Download The Limits of Quantum Computers

The Limits of Quantum Computers
(or: What We Can’t Do With Computers We Don’t Have)
Scott Aaronson
University of Waterloo
SZK
BQP
NPcomplete
So then why can’t we
just ignore quantum
computing, and get
back to real work?
Because the universe isn’t classical
My picture of reality, as an eleven-year-old
messing around with QBASIC:
+ details
(Also Stephen Wolfram’s current picture of reality)
Fancier version: Extended Church-Turing Thesis
Shor’s factoring algorithm
presents us with a choice
Either
1. the Extended Church-Turing Thesis is false,
2. textbook
quantum
mechanics is false, or
That’s
why YOU
should
care
3. there’s an efficient classical factoring algorithm.
about quantum
computing
All three seem
like crackpot speculations.
At least one of them is true!
My Spiel In One Slide
1. Ignoring quantum mechanics won’t make it go away
2. Quantum computing is not a panacea—and that
makes it more interesting rather than less!
3. On our current understanding, quantum computers
could “merely” break RSA, simulate quantum
physics, etc.—not solve generic search problems
exponentially faster
4. So then why do I worry about quantum computing?
Because I’m interested in fundamental limits on what
can efficiently be computed in the physical world.
That makes me professionally obligated to care!
Where Do I Come In?
My work, over the last seven years, has deepened our
understanding of the limitations of quantum computers.
Solved some of the field’s notorious open problems:
- Lower bound for finding collisions in hash functions
- “Direct product theorem” for quantum search
Made unexpected connections:
- Classical lower bounds proved by quantum arguments
- Quantum-state learning algorithm from a lower bound
Plan of Talk
The Gospel According to Shor
Three Limitations of Quantum Computers
- Finding collisions in hash functions
- Solving NP-complete problems with advice
- Finding local optima
Turning Lemons into Lemonade
- Approximately learning quantum states
Summary of Contributions
What Quantum Mechanics Says
If an object can be in two distinguishable states
|0 or |1, then it can also be in a superposition
|0 + |1
Here  and  are complex
amplitudes satisfying
||2+||2=1
If we observe, we see
|0 with probability ||2
|1 with probability ||2
Also, the object collapses to
whichever outcome we see
1
0 1
2
0
To modify a state
n

i 1
i
i
we can multiply vector of amplitudes by a
unitary matrix—one that preserves
n

i 1
2
i
1
0 1





1
2
1
2
1  1  1 

2  12   20
      
1  01   11
2   2   2 
1
0 1
2
We’re seeing interference of amplitudes—the
source of all “quantum weirdness”
2
0
Quantum Computing
A quantum state of n “qubits” takes 2n complex
numbers to describe:

x0,1
x x
n
The goal of quantum computing is to exploit this
exponentiality in our description of the world
Idea: Get paths leading to incorrect answers to
interfere destructively and cancel each other out
Plan of Talk
The Gospel According to Shor
Three Limitations of Quantum Computers
- Finding collisions in hash functions
- Solving NP-complete problems with advice
- Finding local optima
Turning Lemons into Lemonade
- Approximately learning quantum states
Summary of Contributions
The Quantum Black-Box Model
x
f
f(x)You gotta start
somewhere
In this talk, I’ll only care about the number of queries
to a black box, not any other computational steps
But whyGiven
do black-box
Example:
a function f:{0,1}n{0,1},
Almostsuppose
all known
us anything
weresults
want totell
decide
if there’s an x such
that f(x)=1
quantum
algorithms
about the real
world?
are black-box
n
Classically,
~2
queries
to
f
are
needed
Remember
(no quantum
Grover IP=PSPACE?
gave a quantum algorithm thatIP=PSPACE
uses only yet)
~2n/2 queries
[BBBV 1997]: Grover’s algorithm is The
optimal
proof of the
pudding
is in the
Yields “black-box evidence” that quantum
computers
proving
can’t solve NP-complete problems efficiently
Algorithm’s state:

x ,w
x, w
x ,w
x: location to query
w: “workspace” qubits
After a query transformation:

x ,w
x, w  f  x 
x ,w
Between two queries, we can apply an arbitrary
unitary matrix that doesn’t depend on f
Complexity = minimum number of queries needed
2
2
to achieve
 x ,w 
for all oracles f

x ,w
corresponding to
right answer
3
Plan of Talk
The Gospel According to Shor
Three Limitations of Quantum Computers
- Finding collisions in hash functions
- Solving NP-complete problems with advice
- Finding local optima
Turning Lemons into Lemonade
- Approximately learning quantum states
Summary of Contributions
Problem: Find 2 numbers that
are the same (each number appears twice)
28
32
31
69
57
20
49
56
37
68
60
55
12 18 76 96 82 94 99 21 78 88 93 39 44 64
99By
70“birthday
18 94 66paradox”,
92 64 95a46 53 16 35 42 72
66randomized
75 33 93 algorithm
32 47 17 must
70 37 78 79 36 63 40
92
71 28N
85of41
73 63 95 57 43 84 67
examine
the80N 10
numbers
31 62 39 65 74 24 90 26 83 60 91 27 96 35
26 52 88 89 38 97 54 30 62 79 71 84 50 38
[Brassard-Høyer-Tapp
Is that optimal?
20 47 24 54 48 98 23 41 16 40 75 82 13 58
1997] Quantum
Proving a lower
81 34 14 61 52 21 44 22 34 14 51 74 76 83
algorithm based on
bound
90 58 13 10 25 29 11 56 68
12 better
61 51 than
23 77
Grover
that46uses
72
43 69
87 only
97 45 59 constant
73 30 19was
81 open
86 49
1/3
for 548years
85 80N 50queries
11 59 65 67 89 29 86
22 15 17
36 27 42 55 77 19 45 15 53 98 91 87 25 33
Motivation for the Collision Problem

Cryptographic
Hash Functions
Graph Isomorphism:
find a collision in
 1  G  , ,  n!  G  ,  1  H  , ,  n!  H 
Statistical Zero Knowledge (SZK) protocols
What makes the problem so hard?
Basically, that a quantum computer can almost find a
collision after one query!
1
N
x  y
N
 x f x 
x 1
Measure 2nd
register
f x 
2
“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 weren’t sensitive to the fact that
quantum mechanics doesn’t allow these things
[A., STOC’02] N1/5 lower bound on
quantum query
complexity of the
collision problem
[Shi, FOCS’02] Improved to N1/3; also
[A.-Shi, J. ACM 2004] N2/3 lower bound for
element distinctness
[Kutin 2003] Simplifications and
[Ambainis 2003] generalizations
[Midrijanis 2003]
Cartoon Version of Proof
T-query quantum algorithm that
finds collisions in 2-to-1 functions
Suppose it exists by
way of contradiction…
T-query quantum algorithm that
distinguishes 1-to-1 from 2-to-1 functions
Let p(f) = probability
algorithm says f is 2-to-1
Let q(k) = average of p(f)
over all k-to-1 functions f
[Beals et al. 1998] p(f) is a
multilinear polynomial, of
degree at most 2T, in Boolean
indicator variables (f(x),y)
Trivial yet crucial facts:
q(k)  [0,1] for all k=1,2,3,…
q(1)  1/3
q(2)  2/3
The magic step: q(k) itself is a univariate
polynomial in k, of degree at most 2T
Why?
That’s why
Bounded in [0,1] at integer points
1
q(k)
Large derivative
0
1
2
3
. . . . .
. . . . .
k
[A. A. Markov, 1889]:
degq  
N 2 / 5 max2 / 5 dqx  / dx
0 x  N
2 max2 / 5 qx 

 N
1/ 5

0 x  N
Hence the original quantum algorithm must have
made (N1/5) queries
N2/5
Plan of Talk
The Gospel According to Shor
Three Limitations of Quantum Computers
- Finding collisions in hash functions
- Solving NP-complete problems with advice
- Finding local optima
Turning Lemons into Lemonade
- Approximately learning quantum states
Summary of Contributions
The Hunt for the Golden State
Could there be a quantum state | left over
from the Big Bang, such that given any
3SAT instance of size 1,000,000, we could
quickly solve it by just measuring | in an
appropriate basis?
[A., CCC 2004] In the black-box model, no:
there cannot exist any “golden state” for solving
NP-complete problems in polynomial time
Efficient quantum algorithm to solve
SAT using an m-qubit golden state
Suppose it exists
by way of
contradiction…
Efficient quantum algorithm to solve (say) m3
SAT instances, reusing the same golden state
Algorithm to solve m3 SAT
instances with probability 2-m
Guess the golden state!
Replace it by the
maximally mixed state,
i.e. a random m-bit string
To get a contradiction, I now need to prove a directproduct theorem for quantum search:
“If a quantum algorithm doesn’t even have time to solve one
search problem w.h.p., then the probability of its solving k
search problems decreases exponentially with k”
How do I prove the direct-product theorem?
Again using the polynomial method
But this time I need a generalization of A. A. Markov’s
inequality due to [V. A. Markov 1892], which takes into
account not just the first derivative but all higher derivatives
1
0
0
1
2
. . .
m3
. . . . .
2n
[Klauck-Špalek-deWolf, FOCS’04] tightened my direct
product theorem, and also used it to prove the first
quantum time-space tradeoffs
Plan of Talk
The Gospel According to Shor
Three Limitations of Quantum Computers
- Finding collisions in hash functions
- Solving NP-complete problems with advice
- Finding local optima
Turning Lemons into Lemonade
- Approximately learning quantum states
Summary of Contributions
Problem: We’re given black-box
access to a function f:{0,1}nZ
We want to find a local minimum of f,
evaluating f as few times as possible
4
4
3
2
5
[Aldous 1983] Randomized algorithm making 2n/2n queries
[A., STOC’04] Quantum algorithm making 2n/3n1/6 queries
[Aldous 1983] Any randomized alg needs 2n/2-o(n) queries
[A., STOC’04] Any quantum alg needs 2n/4/n queries
My lower-bound proof uses Ambainis’s quantum adversary
method, which upper-bounds how much the entanglement
between algorithm and oracle can increase via a single query
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 constantdimensional grid graphs
Quantum Generosity … Giving back because we careTM
Subsequent improvements:
[Santha-Szegedy, STOC’04]
[Zhang, STOC’06] [Verhoeven,
2006] [Sun-Yao, FOCS’06]
Plan of Talk
The Gospel According to Shor
Three Limitations of Quantum Computers
- Finding collisions in hash functions
- Solving NP-complete problems with advice
- Finding local optima
Turning Lemons into Lemonade
- Approximately learning quantum states
Summary of Contributions
The Lemon
Quantum random access coding
n-bit string,
x1…xn
|
Any one bit xi of
our choice, with
high probability
[ANTV 1999]: | must have (n) qubits—no
asymptotic savings over classical
(Surprisingly, an n-qubit quantum state has no more
“independently accessible degrees of freedom” than an
n-bit classical string)
The Lemonade
“Quantum Occam’s Razor Theorem”
[A. 2006]
|

Upper bound on the sample complexity of “PAC” (Probably
Approximately Correctly) learning a quantum state
Informally: Can predict approximate expectation values of
most measurements on an n-qubit state, after a number of
sample measurements that increases only linearly with n
By contrast, traditional quantum state
tomography requires ~4n measurements
Record so far: n=8
Prohibitive for much larger n
Plan of Talk
The Gospel According to Shor
Three Limitations of Quantum Computers
- Finding collisions in hash functions
- Solving NP-complete problems with advice
- Finding local optima
Turning Lemons into Lemonade
- Approximately learning quantum states
Summary of Contributions
Summary of Contributions
Solved several notorious open problems about the
limitations of quantum computers
Gave evidence that collision-resistant hash functions can
still exist in a quantum world
Proved the first direct product theorem for quantum search
Gave evidence against “golden states” for NP-complete
problems
Solved open problems about classical local optimization
using quantum techniques
Used a quantum coding lower bound to propose a new
learning algorithm, with possible experimental implications
Ten Research Directions I Didn’t
Tell You About Today
Addressing skepticism
Practical simulation of
of quantum computing
stabilizer quantum circuits
[A., STOC 2004] Quantum computers
with
[A.-Gottesman,
Phys Rev A 2004]
anthropic postselection
Grover search with[A.,
finite
Proc. Roy. Soc. 2005]
speed of light
Quantum software copy[A.-Ambainis, FOCS 2003]
Quantum computers with
protection
closed timelike curves
[A., in preparation]
Provably-nonrelativizing
[A.-Watrous, in preparation]
circuit lower bounds
Quantum versus classical
proofs
[A., CCC 2006]
Need to “uncompute garbage” in
Complexity of Bayesian
quantum algorithms
[A., QIC 2003]
agreement protocols
[A.-Kuperberg, CCC 2007]
[A., STOC 2005]
www.scottaaronson.com/papers