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Quantum Computers and Beyond
Scott Aaronson
Associate Professor, EECS
Moore’s Law
Extrapolating: Robot uprising?
But even a killer robot would still be
“merely” a Turing machine, operating on
principles laid down in the 1930s…
=
And it’s conjectured that thousands of
interesting problems are inherently
intractable for Turing machines…
Is there any feasible way to solve
these problems, consistent with
the laws of physics?
Relativity Computer
DONE
Zeno’s Computer
Time (seconds)
STEP 1
STEP 2
STEP 3
STEP 4
STEP 5
Time Travel Computer
S. Aaronson and J. Watrous. Closed Timelike
Curves Make Quantum and Classical
Computing Equivalent, Proceedings of the Royal
Society A 465:631-647, 2009. arXiv:0808.2669.
Answer
Polynomial
Size Circuit
C
“Closed
Timelike
Curve
Register”
R CTC
R CR
0 0 0
“CausalityRespecting
Register”
Quantum Mechanics in 1 Slide
“Like probability theory, but over the complex numbers”
Probability Theory:
Quantum Mechanics:
 s11  s1n   p1   q1 

   
          
s  s  p  q 
nn   n 
 n1
 n
 u11  u1n  1   1 

   
          
 u  u      
nn   n 
 n1
 n
pi  0,
n
p
i 1
i
1
Linear transformations
that conserve 1-norm of
probability vectors:
Stochastic matrices
 i  C,
n

i 1
2
i
1
Linear transformations
that conserve 2-norm of
amplitude vectors:
Unitary matrices
Interference
“The source of all quantum weirdness”





1
2
1
2
1  1  1 

2  12   20
      
1  01   11
2   2   2 
Possible states of a single
quantum bit, or qubit:
0 1
2
1
0 1
2
0
Quantum Computing
“Quantum Mechanics on Steroids”
Where we are: A QC has now factored 21 into
37, with
high probability
(Martín-López
etn al.
2012)
A general
entangled
state of n qubits
requires ~2
amplitudes

Scaling up 
is hard,
 because
 xofxdecoherence! But
unless QM is wrong, theren doesn’t seem to be any
x0,1
fundamental obstacle
to specify:
Presents an obvious practical problem when using
conventional computers to simulate quantum mechanics
Interesting
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
QM—e.g., it could factor integers in polynomial time
But factoring is not believed to be NP-complete!
And today, we don’t believe quantum computers can solve
NP-complete problems in polynomial time in general
(though not surprisingly, we can’t prove it)
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
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