Computational Intractability
As A Law of Physics
Scott Aaronson
University of Waterloo
Things we never see…
GOLDBACH
CONJECTURE:
TRUE
NEXT QUESTION
Warp drive
Perpetuum mobile
Übercomputer
Is the absence of these devices something
physicists should think about?
Goal of talk: Convince you to see the impossibility
of übercomputers as a basic principle of physics
Computer Science 101
Problem: “Given a graph, is it connected?”
Each particular graph is an instance
The size of the instance, n, is the number of
bits needed to specify it
An algorithm is polynomial-time if it uses at
most knc steps, for some constants k,c
P is the class of all problems that have
polynomial-time algorithms
NP: Nondeterministic
Polynomial Time
Does
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31304735342639520488192047545612916330509384
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have a prime factor ending in 7?
NP-hard: If you can solve it, you
can solve everything in NP
NP-complete: NP-hard and in NP
Is there a
Hamilton cycle
(tour that visits
each vertex
exactly once)?
NP-hard
Hamilton cycle
Steiner tree
Graph 3-coloring
Satisfiability
Maximum clique
…
NPcomplete
NP
Graph connectivity
Primality testing
Matrix determinant
Linear programming
…
P
Matrix permanent
Halting problem
…
Factoring
Graph isomorphism
…
Does P=NP?
No.
The (literally) $1,000,000 question
Q: What if P=NP, and the algorithm takes n10000 steps?
A: Then we’d just change the question!
Q: Why is it so hard to prove PNP?
A: Mostly because algorithms can be so clever!
What about quantum
computers?
BQP: Bounded-Error Quantum Polynomial-Time
Shor 1994: BQP contains integer factoring
But factoring isn’t believed to be NP-complete.
So the question remains: can quantum computers solve
NP-complete problems efficiently?
Bennett et al. 1997: “Quantum magic” won’t be enough
If we throw away the problem structure, and just consider
a “landscape” of 2n possible solutions, even a quantum
computer needs ~2n/2 steps to find a correct solution
Quantum Adiabatic Algorithm
(Farhi et al. 2000)
Hi
Hamiltonian with
easily-prepared
ground state
Hf
Ground state encodes
solution to NPcomplete problem
Problem: Eigenvalue gap can be
exponentially small
Other Alleged Ways 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 (thereby solving a
known NP-complete problem)
Protein folding: Can also get stuck at local optima
(e.g., Mad Cow Disease)
DNA computers: Just massively parallel classical
computers!
What would the world actually be like
if we could solve NP-complete
Proof of
Shortest
problems
efficiently?
Riemann
hypothesis with
10,000,000
symbols?
efficient
description of
stock market
data?
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
The NP Hardness Assumption
There is no physical means to solve NP
complete problems in polynomial time.
Rest of talk: Show how
Alright, what can we
say about
this assumption?
complexity
yields
a new
onPNP
linearity of
• Implies, but isperspective
stronger than,
QM, anthropic postselection,
• As falsifiable as
it getstimelike curves, and
closed
initial
conditions physical theory
• Consistent with
currently-known
• Scientifically fruitful?
1. Nonlinear variants of the
Schrödinger Equation
Abrams & Lloyd 1998: If quantum mechanics
were nonlinear, one could exploit that to solve
NP-complete problems in polynomial time
Can take as an
additional
argument for why
QM is linear
1 solution to NP-complete problem
No solutions
2. Anthropic Principle
Foolproof way to solve NP-complete problems in
polynomial time (at least in the Many-Worlds Interpretation):
NP Hardness Assumption
First guess ayields
random
solution.
Then, if it’s wrong,
a nontrivial
constraint
kill yourself! on anthropic theorizing: no
use of the Anthropic Principle
can be valid, if its validity
would give us a way to solve
Technicality:NP-complete
If there are no
solutions,
problems
in you’re out of
luck!
polynomial time
Solution: With tiny probability don’t do anything. Then, if you find
yourself in a universe where you didn’t do anything, there probably were
no solutions, since otherwise you would’ve found one!
What if we combine quantum computing
with the Anthropic Principle?
I.e. perform a polynomial-time quantum
computation, but where we can measure a
qubit and assume the outcome will be |1
Leads to a new complexity
class: PostBQP
(Postselected BQP)
Certainly PostBQP contains NP—but is it
even bigger than that?
Some more animals from the complexity zoo…
PSPACE: Class of problems solvable with a
polynomial amount of memory
PP: Class of problems of the form, “out of 2n
possible solutions, are at least half of them
correct?”
Adleman, DeMarrais, Huang 1998: BQP  PP
Proof: Feynman path integral
Proof easily extends to show PostBQP  PP
A. 2004: PostBQP = PP
In other words, quantum
postselection gives
exactly the power of PP
Surprising part:
This characterization
yields a half-page proof
of a celebrated result of
Beigel, Reingold, and
Spielman, that PP is
closed under intersection
3. Time Travel
Everyone’s first idea for a time travel computer:
Do an arbitrarily long computation, then send the
answer back in time to before you started
THIS DOES NOT WORK
Why not?
• Ignores the Grandfather Paradox
• Doesn’t take into account the computation you’ll have
to do after getting the answer
Deutsch’s Model
A closed timelike curve (CTC) is a computational
resource that, given an efficiently computable function
f:{0,1}n{0,1}n, immediately finds a fixed point of f—
that is, an x such that f(x)=x
Admittedly, not every f has a fixed point
But there’s always a distribution D such that f(D)=D
Probabilistic Resolution of the Grandfather Paradox
- You’re born with ½ probability
- If you’re born, you back and kill your grandfather
- Hence you’re born with ½ probability
Let PCTC be the class of problems solvable in
polynomial time, if for any function f:{0,1}n{0,1}n
described by a poly-size circuit, we can immediately get
an x{0,1}n such that f(m)(x)=x for some m
Theorem: PCTC = PSPACE
Proof: PCTC  PSPACE is easy
For PSPACE  PCTC: Let sinit, sacc, and srej be the initial,
accepting, and rejecting states of a PSPACE machine,
and let (s) be the successor state of s. Then set
f  sacc , b  : sinit ,1 ,


f srej , b : sinit ,0 ,
f  s, b  :  s , b otherwise
The only fixed point is
an infinite loop, with b
set to its “true” value
What if we perform a quantum
computation around a CTC?
Let BQPCTC be the class of problems solvable in
quantum polynomial time, if for any superoperator E
described by a quantum circuit, we can immediately get
a mixed state  such that E() = 
Clearly PSPACE = PCTC  BQPCTC
A., Watrous 2006:
BQPCTC = PSPACE
If closed timelike curves exist, then quantum
computers are no more powerful than classical ones
BQPCTC  PSPACE: Proof Sketch
Let vec() be a “vectorization” of . We can reduce the
problem to the following: given a 22n22n matrix M,
prepare a state  such that
M vec   vec 
Solution: Let
P : lim 1  z I  zM 
1
z 1
Then by Taylor expansion,


MP  M lim 1  z  I  zM  z M    P
z 1
2
2
Hence P projects onto the fixed points of M
Furthermore, we can compute P exactly in PSPACE,
using Csanky’s parallel algorithm for matrix inversion
4. Initial Conditions
Normally we assume a quantum
computer starts in an “all-0” state,
|0…0. But what if much better initial
states were created in the Big Bang, and
have been sitting around ever since?
Leads to the concept of quantum advice…
Useful?
|
Limitations of Quantum Advice
A., 2004
Result #1: BQP/qpoly  PostBQP/poly
“Any problem you can solve using short quantum advice,
you can also solve using short classical advice, provided
you’re willing One
to usecan
exponentially
computation
postulatemore
bizarre,
time to extract
what the advice is telling you.”
exponentially-hard-to-prepare
initial states in Nature, without
Result #2: There
existsthe
anNP
“oracle”
relative to which
violating
Hardness
NP  BQP/qpoly
Assumption
Evidence that NP-complete problems are still hard for
quantum computers in the presence of quantum advice
Concluding Remarks
COMPUTATIONAL
COMPLEXITY
PHYSICS
Prediction: NP Hardness Assumption will eventually be
seen as analogous to Second Law of Thermodynamics or
impossibility of superluminal signaling
Open Question: What is polynomial time in quantum
gravity? (First question: What is time in quantum gravity?)
Links to papers, etc.:
www.scottaaronson.com