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Quantum Computing and the Limits
of the Efficiently Computable
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?
Some would say Alan Turing & friends
already answered this question in the 1930s
No computer program can
infallibly decide the truth or
falsehood (or even the
provability) of arbitrary
mathematical statements!
But what about “merely” searching all possible
proofs with (say) 109 symbols or fewer? Can that be
done in a way that avoids exhaustive search?
This sounds like (literally) a $1,000,000 question:
P=NP?
NP: Nondeterministic
Polynomial-Time
P: Polynomial-Time
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
However, an important presupposition underlying
P vs. NP is the...
Extended Church-Turing Thesis
“Any physically-realistic computing device can be
simulated by a deterministic or probabilistic
Turing machine, with at most polynomial
overhead in time and memory”
So how sure are we of this thesis?
Have there been serious challenges to it?
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”
Ah, but what about
quantum computing?
(you knew it was coming)
Quantum mechanics: “Probability
theory with minus signs”
(Nature seems to prefer it that way)
Quantum Computing
A quantum state of n qubits takes 2n complex
numbers to describe:

x0,1
x x
n
Chemists and physicists knew that for decades, as a
practical problem!
In the 1980s, Feynman, Deutsch, and others had the
amazing idea of building a new type of computer that
could overcome the problem, by itself exploiting the
exponentiality inherent in QM
Actually building a QC: Damn hard, because of decoherence.
(But seems possible in principle!)
Popularizers Beware:
A quantum computer is NOT like a
massively-parallel classical computer!
 

x
 x
x1,, 2 
n
Exponentially-many basis
states, but you only get to
observe one of them
Any hope for a speedup
rides on the magic of
interference
BQP (Bounded-Error Quantum Polynomial-Time): The class
of problems solvable efficiently by aInteresting
quantum computer,
defined by Bernstein and Vazirani in 1993
Shor 1994: Factoring integers is in BQP
NP-complete
NP
BQP
Factoring
P
But factoring is not believed to be NP-complete!
And today, we don’t believe BQP contains all of NP
(though not surprisingly, we can’t prove that it doesn’t)
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!)
So, is there any quantum algorithm for NP-complete problems
that would exploit their structure?
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
Nonlinear variants of the Schrödinger
Equation
Abrams & Lloyd 1998: If quantum mechanics were
nonlinear, one could exploit that to solve NPcomplete problems in polynomial time
1 solution to NP-complete problem
No solutions
Relativity Computer
DONE
Zeno’s Computer
Time (seconds)
STEP 1
STEP 2
STEP 3
STEP 4
STEP 5
“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
If true, would “explain” why adiabatic systems have
small spectral gaps, the Schrödinger equation is linear,
closed timelike curves don’t exist...
Some of My Recent Research
BosonSampling (with Alex Arkhipov):
A proposal for a rudimentary optical
quantum computer, which doesn’t
seem useful for anything (e.g. breaking
codes), but does seem hard to
simulate using classical computers
Computational Complexity of Decoding Hawking
Radiation: Building on a striking recent proposal
by Harlow and Hayden—that part of the
resolution of the black hole information problem
might be that reconstructing the infalling
information from the Hawking radiation would
require an exponentially long computation