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 over the last two decades have deepened
our understanding of physics. That this
represents an intellectual payoff from quantum
computing, whether or not scalable QCs are ever
built.
A Personal Confession While 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

• 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
Ten 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, the limits
of adiabatic computing, search by quantum
walk PART III. Quantum Gravity With a Side of BQP
TQFTs, postselection closed timelike curves,
black holes as mirrors
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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, you could solve
problems that are (probably) intractable 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 BQP/qpoly ? PostBQP/poly A.
2004 Any n-qubit state ? can be PAC-learned
using O(n) sample 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 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 a ground state of a
1D nearest-neighbor Hamiltonian is just as hard
as finding the ground state of any
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
Seems to come tantalizingly close to solving
NP-complete problems in polynomial time! But
Why do these two energy levels almost kiss?
One answer because NP-complete problems are hard!
Van Dam, Mosca, Vazirani 2001 Reichardt 2004
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Quantum Walks
To develop a quantum walk algorithm for spatial
search, algorithmists essentially had to
rediscover the Dirac equation Childs, Goldstone
2004
To develop a quantum walk algorithm for game-tree
search, they wouldve had to rediscover
scattering theory Farhi, Goldstone, Gutmann 2007
A free particle in a 2D box
To develop a quantum walk algorithm for graph
isomorphism, will we need to rediscover some more
physics? Bacon
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PART III. Quantum Gravity With a Side of BQP
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Topological Quantum Field Theory
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
Quantum computers with postselected measurements
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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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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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?)
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A bold (but true) hypothesis linking complexity
and fundamental physics
Encompasses NP?P, NP?BQP, NP?LHC
My Prediction Someday, this hypothesis will be
about as canonical as the 2nd Law or no
superluminal signalling