED
GROUND STATES
(sometimes)
Summary: Reductionism Revised
The limits of computation?
Computer engineering, software
Semiconductors, applied physics
Quantum mechanics, quantum
field theory, general relativity
MATH