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 complexity
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
QCMA/qpoly and QMAlog(2), sometimes even I wonder
whether its all just an irrelevant 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

1. the Extended Church-Turing Thesisthe foundation
   of theoretical CS for decadesis wrong,
2. textbook quantum mechanics is wrong, or
3. 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
Eleven of my favorite quantum complexity theorems
and their relevance for physics PART I.
BQP-Infused Quantum Foundations BQP ? PP, BBBV
lower bound, collision lower bound, limits of
random access codes PART II. BQP-Encrusted
Many-Body Physics QMA-completeness and the
limits of adiabatic computing PART III. Quantum
Gravity With a Side of BQP Black holes as
mirrors, topological QFTs, computational power of
nonlinearities, postselection, and CTCs
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PART I. BQP-Infused Quantum Foundations
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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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QCs Dont Provide Exponential Speedups for
Black-Box Search
I.e., if you want more than the ?N Grover speedup
for solving an NP-complete problem, then youll
need to exploit problem structure Bennett,
Bernstein, Brassard, Vazirani 1997
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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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 Any n-qubit state ? can be
PAC-learned using O(n) sample
measurementsexponentially better than quantum
state 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 II. BQP-Encrusted Many-Body Physics
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QMA-completeness
One of the great achievements of quantum
complexity theory, initiated by Kitaev
Just one of many things we learned from this
theory In general, finding the ground state of
a 1D nearest-neighbor Hamiltonian is just as hard
as finding the ground state of any physical
HamiltonianAharonov, Gottesman, Irani, Kempe
2007
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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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PART III. Quantum Gravity With a Side of BQP
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Black Holes as Mirrors
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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Topological Quantum Field Theories
TQFTs
Witten 1980s
Freedman, Kitaev, Larsen, Wang 2003
Jones Polynomial
BQP
Aharonov, Jones, Landau 2006
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Beyond Quantum Computing?
If QM were nonlinear, one could exploit that to
solve NP-complete problems in polynomial time
Abrams Lloyd 1998
I interpret these results as providing additional
evidence that nonlinear QM, postselection, and
closed timelike curves are physically
impossible. Why? Because Im an optimist.
Quantum computers with postselected measurement
outcomes could solve not only NP-complete
problems, but even counting problems A. 2005
Quantum computers with closed timelike curves
(i.e. time travel) could solve PSPACE-complete
problemsbut not more than that A.-Watrous 2008
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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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A bold (but true) hypothesis linking complexity
and fundamental physics
Encompasses NP?P, NP?BQP, NP?LHC
Prediction Someday, this hypothesis will be as
canonical as no-superluminal-signalling or the
Second Law