The Limits of Quantum Computers (or: What We Can

1
The Limits of Quantum Computers(or What We
Cant Do With Computers We Dont Have)

• Scott Aaronson (MIT)

NP-complete
2
So then why cant we just ignore quantum
computing, and get back to real work?
3
Because the universe isnt classical
?
Fancier version Extended Church-Turing Thesis
4
Shors factoring algorithm presents us with a
choice
Either

1. the Extended Church-Turing Thesis is false,
2. textbook quantum mechanics is false, or
3. theres an efficient classical factoring
   algorithm.

All three seem like crackpot speculations. At
least one of them is true!
5
One-Slide Summary

• Quantum computing is not a panaceaand that makes
  it more interesting rather than less!
• On our current understanding, quantum computers
  could break RSA, simulate quantum dynamics, and
  do other important things, but not solve
  generic search problems exponentially faster
  than classical computers
• 3. In this talk, Ill tell you about some of
  whats known about the capabilities and limits of
  quantum computers

6
Quantum Computing
A quantum state of n qubits takes 2n complex
numbers to describe
The goal of quantum computing is to exploit this
exponentiality in our description of the world
Idea Get paths leading to wrong answers to
interfere destructively and cancel each other
out
7
Shors Result
Quantum computers can factor integers in
polynomial time (thereby break RSA, thereby
swipe your credit card number)
To prove this, Shor had to exploit a special
property of the factoring problem (namely its
reducibility to period-finding)
Ideas extend to computing discrete logarithms,
solving Pells equation, breaking elliptic curve
cryptography
8
But these problems arent believed to be
NP-complete So the question remains can quantum
computers solve NP-complete problems in
polynomial time?
Bennett et al. 1997 Quantum magic wont be
enough
Suppose we throw away the problem structure, and
just consider an abstract space of 2n possible
solutions Then even a quantum computer will need
2n/2 steps to find a correct solution
Note This square-root speedup is achievable, via
Grovers algorithm
9
The quantum adiabatic algorithm Farhi.
Goldstone, et al. 2000 does exploit problem
structure
Hi
Hf
Hamiltonian with easily-prepared ground state
Ground state encodes solution to NP-complete
problem
But it suffers from provable limitations of its
own van Dam, Mosca, Vazirani 2001 Eigenvalue
gap can be exponentially small
10
Another example of a quantum black-box problem
find a collision in a list of numbers
28 12 18 76 96 82 94 99 21 78 88 93
39 44 64 32 99 70 18 94 66 92 64 95
46 53 16 35 42 72 31 66 75 33 93 32
47 17 70 37 78 79 36 63 40 69 92 71
28 85 41 80 10 73 63 95 57 43 84 67
57 31 62 39 65 74 24 90 26 83 60 91
27 96 35 20 26 52 88 89 38 97 54 30
62 79 71 84 50 38 49 20 47 24 54 48
98 23 41 16 40 75 82 13 58 56 81 34
14 61 52 21 44 22 34 14 51 74 76 83
37 90 58 13 10 25 29 11 56 68 12 61
51 23 77 68 72 43 69 46 87 97 45 59
73 30 19 81 86 49 60 85 80 50 11 59
65 67 89 29 86 48 22 15 17 55 36 27
42 55 77 19 45 15 53 98 91 87 25 33
11
What Could We Do With A Fast Quantum Algorithm
For the Collision Problem?
?
Solve Graph Isomorphismby finding a collision in
A. 2002 Any quantum algorithm needs at least
N1/5 queries to find a collision in a list of
size N Shi, Kutin, Ambainis, Midrijanis
Improved to N1/3 (which is optimal)
12
What makes the problem so hard?
Basically, that a quantum computer can almost
find a collision after one query to f!
If only we could now measure twice!
Or if only we could see the whole trajectory of
a hidden variable coursing through the quantum
system!A., Phys. Rev. A 2005
Previous techniques werent sensitive to the fact
that quantum mechanics doesnt allow these things
13
Cartoon Version of Proof
Suppose it exists by way of contradiction
T-query quantum algorithm that finds collisions
in 2-to-1 functions
T-query quantum algorithm that distinguishes
1-to-1 from 2-to-1 functions
Beals et al. 1998 p(f) is a multilinear
polynomial, of degree at most 2T, in Boolean
indicator variables ?(f(x),y)
Let p(f) probability algorithm says f is 2-to-1
Crucial factsq(k) ? 0,1 for all
k1,2,3,q(1) ? 1/3q(2) ? 2/3
Let q(k) average of p(f) over all k-to-1
functions f
14
The magic step q(k) itself is a univariate
polynomial in k, of degree at most 2T
Why?
Thats why
15
Bounded in 0,1 at integer points
1
q(k)
0
. . . . .
. . . . .
1
2
3
N2/5
k
16
Problem Were given black-box access to a
function f0,1n?Z We want to find a local
minimum of f, evaluating f as few times as
possible
4
4
2
5
3
Aldous 1983 Randomized algorithm making 2n/2?n
queriesA., STOC04 Quantum algorithm making
2n/3n1/6 queries
Aldous 1983 Any randomized alg needs 2n/2-o(n)
queriesA., STOC04 Any quantum alg needs
2n/4/n queries
My lower-bound proof uses Ambainiss quantum
adversary method, which upper-bounds how much the
entanglement between algorithm and oracle can
increase via a single query
17
Surprising part Quantum-inspired argument also
yields a better classical lower bound
2n/2/n2 Also yields the first randomized or
quantum lower bounds for local search on
constant-dimensional grid graphs
Quantum Generosity Giving back because we careTM
18
OK, so I accept that quantum computers have
these limitations. Is there any physical means
to solve (say) NP-complete problems in polynomial
time?
19
Famous proposal for how to solve NP-complete
problems Dip two glass plates with pegs between
them into soapy water. Let the soap bubbles form
a minimum Steiner tree connecting the pegs
Other proposals with obvious scaling problems
protein folding, DNA computing, optical computing
20
Relativity Computing
Problem Energy needed to accelerate to
relativistic speed
DONE
Variant Black hole computing
21
Abrams Lloyd 1998 If the Schrödinger equation
were nonlinear, one could exploit that fact to
solve NP-complete problems in polynomial time
One way to interpret this result as additional
evidence that the Schrödinger equation is linear
1 solution to NP-complete problem
No solutions
22
Zeno Computing
Do the first step of a computation in 1 second,
the next in ½ second, the next in ¼ second, etc.
Problem Quantum foaminess
Below the Planck scale, our picture of space and
time breaks down in not-yet-understood ways
23
Scientific American, March 2008
www.scottaaronson.com