Quantum Complexity and Fundamental Physics

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Quantum Complexity and Fundamental Physics

• Scott Aaronson
• MIT

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RESOLVED That the results of quantum computing
research can deepen our understanding of
physics. That this represents an intellectual
payoff from quantum computing, whether or not
scalable QCs are ever built.
A Personal Confession When proving theorems about
obscure quantum complexity classes, sometimes
even I wonder whether its all just a
mathematical game
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But then I meet distinguished physicists who say
things like
A quantum computer is obviously just a souped-up
analog computer continuous voltages, continuous
amplitudes, whats the difference? A quantum
computer with 400 qubits would have 2400
classical bits, so it would violate a
cosmological entropy bound My classical
cellular automaton model can explain everything
about quantum mechanics!(How to account for,
e.g., Schors algorithm for factoring prime
numbers is a detail left for specialists) Who
cares if my theory requires Nature to solve the
Traveling Salesman Problem in an instant? Nature
solves hard problems all the timelike the
Schrödinger equation!
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The biggest implication of QC for fundamental
physics is obvious Shors Trilemma
Because of Shors factoring algorithm, either

• the Extended Church-Turing Thesisthe foundation
  of theoretical CS for decadesis wrong,
• textbook quantum mechanics is wrong, or
• theres a fast classical factoring algorithm.

All three seem like crackpot speculations. At
least one of them is true!
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Rest of the Talk
PART I. Classical Complexity Background Why
computer scientists wont shut up about P vs.
NP PART II. How QC Changes the Picture Physics
invades Platonic heaven PART III. The NP Hardness
Hypothesis A falsifiable prediction about
complexity and physics
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PART I. Classical Complexity Background
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CS Theory 101
Problem Given a graph, is it connected? Each
particular graph is an instance The size of the
instance, n, is the number of bits needed to
specify it An algorithm is polynomial-time if it
uses at most knc steps, for some constants k,c P
is the class of all problems that have
polynomial-time algorithms
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NP Nondeterministic Polynomial Time
Does
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04323398509069628960404170720961978805136508024164
94821602885927126968629464313047353426395204881920
47545612916330509384696811968391223240543368805156
78623037853371491842811969677438058008308154426799
03720933
have a prime factor ending in 7?
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NP-hard If you can solve it, you can solve
everything in NP
NP-complete NP-hard and in NP
Is there a Hamilton cycle (tour that visits each
vertex exactly once)?
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NP-hard
NP-complete
NP
P
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Does PNP?
The (literally) 1,000,000 question
Q What if PNP, and the algorithm takes n10000
steps?
A Then wed just change the question!
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What would the world actually be like if we could
solve NP-complete problems efficiently?
Proof of Riemann hypothesis with ?10,000,000
symbols?
Shortest efficient description of stock market
data?
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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PART II. How QC Changes the Picture
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BQP Bounded-Error Quantum Polynomial-Time
BQP contains integer factoring Shor 1994
But factoring isnt believed to be
NP-complete.So the question remains can quantum
computers solve NP-complete problems efficiently?
(Is NP?BQP?)
Obviously we dont have a proof that they cant
But quantum magic wont be enough BBBV 1997
If we throw away the problem structure, and just
consider a landscape of 2n possible solutions,
even a quantum computer needs 2n/2 steps to find
a correct solution
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QCs Dont Provide Exponential Speedups for
Black-Box Search
The BBBV No SuperSearch Principle can even be
applied in physics (e.g., to lower-bound
tunneling times) Is it a historical accident that
quantum mechanics courses teach the Uncertainty
Principle but not the No SuperSearch Principle?
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The Quantum Adiabatic Algorithm
An amazing quantum analogue of simulated
annealing Farhi, Goldstone, Gutmann et al. 2000
This algorithm seems to come tantalizingly close
to solving NP-complete problems in polynomial
time! But
Why do these two energy levels almost kiss?
Answer Because otherwise wed be solving an
NP-complete problem!
Van Dam, Mosca, Vazirani 2001 Reichardt 2004
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Quantum Computing Is Not Analog
is a linear equation, governing quantities
(amplitudes) that are not directly observable
This fact has many profound implications, such as
The Fault-Tolerance Theorem Absurd precision in
amplitudes is not necessary for scalable quantum
computing
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Computational Power of Hidden Variables
Consider the problem of breaking a cryptographic
hash function given a black box that computes a
2-to-1 function f, find any x,y pair such that
f(x)f(y)
Conclusion A. 2005 If, in a
hidden-variable theory like Bohmian mechanics,
your whole life trajectory flashed before you at
the moment of your death, then you could solve
problems that are presumably hard even for
quantum computers (Probably not NP-complete
problems though)
Can also reduce graph isomorphism to this problem
?
QCs can almost find collisions with just one
query to f!
Nevertheless, any quantum algorithm needs ?(N1/3)
queries to find a collision A.-Shi 2002
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The Absent-Minded Advisor Problem
Can you give your graduate student a state ??
with poly(n) qubitssuch that by measuring ?? in
an appropriate basis, the student can learn your
answer to any yes-or-no question of size n?
NO Ambainis, Nayak, Ta-Shma, Vazirani 1999
Some consequences Not even quantum computers
with magic initial states can do everything
BQP/qpoly ? PostBQP/poly An n-qubit state ? can
be PAC-learned using only O(n)
measurementsexponentially better than tomography
A. 2006 One can give a local Hamiltonian H on
poly(n) qubits, such that any ground state of H
can be used to simulate ? on all yes/no
measurements with small circuits A.-Drucker 2009
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PART III. The NP Hardness Hypothesis
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Things we never see
YES
YES
Warp drive
Übercomputer
Perpetuum mobile
But does the absence of these devices have any
scientific importance?
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A falsifiable hypothesis linking complexity and
physics
Encompasses NP?P, NP?BQP, NP?LHC
Does this hypothesis deserve a similar status as
(say) no-superluminal-signalling or the Second
Law?
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Some alleged ways to solve NP-complete problems
Protein folding
DNA computing
A proposal for massively parallel classical
computing
Can get stuck at local optima (e.g., Mad Cow
Disease)
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My Personal Favorite
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-complete problem)
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Relativity Computing
DONE
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Topological Quantum Field Theories
TQFTs
Witten 1980s
Freedman, Kitaev, Larsen, Wang 2003
Jones Polynomial
BQP
Aharonov, Jones, Landau 2006
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Quantum Gravity Computing?
We know almost nothingbut there are hints of a
nontrivial connection between complexity and QG
Example Against many physicists intuition,
information dropped into a black hole seems to
come out as Hawking radiation almost
immediatelyprovided you know the black holes
state before the information went in Hayden
Preskill 2007 Their argument uses explicit
constructions of approximate unitary 2-designs
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Zeno Computing
Do the first step of a computation in 1 second,
the next in ½ second, the next in ¼ second, etc.
Problem Quantum foaminess
Below the Planck scale (10-33 cm or 10-43 sec),
our usual picture of space and time breaks down
in not-yet-understood ways
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Nonlinear variants of the Schrödinger equation
Abrams Lloyd 1998 If quantum mechanics were
nonlinear, one could exploit that to solve
NP-complete problems in polynomial time
1 solution to NP-complete problem
No solutions
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Closed Timelike Curve Computing
Answer
Polynomial Size Circuit
C
CTC Register
Causality-Respecting Register
R CTC
R CR
0
0
0
Quantum computers with closed timelike curves
could solve PSPACE-complete problemsthough not
more than thatA.-Watrous 2008
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Anthropic Principle
Foolproof way to solve NP-complete problems in
polynomial time (at least in the Many-Worlds
Interpretation) First guess a random solution.
Then, if its wrong, kill yourself
Again, I interpret these results as providing
additional evidence that nonlinear QM, closed
timelike curves, postselection, etc. arent
possible. Why? Because Im an optimist.
Technicality If there are no solutions, youd
seem to be out of luck!
Solution With tiny probability dont do
anything. Then, if you find yourself in a
universe where you didnt do anything, there
probably were no solutions, since otherwise you
wouldve found one
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For Even More Interdisciplinary Excitement,
Heres What You Should Look For
A plausible complexity-theoretic story for how
quantum computing could fail (see A.
2004) Intermediate models of computation between
P and BQP (highly mixed states? restricted sets
of gates?) Foil theories that lead to complexity
classes slightly larger than BQP (only example I
know of hidden variables) A sane notion of
quantum gravity polynomial-time (first step a
sane notion of time in quantum gravity?)
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Scientific American, March 2008
www.scottaaronson.com