Scott Aaronson

1
The Limits of Computation
Quantum Computers and Beyond

• Scott Aaronson
• Associate Professor, EECS

2
Moores Law
3
Extrapolating Robot uprising?
4
But even a killer robot would still be merely a
Turing machine, operating on principles laid down
in the 1930s

5
Is there any feasible way to solve these
problems, consistent with the laws of physics?
And its conjectured that thousands of
interesting problems are inherently intractable
for Turing machines
6
Relativity Computer
DONE
7
Zenos Computer
Time (seconds)
8
Time Travel Computer
S. Aaronson and J. Watrous. Closed Timelike
Curves Make Quantum and Classical Computing
Equivalent, Proceedings of the Royal Society A
465631-647, 2009. arXiv0808.2669.
9
Quantum Computers
10
Quantum Mechanics in 1 Slide
Like probability theory, but over the complex
numbers
11
Interference
The source of all quantum weirdness
Possible states of a single quantum bit, or qubit
12
Quantum ComputingQuantum Mechanics on Steroids
Where we are A QC has now factored 21 into 3?7,
with high probability (Martín-López et al.
2012) Scaling up is hard, because of decoherence!
But unless QM is wrong, there doesnt seem to be
any fundamental obstacle
A general entangled state of n qubits requires
2n amplitudes to specify
Presents an obvious practical problem when using
conventional computers to simulate quantum
mechanics
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 QMe.g., it could factor integers in
polynomial time
13
But factoring is not believed to be
NP-complete! And today, we dont believe quantum
computers can solve NP-complete problems in
polynomial time in general (though not
surprisingly, we cant prove it)
Bennett et al. 1997 Quantum magic wont 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 Grovers algorithm!)
So, is there any quantum algorithm for
NP-complete problems that would exploit their
structure?
14
Quantum Adiabatic Algorithm(Farhi et al. 2000)
Hi
Hf
Hamiltonian with easily-prepared ground state
Ground state encodes solution to NP-complete
problem
Problem Eigenvalue gap can be exponentially
small
15
Some of My Recent Research
BosonSampling (with Alex Arkhipov) A proposal
for a rudimentary optical quantum computer, which
doesnt 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 Haydenthat 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